Polite reminder that 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:
What are you studying this summer?
Have you come across any interesting problems, theorems, articles or books lately?
Are you open to the suggestion of starting an /mg/ reading group, where we'd go over a paper or textbook together?
Other urls found in this thread:
yutsumura.com
arxiv.org
warosu.org
warosu.org
warosu.org
warosu.org
warosu.org
en.wikipedia.org
en.wikipedia.org
en.wikipedia.org
en.wikipedia.org
twitter.com
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?
...
>Polite reminder that 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
I would be interested in a reading group depending on the textbook since I am but a mere brainlet
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.
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]
Actual math in /mg/!? What heresy is this!? Appreciate the historical post user
>What are you studying this summer?
I've been reading Johnstone's book on topos theory, Atiyah's book on topological K-theory and Brown's book on cohomology of groups.
>Have you come across any interesting problems, theorems, articles or books lately?
Technically it is none of these, but I must shill for the Mitchell-Bénabou language. You can just take an arbitrary topos, and you can use a language almost that of set theory there!
Just a guess but what if you considered elements of the form [math]a\otimes1[/math], and trying restrict the isomorphism to these?
Do you guys know this site btw? yutsumura.com
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]?
I came across this arxiv.org
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 Veeky Forums
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.
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...
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?
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.
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.
[math] \int_{-\infty}^{+\infty} e^{-x^2} = \sqrt{pi} [/math]
Convert to polar coordinates
Thanks, I'll give that a shot.
>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?
>trig final
You have to be 18 to post here.
All you need is [math] \text{sin}^2x + \text{cos}^2x = 1 [/math] mate.
[math]\sqrt{\pi}[/math]
I'm 24 and going back to school.
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?
>quantitative chem
You're confused. This is the math general.
>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.
Yes.
I know. but its just for anybody to stardarize at what level i am.
standardize*
The history lesson is appreciated. Where did you copy this from?
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)?
You're both so very right...
>bunked
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 Veeky Forums 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!)
Read what you just wrote again user. But read it slowly this time.
Very nice. I haven't read any old maths manuscripts myself. This is interesting.
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:
warosu.org
Where I pretty-much derive the cubic:
and the quartic:
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):
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.
Why stop at the quartic (you mean, biquadratic)? Let's see the formula(s) for the roots of the quintic!
does f(0,0)=0 and f(x,y)=1 for all (x,y)!=(0,0) work?
>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?
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
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.
did your calc 3 cover differential equations? it's like calc 3.5, more of the same.
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.
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
>It claims to be a proof of the Riemann Hypothesis
It claims to be a proof of something false then.
>something false
proof?
Do you have proof that it's false? If not then please kys.
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.
Of what?
>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.
>>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.
>Of what?
RH's falseness
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.
Right and wrong. Those mathematicians that dislike the supposed "lack of rigor" in physics should also reject statements proven assuming generalized RH/CH.
RH/CH?
Please forgive me I'm an undergrad.
Assuming anything that is not proven (except axioms lol) cannot yield a proof. Any mathematician thinking otherwise is an idiot.
Could this potentially lead to a substantial proof though?
>RH/CH?
RH = Riemann hypothesis
CH = continuum hypothesis
Assuming anything without proof can only be used to prove that it is wrong.
However interesting the content, the formatting of your post is dreadful.
No. The limit to (0,0) from any direction would be 0 in that case, not 1.
I need to get out less. I'm neglecting math too much.
>The limit to (0,0) from any direction would be 0
why?
Because that is the value at (0,0), duh.
That's not quite how limits work, here have a read:
en.wikipedia.org
absolutely. ability to think critically and abstractly will be vastly improved. capacity for problem solving increase a ton as well.
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?
The value of a function at the point you're taking the limit to doesn't affect the limit, here have a read:
en.wikipedia.org
Is this what you want?
>Calculate the limit if you don't believe me. It's 0.
>It's 0.
The fuck?
why is this such a niche field??? it seems like the purest form of mathematics possible
It was subsumed by category theory
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.
>The fuck?
High schoolers don't understand delta epsilon yet.
It's not niche at all.
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.
great math post
see
...
Back to high school, weeb.
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].
Do you not know how to read? Re-read
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.
general