/mg/ - math general

Can you gain much from a more rigurous study of calculus instead of the "engineering" approach?

>tfw no satisfactory definition of mathematics

Now that the 'applied math isn't math' meme has been debunked can we have some healthy discussion about control theory, preferably without the autismic high schoolers trying to convince everyone they're the math police?

>>9076050
>Polite reminder that >>>/sci/sqt exists and there is no need to duplicate it in here. Please do try to look up the answer to your question elsewhere before posting it (especially if you're asking for textbook recommendations). Anyway:

how to kill an already dead thread 101

>>9076050
I would be interested in a reading group depending on the textbook since I am but a mere brainlet

>>9076080
Appreciation for the subject

In his Ars Magna, 16th century Italian algebraist Cardano derives for the first time, (a version of) the general solution for the cubic. In order to do this, Cardano is obliged to use very old machinery and thought processes, which leads to an onerous task.

In particular, Cardano does not expressly treat of our modern version of the general cubic equation (or the monic analogue with leading coefficient of one, as you like), [math] ax^3 + bx^2 + cx + d = 0 [/math], being obliged instead to consider /several specific cases/, such as, say, [math] N + x^3 = ax [/math] or [math] N = ax^2 + bx + x^3 [/math], where [math] N [/math] is the constant term. For Cardano, an equation must have (potentially) /non-zero stuff/ on either side, which obviates today's general form. Cardano's restrictions on the forms which he can write (which give rise to the multiplicity of cases which he is obliged to consider) therefore invite a few counting problems which are disctinct from his real goal of solving the cubic, and which can be generalized unto themselves.

The purpose of these few posts, which are not meant to be taken too seriously, is simply to present a few simple counting formulas which correspond to a complete treatment of what Cardano was about in this "enumerating" situation, without going into the substance of Cardano's algebraic problems. Readers who find this dull are invited to skip a bit until they find posts more their speed.

-The rules of the game go like this: begin with the general form of a univariate polynomial equation of degree [math] n-1 [/math]. Such an object has precisely [math] n [/math] terms on its LHS, and a zero on the RHS. Throw out the zero term, the RHS. Each side of the equation must have at least one of the [math] n [/math] terms already enumerated on the LHS.

The basic question is, for a given degree of polynomial (or, number of LHS terms), how many of "Cardano's equations" are there? We answer this and a few related questions by deriving and defining three (sequences of) numbers.

In the first place, all conceivable equations can be given, with multiplicity, by a series of multiplied permutations. Let [math] 0, 1, 2, 3, ... [/math] be shorthand to denote the distinct objects the constant, linear, quadratic, and cubic term, and so on. What is of interest just now is not a polynomial's degree, but how many things are available to work with.

By Cardano's restrictions, we must have at first, a minimum of two terms to work with - an equation with at least one term on either side. A series will be defined, ranging from two (things, terms) up through [math] n [/math] itself (all the terms), with dummy [math] k [/math]. In each case, precisely the [math] k [/math]-combination of [math] n [/math] objects is what is called for. And in each such case, there are [math] k - 1 [/math] possibilities for placement of the equality sign, with the rest being filled in by addition signs (Cardano does not write subtractions in his forms, and for our part we're unconcerned with the sign of a given coefficient). All of this put together gives rise to the first sequence, the [math] n [/math]th Cardano numbers of the first kind, which are given by

[eqn] C_{1_{n}} = \sum_{k=2}^{n} ( _n P_{k} ) (k-1) [/eqn]

where [math] ( _n P_{k} ) = \frac{n!}{(n-k)!} [/math] denotes the number of [math] k [/math]-permutations of [math] n [/math] objects.

>>9076050
Let [math]A,B[/math] fin. gen. [math]\mathbb{R}[/math]-algebras.

Does [math]A{ \otimes _\mathbb{R}}\mathbb{C} \cong B{ \otimes _\mathbb{R}}\mathbb{C}[/math] imply [math]A \cong B[/math] ?

This is all well and good as an exhaustive counting tool, but clearly it doesn't quite answer the real question, "how many distinct equations are there, really?" And why not? Because [math] C_{1_{n}} [/math] counts equations with multiplicity, counting equivalent equations such as [math] 1 = 2 + 0 + 3 [/math] and [math] 3 + 0 + 2 = 1 [/math] separately. We therefore need to account for and eliminate multiplicity, in the ensuing (sequence of) number(s).

The example was chosen specifically to call attention to three aspects of equations involving sums: an equation which merely has its sides flipped is equivalent to the first one, order of addends does not matter to the sum, and when an equation, or a /string of information/ of this type is presented, it specifically entails /its own mirror image, or reverse/ as an equivalent expression. In other words, the symmetry of the equality relation and the commutativity of the addition operation attach, and consequently an equation's "mirror-image" is also equivalent to itself. These are what must be accounted for.

Now by way of a concrete example: suppose that one looks at quartics, so that the available five terms to work with are [math] 0, 1, 2, 3, 4 [/math]. Again, the number of terms to be considered ranges in cases from two up through [math] n [/math] (five). Prerequisites of judging two equations to be equivalent in our situation are that they should have the same number of terms on both sides, and symmetrically about the equality sign (on either side). The mere partition of the business into cases ranging from 2 through [math] n [/math] immediately addresses the former, a later subtlety will address the latter.

Choose some specific [math] k [/math]. What is the activity that is to take place?

-A /[math] k [/math]-combination/ of the [math] n [/math] elements is to be selected.
-For each such [math] k [/math]-combination, the combination is to be further partitioned by placement of the equality sign, with at least one element on both sides. These will be the [/math] j [/math]-combinations of the given [math] k [/math]-combination(s). Order is no longer pertinent due to commutativity, hence combinations.
-Note that to consider the [math] j [/math]-combination of terms on one side is simultaneously to consider same on the other side. And this precisely because combinations are symmetric (visualize Pascal's triangle). Thus, simply running through once with attention paid to one side is enough, and does not occasion any double-counting. This and the previous point deal with the above "subtlety".
-nevertheless, a factor of 1/2 must yet be introduced apart from the above. And this precisely in order to do away with each equation's mirror-image, which had not yet been done above, as a distinct consideration (and not to be confused with the other points).

The result of these considerations leads directly to the [math] n [/math]th Cardano numbers of the second kind, which produces Cardano's forms for a polynomial of appropriate degree, without double-counting. It is given first as per the above, and readily rearranges to a nicer form

[eqn] C_{2_{n}} = \sum_{k=2}^{n} \bigg( {n \choose k} \frac{1}{2} \sum_{j=1}^{k-1} {k \choose j} \bigg) = \sum_{k=2}^{n} \bigg( {n \choose k} (2^{k-1} -1) \bigg) [/eqn].

The [math] n [/math]th Cardano numbers of the third kind, then, are simply given in terms of the above just identified, and refine further to do away with cases, included with a particular degree but not including that polynomial's leading term (we don't necessarily want to enumerate quadratics while enumerating cubics):

[eqn] C_{3_{n}} = C_{2_{n}} - C_{2_{n-1}} [/eqn]

>>9076399
>>9076449
>>9076560
>>9076590
Actual math in /mg/!? What heresy is this!? Appreciate the historical post anon

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At this point, it is worth making a short table, just to get the concrete flavor of things. As you'd expect, the first sequence blows up (due to being based upon permutations, which also blow up), and although the latter sequences also increase, they are held in check to a good extent by their combination-terms, but the power terms push the business ever upwards The nth Cardano number of the second kind greatly improves upon the multiple-counting of the first, and the items highlighted red are those most immediately pertinent to Cardano, representing counts on the cubic and quartic cases, from his vantage point (though a cursory look suggests that even Cardano couldn't be fucked to do several dozen of these as the quartic numbers suggest, instead bringing his algebra to bear on families of cases):

[math]

\begin{array}{cccc}
n & C_{1_{n}} & C_{2_{n}} & C_{3_{n}} \\
2 & 2 & 1 & 1_{def.} \\
3 & 18 & 6 & 5 \\
4 & 132 & \color{red}{25} & \color{red}{19} \\
5 & 980 & \color{red}{90} & \color{red}{65} \\
6 & 7830 & 301 & 211 \\
7 & 68502 & 966 & 665 \\
\end{array}

[/math]

This effectively ends an exercise I'd set myself some time ago (and got stuck on), though I still want to check some other bits for myself. Interestingly, "the number of cases" that Cardano considers at various points in his text do not seem to match perfectly with the numbers given above, but at least this information puts me in a good position to judge Cardano's work more systematically, which is what I was really after (in Ars Magna's Chapter II, Cardano enumerates equations in a quite goofy way by our standards).

Other exercises I leave myself include to prove that such-and-such is always less than such-and-such (induction), and to double check my notion of the number of the third kind by a kind of justification (I have written a bit hastily here but I think my intuition is good, I initially went about this 'justification' in a way which was demonstrably false).

Is there a known example of a function [math]f:\,\mathbf R^2\,\longrightarrow \, \mathbf R[/math] such that [math]\forall \theta \,\in\, \mathbf R,\, \lim_{x \,\to\, 0} f\left( x\,\cos\,\theta,\, x\,\sin\,\theta \right) \,=\, 1[/math] but [math]f[/math] is discontinuous at [math]\left( 0,\, 0\right)[/math]?

>>9076050
I came across this https://arxiv.org/pdf/1703.03827.pdf

It claims to be a proof of the Riemann Hypothesis, still reading it over, haven't gotten around to checking the work. What do you guys think about it?

if you don't know what a hyperloglog is and how it works, you need to leave this thread and leave /sci/

Hello everyone, i dont understand this.
I tried looking for help elsewhere online, but none of the explanations really helped.

Determine whether the function f (x) = x2 from the set of integers to the set of integers is
one-to-one.
Solution: The function f (x) = x2 is not one-to-one because, for instance, f (1) = f (−1) = 1,
but 1 = −1.
Note that the function f (x) = x2 with its domain restricted to Z+ is one-to-one. (Technically,
when we restrict the domain of a function, we obtain a new function whose values agree
with those of the original function for the elements of the restricted domain. The restricted
function is not defined for elements of the original domain outside of the restricted domain.) ▲

This is discrete Math.

>>9076876
I'm assuming x2 is x^2.

A one to one function is a function that is bijective - that is, no two numbers in the domain map to the same number in the range, and every number in the range is achieved by some number in the domain.

x^2 is not one to one because 1^2 = (-1)^2 = 1. In fact, x^2 = (-x)^2 for all x, not just for 1.

Ok, that makes a bit more sense!
Thank you very much, ive been studying Discrete Math from Rosen's textbook, and it tends to be very dry sometimes.

Oh, but one thing, isnt it supposed to be injunctive? The book says that one to one is an injunction...

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I've been stuck on this integral for a while (pic related). I wanted to learn about the Lambert W Function and I came across this formula on the Wikipedia page. I've been trying to figure it out for the better part of a week but I can't seem to get anywhere near to figuring it out. Can anyone help/ give me a hint?

>>9076908
You mean injective?

It's kind of confusing, one to one can mean either injective or bijective depending on the context... but the example above wasn't injective so it's the same idea.

>>9076876
take this crap to the stupid questions thread. this is literally an immediate application of a definition that you evidently haven't tried to understand.

>>9076920
[math] \int_{-\infty}^{+\infty} e^{-x^2} = \sqrt{pi} [/math]

Convert to polar coordinates

>>9076935
Thanks, I'll give that a shot.

>>9076117
>the 'applied math isn't math' meme has been debunked
It's not a meme and its hasn't been "debunked". There's nothing to debunk in the first place since it's a fact. Applied math isn't math.

i have a trig final on wednesday

what formulas should i put on the 2 3x5 notecards my professor is allowing?

>>9077036
>trig final
You have to be 18 to post here.

>>9077036
All you need is [math] \text{sin}^2x + \text{cos}^2x = 1 [/math] mate.

>>9076935
[math]\sqrt{\pi}[/math]

>>9077043
I'm 24 and going back to school.

>>9076050
Ive been lurking this board now for some time. I was trying to learn Calculus this summer. I just finished my quantitative chem, any recommendation?
Also im supposed to take it next semester. So can anyone help?

>>9077056
>quantitative chem
You're confused. This is the math general.

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>>9076050
>What are you studying this summer?
C, Pthon, Embedded Hardware

>Have you come across any interesting problems, theorems, articles or books lately?
www.gold-saucer.org/math/lebesgue/lebesgue.pdf


>Are you open to the suggestion of starting an /mg/ reading group, where we'd go over a paper or textbook together?

Yes. With whiteboards.

>>9076534
Yes.

>>9077058
I know. but its just for anybody to stardarize at what level i am.

>>9077068
standardize*

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>>9076399
>>9076449
>>9076560
>>9076590
>>9076774
The history lesson is appreciated. Where did you copy this from?

>>9076166
How can you solve differential equations if you're a finitist? Shouldn't you replace that with difference equations? Or is this yet another case of play-pretend just like Wildberger, where they claim to be finitists but use mathematical tools that depend on some conception of infinity anyway (like limits)?

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>>9076933
>>9076934
You're both so very right...

>>9077033
>bunked

>>9077074

New posts and derivations freind-o. I have a special interest in the history of mathematics. I am not learned in other topics and admittedly the math itself is pretty low, but I dig this stuff now that I have time for it. The above posts represent my own simple and original questions relating to historical items.

Previously on /sci/ I've covered the Rhind Papyrus and the basic derivations of the cubic and quartic, albeit with simple caveats. If you want I'll link this autism. Cardano's Ars Magna invokes about twenty of Euclid's props (I've fully understood them recently), so that I now feel in a good position to move through the rest of Cardano's text, I feel like I have the appropriate vocab and toolkits ready.

I also did a few dumb-easy lemmas about Euler Bricks and Perfect Cuboids (do they exist?), and I've been meaning to pick that back up as well. In the course of this, I realized that there was an important relationship to square triangular numbers, which are explicated in a paper due to Nyblom, and likewise in "Pythagorean Triangles", a very short and elementary, yet tantalizing text by Sierpinski. The course of this has caused me to better appreciate Sierpinski's general power-ranking.

Future projects include the Elements proper and Disquisitiones Arithmeticae (I got stuck on page two!)

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>>9076810
Read what you just wrote again anon. But read it slowly this time.

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>>9077109
Very nice. I haven't read any old maths manuscripts myself. This is interesting.

>>9077122

I'm itching to stroke it so I'll take this as licence.

My initial movement through the Rhind papyrus is here. This groping was the basis for 80% of the item's current wiki:

https://warosu.org/sci/thread/S8027713#p8027824

Where I pretty-much derive the cubic:

https://warosu.org/sci/thread/7529602

and the quartic:

https://warosu.org/sci/thread/7613239

After this I slightly scratched resolvents, but again got stuck and then real life happened for some months.

An old Euler Brick thread once I got going on lemmas (apparently I did two):

https://warosu.org/sci/thread/S7496579

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Forgive me if this is a /SQT/ question, but I'm after some good books on the latter half of Calculus as well as complex numbers/analysis and vectors.

Particularly from a physics/engineering standpoint.

My uni course descriptions say:
Calc 3:
>polar forms, parametric equations, and vectors.
>will also cover indeterminate forms, improper integrals, and sequences and series.

Calc 4:
>multivariate and vector calculus, moments and centroids, surface area, volume, line and surface integrals including the theorems of Green, Stokes, and Gauss.

Each are 16 week classes squeezed into 8 so you can take both in one semester.
I'm a bit concerned and would like to get a headstart so I don't fall permanently behind because I got stuck on one or two lessons.

>>9077141
Why stop at the quartic (you mean, biquadratic)? Let's see the formula(s) for the roots of the quintic!

>>9076810
does f(0,0)=0 and f(x,y)=1 for all (x,y)!=(0,0) work?

>>9077083
>that depend on some conception of infinity anyway
He doesn't really have anything against a "concept of infinity" if it can be specified finitely.

Just signed up for ordinary differential equations, linear algebra 2, statistics and numerical analysis for my next semester math classes. Physics 2 for my non-math class.

Anything I can expect from this?

>>9077193
ODEs is easy, stats is easy, linear algebra 2 is easy (if you went to a good school, all the good content from upper div 2 would have been put in upper div 1), and numerical analysis is easy. i'm not good at physics, but a class called "physics 2" is probably easy

>>9077204
Nice. Hopefully it will leave time to self study. The calculus class I took last semester was calc 3 so how does that compare to ODEs in terms of difficulty. I know that now that the calc sequence itself is over ODE is a whole new thing but I just want to know in terms of difficulty how it compares.

>>9077209
did your calc 3 cover differential equations? it's like calc 3.5, more of the same.

>>9077212
No, calculus 3 was just generalizing calc 1 and 2 into [math] R^n [/math]. We didn't cover differential equations.

We did see some differential equations in physics 1 because first order linear ordinary differential equations popped up but I remember we derived a general solutions for those so I suppose the actual ODE course will not linger so much on that.

>>9077219
is it even an upper div course? my school had six lower div math courses: single variable calc 1 & 2, multivariate calc 1 & 2, linear algebra, and differential equations.
in either case, whether it's your first diff eq course or ODEs, they're pretty easy to work with. PDEs are in another ballpark though

>>9076821
>It claims to be a proof of the Riemann Hypothesis
It claims to be a proof of something false then.

>>9077235
>something false
proof?

>>9077235
Do you have proof that it's false? If not then please kys.

>>9077234
What is the distinction between an upper div course and a lower div course? Anyways, I think it is a lower div course given that it is the first differential equations class I take and it is a sophomore course.

All the serious classes seem to happen next semester. Everything from algebra, topology and analysis so perhaps that is what you mean by upper div courses.

>>9077237
Of what?
>>9077238
>Do you have proof that it's false?
I do.
>kys
What does this mean? Sorry, I'm not familiar with reddit "culture".

>algebraic statistics for computational biology
op af
First section first example it tells you how to do EM by transforming your loglikelihood into a polynomials in terms of groebner basis, bypassing the need to do that idiotic E/M step.

>>9077247
>>Do you have proof that it's false?
>I do.
Post it then

>kys
What does this mean? Sorry, I'm not familiar with reddit "culture".

nice b8 m8

>Post it then
I will, if you rewrite your post without redditry.

>>9077247
>Of what?
RH's falseness

>>9077036
sin^2x+cos^2x=1
sin(2x)=2sinxcosx
sin(x^2)=(1-cos(2x))/2

Those are the only three I remember and have ever used.

>>9076821
Right and wrong. Those mathematicians that dislike the supposed "lack of rigor" in physics should also reject statements proven assuming generalized RH/CH.

>>9077267
RH/CH?
Please forgive me I'm an undergrad.

>>9077271
Assuming anything that is not proven (except axioms lol) cannot yield a proof. Any mathematician thinking otherwise is an idiot.

>>9077279
Could this potentially lead to a substantial proof though?

>>9077271
>RH/CH?
RH = Riemann hypothesis
CH = continuum hypothesis

>>9077285
You're responding to copypasta

https://warosu.org/sci/?task=search&ghost=&search_text=Assuming+anything+that+is+not+proven+%28except+axioms+lol%29+cannot+yield+a+proof.+Any+mathematician+thinking+otherwise+is+an+idiot.

>>9077285
Assuming anything without proof can only be used to prove that it is wrong.

>>9077141
However interesting the content, the formatting of your post is dreadful.
>>>/r/eddit

>>9077188
No. The limit to (0,0) from any direction would be 0 in that case, not 1.

>>9077287
>>9077288
I need to get out less. I'm neglecting math too much.

>>9077292
>The limit to (0,0) from any direction would be 0
why?

>>9077295
Because that is the value at (0,0), duh.

>>9077299
That's not quite how limits work, here have a read:
https://en.wikipedia.org/wiki/Limit_(mathematics)

>>9076080
absolutely. ability to think critically and abstractly will be vastly improved. capacity for problem solving increase a ton as well.

>>9077301
That's exactly how they work. Calculate the limit if you don't believe me. It's 0.

is there a field of math dedicated to the "pure" study of operations? would it just be a subset of abstract algebra?

https://en.wikipedia.org/wiki/Operation_(mathematics)

>>9077305
The value of a function at the point you're taking the limit to doesn't affect the limit, here have a read:
https://en.wikipedia.org/wiki/Limit_of_a_function#Functions_of_more_than_one_variable

>>9077310
Is this what you want?

https://en.wikipedia.org/wiki/Universal_algebra

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>>9077305
>Calculate the limit if you don't believe me. It's 0.
>It's 0.
The fuck?

>>9077316

why is this such a niche field??? it seems like the purest form of mathematics possible

>>9077327
It was subsumed by category theory

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>>9077193
Physics 2 is usually electromagnetism, right?
The hardest stuff that'll show up there is gradient operators and three dimensional vectors.

You'll have vector fields for electric fields and magnetic fields, and two electric and magnetic transverse waves coming off of particles with third vector descriptions.

Maybe brush up on your sinusoidal wave formulas? Asin(kx-wt) stuff? That might come up a lot. Young's Double Slit experiment proving wave-particle duality is based on the fact that the light from two slits interfere or reinforce one another based on how their phases shift as their respective distances to the screen change.

And if you cover that you'll also cover constructive/destructive interference for all kinds of other waves, like dead spots in radio broadcast areas or perhaps the angle of reflection or refraction of light traveling in a prism, which reflects internally if the angle of incidence is too great, but refracts out of the prism if the angle is small enough.

>>9077325
>The fuck?
High schoolers don't understand delta epsilon yet.

>>9077327
It's not niche at all.

>>9077325
Limits of 0 are the cornerstone of calculus.

As the separation between two points composing a line approach 0, you get the instantaneous slope: The derivative.

As the width of columns under a curve approach 0 and thus the number of columns approach infinity, the accuracy with which the sum of the columns' areas approximate the actual area under the curve approaches the true value: The integral.

>>9077335
great math post
>>>/sci/pg

>>9077335
see >>9077340

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>>9077292
>>9077299
>>9077305
>>9077339

>>9077112
Back to high school, weeb.

>>9077315
Alright, give me an [math]\left( \varepsilon,\, \delta\right)[/math] proof of [math]\lim_{x \,\to\, 0} f\left( x \right) \,=\, 1[/math] where [math]f:\, x\, \longmapsto\, \begin{cases} 1 \qquad\text{if } x \,=\, 0 \\ 0 \qquad\text{otherwise} \end{cases}[/math].

>>9077346
Do you not know how to read? Re-read >>9077188

>>9077340
>>9077342
links to damn near every thread because every thread has an incidence of "pg" in it, brainiacs.

Also, I was answering a question about what to expect from physics 2, math-wise, since math nerds usually can't handle much beyond rote calculation and proofing.

Physics 2 is pretty straightforward.
Lots of algebra.
Some calculus.
Vector fields out the ass. Dot and Cross Products abound.
Trig should be good if the class goes into waves.
Trig should be very good if the class goes into geometric optics from waves.

>>9077353
>>>/sci/physics general

>>9077356
Strike two.

>>9077352
[math]\left( \varepsilon,\, \delta \right)[/math] proof needed.

>>9077358
No one made any claim about the function you posted

>>9077353
>>9077356

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>>9077346
>all this confusion over what the limit at (0,0) is because he misread what function that anon defined
This is such a cute mistake. Can I hug you?

>>9077371
>>9077371
>Can I hug you?
I'm not who you're quoting but I'd still like a hug.

>>9077371
Same here >>9077377

>>9077371
It's more than just misreading, see this post >>9077299 for a fundamental misunderstanding of what limits are

>>9076050
>Are you open to the suggestion of starting an /mg/ reading group, where we'd go over a paper or textbook together?
As long as it doesn't involve d*scord or other similar garbage.

>>9077399
lame

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>>9077401
Someone is jealous.

>>9077371
This is the last time I am going to repeat it.

>>9077315
>The value of a function at the point you're taking the limit to doesn't affect the limit
Give a formal proof, or fuck off back to Rddit, >>>/a/ or whatever shithole you came from.

>>9077384
Maybe he meant "that is the value of the limit at (0,0)" while he was thinking of the misread function.

>>9077413
Repeat what?

>>9077413
>Give a formal proof, or fuck off back to Rddit, >>>/a/ or whatever shithole you came from.
It's true by definition, here have a read:
https://en.wikipedia.org/wiki/Limit_of_a_function#Functions_of_more_than_one_variable

>>9076821
Vladimir Blinovsky is a known fraud. I don't understand why arxiv staff haven't deleted his submissions and banned his account yet.

>>9077424
That's not even what it says.

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>>9077442
That is exactly what it says, what about the delta-epsilon formulation is confusing you? It may seem like a lot of symbols at first glance but you'll get used to it eventually

>>9077453
>for [math]\frac12 \,>\, 0[/math], there exists [math]\delta \,>\, 0[/math] such that [math]\forall \left( x,\, y \right) \,\in\, \mathbf R^2,\, \left\| \left( x,\, y \right) \right\| \,<\, \delta \,\Rightarrow\, \left\| f\left( x,\, y \right) \right\| \,<\, \frac12[/math]
Which [math]\delta[/math]?

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>>9077299
do you even study math?

>>9077305
you are assuming the function is continuous.
it clearly isnt.

>>9077573
>it clearly isnt.
Prove it.

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>>9077573

you specifically
>>9077610

>>9077614
What are you even on about? Prove that the function isn't continuous or fuck off.

>>9077650
Burden of proof is on you since you made the claim.

>>9077651
I didn't make any claims.
>burden of proof
There is no such thing in math. You can fuck off to >>>/lit/

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>>9077655

>>9077656
You might be retarded. Fucking weakling.

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>>9077658

>>9077666
So you can't prove that the function is discontinuous?

>>9077678
You were the one to claim otherwise. Instead of posting proof you doubled down on your stupidity. It's reasonable to assume your posts are just bait.

>>9077682
>You were the one to claim otherwise
I wasn't the other poster, I thought I made it pretty clear.
>It's reasonable to assume your posts are just bait.
It's reasonable to assume you can't prove that the function is discontinuous.

>>9077682
>It's reasonable to assume your posts are just bait.
and yet people have been angrily replying for hours upon hours
as expected of math general autists

>>9077703
>as expected of general math autists
ftfy

Summer truly is the worst season.

>>9077699
>It's reasonable to assume you can't prove that the function is discontinuous.
[eqn]f\left( \frac1n,\, 0 \right) \,=\, 1 \,\xrightarrow[n \,\to\, \infty]{}\, 1 \,\neq\, 0 \,=\, f\left( 0,\, 0 \right)[/eqn]
Now go on with your "I WAS JST PRTENDING XDD," and go back.

>>9077678
>prove that the function is discontinuous
preimage of {0} is an open set

>>9077739
This isn't a "proof". Now I'm pretty fucking you're just baiting.

>>9077158
Try math methods by Arfken Weber, not the best book but it has what you need.

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ls there something like trigonometric functions(sin,cos,tan) but defined out of other closed functions instead of a circle like an ellipse?

Can anyone help a retard?

"Let ODD-ODD be the language of strings, over the alphabet {a,b}, that contain an odd number of a’s and an odd number of b’s.

Let ODD-ODD be its complement.

Prove that PALINDROMES⊆ODD-ODD."

The wording of this is throwing me off as a new person trying to teach myself maths. Isn't the complement of a set everything that isn't picked in the set? So out of {a,b} if {a} is chosen then {b} is the complement?

So what exactly is being asked? An odd number of a's and b's put together can never make a palindrome because one must be even. How would I "prove" this? Induction? Contradiction?

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>>9076050
>What are you studying this summer?
I'm trying to prove that Peano arithmetic is inconsistent. I think I'm pretty close to some good results.

How do you select exercises to do?

>>9077800
>Can anyone help a retard?
If you rewrite your post without reddit spacing.

>>9077810
Can anyone help a retard?
"Let ODD-ODD be the language of strings, over the alphabet {a,b}, that contain an odd number of a’s and an odd number of b’s.
Let ODD-ODD be its complement.
Prove that PALINDROMES⊆ODD-ODD."
The wording of this is throwing me off as a new person trying to teach myself maths. Isn't the complement of a set everything that isn't picked in the set? So out of {a,b} if {a} is chosen then {b} is the complement?
So what exactly is being asked? An odd number of a's and b's put together can never make a palindrome because one must be even. How would I "prove" this? Induction? Contradiction?

>>9077800
>An odd number of a's and b's put together can never make a palindrome because one must be even.
What are you trying to say here? Not only does a palindrome not have to be even (aba), but also that is irrelevant since odd+odd=even.
>Isn't the complement of a set everything that isn't picked in the set?
Yeah, but that's a definition for sets. Although I'm pretty sure the definition of a language complement is similar in kind - all strings in the language's complement are not in the language. So the following gotta be wrong:
>Let ODD-ODD be its [ODD-ODD's] complement.

>>9077801
It has already been proven consistent anon.

>>9077835
But palindromes can never be an odd amount of a's and an odd amount of b's. One has to be even, e.g.:

aba
abbba
aabaa
ababa
bbbaaabbb
aabbaa

I guess what i'm asking is; am I meant to show that no palindrome exists that is of odd a's and odd b's / how would I prove that / is that what it's asking?

>>9077842
https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof

>>9077847
>Not only does a palindrome not have to be even
sorry, reading comprehension problems - I thought "one" was referring to "palindrome" not to "a or b".
>I meant to show that no palindrome exists that is of odd a's and odd b's / how would I prove that
You need to show that every palindrome is of the form (word rev(word)) (obvious evenness of every letter's count) or (word [a or b] rev(word)) (only one letter's count is odd)

>>9077842
I know for a fact that it's inconsistent, now I'm just trying to prove it.
>>9077849
Did you even read this page yourself?

>>9077861
Thanks! I'll look into how I might achieve this. I'm confident in solving a solution but proving it is a whole new game for me which i'm finding incredibly difficult.

>>9077801
You're Russian aren't you?

>>9077937
I don't know, am I?

>>9077942
дa

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What are your favourite maths related youtube videos, /mg/?
https://www.youtube.com/watch?v=wsQWLaUn7jo

>>9077945
Wait. Really? Truly?

>>9077772
Looks neat. Thank you.
I balk at the price, though.
I'll have to shop around but a first search says $60 is the cheapest I can get it.

I was about to start reading Calculus for Dummies out of desperation.

>>9077473
None, because the limit isn't 0. Please learn to read before attempting basic limit questions.

>>9077796
sinh and cosh

>>9078068
You can download it online pretty easily, hell just google the name and the first pdf you can download from

>>9078100
Oh damn, that worked. Thank you.

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>>9077193
Expect the U(1) gauge group.

>>9078069

cf
>>9077292
>>9077299

>>9078092
What shape are they based on?

>>9078294
>What shape are they based on?
They are called the hyperbolic functions so... I think perhaps they are based on squares.

>page 10
Don't die on me baby!
(Leibniz > Newton.)

>>9077863
What's wrong with Gentzen's proof?

>>9079825
There is nothing "wrong" with it, it just doesn't prove that there are no contradictions in PA.

>>9077937
Why do you think so?

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>>9079871
Same

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I can not tell if this is meant as a stab at certain people or not

>>9079894
it's a trick question, because "decisions" don't exist and who cares about poor people anyway?

t. economist

official /mg/ notepad
https://etherpad.net/p/KJ6LvtHFZs

>>9076166
>high iq
>aderall

I though being smart rids you of attention span related problems? :thinking:

>>9079930
>https://etherpad.net/p/KJ6LvtHFZs
The fuck is this?

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>>9076810
>[math]\mathbf{R}[/math]
No such thing.

ewwwwwwwwww, why would you want to study math lol

>>9080199
Because math is true and truth is beauty anon. Why do you hate beautiful things?

>>9080199
Because math can be shown to be homotopy equivalent to anime.

>>9077327
Just applied model theory

>>9077247
>reddit "culture"
If you don't know what kys means you haven't been here for more then a week.
Lurk before posting or go back to R*ddit.

>>9077327
All mathematics is niche. Reality check: less than 1% of the world's population knows calculus.

>>9080442
AND at least half of that 1% think calculus is DIFFICULT.

>>9080422
>If you don't know what kys means
I don't. Since I'm not familiar with reddit "culture".

>>9080649
As I said "kys" has been part of 4chan culture for many years.
If you don't know what it means you are a redditor (or underaged) and you need to continue lurking.

Someone told me ALL polynomials with real number coefficients can be factored into a product of linear factors and irreducible quadratics.

Is this correct? This isn't self evident to me.

>>9080718
Yes, it's a property of the Fundamental Theorem of Algebra and the Complex Conjugate Root Theorem:
https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem

Basically a polynomial P(x) has a root "c". Either c is Real or not real (i.e. Complex with imaginary component).
-If c is real then P(x) = (x - c)*P2(x), where P2(x) is a polynomial of smaller degree.
-If c isn't real then by the Complex Conjugat Root Theorem (and the quadratic equation), P(x) = (ax^2 + bx + d)*P2(x) for some a, b, d which are easily derived from c and its conjugate.

And you can keep factoring P2(x) until it's too small to factor any more (i.e. until it's linear or quadratic).

Given a number [math] n [/math] and its prime factorization [math]p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_k^{\alpha_k}[/math], how can I find the largest divisor [math] d \, \vert \, n \, \text{such that} \, d < \sqrt{n} [/math]? it's computationally infeasible to list all possible divisors

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Is it true that a believing heart is your mathematics?

>>9081006
Easy man, consider the number
[math] d = \prod p_i^{\lfloor\frac{a_i}{2}\rfloor} [/math]

I'll let you figure out the special cases. Now read my meme >>9081800

>>9081809
>Easy man, consider the number
>[math]d = \prod p_i^{\lfloor\frac{a_i}{2}\rfloor}[/math]
that wont work homie
your meme sucks btw

>>9081829
>that wont work homie
I didn't say that was the solution. I said you should consider it.
>your meme sucks btw
Well, no surprise a brainlet cannot appreciate true magic.

>>9081832
>I didn't say that was the solution. I said you should consider it.
i considered it, it didnt work

>>9081836
>i considered it, it didnt work
You are not considering it right. You should not consider it as the solution, you should simply consider it. See what happens.

There are three questions you need to answer to understand your problem

1) Are the any cases in which my [math] d [/math] is the solution? Why or why not?
2) Are there any cases in which it isn't the solution? You already found out this is the case so... why?
3) How close/how far can my [math] d [/math] be from the real solution

And then as I said, you should consider all the cases that will pop up. But to do this you must have a believing heart because. I want to make a world where everyone can smile with each other. A believing heart is everyone's magic! NOCTU ORFEI
AUDE FRAETOR!!!!!!!!!!!!!!!!!!!
SHINYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
ARUKKU!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1

>>9081800
>all those chink hacks listed
Kill yourself my man. Only Europeans and some Elevens can do mathematics.

>>9081879
>racism

Ah, poor soul. You are forgetting Phasansheer Shearylla. I have wished upon the Grand Triskelion for a world in where EVERYONE can smile with each other. No discriminations and no suffering!

A believing heart has always been the source of mathematics, but if you cannot bring yourself to believe in the believing hearts of other people then you may never achieve a truly believing heart yourself and therefore you will never achieve great mathematics.

And even if the future I chased with all my might is an illusion... I can never fly if I'm always looking down-I shout out to the sky! Embracing the flower that's grown in this heart, still unable to give up. I'll step down a new path... and into a new future!

>>9081893
You need psychiatric help.

>>9081902
You are not understanding my point. I want to see the new world that will expand once I reach the shooting star I have been chasing so frantically.

Let's search for a continuation to this story that we can't bare waiting for! Racism is wrong because the future doesn't exist for one person alone, or even one race alone. The scenery may change as we proceed, but we'll never forget this twinkling sky we call mathematics, and the treasures we have found along the way!

>>9081906
see
>>9081902

>>9081921
How does it feel to not have a believing heart?

does anyone have a god tier list of math books for a rigorous understanding on every topic?

>>9081988
Bourbaki.

>>9082045
Name a more comprehensive collection on mathematics monographs with the same level of rigour.

>>9081988
Why do you want it if both of us already know you won't work through them?

>>9082045
>>9082055
Nice moronic posts. Every retard can quibble. Either contribute something or fuck off, brainlet.

>>9082061
The post claiming Bourbaki covers every topic doesn't exist you stupid nigger. There is no collection of monographs that covers every topic in mathematics.

>>9082071
See >>9082050
>>9082067

>>9082079
No one claimed it exists. The point was that Bourbaki is the maximal solution to that guy's question. Fucking autists and their inability to read between the lines.

>>9080717
>As I said "kys" has been part of 4chan culture for many years.
I don't consider subreddits like /v/ or /b/ to be "4chan culture".
>If you don't know what it means you are a redditor (or underaged)
I am neither.

oh no nash is dead I had forgotten/not noticed ;_;

>>9082226
>get a nobel prize for studying best strategies
>die because you rode a car without using the seatbelt
And who am I quoting? Moses at the gate. He must have had a laugh.

>>9080383
This is why Grothendieck became a hermit.

>>9082249
>reddit spacing
Your kind isn't welcome here.

>>9082273
see >>9082266

>>9082284
>reddit """culture"""
see >>9082266

Ok can someone tell me if this proof is correct?

>Let [math]G[/math] be a finite group of odd order. Show that every element of [math]G[/math] is a square.

Suppose otherwise. Then [math]\exists g\in G[/math] which is not a square. Consider the group generated by [math]g^{-1}[/math]. It must have odd order since [math]|G|[/math] is odd. Then for some [math]k\in\mathbb{N}[/math], we have [math](h^{-1})^{2k+1}=e_G\implies h=(h^{-k})^2[/math], contradiction.

>>9082307
accidentally switched from [math]g[/math] to [math]h[/math] at the end

I'm 21 and I'm a NEET, but I'm already tired of this life. Is it too late for me to start a math degree?

>>9082320
Depends. If you're in the US or anywhere you need to pay (exorbitant) tuition fees, you're better off without a degree. In those places college has become a debt slavery scheme/scam. Just self-study.

>>9082319
>>9082313
ok how about this one (generalisation to group of any order).
>Let [math]|G|=n[/math] and [math]gcd(k,n)=1[/math]. Then the function [math]\phi:G\to G[/math], [math]g\mapsto g^k[/math] is surjective.

Suppose otherwise, and let [math]h\notin \phi(G)[/math].We have that [math]\exists a,b: ak+bn=1\implies h=h^{ak+bn}=(h^{a})^k[/math].

>>9082259
as well as an abel prize a few days prior

>>9082334
No, in my country college is free and passing in the national exam to get to a good university is not that hard. What I'm worried about is my age, people generally start college with 17 or 18, but I've wasted all those years and I'm already 21, I think I'm already too old to seek an academic career, but at the same time there's nothing else I could do other than mathematics.

>>9082340
Let [math]|G|=n[/math] and [math]\gcd(k, n)=1[/math]. If [math]h\in G[/math] is arbitrary, then, by Bezout's lemma, there are integers [math]a, b[/math] such that [math]ak+bn=1[/math], and so [math]h=h^{ak+bn}=h^{ak}=(h^a)^k[/math]. No need for counterassumptions this time either.

>>9082344
Oh right!

>>9082354
[math] h = h^{ak+bn} = h^{ak} = (h^a)^k [/math]
pls work

>>9082259
>>9082354
A literal avatarfag ? Is this the yukariposter's new guise ?

>>9082367
Just report every single post. That retard got banned pretty quick.

>>9082352
There are people in their 30s and 40s going to college to study a new subject. You're too self-conscious about your age for some other reason I'd wager, and your academic career is just an excuse.

>>9077339
>approach "infinity"
No such thing.

>>9082352
In my school there's someone over 30 while most of us are 21 and another one has a daughter.

Isn't Pi evidence that math as we know it is flawed and that it should be redone? Nobody knows the exact value of Pi, the trillions of numbers after the decimal place cannot be calculated. Math is just "approximate" and that's fucking bullshit, I want a system of calculating numbers that works.

For the sake of absolute precision, precision that you would need to calculate being able to shoot a thread into a needle from a thousand miles away and, more realistically, for space travel, math just doesn't work.

>>9082495
Gödel already proved that you can't actually prove anything in mathematics.

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>>9081800
Truly mathematics is the most magical of disciplines.

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Let [math] \mathbb{P}^n(K) [/math] be the [math] K [/math]-valued points of an [math] n [/math]-dimensional projective space over a field [math] K [/math]. For [math] A = (a_{ij}) \in GL(n+1, K) [/math] we define an automorphism [math] {\varphi}_A [/math] of the polynomial ring [math] R = K[x_0, x_1, \dots, x_n] [/math] by
[math] {\varphi}_A (x_i) = \sum\limits_{j=0}^{n}a_{ij}x_j [/math] for [math] i \in \overline{0,n} [/math]
We have that [math] G({\varphi}_A) = \mathbb{P}^n_K [/math] and [math] {\varphi}_A [/math] determines a projective transformation
[math] f_A = ^{\ \ a}{\varphi}_A : \mathbb{P}^n_K \longrightarrow \mathbb{P}^n_K [/math]

Prove that:
[math] 1^{\circ} [/math] [math] \forall \alpha \in K^{\times} = K \setminus \{ 0 \}[/math], [math] f_{\alpha A} = f_A [/math]
[math] 2^{\circ} [/math] [math] \forall A,B \in GL(n+1, K) [/math], [math] f_{AB} = f_B \circ f_A [/math]
[math] 3^{\circ} [/math] Let [math] A^{-1} = (b_{ij}) [/math]. Then [math] f_A \left( (a_0:a_1:\dots :a_n) \right) = \left( \sum\limits_{j=0}^{n}b_{0j}a_j : \sum\limits_{j=0}^{n}b_{1j}a_j : \dots : \sum\limits_{j=0}^{n} b_{nj}a_j \right) [/math]
[math] 4^{\circ} [/math] There exists a unique projective transformation [math] f [/math] mapping distinct points [math] P_1,\ P_2 [/math] and [math] P_3 [/math] on [math] \mathbb{P}^1_K [/math] to distinct points [math] Q_1,\ Q_2 [/math] and [math] Q_3 [/math] (in that order)

>>9082504
Retardedly wrong.

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>>9082495
"real" numbers don't really exist.

>>9082495
>Isn't Pi evidence that math as we know it is flawed and that it should be redone?
No.

>>9076050
Given
[math] (-1)^{n} < \sqrt n\ \epsilon [/math]
If I square both sides do I need to change the inequality? Since the LHS could be negative

>>9082266
Sure thing newfag, you know best.

Now LURK MOAR.

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>>9082383
>tfw you have so many humanities students in your classes recently who love "alternate mathematics" without infinity or other "hard" subjects you can't tell if /mg/ is memeing you or not

>>9082495
Your knowledge of the term "math", just like your overall knowledge of math is very limited.

So stop spouting bullshit about "math not being suited for space travel" because it "only approximates" (which is the basic principle of engineering) this is non-sensical.
I get that some finitists have valid points, but you don't, since it seems improbable that you have completed a course in calculus and comprehended it's content.

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>>9082495
Irrational numbers outnumber rational numbers.
They are the norm, not the exception.

>>9082645
Please tell me you're joking anon

>>9082655
I dunno. I'm just a student.
Infinity is used in all of my serious classes.
But we have some students and one professor who are extremely anti-infinity and study the work of some dude, I forget his name, who is remodeling math around not having infinity.

It seems like they replace infinity with several other assumptions, which rubs me the wrong way.

The thing these students typically have in common is they're Environmental Science/ Environmental Engineering/ Meteorology, etc, tree hugging hippy types with minors in intersectional social justice and similar topics. A lot of em think "white man's math" is "probably wrong" and needs to be reviewed by "other cultures" so those other cultures can make their own contributions.

>>9082652
>Irrational numbers outnumber rational numbers.
How can something nonexistent outnumber something which exists?

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>>9077299

>>9082690
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

>>9082713
Don't bother. He's obsessive about posting how infinite things don't exist. You will be wasting your time if you keep replying.

>>9076935
Okay I finally got around to doing that and I'm left with

[math]\left (\frac{1}{2} \right ) W\left ( 2cot^2\left (\theta \right ) \right )sec^2\left ( \theta \right ) = r^2[/math]

I'm not really sure where to go from here.

>>9082668
>It seems like they replace infinity with several other assumptions
Such as?

>>9082713
It presupposes the existence of "real" numbers, which simply don't exist.

Should I just remove the axiom of "infinity" or should I outright negate it?

>>9082776
Abnegate urself faggot. Inaccessible cardinals exist.
Deal with it.

>>9082808
Sorry, we don't discuss such fairy tales here. This is a Math thread.

>>9082808
>Inaccessible cardinals exist.
Prove it.

Is there any language (aside from english) that I should learn if I aim to do research?

>>9082855
Russian ))

>>9082855
The language of mathematics.

>>9082855
French & German. (Some would meme Chinese too, but I disagree. China's scientific output is meagre compared to its population, and most of it is low quality. Very intense plagiarising going on there.)

>>9082859
And this >>9082857
Russians have a very good maths tradition and they're pretty insular.

>>9082855
If you have to pick one then French

What is the most occult type of mathematics?

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>>9083047
IUTeich

>>9082861

I've visited a local math-only library a few times recently, and I was impressed by how many Soviet/Russian math journals there were that I had never heard of.

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>>9082855
Maple and Mathematica

Is it true that a believing heart is your mathematics?

>>9083952
this thread is cancerous enough without you trying to force your pathetic non-meme every single day
kill yourself

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>>9083961
>kill yourself
But why? You don't know me. You don't know my story. Please allow me to fix that

I am a lonely PhD student. I do not have friends, I do not have gf. All I have is anime and mathematics, but despite all of that I had been very depressed lately but then an anime appeared in front of me: Little Witch Academia. I had never watched the previous material there was so for me it was completely new.

Week by week, day by day, I came to identify with Kagari Atsuko (Akko for short). The way she was clearly talented but yet no one recognized her for what she was inside really resonated with me. She is like an anime female version of me except for that one thing: her believing heart. Even though she and I shared the same struggle there I was depressed while she had a good fucking attitude about everything! Even in episodes where I would cry for her because of all the shit people put her through, she would never cry with me. She kept that beautiful smile all the way. That was the difference, she had a believing heart that I didn't. That was my missing piece. My missing spark! So I learned from her. I started to believe in myself more for one but then the message against elitism of Little Witch Academia really hit me deep:

I no longer make threads about mathematicians being superior because Akko is right. We need a world in where EVERYONE can smile with each other. Even if they are a mathematician and a biologist, they should be able to see each other eye to eye and smile together! That is what I want for me and for everyone! And that is why I made my image. So that other people will see the value of Little Witch Academia. It saved me so... why can't it save you?

A believing heart is your magic, my friend! A believing heart is everyone's magic!

>>9083976
is this the new MLP

oh I think I get entropy of information now.

It is the average of bits needed to represent the likelyhood of each outcome, weighted by probability of the outcome ?

realizing that averaging, expectation, normalizing, etc is all the same operation just with different methods of weighting feels pretty big to me, but maybe I am just dumb.

>>9084013
No, it is better. "Friendship is magic" is a right idea but it is incomplete. Friendship is part of magic (Phasansheer Shearylla) but it is not all of it. The purest form of magic is a believing heart!

The world could be shitty, life could be shitty, everything could be shitty. But if you only have a believing heart then you can change it. If you allow me to quote the second opening:
Donna toki mo shinjiteiru yo kono sekai wo, jibun no yume wo!

Or as we would say in english: I'll never stop believing in this world or in my dreams!

You see, you need to believe. Not only in yourself but also in your dreams and even more importantly in this world! You need to believe in this world's potential for greatness. You need to truly believe that even in life is shitty then life will allow you to mold it and make it better. This uncontrollable feeling is the signal of a new beginning! Our feelings are still in a deep sleep, so lets wake them up!

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>4chan fails to answer a really fucking simple question
>try to ask question on reddit, deciding to give it a chance
>question has to pass moderation to be publicly viewable
>moderator tries to answer the question themself
>they get it completely wrong because they assumed that the question was wrong (lolwut)
>they think they've answered the question so they won't make the question public
Reddit makes me want to throw bricks at things.
But it's cold outside so I won't do that.

Anyway, say that I have a spring of zero size attached to a wall, and a stiff rod attached to that spring.
What's the formula for how much that rod will bend away from the normal of the wall?

I tried to figure it out, but I kept on getting stuck at x/cos(x) = # (x is angle, # is a known number)

>>9083952
Fuck off. You're cancer.

>>9084099
>>>/physics/ general

>>9084112
Shinjiru kokoro ga anata no sugaku?

>>9084115
It's question with hard maths and simple physics.
If I ask a physicist, they'll mistake the question for a different one with hard physics, no matter how fucking hard I try.

Seriously, it's like they're trolling with how impossible it is to get physicists to interpret that question correctly, literally all of them will give an answer for a bending rod.

/mg/ reading group thread here
>>9084140#

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Lurkin
>Hard

>>9082056
i promise to work through them

>>9081006
take logarithms
it turns into a special case of knapsack problem
you might want to read about existing algorithms for this and see if they are fast enough for your liking

>>9084099
>try to ask question on reddit
Redditors aren't welcome here.
see >>9084115

>>9084289
>try to ask question on reddit, deciding to give it a chance
Pretty sure a redditor doesn't "give reddit a chance".
Also, I gave a reason I didn't ask the physics general.

>>9084294
A redditor is anyone who uses the website "reddit.com" in any capacity. You are a redditor and your kind isn't welcome here.
see >>9084115

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So is this a (maybe oversimplified, but still) way of understanding the updating of the weight representing the slope in gradient descent for linear regression?

The update is given by
[math] w_1 \leftarrow w_1 + \alpha(y-h_w(x))*x [/math]
where (x,y) is a data pair, and h_w(x) is a line [math]w_1x + w_0[/math] being fitted

And what I am wondering about is whether this is a sensible way of understanding how positive an negative values of the "error vector" ends up working as hoped:
there are four cases, barring a perfect fit: [math]h_w(x)[/math] might be too high (higher than y), so that the parenthesized term is negative,
or it might be too low, so that it is positive.
x might also be negative, or it might be positive.

case 1: [math]h_w(x)[/math] too large, x positive. This is (a) in the image.
In this case, the second term in the update is negative, which results in a decrease in slope, which again adjusts (a) downwards.

case 2: [math]h_w(x)[/math] too large, x negative. This is (d) in the image.
In this case, the second term in the update is positive, which results in an increase in slope, which again adjusts (d) downwards.

case 3: positive delta*positive x, (c), increase slope (c upwards)

case 4: positive delta*negative x, (b), decrease slope (b upwards)

I am just having issues imagining the effect positive or negative values of X might have on the direction the weight is adjusted in, without mapping out each case like this, if it is even the right way of thinking of it.

What stronger statement should I prove to gain the negation of RH as a trivial corollary?

>>9084924
What's with all these retarded questions you're asking? Why are you even here for fuck's sake?

>>9084115
If you can't answer the question, just say so dummy, instead of spamming

>>9082202
This is some pretty good bait

>>9084246
>a special case of knapsack problem
based
thank you anon

>>9076050

does [math]x^a + x^b = O(x^c)[/math] for some nontrivial A, B and C?

this isn't fermat's last theorem is it?

>>9085022
Very rude.

constructivism

rational analysis

natural analysis

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Is there any real reason to remove axioms instead of negating them?

>>9076166
I have 130 and I don't do any of those memes.

>>9085888
What's the difference? I'd imagine instances where you can prove an axiom using the other axioms are few and far between, and otherwise it sounds like you're talking about the same thing.

>>9085946
>What's the difference?
The theorems you prove may or may not hold in other systems.

>>9085955
Yes. Systems with different axioms. But how is "removing" an axiom different from "negating" it? In both cases, you would have to throw out all proofs which use the axiom.

>>9085970
God doesn't exist vs we don't know that God exists.

>>9085979
"God doesn't exist" is (if unproven) an axiom. One that just happens to use negative terminology. The fact that "not" is used does not make it any less formally assertive.

"We don't know (yet) that God exists" is, you could say, a negated axiom (or at the very least not an axiom). Well I guess there are SOME assertive truths in there, but they are very minor and not particularly constructive.

>studying numbers
holy shit this is autism
why would you do this

>>9085970
>But how is "removing" an axiom different from "negating" it?
The things you can prove change. Negation of excluded middle implies the negation of AC. This is pretty different from AC being independent.

>>9077052
I respect your dedication, anon. I hope it goes well.

>>9085993
Oh, I see, so you're saying that Axiom "X is Y" is replaced with a new Axiom: "X is not Y."

I guess in certain situations some of the old proofs can be modified into new complement proofs. But I think most of the time things would be much muddier and you'd still just be throwing out most of the old proofs completely out the window (and replacing them with completely new proofs).

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Dear /Math general/ , I'm in a math competition and need to get good at number theory (divisibility, mod equations and that sort of stuff), combinatorics (semi advanced stuff), geometry puzzles and problems (prove said square is part of x circle using cyclical squares), and algebra (certain systems of equations). The stuff is at math olympiad level difficulty and if I can win I can get a scholarship. Anyone have any good resources on this (I know I'm asking for entry level things but that's what will be on the test).
thank you in advance

can someone explain induction on the reals as if I was a total brainlet?

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>>9086092
>the "reals"
No such thing, anon.

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>>9086100
y-yes, of c-course...

b-but let's just play a fantasy for a moment and assume they exist.

>>9086092
Easy. Put the real numbers on a list and then define the succesor function of that list.

>>9086110
How does one define a "successor" function from the empty set to the empty set?

What do I have to prove to show once and for all that assuming the existence of "real" numbers leads to a contradiction? I want to prove a meta-theorem of some sort.

>>9086115
[math] f(1) = \{ \} [/math]
[math] f(S(x)) = \{ \}, x \in \mathbb{N} [/math]

>>9086127
Consider these
Definition: Infinity is the biggest thing there is.
Theorem: The natural numbers are infinite

Now assume the real numbers exist. It is possible to prove that the set of reals is bigger than the set of naturals and this contradicts the fact that infinity is the biggest. You found something bigger than the biggest and that is a contradiction.

Q.E.D.

>>9086128
>[math]x \in \mathbb{N}[/math]
So [math]x \in \emptyset[/math]?

>>9086133
>"Infinity" is the biggest thing there is.
This isn't even close to being well defined.
>The natural numbers are infinite
This is inconsistent with the negation of the axiom of "infinity".

>>9086135
Jesus fucking christ if the naturals do not exist then what exists? Please show me a number you agree exists.

Frank Stenger, a mathematician from the University of Utah, claims to have proved the Riemann Hypothesis.

https://arxiv.org/abs/1708.01209

>>9086138
>This is inconsistent with the negation of the axiom of "infinity".
If you are negating the axiom of infinity then what the fuck are you asking for a contradiction then? The reals obviously contradict the negation of that axiom you fucking retard.

>>9086139
I didn't claim the naturals do not exist, I merely stated the obvious fact that the "set" [math] \mathbb{N}[/math] doesn't exist.

>>9086149
>>9086149
>>9086149
NEW
>>9086149
>>9086149
>>9086149

>>9086143
I want to have the choice of either negating it or not. It's clear what to do when it's false, but I need some other options for my friends who still believe in fairy tales.

>>9086140
Kek Hi Frank. Neat idea there, see you in 5 years when either we will all be mocking you for being retarded or sucking your cock.

Also, very nice restraint by not calling G the Frank function because we all know how these things go. I bet that if your proof turns out to be true then people will start calling it the Frank Zeta Function or something like that. Good luck.

>>9086069
Titu's books are standard for USAMO/IMO training.

>>9086150
Homotopy Type Theorist detected

>>9086127

Part of the Wildberger argument is that things like pi literally cannot exist because the universe has finite "memory"

>>9086449
The "type" of naturals doesn't exist either.