Stupid Questions Thread /sci/ what is the purpose of capital

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I've used it in a class on number theory

Also in complex analysis (Riemann zeta function)

>>7943466
It is the product of a series.

PI,n=1, 10 = 1*2*3*4*5*6*7*8*9*10

This here's already up
>>7943466

>>7943486

I'm aware of that, I was just wondering what it's used for.

>>7943485
Ah that's interesting. I definitely should pick up a book on complex analysis, I've been wanting to for a while.

>>7943466
it's a very compact notation, I'm an undergrad and I've used it a lot, so it surprises me you haven't.

Sometimes people will avoid it by writing

x1 * x2 * ... * xn

instead, though

>>7943493
you're being memed. look at the expression closely.
also, you went through undergrad math without complex analysis?
???

>>7943466
>>7943492
Did I link to yourself? Gosh, that's retarded. Pardon me, I meant >>7930606

Products are not very interesting from a "theoretical" point of view because you can predict what will happen by taking the log, which gives you a series, and using your standard calculus toolbox to study that series instead.
Now, there are some interesting formulas involving products, for example Wallis' formula [math]\displaystyle \frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{1}{1-1/4n^2}[/math]

>>7943493
As this anon says >>7943503, how the hell did you not do complex analysis?
Also, getting to the point of the Riemann zeta function is not going to be particularly fast, although it's not hard to prove the Euler formula for the number of primes is equal to the zeta function.

>>7943485
>memechizuki
>[math]\frac{1}{12}[/math] meme: [math]\Pi[/math]-edition
GENIUS

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>>7943493
> getting memed this hard
Anon, please, don't hurt yourself

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Say you have to solve the differential equation f’(x)=f(x) with f(0)=1.
You naturally make the ansatz

[math] f(x) = 1 + c_1 \, x + c_2 \, x^2 + c_3 \, x^3 + \dots [/math],

[math] f’(x) = c_1 + 2\,c_2\, x + 3\,c_3\, x^2 + \dots[/math].

Comparing coefficients, this implies that the solution to f’(x)=f(x) must have, for example [math] 3 \, c_3 = c_2 [/math]. In fact all coefficients are determined this way, by the recursive relation

[math] \dfrac { c_{n+1} } {c_n} = \dfrac {1} {n+1} [/math]

With the polynomial q(n) := n+1, this means
[math] c_n = \frac{1} { \prod_{k=1}^n q(k) } c_0 = \dfrac {1} {n!} [/math]
and hence
[math] f(x) = \sum_{n=0}^\infty \dfrac {1} {n!} x^n [/math].

Such an approach to solve a differential equation will often look like this. A whole lot of function have series coefficients [math]c_n[/math], such that

[math] \dfrac { c_{n+1} } {c_n} = \dfrac {p(n)} {q(n)} [/math]

where p and q are some polynomials. Any (arbitrary product of) polynomials in n can be written as a product of terms [math] (a_i-n) [/math]. They are the solutions to differential equations with recursive character:

[math] \dfrac {d}{dz} D_b f(z) = D_a f(z) [/math]

with

[math] D_b := \prod_{n=1}^{q} \left ( z \dfrac {d} {dz} + b_n-1 \right) [/math],

[math] D_a := \prod_{n=1}^{p} \left( z \dfrac {d} {dz} + a_n \right) [/math].

is solved by

[math] {}_p F_q [a_1,…,a_p; b_1,…,b_q] (z) := \sum_{n=0}^\infty c_n z^n [/math]

with

[math] c_n = \prod_{m=0}^{n-1} \dfrac{1} { (1+m) } \dfrac { \prod_{k=1}^p (a_k+m) }{ \prod_{j=1}^q(b_j+m) } [/math]

E.g. a1 = a2 = b1 = 1 characterized some simple differential equation that characterized the log, as

[math] \dfrac { \prod_{m=0}^{n-1} (a_1+m) } { \prod_{m=0}^{n-1} (b_1+m) } = \dfrac {n!} { (n+1)! } = \dfrac {1} {n+1} [/math]

and then

[math] {}_2 F_1 [1, 1; 2](z) = \sum_{n=0}^\infty \dfrac {1} {n+1} z^n = \dfrac {1} {(-z)} \sum_{k=1}^\infty \dfrac{ (-1)^{k+1} } {k} (-z)^k = - \dfrac {1} {z} \log(1-z) [/math]

>>7943466
If I'm talking about prime factorizations, I'm much more inclined to write [math]\prod_{i=1}^n p_i^{g_i}[/math] than typing it out with ellipses. You use it the exact same way you'd use sigma notation for sums, and it baffles me that this is so mysterious to you.

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>>7943550
Oh, the product in the f'(x)=f(x) problem should run from 0 to n-1.


Tangent: If you replace that factorial'ish product with Gammas you get the Meijer-G, which captures just about any function

https://en.wikipedia.org/wiki/Meijer_G-function

Anyway, there is also the Weierstrass factorization theorem, giving e.g.

[math]\sin (\pi z) = (\pi z) \cdot \prod_{n=1}^\infty \left( 1 - \left( \dfrac {z} {n} \right)^2 \right)[/math]

In path integrals, you need

[math] \lim_{ N \to \infty} \prod_{n=0}^{N-1} \left( 1 + \frac {x_n} {N} \right) =
\exp \left ( \lim_{N \to \infty } \frac {1} {N} \sum_{n=0}^{N-1} x_n \right) [/math]

And if you have autism, consider this identity

[math] \prod_{m=1}^\infty \left( 1 - q^{2m} \right) \left( 1 + w^{2} q^{2m-1} \right) \left( 1 + w^{-2} q^{2m-1} \right) = \sum_{n=-\infty}^\infty w^{2n} q^{n^2} [/math]

And finally here is an product to sum translation I came up, that enabled me to tackle some tough infinite product problem on MO

[math] \prod_{k=K}^\infty b_k = \lim_{ n \to \infty } a_n = \prod_{k=K}^{M-1} b_k + \sum_{n=M}^\infty ( b_n - 1 ) \, \prod_{k=K}^{n-1} b_k [/math]

I got one.

Since an electric motor is just an alternator or induction generator 'run backwards' why can't electric cars use their motive system to self-charge, say going down a hill or coasting to a stop? Converting the momentum of the car into usable electrical energy?

Also, could I use an alternator off a large truck modify it a bit and apply a current to it to propel a golf-cart or something similar?

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>>7943550
>>7943609
Nice posts, anon. Where did you learn about the Meijer G-function?

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>>7943609
This is a fantastic resource. Thanks for sharing it with me. I'll be sure to investigate this a bit more.

As for the other anons wondering why I haven't found this yet is because I have a mathematics minor. I'm majoring in physical chemistry.

Thanks everybody.

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>>7943615
Not sure, probably because Mathematica implements it and I encountered it as a solution to a differnetial equation there.
The inbetween between the two is the Fox–Wright function
(https://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function)
I always found the integrand in

[math] G_{p,q}^{\,m,n} = \frac{1}{2 \pi i} \int_L \dfrac { \prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma (1 - a_j +s) } { \prod_{j=m+1}^q \Gamma(1 - b_j + s) \prod_{j=n+1}^p \Gamma(a_j - s) } \,z^s \,ds [/math]

a little disturbing. Then some time ago, I came across the booklet "Quantum Calculus" by Kac, which on the side gives a motivation for the Hypergeometric function (a version of what I wrote down above).
And when you e.g. look at
[math] \dfrac { \Gamma(n+m+1) } { \Gamma (m+1) } = \dfrac { (n+m)! } { m! } = \dfrac { \prod_{j=1}^{n+m} j } { \prod_{i=1}^m i} = \prod_{j=m+1}^{m+n} j = (m+1) \cdot (m+2) \cdots (m+n). [/math]

or
[math] \dfrac { \Gamma(4+5) } { \Gamma(4) } = 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 [/math]

you lose your fear.

>>7943613
They are less useful than sums in a way, as existence of products seems easier to check. Or maybe it has to do with how function spaces are (approximated by) vector spaces.

There is, btw., a whole theory of "product calculus", if you will.
https://en.wikipedia.org/wiki/Product_integral

What you write also makes me think of Euler products such as the identity

[math] \sum_{n=1}^{\infty} \dfrac {1} {n^{s}} = prod_{p} \dfrac {1} {1-p^{-s}}[/math]

I think the more general theory of zeta functions wouldn't be able without definitions of products, when e.g. looking at
https://en.wikipedia.org/wiki/Arithmetic_zeta_function#Definition

In many cases I think products just arise via
[math] \exp(\sum_n a_n) = \prod_n \exp(a_n) [/math]
so that they don't seem to magical anymore.

Here's one product I find odd:

[math] \prod_{k=1}^n \left(1+\dfrac {1} {k} \right) = n+1 [/math]

This thread should be about dependent products types, thought.

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>>7943631
But I want to point out that the hypergeometric function is more useful than the Meijer-G (or the yet more general Fox H-function, or q-analogs of it).
I personally masturbate to the Mittag-Leffler function

[math] E_{\alpha, \beta} (z) = \sum_{k=0}^\infty \dfrac {z^k} {\Gamma ( \alpha \cdot k + \beta) } [/math]

and not only because the guy looks badass, but because [math] \alpha \mapsto E_{\alpha, \beta} (z) [/math] interpolates between [math] \dfrac{1}{1-z} [/math] and [math] \exp(z) [/math].

How much math do you need to understand basic physics? I dropped out of high school and the last math class I took was pre calc.

>>7943745
All you need is trig for HS-tier physics, yes, even AP Physics. Once you go higher up, precalc will do nowhere near good.

>>7943745
find out >7941629

>>7943745
srry >>>7941629

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>>7943751
Do you understand this?

>>7943758
a.k.a. a basis transformation in C^4

>>7943751
I'm fairly certain I'll need to go above high school level so I'll start learning calculus and see how far that takes me.

>>7943753
>>7943755
Thank you that looks like it will be very helpful to see what I need to know.

>>7943758
>simple, HS-tier physics
>quantum mechanics of systems of several particles

Come on, now. For basic, introductory physics at the undergraduate level (i.e. a first course in the subject at the level of a physics major) you really only need a solid grasp of multivariable calculus, linear alegebra, and basic ODEs/ODE methods. Everything else will be self-contained in any decent physics text about the subject you're learning.

>>7943669
Interesting. I'll take a look at it.

>>7943466
Denoting a product, I have never seen it used.

Denoting complexity classes, the symbol is everywhere in logic, from recursion theory to model theory to descriptive set theory.

Products:
ex. [math] \prod\limits_{n = 1}^m {{a_n}} = {a_1}...{a_m} [/math]

Disjoint unions:
ex. [math] TM = \coprod\limits_{p \in M} {{T_p}M} [/math]

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>>7943758
What the fudge is this shit and how do I learn it?

>>7943796
quantum entanglement

It's nothing special. Just an (informal) tensor product of hilbert spaces.

>>7943758
>QM is HS-tier
Can you not bash physics for at least one day, mathfags?

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>>7943793
That's the sum, though.
>>7943466
What I wanted to get at initially:

If X has two terms a,b : X and Y has three terms u,v,w : Y, then X+Y has five terms
[math] \nu_{ l} (a), \ \nu_{ l} (b), \ \nu_{ r} (u), \ \nu_{ r} (v), \ \nu_{ r} (w) \ :\ X+Y [/math]

X x Y has six terms
[math] \langle a,u \rangle, \ \langle a,v \rangle, \ \langle a,w \rangle, \ \langle b,u \rangle, \ \langle b,v \rangle, \ \langle b,w \rangle \ :\ X \times Y [/math]

X->Y has eight terms
[math] a \mapsto u \ and \ b \mapsto u, \ a \mapsto u \ and \ b \mapsto v, \dots \ :\ X \to Y [/math]

and these have no terms else. So clearly, at least for types where a cardinal assignment into the natural numbers makes sense, we have

[math] | X+Y | = | X | + | Y | [/math]
[math] | X \times Y | = | X | \times | Y | [/math]
[math] | X \to Y | = | Y | ^{ | X | } [/math]

where +, x, -> on the left are type constructors and on the right are the basic arithmetic operations.

For the depended sum and product, the above can be viewed as special cases for depended types:
If x:A and B(x) is a family of types varying over A (the B’s are fibers over A), then the sum type [math] \sum (x:A). B(x) [/math] (disjoint union of fibers) and the product type [math] \prod(x:A). B(x) [/math] (space of sections) fulfill

[math] |\sum(x:A). B(x) | = \sum_{x:A}| B(x)| [/math]
[math] |\prod(x:A). B(x) | = \prod_{x:A}| B(x)| [/math]

For A is a type of only two terms l,r :A and B( l ) := X, B( r ):=Y, we get
[math] \sum(x:A). B(x)=B( {l} )+B( {r} )=X+Y [/math]
For B being constant (think of how the fibration looks like), we get
[math] \sum(x:A). B(x)= \sum(x:A). \,B=A \times B [/math]
[math] \prod(x:A). B(x)= \prod(x:A). \,B=A \to B [/math]

>>7943796
That's vectors (a.k.a. Bell states) in terms of other base vectors, each the tensor product of 2D vectors (that are somehow physically realized.) These vectors are interesting to physically produce because they maximize a property (entanglement).

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>>7943854
The important line her is supposed to be

[math] |\prod(x:A). B(x) | = \prod_{x:A}| B(x)| [/math]

where we consider the product over a formula (quantity) to arise as an incarnation of a function space. Notes in pic related also relevant, although I don't want to make a simple thing seem too fancy.
My point is that it's natural.

>>7943844
prereqs?

>>7943885
linear algebra, calculus, and idea what a (quantum) state is and which physical system exhibit simple ones (photon orientation, trapped particles, ...)

>>7943885
linear algebra, multivar calculus, and vector calc

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>>7943897
>>7943900
>tfw in calc 1

>>7943914
This is /sqt/, if you need help on Calc I right now, ask away.

>>7943466
It's like sum notation but it multiplies instead

>>7943916
My class is basically going at down syndrome pace so it's all easy

>>7943916
Is trying to learn this stuff futile while only in calc 1? http://people.math.aau.dk/~raussen/INSB/AD2-11/book.pdf

>>7943939
If your math knowledge is only as far as calc I, forget it. Save it for later. This stuff deals with a lot of vectors, matrices, and sets.

>>7943897
Any old book on linear algebra won't cut it, you'll need a QM book that goes through the formalization of QM and dives a bit into the framework, i.e. hilbert space. A really good grasp of that is necessary and it's not the easiest if you're not used to it.

bumpu-desu!!

>>7943939
No. This is fine, this isn't actual differential geometry: It's a rigorless, soft introduction to surfaces in the way you use them in calculus. Absolutely fine.

The energy momentum tensor of the EM radiation field is
[eqn]T^{\mu\nu} = F^{\mu\sigma} F^\nu_\sigma - g^{\mu\nu} L[/eqn]
where
[eqn]L = -\frac{1}{4} F^{\mu\nu} F_{\mu\nu} \, .[/eqn]
Now I'm just starting on classical field theory, but from what I've gleaned in QFT (and QED in particular, where this comes in) is that [math]L[/math] in that form appears quite often (i.e. in the Lagrangian density). I'm not sure where its form comes from though, because the book I'm going through just sort of throws it out there without any derivation. I can understand the EM tensor product from a qualitative standpoint, but how does the factor of [math]-1/4[/math] show up?

>>7944202
I think it's just a convention. If you chose a different coefficient and changed the couplings of the field to other fields as needed, there would be no physical difference, just a rescaling of the field value.

>>7943466
I've used it in my stats work.

>>7944209

Alright, but then could you point me to a source that explains why this convention is such?

I have 98 dof and its a student's two tailed test. What am I supposed to think of a t value of 7~ compared to a t value of 51~ other than that theyre both significant?

>>7944202
Cause that's the trace of 4 values and depending on your choice +++- or ---+.

This is probably just to make it into a positive and average value.

>>7944233
>>7944209

Nevermind, found some resources.
http://www.maths.tcd.ie/~cblair/notes/432.pdf
http://www.reed.edu/physics/faculty/wheeler/documents/Electrodynamics/Class%20Notes/Chapter%203.pdf

For anyone else curious, basically it has to do with the fact that
[eqn]F^{\mu\nu} F_{\mu\nu} = 2( \mathbf{E} \cdot \mathbf{E} - \mathbf{B} \cdot \mathbf{B} ) \, .[/eqn]
Should've just checked the math out myself.

>>7944202
Given a gauge group [math]G[/math] and its Lie algebra [math]\operatorname{Lie}G[/math] the gauge field strength is defined as
[math]
\mathcal{F}_{\mu\nu} = \partial_{\mu}\mathcal{A}_{\nu} - \partial_{\nu}\mathcal{A}_{\mu} + [\mathcal{A}_{\mu},\mathcal{A}_{\nu}]
[/math]
where [math]\mathcal{A}_{\mu} = s^{*}_{\mu}\omega[/math] is the pulled-back section of the connection 1-form [math]\omega \in \Lambda^1(P) \otimes \operatorname{Lie}G[/math] on the principal bundle [math]P = (P,M,G)[/math] of the action of the gauge group [math]G[/math] on the manifold [math]M[/math]. From this perspective we can derive the gauge transformations of [math]\mathcal{A}[/math] under the action of [math]G[/math] as coordinate transitions
[math]
\mathcal{A}_{\mu} \rightarrow g_{\nu\mu}\mathcal{A}_{\mu}g_{\nu\mu}^{-1} + g_{\mu\nu}^{-1}dg_{\mu\nu}
[/math]
where [math]g_{\nu\mu}[/math] are the coordinate transition mapes on [math]P[/math], which take values in [math]G[/math].

Alternatively we can consider [math]\mathcal{A}[/math] as a Lie algebra-valued 1-form on [math]M[/math] and the field strength tensor as its anti-symmetrized derivative [math]\mathcal{F} = d\mathcal{A}[/math]. In this form we can describe a gauge invariant scalar [math]\operatorname{Tr} \mathcal{F}*\mathcal{F}[/math], where the trace is taken over [math]\operatorname{Lie}G[/math] and where [math]*[/math] is the Hodge dual operator.

The [math]-1/4[/math] comes from normalization.

>>7944244
the p-value is the probability of observing your data given that the null hypothesis is true

usually you're comparing means when using student's test so the null hypothesis is that the means of two distributions are the same. a low p-value indicates that there is a low probability of observing your data given that this is true and equivalently a low likelyhood of the null hypothesis.

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>>7943485
Hahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha


Hey this guy made the same joke u did

>>7943854
Disjoint union is the sum. I like it how the duality is easy to remember from the disjoint union symbol. Just do to the arrows what has been done to pi!

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How can a variable approach infinity, does it just get arbitrarily large and then we agree that it's basically infinity? I thought infinity was a direction and not a number. I'm not trolling, I'm done with calculus and I still don't understand how that shit makes any sense.

>>7944543
Infinity is not so much a direction as it is a pattern of incremental increases or decreases. When we consider the behavior of a variable ad infinitum, we hope to discover some pattern that is present no matter the extreme to which the variable is taken.

If f(x) asymptotically approaches 0 as x goes to infinity, then a slight increase in x should always bring f(x) closer to 0, so long as x is sufficiently large.

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>>7944543
I take one for the notebook:

In calculus/analysis, infinity isn't used as an entity (like a number), but instead
>limit n to infinity
means
>for whatever m you choose (arbitry), there is such and such, that such and such

For example you may consider a sequence [math] s_n [/math] given by
1/2, 1/4, 1/8, 1/16, ...
then the "limit of n to infinity" is the number y=0.
Why? Because for all real numbers [math] \varepsilon [/math] bigger than zero, you can find a natural number m, so that for all numbers n after that, you have that [math] s_n [/math] became smaller than [math] \varepsilon [/math].
For example, choose the small number [math] \varepsilon = 0.0003 [/math]. The sequence becomes forever smaller than that after, say, m = 4000. Indeed, for any n after 4000, the number [math] s_n [/math] is smaller than 1/4000^2, while already 1/4000=0.00025<0.0003.

So the limit to infinity is formalized as something to do with
>for arbitrary high index values, the thing itself is still restricted.

Coming back to the definition:
The limit of n to infinity of a seqeunce [math] s_n [/math] is y, if for all real numbers [math] \varepsilon [/math] bigger than zero, you can find a natural number m, so that for all numbers n after that, you have that the difference (here given by the distance on the real number line) between the value [math] s_n [/math] and this y became smaller than [math] \varepsilon [/math].

In formulas

[math] \lim_{n \to \infty} s_n = y[/math]

iff

[math] \forall ( \varepsilon \in {\mathbb R}_{>0} ) . \, \exists ( m \in {\mathbb N} ) . \, \forall ( n \ge_{\mathbb N} m) . \, | s_n - y \, |<\varepsilon [/math]

Another example: Consider again the sequence [math] s_n [/math] given by
1/2, 1/4, 1/8, 1/16, ...
and create a new sequence

[math] S_m = \sum_{k=1}^n a_n [/math]

which has members
1/2, 1/2+1/4, 1/2+1/4+1/8, ...

You can prove that with y=1 the above formula regarding [math] \forall ( \varepsilon \in {\mathbb R}_{>0} ) [/math] holds.

arbitrary cartesian products, or pi types

more generally, an arbitrary categorical product over an index set

>>7944431
Thanks

How do you prove that two numbers have a common denominator?

>>7945219
You trolling brah?

>>7945265
no, just retarded

>>7943466
Surprised you haven't seen this, I first encountered it in a freshman discrete math class.

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How can C be a field if not all complex numbers can be compared?

>>7945276
A field need not be ordered you stupid gayfag

>>7945276
A field is a commutative ring with unity in which every nonzero element is invertable.
The complex numbers satisfy this definition as a ring under standard addition and multiplication.

How is group theory applicable in physics? I only took an intro class but I heard there's some sort of theorem that lets you convert between groups and fields or something, which would presumably be helpful in QFT, right?

>>7945426
Literally right here
>>7944382

>>7945426
Quantum mechanics is basically a combination of linear algebra and group theory. The spin group [math]Sp(2)[/math] is a universal double cover of [math]SU(2)[/math], and as such a representation on [math]SU(2)[/math] can be extended to a double-valued representation on [math]Sp(2)[/math]. This allows us to study spins as a matrix group with action on the spin sector of the particle Hilbert space [,ath]\mathcal{H}[/math].
There are many things we can do with this, like using Shur's lemma to see if our representation is irreducible (nice), and use group theoretic selection rules (aka the Clebsch-Gordan table) to decompose the spin sectors of an N-particle Hilbert space [math]\mathcal{H}^{i_1} \otimes \dots \otimes \mathcal{H}^{i_n}[/math] into a direct sum of spin tuplet sectors [math]\mathcal{H}^{j_1} \oplus \dots \oplus \mathcal{H}^{j_n}[/math].

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How come instead of sigma and phi we dont use a symbol for iteration over some arbitratry operator?

>>7943466
If you go far enough in complex analysis you will study infinite products by a second class. My class is supposed to but my prof is notorious for not covering everything he wants to.

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>>7945426
I don't quite know where to draw the line that supposedly separates the math of theoretical physics and group representation theory

This may be nice:
http://arxiv.org/pdf/0810.1019.pdf

>>7945451
If [math]S_k[/math] is a sequence of elements of a monoid [math](M,*)[/math], then you may define

[math] \sum_{k=1}^n\ S_k [/math]

as [math] e\in M [/math] for [math] n=0 [/math] and else

[math] \left ( \sum_{k=1}^{n-1} \ S_k \right)\ *\ S_n [/math].

The product [math]\Pi_{k=1}^n[/math] is then a special case of this for * the multiplication of numbers and e=1.
Function concatenation is another good one.

To namedrop some more, it makes me thing of F-algebras and initial ones
https://en.wikipedia.org/wiki/Initial_algebra#Use_in_computer_science

>>7943520
Perhaps he either took an entirely pure math undergrad or pure math with some double major.

>>7945499
To elaborate on the first comment: Field theory is basically infinite dimensional representation theory of the groups involved in our favorite notion of spacetime invariance. That's the guideline we use to set up local Lagrangians, and thus dynamics of mechanics.

>>7945451
Because ∑um and Πroduct.

>>7945505

Neat, thanks!

>>7945451
Pretty sure Π is pi, not φ.

Could anyone recommend for some good English language material for Linear Algebra 1&2 or just vectors? I am going over my notes and I think it would help me.

>>7945499
take your pedophile cartoons back to >>>/a/

It is widely said that in physics we are just approximating. It isn't exact, like pure mathematics. But could there not exist a perfect/exact mathematical description of the universe? The "axioms" of the universe so to speak?

In such a case, if we came up with a theory of everything, would it be possible to confirm whether these are indeed the axioms of the universe, rather than just a model that happens to be very accurate?

Also, what would be the difference between this abstract mathematical model of the universe and the universe itself? Would they not be precisely the same thing? After all we only consider the universe to be "physical" rather than abstract because we are inside it, which is a subjective bias.

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I was taught the squeezing theorem last week but I'm confused as to how you're meant to use it properly. My course book says its useful for limits of sequences involving sin(n), cos(n) and (-1)^n.

Would I be correct to use it for these two questions? C because it contains -1^n and E because it contains sine. My confusion stems from my answer to E being 0.

>>7945888

no

>>7945888
>But could there not exist a perfect/exact mathematical description of the universe?

Yes. M-theory is an example of something that intends to do this. A hypothetical non-perturbative theory that accounts for all known phenomena.


>In such a case, if we came up with a theory of everything, would it be possible to confirm whether these are indeed the axioms of the universe, rather than just a model that happens to be very accurate?

No. There could also be something about the universe we have never observed (for many possible different reasons) and therefore we never felt the need to account for it in our theory. You would have to be a literal god to confirm the theory is exactly correct.

>>7943466
what kind of shit undergrad mathematics major are you doing OP? you see products in intro statistics when you do maximum likelihood estimators. also, you see it in topology for doing cartesian products. also, you see it in algebra for permutations. this semester, ive used them for a Queueing networks course where we learned about Jackson networks, which have a product-form.

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>>7943466
How to solve pic related. I rewrote the problem and incorrectly got the right answer.

Need help seeing where to go next.

>>7946181
take your pedophile cartoons back to >>>/a/

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I have a exam tomorrow early morning 8:30 (have to get up at 7:00). Should I go to bed at 5 to sleep for two hours or better don't sleep because it could make me more tired?

>>7946190
what?

>>7946191
Go to sleep for two hours. You'll initially feel more tired when you get up, but you'll need some sleep to get through the test and day.

>>7946195
Ok, thanks for the advice.

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>>7946181
Also this problem as well.

>>7946200
1) don't sleep for more than 40mins to an hour.
2) have coffee/creatine/ALCAR/modafinil immediately before you sleep
3) do it as close to your exam location as you can while:
i) guaranteeing you will wake up
ii) guaranteeing the comfort required to sleep

>>7945935
Consider
[math]a_n=\frac{\sin(n)}{n}[/math]. The [math]\sin[/math] function has a maximum of [math]1[/math] and a minimum of [math]-1[/math].
[math][/math]
We can then write
[eqn]0\leq |\sin(n)|\leq 1[/eqn]
Then
[eqn]0\leq \left|\frac{\sin(n)}{n}\right|\leq \left|\frac{1}{n}\right|[/eqn]
But we know [math]\frac{1}{n}\rightarrow0[/math] as [math]{n\rightarrow\infty}[/math], so by the squeezing theorem, [math]a_n\rightarrow 0[/math].

Similarly, [math]0\leq|(-1)^n|\leq 1[/math] and you can squeeze a limit there.

>>7946212
>>7946181
L'hopital's rule in one variable (or two) works

>>7945501
>complex analysis not pure math

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pls help with pic, how do I go about this?

theory is here:
http://www.pdf-archive.com/2016/03/21/gp04/

I have 50 days to learn the material for AB calculus and as it stands I know up to the power rule. I started at the beginning of february.

d-do you think I can do it

Calc II question
How would you prove or disprove that if [math]a_n\geq 0[/math] and [math]\sum _{n=0}^{\infty } a_n[/math] is convergent, then [math]\lim_{n\to \infty } \, n a_n=0[/math]?
I admit I haven't thought a lot on this but I couldn't find a counterexample that I could easily show works by hand.
Similarly, I don't know how to go about it formally/generally. I'm guessing it's something not to complicated with the fact that [math]\lim_{n\to \infty } \, a_n=0[/math] and messing around with ε, but I don't know where to start.
I have a feeling it's true just because I can't think of an [math]a_n=n^p[/math] that satisfies the criteria but doesn't tend to 0 when multiplied by n.

What the hell are those american Math course names? Math xyzh? How am I supposed to have an idea what it is???

>>7943466
I remembered an important formula where it's actually used.

https://en.wikipedia.org/wiki/Chain_rule_(probability)

>>7946707
Thank you, so you can somewhat just do it by inspection with no proper arithmetic? I'm nervous because it doesn't seem rigorous enough.

>>7947190
This is pretty rough and I can't be bothered to tex it, but look at the partial sum from, say n=N to n=2N. this has to converge to zero in the limit as N->\infty because all of the a_n do. but since this is a partial sum, it must be bounded by the length of the sum times the greatest element. Since we are limiting, we can assume that the a_n are monotonically decreasing and thus the largest element of the partial sum is a_N. So, the partial sum from n=N to 2N is bounded by (2N-N)*(a_N) = N a_N. And we already (sort of argued that this goes to zero in the limit. Note that this shit would not fly in an analysis class but for calc 2 I think it should be fine. Very cool problem for a calc 2 class btw. you must have a good prof.

>>7947177
Please rspeond

I have a good work ethic for this and I've made great progress thus far, what do you all think?

>>7947177
>taking any less than 1 day to learn that material

>>7947336
But all of the shit for the exam, not just understanding theory, but knowing certain algebraic tricks/methods, knowing what the question asks quickly because it's timed I think, etc.

>>7947190
Well you know that the series [math]\sum_{n=1}^\infty \frac{1}{n}[/math] diverges, so [math]a_n < \frac{1}{n}[/math] for sufficiently large [math]n[/math].
So [math]0\leq na_n<1[/math] is bounded. What do you make of this?

>>7947177
>>7947333
If you put in the work you should have a good shot at it. Most of the things in calc I are just slight variations on derivatives and integration, like volume and surface area.

>>7947305
I mean, the inequality I showed is a standard inequality since |sin| is always at most 1, not really much proof needed, and also 1/n->0 is pretty standard, so maybe you want to show the epsilon proof that 1/n tends to 0, but other than that you're just using the squeezing theorem.

>>7947357
>so [math]a_n < \frac{1}{n}[/math] for sufficiently large [math]n[/math].
u wot m8 ?

>>7945935
You have all right to be puzzled because e) actually does not have a limit. For c), the squeeze theorem works

>>7947357
Umm, my only idea is
[math]0\leq \text{na}_n^{1/n}<1\Rightarrow \lim_{n\to \infty } \, \text{na}_n^{1/n}=L<1\Rightarrow \sum _{n=0}^{\infty } \text{na}_n=C(\in \mathbb{R})\Rightarrow \lim_{n\to \infty } \, \text{na}_n=0[/math]
But doesn't taking the limit of something, excuse my lack of terminology, turn [math]<[/math] into [math]<=[/math]? And I think [math]L < 1[/math] is required for the criterion I used?
>>7947320
I think I sort of get what you're saying, I'll have to read through it once or twice more to be sure though.
It's a sort of bonus problem for extra points, and is marked as one of the harder ones out of the available ones. Another one off that list was giving the circumference and area of a sort-of Koch Snowflake (started with a regular hexagon instead of a triangle though).

>>7947432
There are only going to be finitely many elements where [math]a_n\geq \frac{1}{n}[/math], otherwise, the series will not converge

>>7947437
Yeah it does lmao. [math]\sin \frac{\pi n}{2} = (-1)^{n}[/math]

>>7947190
The proposition is false.

Consider [math] \sum_{n=1}^\infty a_n [/math] where [math] a_n = 0 [/math] is [math] n [/math] is not a square (i.e. [math] n = k^2 [/math] for some integer [math] k [/math]), [math] a_n = \frac{1}{n} [/math] if [math] n [/math] is a square.

Then the sum is equal to [math] \sum_{k=1}^\infty \frac{1}{k^2} [/math], so converges.

However, [math] \lim_{n \rightarrow \infty} n a_n \neq 0 [/math] because the limit of the subsequence [math] n^2 a_{n^2} [/math] is 1.

bumpu desu

>>7947830
Yeah no, take [math]u_n = \frac{1}{n^2}[/math] if n is not a square and [math]u_n = \frac{\ln n}{n}[/math] if n is a square. Then [math](u_n)[/math] is summable but it is strictly greater than 1/n for infinitely many n.
As a general principle, you do not want to make pointwise assertions on a sequence based on average facts

>>7946212
This one is easily doable using squeeze theorem.

[eqn]-|x| ≤ x ≤ |x| \\
-1 ≤ \frac{x^2 - y^2}{x^2 + y^2} ≤ 1 \\
∴ -|x| ≤ \frac{x^3 - xy^2}{x^2 + y^2} ≤ |x| \\
\lim_{(x,y) \to (0,0)} -|x| = 0 \\
\lim_{(x,y) \to (0,0)} |x| = 0 \\
∴ \lim_{(x,y) \to (0,0)} \frac{x^3 + xy^2}{x^2 + y^2} = 0[/eqn]

For a really small volume of liquid which method of delivery is better, a pipette with a smaller range or a pipette with a larger range? I have data from a lab showing small liquid delivery (<0.1mL) being more accurate with a pipette with a larger volume range, however I feel like it should be more accurate with a pipette with smaller volume range. Thoughts?

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>>7946212
>>7948936
>me in charge of not making mistakes

Last line should be
[eqn]\lim_{(x,y) \to (0,0)} \frac{x^3 - xy^2}{x^2 + y^2} = 0[/eqn]

>>7948822
well i just conjectured it tbqh, didn't really want to think it through

Are these sufficient definitions for closed and open sets in a metric space (M,d)?

A set [math] S\subset M[/math] is open in [math]M [/math] if
[eqn] \forall x \in S, y\in M: \exists \varepsilon > 0: d(x,y) < \varepsilon [/eqn]
A set [math] S \subset M [/math] is closed in [math] M [/math] if
[eqn] \forall x\in S, \varepsilon>0: \exists y\in S: 0<d(x,y) < \varepsilon [/math]

>>7949197
No, that's bullshit.

>>7943503
>you're being memed.
What's the meme?

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>>7948946
try the hospital rule
*wee woo wee woo*

>>7949266
[math]e^{\frac{1}{12}}[/math]

>>7949269
It was literally just a sign I typed wrongly by mistake in the last step.

I'm sure I got the rest of the working right.

I already posted this in the other thread but to with no answers, please help me.

Im loaning 600000 dollars
the rate is 4,5%
And im supposed to pay back the loan on 20 years

My first question, how much am i going to amortise the first month?

second question, how big is the interest expense the first month?

thanks in advance

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Due to shit Italy bureaucracy I'm going to delay my graduation (Chemistry Bsc) by, most likely, 1 year.

Is this as bad as it seems? I will get my shitty and useless Bsc at 24 years old (the more you know: HS in Italy ends when you're 19).
Is this as bad as it seems? I won't ever get a good phd position, am I right? I womt ever get a decent job in a decemt lab, am I right? is it too late for me?

I want to kill myself.

Is (f+g)'(x) = f '(x) + g'(x) always?

2(2^(n-1) - 1) + 1

Can someone tell me why this simplifies to 2^n-1?

Or at least what I can Google/learn to figure it out. I have no idea what terms to use.

>>7949473
x^n * x = x^(n+1) by the definition of exponents
So 2^(n-1) * 2 = 2^n

>>7949464
With the standard definition of function addition and differentiation.
Oh, and if f and g are differentiable.

>>7949464
Sum rule of differentiation says yes.

>>7949499
>>7949501
Thanks mates.

>>7949491
Thanks

[eqn]2(2^{n-1} - 1) + 1 = 2·2^{n-1} -2·1 + 1 = 2^n - 1[/eqn]

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>>7949464

>>7949512
Meant for >>7949473

>>7949512
>>7949525
Thanks

>>7949507
You're welcome.
If you want a peek at something cool, look up linearity.
The derivative (and integral for that matter) is linear, meaning that if D is the derivative operator, f and g are differentiable functions, and c is any constant,
D(cf+Cg) = cD(f)+cD(g)

>>7949258
D:
How so?

>>7949405
1. That's clearly homework, so maybe that's part of why nobody is answering.
2. You're not even asking a well defined question. The statement "the rate is 4.5%" tells us nothing. What is it? 4.5% per second? Per year? Per month?

>>7947830
an = 2/n converges :^)
Please rely on your definitions

are there any other basic operations (other than addition and multiplication) that have capital letter notation?

>>7949560
The sum of 2/n does not converge.
>>7948089
This is a clever counterexample.

>>7949624
Sure it does
an=2/n=2/n+0=2/n+n-n
Now simply apply the Riemann conditional convergence theorem. The above series converges to 0.

>>7943466
take a topology course (actual point-set topology, not some metric spaces crap) and you'll see. it's not "useless cool sorta thing", how else would you express an arbitrary product of sets ?

>>7943486
isn't that just the factorial?

>>7950010
... That uses the exact definition of the factorial in the naturals

I haven't touched combinatorics in years and this interesting puzzle came up the other day:
You number each card in a deck of cards 1 through 52. You shuffle the deck and draw 10 cards. What are the chances they will be in ascending order?
Any help would be appreciated, this has stumped me.

>>7950097
1/10!

I'm trying to gather some data about the weapons used in a video game in tournaments. The data sets I recorded were the number of times a weapon appeared on a team, how many times that weapon was used on the winning team, and the total matches played.

I planned to create a formula with these values that would give me a numeric ranking, based on how often the weapon is played and how often it wins. The end result was the formula

100(W/A)(A/(2M))

Where W = wins, A = appearances, and M = total matches. The 100 is just to turn it into a whole number. However, I then realized that simplified, the equation was 50W/M, which didn't account for the appearances at all.

What would be a good way to restructure this equation so that it accounts for both the weapon's overall win rate, as well as how often it gets played?

>>7950104
Is that right? Does it not matter how many cards are in the deck?

Do you guys have any go-to websites for scientific definitions of words (sans wikipedia).

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If the kernel of this integral equation is squared, is it still linear or does it only depend upon u(x)?

[math] d \vec{l} = dy \hat{y} + dz \hat{z} [\math]

>>7950220

[math] d \vec{l} = dy \hat{y} + dz \hat{z} [/math]

>>7950220
>>7950222

How do I make it work?

>>7950222
>>7950228

okay... it worked on the 2nd try, but Idk why I couldn't see it for a while.

Hi /sci/, tell me if this proof for [math]\mathbb{R} * \sim \mathbb{R} * \texttimes \mathbb{R}[/math] is correct.

We have [math] G={ \left \begin{array}{ccc} a & c \\ 0 & b \\ \end{array} \right | a,b \in \mathbb{R} * , c \in \mathbb{R} } [/math], and [math] K={ \left \begin{array}{ccc} 1& c \\ 0 & 1\\ \end{array} \right | c \in \mathbb{R} } [/math] ,where [math] K \triangleleft G [/math] .

Let's consider the determinant function [math] \mathrm{det} : G \shortrightarrow \mathbb{R} * [/math] , where [math]ker( \mathrm{det} ) = K [/math].
By the isomorphism theorem, we have [math] G / K \cong \mathbb {R}* [/math].

Now let's consider [math] \varphi : G \shortrightarrow \mathbb{R} * \texttimes \mathbb{R} * [/math], where [math] \varphi \left A \right = ( a , b ) [/math], for some [math] A \in G [/math]. We have [math] ker \varphi = K [/math]. Again, by the isomorphism theorem, [math] G/K \cong \mathbb{R} * \texttimes \mathbb{R}* [/math].

By transitivity we have [math] \mathbb{R} * \cong \mathbb{R} * \texttimes \mathbb{R} * [/math] , and hence [math] \mathbb{R} * \sim \mathbb{R} * \texttimes \mathbb{R} * [/math].

Wonder what this LaTeX abomination is gonna look like.

>>7950295
Fucking hell, can someone tell me how to matrix on /sci/?

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Any tips on this proof? I keep trying to apply the definition but get stuck. Pic related

>>7950316
you do it right. 4chans parsers are just incredibly shit and have a fuckton of bugs

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I don't understand the part where it says 3 = 3a/a

How did they get that assumption? Is it like converting an mixed fraction to an improper fraction?

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can somebody tell me why the shit separation of variables actually works?
i know differentials aren't actually fractions or objects you can manipulate like that
i was also told you can use separation of variables for partial differential equations, which, given my current definition of separation of variables, makes no fucking sense whatsoever

what's going on?

Say I have a standard conic equation. How can I represent it with a matrix? And how can I use the characteristic polynomial of said matrix to find ouch what type of conic it is?

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>>7950723
Actually, I think I'm right. Ignore that.

This is confusing me now.
After the Multiply the Reciprocal step, 3x times by x equals 3x^2. I struggle to understand why the extra x made the 3x an exponent.

>>7950683
What is that div operator? I've only encountered the modulus so far.

>>7950723
.... Oh anon :(
3=3*1, but 1=a/a if a=|=0, so 3=3a/a.
Maybe you're retarded?

>>7950768
Yes I think I may have some form of retardation in maths, in some way. I do these problems and always get some step wrong even though I know how to do that kind of step, it just doesn't come to me when I see it in the equation. My partner, who is more adept with maths than I, has got fed up with me, hence why I'm asking here now.

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>>7950295
>>7950316
>leltex
Just try typing out all the shit without the [math] or [eqn] tags, and let some autist fix it for you.

>>7950730
So you would have something like
If dy/dx=f(x)g(y), then you have that
int(dy/g(y))=int(f(x)dx)
If you don't want to use shitty notation, this really says
int(1/g(y) * dy/dx * dx)=int(f(x)dx)
But from the chain rule,
d/dx(H(y)) = h(y)dy/dx where h(y)=1/g(y) and h is the derivative of H with respect to y.
So you're just working in reverse from the equation
H(y)=F(x).
There's nothing really wrong here. The notation is just shit in terms of rigor, but helpful for intuition.

>>7950746
Oh anon, it's me again :(
You REALLY have some work to do :(
The fucking definition of x^2 is x*x. so...
(3x)*x=3(x*x)=3(x^2) or just 3x^2 since real number multiplication is associative and by the above definition.

>>7950783
I don't mean to be mean. I'll call you retarded but I know it's tough.
With simplification, it's very common to multiply by 1 or add 0 to change an equation. So you might multiply by a/a or add a-a to an equation. Watch out for that or try that sometime. It might help out.

>>7950784
I'm on mobile without a LaTeX extension. Do you fags really need it translated to understand it?

>>7950730
http://www.theshapeofmath.com/oxford/physics/year1/calc/sepdiff

>>7943780
For basic introductory physics all you need is basic differential and integral calc. Once you get into classes like Classical Mechanics and E&M is when you start needing ODEs/multivariable calc/complex variables and then linear algebra doesn't come into play until Quantum usually.

>>7950799
I can't even tell it was supposed to be a matrix. What do you think?

>>7943466
How can you possibly go trough undergrad without this what type of shit school are you at

>>7950809
Maybe he hasn't taken analysis yet and just thinks he's done most of undergraduate mathematics, and maybe his intro to proofs class didn't cover it? But honestly he should have encountered it just by screwing around on wikipedia

>>7950760
the quotient

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Can someone help me understand the calculation steps of this very basic physics problem?

I tried to do it myself but I just can't get the same result. I have no idea how at step (5) they arrive at (m1-m2)*v

Someone please explain it to me and maybe give a tip for problems with two unknowns in general? Thanks.

m1 is the only given value and it's 0.3kg

>>7950867
Oh. Then that's pretty easy. You apply the division algorithm. Note that a =sn+r, b =tn+q, and a+b = pn+w for 0<=r, q, w < n. Now, we have that w=r+q<n.
a+b=(a+b)
sn+r+tn+q=pn+w
sn+tn=pn
n(s+t)=np,
s+t=n
equivalently,
a div n + b div n = a+b div n
Also you might want to note that s, n, t, etc. are unique but I don't think I used that.

>>7950899
Me again. Make it more rigorous in your actual homework. That's just to show you the general idea.

>>7950883
velocity is a vector, you must assign both a direction and a magnitude to it. Pick a coordinate system which has (for example) the x-axis along the horizontal--where each of these velocities lie. Choose the rightward direction to be (+) and the leftward direction to be (-), then assign these directions to the velocity vectors shown. You're now ready to perform vector sums of the momentum vectors. Do this consistently and you get the steps shown.

Whatever happened to that guy that used to come to this board and optimize our images for us. I miss him

>>7950784
>and let some autist fix it for you.
That autist is me. :^)
#MakeSciGreatAgain #DeportPlaintextMaths

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Pic related: I don't understand how to solve 2x = 5x, you subtract 2x from 5x to get 3x.
If x was 4, it's like saying 2*4 = 5*4.. doesn't make sense to me.

Shouldn't it be solved by going:
2x/2 = 5x/2
x = 2.5x

>>7950795
I understand now. If I wrote fully out as 3 * x * x it's like they grouped the two x's together to form the x^2.

>>7951047
You can't use x = 4 in that equation because x = 4 is not a solution. The way to find the solution is to subtract the 2x from the LHS to get a 0. Then you know x=0.

>>7951047
When it says that [math]2x=5x,[/math] don't view it as stating that all [math]x[/math] values satisfy this equation. View it as, "what are the possible [math]x[/math] values that makes the equation true?"

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Number 24. I got x =2, y=4, z=6. But I was thinking that I don't know for sure if the sides are equal to each other. I basically just assumed x+2=4 and x+y=z (so 2+y=z). I'm unsure that those are safe assumptions. Not sure if I'm supposed to solve this based on some tangent of a circle relationship or what. I really just wanna know the solution 100%.

>>7951137
Never mind. Figured it out. Two tangents from the same external point are equal. So x+2=4. And the rest of them equal z.

If I have a Noetherian ring, pick any of its ideals and want to prove there are only finitely many minimal prime ideals containing the ideal, how do I proceed?

I have assumed there exists atleast one ideal for which there are infinitely many minimal primes containing it. Then I get a non-empty set of ideals with infinitely many minimal primes, which by necessity has a maximal element due to the ring being Noetherian. Now, any of these maximal elements can not be a prime ideal themselves, since otherwise they'd be their own and only minimal ideals, and not in the set contradicting the maximality.

Now, choosing any of the maximal elements from the set, I know there exist elements [math]a, b[/math] such that neither of them is in the ideal, but their product is. What I should now do, atleast according to Eisenbud, is to show that every prime ideal containing the chosen maximal element must contain the ideal generated by the chosen ideal and [math]a[/math] or [math]b[/math].

The conclusion is immediate from this, but I just can't seem to make the final step forward.

>>7951173
If the prime contains the maximal element, then it contains ab, hence a or b.

I can't conceptualize what about n!/k!(n-k)! makes it work

I got a high C in my college Pre-Calc class and a B in my high schools honors physics class by being a shitty student. If I spend 8 hours/day doing math and physics for the entire summer, should I be equipped enough to get A's through maybe Calc 3? Will that be enough time to really get good at math.

>>7951378
Of course. That's easy. The hard part is doing 8 hours a day of math during summer break.

>>7950152
Wikipedia...
Depends on the subject
For math I use proofwiki

It stands for 'the product over'. Similar to big sigma, but the operation is multiplication.

>>7951368
deez

>>7951411
it gets a lot easier whenever you don't have many friends and will be without Internet.

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>>7951201
Yes, of course. How silly of me. Thx bb

>>7951468
Thanks me lad

>>7951378
You could study for a total of 8 hours this summer and be well prepared

how do you get the improper integral of an absolute function?

>>7951689
Do you mean the absolute value of a function? Integrate piecewise, basically.

>>7950132
You can ignore unnumbered cards, they don't actually change the probability of drawing a card whose number isn't smaller than the previous one.

>>7951368
Pls respuiond

>>7951823
Think of it like this. You have n objects you can permute in n! different ways. You choose k elements, so n-k remain untouched, there are (n-k)! ways to choose the k elements. Finally, there are k! different orders in which the elements can be chosen.

Because of Lagrange's mean value theorem we know [math]f(b)-f(a)=(b-a)f'(c)[/math], so if b=a+h then [math]f(a+h)=f(a)+f'(c)h[/math]. I define g(h) such that [math]c=a+g(h)h[/math], with [math]g(h)\epsilon (0,1)[/math] so I can write [math]f(a+h)=f(a)+f'(g(h)h)h[/math].

What I am told to prove is that if [math]f''(a)[/math] is continuous, there is a [math]\varepsilon>0[/math] such that for every [math]h\epsilon (-\varepsilon,\varepsilon ), h\neq 0[/math], g(h) is unique. Also I have to determine whether the converse is true.

I don't really know where to start, so any help is appreciated.

>>7952025
Fuck, it's supposed to be [math]f(a+h)=f(a)+f'(a+g(h)h)h[/math] , sorry.

>>7943466
I have a summer vacation to get gud at maths for my Master's. What do I need to understand convex analysis for example? Or regression analysis.

I mean, what background do I require to tackle this level of maths? Is all of Khan enough, or are there better resources? I know some maths, but it doesn't feel intuitive to me, nor do I feel confident in my skills. Finishing my CS undergrad this year

How to construct a "circle" with countably infinitely many points?
What are some properties of an object like this? Is it not connected?

>>7943613
Prius' do this.

>>7943485
Let's make this a new meme! [eqn]\prod_{n=1}^{\infty} e^n = \frac{1}{e^{\frac{1}{12}}}[/eqn]

Hi,

I'm starting to inspect the basic "d / dx" a bit which denotes that the function which follows from it is a derivative.

If it is true that the derivative of x^2 is 2x, why is the transition from one integer x to x + 1
2x + 1
and not 2x?

for example, 9^2 = 81, and 8^2 = 64.
81 - 64 = 17, but (2)(8) = 16.

I notice that each are always one less than the 2x it describes.

Thanks for your help!

>>7953043
which denotes the derivative of a function which follows from it***

I really need to watch how I word things. It is the most important in math. Sorry!

>>7953043
What happens as real numbers approach 9? Think of it like a sequence

How do I get past burnout with my subject?

Math has been my primary hobby for a long time but lately I have no interest in doing any at all.

>>7953068
Start doing challenging things. Everything will eventually become boring if you know it all.

>>7953053
But isn't the statement saying
>for every small change in x, the following function changes by this amount
?

So assume going from one integer to the one that immediately follows from it is as small as you can get. So to edit the sentence as so:

>for every small change away from 8 (now 9), the following function changes by 2x, that is (8)(2), or 64 + 16

>>7953043
Think of limits. As [math]x[/math] approaches [math]c[/math], where is the function heading towards?

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Can someone explain this shit to me, including all steps?

>>7953083
Radioactive decay.
[eqn]-\frac{dN}{N} = \lambda \; dt[/eqn]

>>7953083
ln(N(t)/N)=-lambda*t

Just take the natural log of both sides

>>7952610
Why wouldn't that be a polygon? I guess that's my stupid question for the thread

>>7953095

>Just take the natural log of both sides

i dont even know what that means :(

>>7953102
Natural logarithm: [math]\ln x = e^k[/math]
Or in terms that may make sense, logarithm with [math]e[/math] as its base.

[math]\log_e x = \ln x[/math]

>>7953077
>As 8 approaches 9, the function (64) is heading towards 81, but not quite there because approaching it means it hasn't reached it yet? So since it's not there, 80 instead of 81?

I'm still very confused.

>>7953102
The natural logarithm is a function denoted by ln(x). In any elementary algebra book (or wikipedia) you will see the property which states that ln(e^(x)) = x for any x. Thus to solve an equation for x when you have an e^(x) term, you would be wise to isolate e^(x) by dividing/distracting everything over to the other side of the equation so you can then take then natural log of the e^(x) side and obtain an equation in terms of x. However remember that what you do to one side of the equation you must do to the other to maintain equality so you must also take the natural log of the non e^(x) well. Hence you are 'taking the natural log of both sides' to arrive at an equation that satisfies the problem.

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>>7953107
the derivative graphically is known as the tangent or the rate of change

for x^2 the tangent or first derivative (d/dx) at any point is 2x

now if you visualize this it makes sense that you would not get the same value as going from 8^2 to 9^2 with just using the derivative because the function itself only follows the tangent line for so long

pic related is the visualization for the first derivative

>>7953114
>>7953104
>>7953095
>>7953088

thanks guys

i know for an equation like x^k=A, there will be a k number of roots. How do you go about finding them without a calculator though? Tedious or not, I'd like to know how to solve by hand.
For example, given an equation like x^25+3x+24=0 (arbitrary numbers of roots, constants, etc.), is there a method to deduce all roots (or at least real). I'm thinking of like a super quadratic formula

>>7953245
[eqn]x^k = A \iff A^{\frac{1}{k}} = x[/eqn]

>>7952986
If it works like this, then does the following apply as well?

Let
[eqn]f(x) = \sum_{n=1}^{\infty} \frac{n}{a^n} \sin{a^n x} = \sum_{k=1}^{\infty} \sum_{n=k}^{\infty} \frac{1}{a^n} \sin{a^n x}[/eqn]
where a > 1. E.g. a = 5.

Then
[eqn]f'(x) = \sum_{n=1}^{\infty} n \cos{a^n x} \\
f'(0) = \sum_{n=1}^{\infty} n \cos{0} = \sum_{n=1}^{\infty} n = -\frac{1}{12}[/eqn]

>>7953258
Well.. [math]n \cos{0} \righarrow n \cdot 1 \rightarrow n[/math] thus completing the meme. Therefore, yes. (Assuming this is [hopefully] in radians)

>>7953261
Oh, good. Then on a semi-related note, I'm reposting this question from the other SQT:
>>7944645
>>7944665

>radians
Well, duh. If it doesn't have the degree symbol, it's in radians by default.

>>7952610
>>7953101
It's called an apeirogon.

What makes the hyperbolic plane so intuitive for visualizing infinitely extending objects?
I only know it's hyperbolic. Why does that lend itself so well to the inspection of infinity?

>>7953314
It's not really clear what you mean so it's hard to say. I guess you could say the disk model of hyperbolic space (https://en.m.wikipedia.org/wiki/Poincaré_disk_model) gives some idea of how objects can "extend to infinity" as you can see lines that have a finite length (for the euclidean distance) but infinite length for the hyperbolic metric, maybe that's what you were talking about ?

I'm going to be a sophomore in fall and I have only completed pre Calc. I need like 8 more math classes Calc 1-4 and then some specialized stuff. Am I way behind? I've been focusing on gen ed stuff and knocking out business classes so I can get into my business school.

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"f is a polynomial function of degree k if f can be presented in the form..."

I don't understand this, I didn't think the coefficients mattered in degree of a polynomial. This I think says (let's say k is 5) "the fifth coefficient (the coefficient of the term who is raised to the fifth power) plus the 4th coefficient plus the third, etc" and that means for a polynomial to have a degree of 5 it must look like (cn^5+cn^4+cn^3+cn^2+cn^1+cn^0). But I thought for example (3x^5+7x^2) was a polynomial of degree 5.

Oh, but I guess you can have c=0 for some of those and then my example would fit. Except not, because the book says c can't equal 0.

>>7953563
My dad was in the same situation as you, he has a bachelor's in CE. He managed to complete it in 4 years by taking summer courses.

>>7953769
Shit dude. I completely forgot to mention that I'm going for a math minor and not a major. Kind of the most important info that I left out.

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>>7953766
Hee it is again. How is that a polynomial of degree n? What does a sub not even mean in this context?

>>7953766
>Oh, but I guess you can have c=0 for some of those and then my example would fit. Except not, because the book says c can't equal 0.
It says [math]c_n \ne 0[/math]; all of the others can be 0. You should work on your reading comprehension. Clearly if the nth coefficient is 0, then the polynomials doesn't have degree n.

>>7953833
Shit, I meant [math]c_k \ne 0[/math]. Who uses n for the variable?

"In a probability experiment with a finite number of outcomes, every possible outcome will be obtained in a finite number of trials."

Is there a name for this 'conjecture?'
Is this statement true?

>>7953833
Well yeah clearly. But I mean, why do they make the nth coefficient correspond to the exponent of that term? It doesn't make any sense.

>>7953863
>Is there a name for this 'conjecture?'
Gambler's fallacy
>Is this statement true?
No

how do I solve f(x) = x^x ?

>>7953975
Depends on f.

>>7953996
I guess I should rephrase.

how would I solve an equation like x^x = k, where k is a normal integer?

>>7954004
that was literally the topic of a thread at the weekend, search the archive

>>7953947
So it can't be proven that the amount is not infinite?

>>7953766
Okay and this one, what's the difference between c sub k and c sub i

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Create a turning machine to compute:

[math]
f(x) = x \mod 3 = 1, \text{ Rewrite tape to contain a 1 followed by infinite blanks.}\\
f(x) = x \mod 3 = 2, \text{ Rewrite tape to contain a 2 followed by infinite blanks.}\\
f(x) = x \mod 3 = 0, \text{ Rewrite tape to contain a 0 followed by infinite blanks.}\\
\\
\text{All tapes begin with } \bigtriangleup \\
x \text{ is a natural number in unary} \\
\not{b} \text{ is the blank symbol}
[/math]


Did I do it right?

are higher level, published theoretical CS papers/books based on ZFC?

I know that for the derivative of x divided by x, we can substitute the derivative of log x but is there something similar for x divided by its derivative?

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Hi guys, my first time running a study, having trouble with actually conducting the study. Skip to the $ if you don't want backstory.

I want to study the delay rate of buses in my city X received calls to customer service, tweets, and fb posts There's an API endpoint that tells me if the buses are ahead or behind schedule and updates every 2-3 minutes, but only tells me this when I ask for information about a specific bus stop. I'm also given the entire log of received calls, tweets, and fb posts, which are already labeled as a negative experience, positive experience, testimonial, or neutral. There are 8804 stops, and 247 bus routes (494 for both directions). 10000 API calls per day.

$
The problem is that observing one bus stop for one day would cost 480 API calls. So 20 bus stops will be monitored. I came to this number by:
>24 hours * 60 minutes = 1440 minutes / day.
>1440 minutes / 3 minute updates = 480 API calls to monitor 1 bus stop for 24 hours

I've thought about sampling:
>Method 1
I observe 20 bus stops, get their bus delay level every 3 minutes.
>Method 2
I observe 40 bus stops by alternating the stops that I watch every three minutes. So at 10:00 AM, I get the delay level for stops A1, B1, C1, etc. Then at 10:03 AM, I get the delay level for sensors A2, B2, C2, etc. Then at 10:06 AM, I get the delay level for sensors A1, B1, C1, etc. I think I can interpolate the 6 minute difference for each bus at each stop.
>Method 3
Use method 2, but stop from 12 AM to 7AM. This should give me more stops to observe.

I think method 1 would give me a more accurate sample set because I won't need to interpolate any data in between the points of delay for each bus that visits the sampled stops. However, method 2 gets a wider overall set of stops.

I still think that ~50 stops isn't a lot to work with and compare with customer service interactions. Is there a better way to conduct or do you guys think that this study is doa?

Thank you guys in advance.

I'm a medfag with no background in computing or mathematics. I have an opportunity to study a large epidemiological database using R. I don't know the first thing about writing loops etc. To go from no background to competent (not relying on my supervisor daily), what kind of task is that?

>>7954680
To become competent I say it would take you a year to two years.

To simply get comfy with programming I say it would take you 6 months.

To get comfy with the basics I'd say it would take you two weeks to two months.

>>7954694

Don't mind me, I just migrated from an old machine where the quick-reply 4chan capability failed for some reason-maybe I turned it off. Anyway I'm not replying to you, just proving out that I can preview LaTeX again via the above, before posting it in this Balinese puppet show. Test, test, toast, test.

[math] \displaystyle \int\limits_{tomato}^{potato} "46x" dx + 2 = A, [/math] , where "A" is a very special Down babby.

>>7954851
Leibniz notation is the pure defintion of eyecandy

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>>7943466
How can I estimate this? My book doesn't give any hint, and I'm too tired to think properly.

>>7954924
>implying can think if not tired.

>>7954924
well w seems to have a critical point when t=0 so

>>7954928
Not really, he only implied that if he can think then he is not tired

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>>7954930
How so? It would be zero.

Here's a similar example with an answer. I don't know what this problem wants from me. Its bad enough the partial derivative is ugly.

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>>7954937
It was not 0, it was 0.838

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>>7954987
I got the answer, but I still dont understand. Pls halp.

>>7954987
0.838 is 0 +/- 1

>>7954851

one more piece of latex testing on this machine:

[math]

\displaystyle

\begin{bmatrix}

W & E & L & C & O \\
M & E & T & O & T \\
H & E & N & E & X \\
T & L & E & V & E \\
L & S & E & G & A \\

\end{bmatrix}

[/math]

>>7956126
najs latex, anon-ku

Can I do a chemistry major with poor math?

>>7956329
Yes all you need is basic multivariable calc and linear algebra which anybody can get through

>>7956335
Good to know

Trig Identity problem.

This problem uses degrees.
How do I get sin75 to look the same when using the sum of sin and half angle for sin.
When I use sin (45+30) I get sqrt(3)-1/2sqrt(2).
When I use sin (150/2) I get sqrt[2sqrt(3)]/2.
Both sides can be changed but I struggle to make them appear the same.

>>7956780
wew lad

>>7956803
I've tried using exponent notation instead but I still cant figure out how to make them appear the same :((((((((

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What? How the fuck?
What does he mean with "pattern of coefficients"?

>>7957025
Isn't the author basically telling you that you should know what the coefficients mean. Like observing the +'s and -'s?

>>7957107
>>7957025
Also horner

>>7957025
plug -1 in as x.

3 + 1 - 10 + 2 + 4 = 0, as desired so now you know (x + 1) is a factor of the equation so divide it out

>>7957107
>>7957112
Yes but how does he just guesses that -1 is a root just by looking at the coefficients?

It's useful in calculating impedance.

>>7957120
that should always be your second guess if you have no idea where to start, second only to 1 of course

How does one divide a decimal by a decimal using the European style of long division? I can't find anything outside of division of two whole numbers using this method.

>>7957188
I mean that I know how to divide it. But I can't explain where one figures out how to place the decimal point correctly.

>>7943613
It's called regenerative breaking.
All electric cars do this

>>7946181
The limit exists, yeah, but I don't think the answer is 1/28?

>>7957490 here, never mind. It is.

>>7957490
it is, dumbo

>>7957207
Multiply both numbers by 10 until they're whole numbers. Hopefully you should understand why this doesn't change their quotient.

why can 1/sqrtx be written as x^(-.5)?

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Capital Pi is used in Computer Engineering / Logic to describe the canonical form of maxterms of various logic diagrams i.e. Karnaugh maps, truth tables, etc

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are there any sites where I can find only concepts, without mathematic explanations or references, about astronomy and biology? or even some blogs you know if they're good, I don't mind.

>>7954207
I'm from the g sqt. I don't understand your notation or even what that chart is meant to be. An instruction needs 5 parts, if in state x and encounters symbol 0, set the state to y, write symbol 1 and move left for example.

Posted in the other question thread and got no reply... probably not the most interesting problem, but...
For euler's equations for angular velocity, without forcing the solutions can be thought of as the intersection between the two ellipsoids given by the equations of the conserved properties

[eqn]
E = \tfrac{1}{2} A \omega^2_1 + \tfrac{1}{2} B \omega^2_3 + \tfrac{1}{2} C \omega^2_3 \\
H^2 = A^2 \omega^2_1 + B^2 \omega^2_2 + C^2 \omega^2_3
[/eqn]

But when there's forcing, as shown below, this obviously ceases to be true. I suspect that the solutions now lie on the surface of a cone, but I've fittered around, and have no idea how to prove this.

[eqn]
A \frac{d\omega_1}{dt} + (C-B) \omega_2 \omega_3 = -k \omega_1 \\
B \frac{d\omega_1}{dt} + (A-C) \omega_2 \omega_3 = -k \omega_2 \\
C \frac{d\omega_1}{dt} + (B-A) \omega_2 \omega_3 = -k \omega_3
[/eqn]

>>7959665
It's a state diagram, the circles are the states and the transitions are the state changes with the instructions on them

example:
State 'S' is the start state, the transition means, "when a triangle is read, write a triangle, move Right", then change state to "1".

>>7959117
Well it's still the product of all the terms.

>>7959101
Because [math] \frac{1}{ \sqrt{4} } = 4^{-.5} [/math]

>>7959683
your question is badly posed. are you trying to solve those coupled differential equations or what?

>>7946181
[eqn]\lim_{(x,y) \to (2,0)} \frac{1 -\cos{y} }{7xy^2} = \lim_{y \to 0} \frac{1 -\cos{y} }{14y^2}[/eqn]
This is literally a single variable calculus question disguised as a multivariable calculus question.

Who the fuck sets these kind of questions?

>>7961179
>these
this*

>>7961179

lol

fucking first year math does it, reminds me of all the shit I had to do.

I hate in physics when half the question is a unit conversion problem, that just boils my piss, like hoenstly it doesn't prove anything

didn't get an answer in another thread so ill ask again. whats a good resource to learn integration by parts that i dont have to buy

>>7961946
patrickjmt

>>7961946

What's to learn? It's pretty clear cut t.bh. Anything like

[eqn] xe^{x}[/eqn]

or a function multiplied by another function in general is a good candidate for IBP

>>7959750

These ODEs describe the rotation of a body, and my aim is to find the range of possible motion.

They cannot be solved analytically, thus the need to think about the problem from another perspective, and I feel a geometrical approach is best, as it was very effective in the unforced problem.