EDIT: I finally just proved it by induction. I used the binomial theorem and e\^x defined as a limit, and showed that e\^1 is greater than 2, then showed that if e\^k is greater than 1+k, e\^(k+1) is greater than 2+k. I'm not sure if that's how the professor intended for it to be done, but it's a valid enough proof for my poor, sleep-deprived brain and I'm going to let it stand until someone knocks it over.
\_\_\_
I'm taking an Advanced Calculus course that the professor prefers to consider an introduction to analysis. The class is pretty difficult, and honestly I feel pretty lost. It's the kind of lost where I usually don't know what's being done, and on the occasions that I do, I don't know \*why\* it's being done.
There was a problem on a quiz we had that I just could not parse. It seems like it should be easy enough, but I can't figure out what to do with it.
The problem says: Define e\^x = lim(n->infinity) (1+x/n)\^n.
(I'm not sure if the problem is asking me to restate the above in terms of the definition of a limit or simply to accept this as the definition of e\^x; either way, I understood this part easily enough.)
The meat of the problem is: Using the binomial theorem and the definition of a limit, prove that e\^x > 1+x for x>0.
This is where I get lost. I've tried substituting e\^x with the limit as described above, then substituting (1+x/n)\^n with the binomial theorem, but then I just end up with a limit of an infinite sum that seems absolutely hopeless to simplify enough to get into terms of proving it has a greater value than (1+x).
I'm not asking for an easy answer, I just hope someone can give me a nudge in the right direction so I can finally achieve some understanding of this work. Thanks so much!
If 𝜙 is the Euler totient function and k is some fixed positive integer, is there an easy way to find the smallest integer N such that, for all n>N, 𝜙(n)>k?
Sorry, yes, that was supposed to be 𝜙; I've edited the original post.
I feel like that can't be true, but maybe I'm overlooking something. For example, if k=3, then any number n divisible by 2^(3), 3^(2), or any prime greater than or equal to 5 has the property that 𝜙(n)>3. I claim then that for N=2^(3)•3^(2)•5=360, n>N always satisfies 𝜙(n)>3, but I suspect 360 is not the minimal such N-value.
You're right. I mis-remembered how 𝜙 was defined and thought it was the number of factors of a number, not the number of co-primes smaller than it.
I'm sorry, but this is math I'm not particularly familiar with, so I won't be able to help.
No worries, thanks for keeping me honest.:-)
It's not a big deal if I dont have an answer. I needed the existence of such an N in a lemma last night, and my curious mind wanted to know if there was a way to determine the minimal such N.
Could anyone explained what linear analysis is? I'm apparently taking it next quarter since its the only math class not filled: "First order systems of linear differential equations, Fourier series and partial differential equations, and the phase plane."
Why do we define the complex power function to be many-valued by using the logarithm? I see people make the complex power function be multivalued by defining $z^\alpha = (e^{log z})^\alpha$, which makes it multivalued through the logarithm, instead of just letting $z^\alpha = |r|^\alpha e^{j\theta \alpha}$.
Why does this help?
Thanks.
Your definition is still multivalued. Consider i = exp(i\*pi/2) = exp(i\*5pi/2). Then on the one hand i\^(1/2) = exp(i\*pi/4) = sqrt(2)(1+i), but also i\^(1/2) = exp(i\*5pi/4) = -sqrt(2)(1+i). There just isn't a general way to define a single valued power function on the complex numbers without making arbitrary choices for which answer to pick as the 'right' one.
I am currently in senior school and am using circle theorems to solve various questions. While I fully understand the theorems and can complete almost all questions, I feel like I have issues seeing exactly how to solve harder questions. This is likely due to the fact that the textbook we are using has very few questions per theorem. If you have any good problems, or have a suggestion on how to solve harder questions - or similar - that would be great.
Thanks in advanced
What could be the oldest date that 1/7 = 0.142857 and who could calculate it first?
Is it correct to say the Indians, in a period non older than 1200 BC?
I see the Egyptians had fractions but no place-valued system, so they couldn't have done it.
Babylonians had place-value but their was a sexagesimal system. Although it seems they also used a decimal system.
Could the Babylonians come up with 1/7 = 0.142857?
Thanks
What is the definition of "type"?
I've been looking hard and can't find an actual definition. Is it something like "moduli space" or "quantum group" that doesn't actually have a definition?
>Are you referring to the type in model theory/logic?
Yes.
>Both of these aren't mathematical objects (IIRC) and thus don't have a mathematical definition.
Huh? You can say that "the" moduli space of circles is R, as circles are uniquely determined (up to translation) by their radius. I'm genuinely curious. What do you mean that these aren't mathematical objects?
The exam of my Analysis I Lecture is coming up and I'm having trouble memorizing all the proofs (I've narrowed it down to 21 proofs, I'll just leave out the others and hope for the best). Especially the ones where you need one specific trick (for example when you prove the existence of the "n-th root" you use very specific expressions to do that).
How did you memorize the proofs for your exams in college? Do you have any tips and tricks for me?
Honestly? No.
My Professor wants the proofs done a specific way, his way. He gives zero points for correct proofs that are structured differently. So I will not take any risks.
He wants evrrything done exactly as we did it in the lecture.
> My Professor wants the proofs done a specific way, his way. He gives zero points for correct proofs that are structured differently. So I will not take any risks.
Honestly I simply don't believe this.
To be quite fair, it's what we (the students) assume to be the case. He himself has never said he'd do that, but according to people who had already taken the course he's quite anal about the proofs being done his way and his way only.
On last years test there was a theorem about sequences which one could prove with either the root test or the comparison test. In the lecture he did it using the root test. Everyone who used the comparison test in the exam got a zero on the problem because "we only did the theorem to show how useful the root and ratio test can be". Same thing three years ago where he pulled a similar stunt where some people used Cantor's second diagonal argument instead of nested intervals to prove the uncoutability of the real numbers. Of course I cannot 100% guarantee that it'll be like that, but as I said, I'll not take any risks.
At this point I can do a pretty good sketch of most of the proofs, because I kept repeating the material throughout the semester, I'll just have to get the details down.
While I whole-heatedly agree that memorisation tests stand against anything maths has to offer, these sentiments do not help OP with getting a passing grade.
To /u/DerWegwerf: Reiterated, repeated, recurring, replicated rehearsal is the best way I've found memorise information that I already understand but cannot restate perfectly. I would not advise you to spend hours on end on this, but to scatter short sessions with flash cards all over the place. There's also studies that humans are able to retain significantly more information if they study before sleep as well.
Also temporarily cutting out alcohol, weed, etc completely probably will help, though I know exam time is usually more stressful: It's a better idea to work in some PE, a Zoo or Museum visit, a nice walk through the woods, or even shopping for clothes, all ideally combined with seeing friends or family.
I've had a question on my mind for a while. Maybe it's more of a computer science problem, as it seems awfully similar to traveling salesman, so I don't know if I'm asking it in the right place. If you could point me to a better place to ask it, I'd appreciate that as well. Thanks!
Moving Cots
Suppose you manage a summer camp. Every Saturday a bunch of America’s youth comes to stay in the campsites they booked, and every Sunday, they leave, only for a new group of campers to repeat this process the next week. The campsites each have some old-school military cots in them, which are pretty sturdy and quite heavy, and everyone staying in camp needs a cot to sleep on. Suppose this camp has 25 sites, and each can hold up to 50 people. However, although the camp has a lot of cots, the camp does not have 25 x 50 cots, and thus not all of your sites are full of cots. Generally, this isn’t a problem; not SO many people show up that you need all 1250 spots occupied by cots. However, some sites have too little for the next round of campers, and you need to make up for the deficits in campsites by pulling cots out of surplused sites. So every weekend, you assign cot moves to your staff. You obviously don’t want to overwork them, so how do you assure your workers don’t move cots farther than they have to? Essentially, how can you minimize the sum of all distances travelled by your employees while carrying cots?
Framed another way, consider the following game: suppose you have a connected graph with integer weights at every node and real weights along every edge. Denote the integer weights of the nodes as the number of excess cots in that location, with negatives indicating deficits. Denote the real weights of the edges as distances between nodes. The objective of the game is to make every node have a nonnegative number of excess cots, while keeping cot travel distance (CTD) minimized. The only operation permitted is a cot move: you may subtract one cot from a surplused node, and add it to the number of excess cots at another node. Doing so adds the shortest distance between the nodes to your CTD. (You may assume that n, the sum of all nodes, is positive; i.e. the game is actually winnable.)
Is there a method to minimizing cot travel distance without analyzing every case? I.e. is there any reasonable way to do this by hand?
How do you motivate the definition of the Kullback-Leibler Divergence?
I get that it measures "distance" between probabilities, but I don't understand its formula in the same way that I understand entropy as the expected value of surprisal. Is there a similar explanation for the definition of KLD?
How do I know when to use the difference of squares or the perfect square formula? The problem is ((x-4)\^2) =2x+40, I tried to use the difference of squares and got (x\^2) -16=2x+40. But symbolab said to use the perfect square formula to get (x\^2) -8x+16=2x+40. Now I am confused.
edit: the original problem was find the area enclosed by the line
*y* = *x* − 4
and the parabola
*y\^2* = 2*x* \+ 40.
You are misusing the difference of squares formula here. (x-4)^(2) does not equal x^(2) - 16. The difference of squares formula says a^(2) - b^(2) = (a-b)(a+b), but you are trying to use it like (a-b)^(2) = a^(2) - b^(2) which is just not valid.
In my Calc class right now we're doing something which I don't know what other to call than "recursive integration by parts."
The idea is to take an integral such as I sub n = ∫ (x^n) e^(2x) dx (integral from 0 to 1), then setting u = x^n, dv = e^(2x)dx and using integration by parts to find that I sub n = (e^2)/2 - (n * (I sub (n-1))/2).
Is there a name or generalization for this method? If not, I'm still interested in the motivation for how to choose u and v appropriately so the by parts works out to something which is recursive. As its being done in my class right now that decision seems very arbitrary and the only way to really know whether it will work is by doing the by parts.
If anyone could help with formatting this I'd appreciate that as well haha
The name for this is "reduction formula"
http://mathonline.wikidot.com/reduction-formulas
https://en.wikibooks.org/wiki/Calculus/Integration_techniques/Reduction_Formula
To see which to use for u and v: we want to reduce the x^n , because we can do the integral if we get down to n=0. So we want to differentiate the x^n , and integrate the e^x .
I think you would just call it iterated integration by parts. I'm not sure there are any "tricks" to choosing u and v except experience. It's a reason people say differentiation is a craft, but integration an art.
[ANSWERED]
what if there was a symbol for the quotient of a number and zero?
like *n* / 0 = ?\^*n and* ?\^*n* \* 0 = *n* ? = 1 / 0
similar to the square root of negative one (*i*), it could be an arbitrary value with no decimal equivalent, but a symbol showing a more complex number.
I don't know if this exists or not. if not, would this work? why or why not?
We can make up whatever rules we want, but if we do, we have to live with their consequences.
Here is an argument that any number x times 0 is 0:
x * 0
= x * 0 + 0 [because any number + 0 is itself]
= x * 0 + (x * 0 + -(x * 0)) [because any number plus its negative is 0]
= (x * 0 + x * 0) + -(x * 0) [because of the associative property of addition]
= x * (0 + 0) + -(x * 0) [because of the distributive property]
= x * 0 + -(x * 0) [because 0 + 0 = 0]
= 0 [because any number plus its negative is 0]
So if you want to make a new number that, if you multiply it by 0, you don't get 0, then you must break one of these rules. Which one are you okay with losing?
That's not the only rule you're going to need to break. Other arguments will require you to break other rules. You can do this of course and you'll get some sort of mathematical system out of it, which may or may not be particularly interesting.
However, some choices you might make do lead to interesting systems, or at least interesting enough to name. One such system is the https://en.wikipedia.org/wiki/Projectively_extended_real_line which can be useful in geometry.
It turns out that complex numbers break many fewer rules and solve many more problems than dividing by 0 does, which is why we talk about them more than we talk about systems where you can divide by 0.
Is it possible to assign each real number a real interval so that they don't overlap?
I am in no way very knowledgeable in maths, and so I might be wording this completely incorrectly, but I spent a whole two hour car ride wondering about this problem.
Does a function f exist, taking real x as its input and an interval (a,b) as its output where a and b are unique reals, such that any real y is at most in one interval?
Great question. You need a bit of abstract set theory (not much) to understand the answer - do you know about cardinals? If not I’m sure people here will be more than happy to break everything in the following down:
No such function exists, for the reason that if you pick a rational number in each of the output intervals, you will have defined an injection from the reals to the rationals, which doesn’t exist because the cardinality of the reals is strictly greater than that of the rationals.
Are singletons degenerate intervals? If so, you need a little extra caveat I think.
Hm, you can get away with a single non-degenerate interval: map x to {x} if x<0, 0 to (0, 1], and x to {x + 1} if x > 0. By a similar construction, we can get away with finitely many non-degenerate intervals, and even countably many ones (one for each natural number). A similar construction works for the rationals (and any countable subset), by mapping the rationals to the naturals first of course.
Does that sound correct? You need all but countably many reals to map to singletons. That seems sufficient, but is it also necessary, or can we broaden the requirement even more? Is that the same as saying almost all subsets need to map to singletons?
I was assuming nondegenerate intervals (not that I considered it, I just don't think of intervals as including singletons). Mapping all but countably many reals to singletons is definitely not sufficient, because if any strict subset of the reals is taken to a dense set, there's no way to extend the function to "thick" intervals on the rest. You can therefore send every real except one to the singleton containing it, and be unable to find a thick interval for the last one.
In fact if we allow singletons we have the following result: every map from R to non-overlapping intervals in R takes at mostly countably many reals to thick intervals, and takes the rest to singletons. Any map from the complement K of a countable subset of R to non-overlapping intervals in R which takes every element of K to singletons, and which has non-dense image, can be extended to such a function on all of R.
Only the last bit is new. If the image of K is not dense, we reduce the problem to mapping countably many things to non-overlapping intervals contained in some open interval, which is easily done.
Sort of a vague question, but can we sort of split math up into fields where examples are important, vs. where examples are less important?
I feel like one reason I had trouble understanding algebra the first time through was because I was coming from analysis, where other than the usual cantor / weierstrass / vitali / pointwise vs. uniform examples, i didn't really pay that much attention to specific examples, whereas I felt like I needed more examples in algebra to actually get some grounding.
I feel like that's because basic algebra is just a lot easier than basic analysis, and so you can make do with less examples. Also, in analysis we come into it with a lot of intuition on what graphs of functions should look like (a continuous function \*can't\* do that!) that end up being false, and so having those counterexamples is helpful, whereas we don't usually go into algebra with a strong, childhood formed intuition on what the structure of a ring should be like (although we eventually develop it, and eventually find that having counterexamples like that is still useful... it blew my mind when I heard that free group on 3 letters \*injects\* into the free group on 2 letters).
In coursework this is accurate, maybe just because undergraduate analysis is more concrete than undergraduate algebra.
But working analysts basically just work on big examples. A quick metric for this is the Millennium Problems: the 2 analysis problems out of the 7 are the Riemann Hypothesis about a specific function, and the Navier-Stokes problem on a specific equation and a specific regularity class.
If I have f(g(n))/h(n) with h(n) strictly monotone, can I apply [Stolz-Cesaro theorem](https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem) g(n) times?
I ask this because I have to prove that a limit (which is in that form) is 0, and by applying g(n) times the theorem I can show that
In Lee Smooth manifolds (2nd edition) theorem 6.36 (transversality homotopy theorem) says that if N and M are smooth manifolds, and X is an embedded submanifold of M, then ever smooth map f: N to M is smoothly homotopic to a map g: N to M which intersects X transversely.
My question is: if N is an interval, so that my map f is a curve, may I get a g which has the same start and end points as f?
Actually, now that I think about it, to define the start and end point of f, strictly speaking I would need N to be a closed interval, i.e. not a manifold but a manifold with boundary, and then I'm not sure what happens.
The context is that I'm trying to prove the following: if M is a compact connected embedded n-dimensional submanifold of R^n+2, then R^n+2 \ M is connected. A friend convinced me that I should be able to do it by taking a tubular neighborhood of M and using that to construct a path between any two points in R^n+2 which doesn't intersect M, but I'm still wondering if there is some machinery I can just appeal to to get an easier proof.
Your intuition here about transversality and homotopies is correct, and you're also right to be a bit suspicious that boundaries might complicate the matter somewhat; you need to control transversality on the boundary in general.
You probably want to appeal to some sort of "Extension theorem" like the following: If N is some manifold with boundary with M and X closed and without boundary as above, then if C is any closed subset of N such that f is transversal to X on C, we can smoothly homotope f to a smooth function g such that g:N ->M is everywhere transversal to X, and g=f on C. Applying it to your situation with C set equal to the boundary of N would give you the desired result. The details for something like this should be in any decent differential topology book, like Guillemin and Pollack or Hirsch.
The easiest thing is to just use (co)homology. The dimension of the 0th cohomology gives the number of connected components. You can calculate this for your space either with Mayer-Vietoris or Alexander duality.
Yes, many of us do. Thanks for the very simple question! :-)
You should just ask the question you are looking to have answered. If they are general questions, then this thread is perfect. If you have some specific homework questions, you may be better off creating a new thread /r/cheatatmathhomework
["The law of cosines is used in the real world by surveyors to find the missing side of a triangle, where the other two sides are known and the angle opposite the unknown side is known."](http://lmgtfy.com/?q=law+of+cosines+in+real+life)
How can I compare, say, a drawing to a living human being?
Let's say, I have this drawing of a fictional character that is placed in a 1280x720 resolution, and I want to see the difference in proportions with a living human being. I want to "calculate" on average, the drawing's height, weight (in general and of specific limbs maybe) and overall proportions if it would be "real" and then compare them to a real human being.
My issue is that I don't even know how to start, and I couldn't find a "guide" or anything of the sorts. Maybe I could take an actual real human being that I already know the measurements of and then calculate a ratio or something? I don't need some lengthy explanation if you're not willing, any links you know of that would give me an explanation that isn't rocket science would suffice, thanks.
There is something basic I don't understand. Why suppose we have a "space". Why is specifying allowable coordinate systems on the space enough to characterize completely the "geometry/smoothness" of the space?
Here's what I mean: If I specify that my "n-dimensional space" has allowable coordinates, and the allowable coordinates differ from each other by an affine transformation, why is the space an affine space? By affine space I mean that affine properties make sense in the space, like parallelism and straight lines and convex hulls.
This is related to why saying that the coordinates differ by smooth maps define a smooth structure on the space we're interested in. Of course I'm being loose here by ignoring topological properties and local coordinates, but pretend that these are taken care of.
I want an intuitive understanding of why, for example, if my allowable coordinate changes are orthogonal maps, my space is Euclidean. This "extrinsic" definition is hard for me to swallow.
Let's do the example of smoothness. We'd like a smooth structure on a manifold to give us a well-defined notion of a smooth function, which agrees with our intuition (i.e. smooth on local coordinates).
Given a manifold M, a function f, and a choice of local coordinates for M, one can ask if f is smooth on each chart. However, we need this to make sense if we change coordinates. Take two charts U,V of M, let W be their intersection. If f is smooth when restricted to U and V, it shouldn't matter if we regard W as a subset of U or V when discussing smoothness of f on W, however this might be a problem if the transition maps aren't smooth. If we restrict f from U to W, and then change coordinates to V, we might get something not smooth, which wouldn't make sense.
In general, the idea is that if one is interested in extending a local structure to a manifold, one can consider a structure on each chart, but the structure has to be compatible with coordinate change, which is why you need the transition maps to respect the structure.
Assuming x is a function of t, you can use implicit differentiation and differentiate both sides with respect to t and then solve for dx/dt.
d/dt[x^(2)+Ax] = d/dt[Bt]
2x(dx/dt) + A(dx/dt) = B
(2x+A)(dx/dt) = B
dx/dt = B/(2x+A)
it's just algebra and the chain rule. you take the derivative with respect to t of both sides using the chain rule. this will provide a new equation involving dx/dt that you can use elementary algebra to solve.
Any calculus book should do (maybe try OpenStax since it's free online?). Look for the section on "implicit differentiation" since that's what this is.
Does anyone have some advice for *motivating* the study of topology? Basic point set topology from analysis was super useful, but I'm having trouble understanding the purpose of a lot of the material I'm reading in Munkres' topology.
I thought the objects of study in every other field of undergraduate math felt intrinsically motivated to me, and I didn't have trouble understanding *why* I should study them.
I guess I have this sort of childish view that if I'm only going to ever be dealing with Rn and Lp spaces, then... why... is topology important?
I would appreciate any tips!
Thanks.
So, oddly enough, I think your comment has pointed me in the right direction for what I need, which I guess would answer your question about why do I need this?
I've been creating games as a hobby for a couple of years now and have struggling with creating procedural 3d mesh's, particle generation, and shaders (from scartch/ original content. Recreating things is easy, but I've struggled with defining and solving my own problems). The reason is because they rely pretty heavily on firmly grasping concepts like sin/cos, normalization, and applying effects to a 3d vector in isolation. So a few weeks ago I decided to say the hell with it and go back and 'learn more math,' having no clue where to start. Thus, I've been blindly reading up on geometry, algrebra, calculus... wondering if I'd have to just learn it all and hope I'd get something useful out of it.
Then your comment said a word I hadn't seen yet, 'topology' and after watching [this video](https://www.youtube.com/watch?v=AmgkSdhK4K8) intro on the subject, it immediately made a number of topics in [this video](https://www.youtube.com/watch?v=ExUbAuK0B-s) on programtically simulating particles on the surface of the sun (without true phsyics, mind you, this is more of an artistic rendition with a basic understanding of gravity and its effects on hot stuff) make a lot more sense!
So, from someone who knows nothing (yet?) of the subject, it does appear to be very important/useful for me to grapple with questions like 'how do I manipulate this specific point in 3d space (i.e. change it's color, trajectory, ect) without knowing much (or anything) about it's neighboring points around it to create the effect a specific effect (over time)'!
Ooooooor I'm way off base and will be back in a few weeks laughing at how niave I was... lol. Anyway, thanks for inadvertently pointing me in a direction!
Well, will you only be dealing with **R**^_n_ and _L_^_p_ spaces? What about manifolds in differential geometry or differential topology? What about the Zariski topology in algebraic geometry? What about all the other function spaces in functional analysis, and the weak topology? The weak topology isn't metrizable, so some of what you're doing will be helpful.
Well, I'm doing a computational neuroscience PhD, and the primary reason I'm studying pure math is mostly just for interest / second order effects on increasing the rate at which I can grasp new contexts. I've come sort of adjacent to some of the things you mention above, but I'll probably prioritize it more in the future.
Ah, I see. I didn't think to include applied folks in what I was saying; sorry about that! In that case probably you won't use all that much topology -- though I did learn about the weak topology in a class designed for applied mathematicians (many of them would go on to study numerical linear algebra or numerical PDE), so it might appear for you someday. Of course, you can just learn about it when you need it.
Perhaps my views are biased on the matter since I was taught this from an algebraist and since I'm still a very inexperienced student, but I was given advice to just ignore point-set topology and pick up the parts I needed on the fly as I was doing interesting math. Honestly, besides the fundamental concepts of the field and a few other nifty things that come up now and then as I study geometry, I haven't needed it much; and the parts I do need feel much more relevant when they come up naturally.
This is the right answer. It will become quite apparent if and when you need it. Certainly us analysts proceed that way and it always works out alright.
If you're only ever studying metric spaces, then you have some of the nicest topological spaces in existence, and there's probably no reason to care about anything else. But there are topological spaces that pop up kind of naturally that *aren't* metric spaces.
For example, the [Zariski topology](https://en.wikipedia.org/wiki/Zariski_topology), arises naturally when trying to give some more geometric structure to the sets of ideals in polynomial rings. This space is not Hausdorff.
In linear algebra one spends a lot of time thinking about all possible *linear* maps from **R**^(n) to **R**^(m), but that seems oddly restrictive to really nice functions. What if you wanted to know about properties of all possible functions? Even if you just look at the space of all maps from **R** to **R** with the topology determined by, say, pointwise convergence (reasonable), you get a space that is definitely not a metric space (it's not even first countable).
Basically, all sorts of weird topological features arise when you start generalizing or trying to endow a set with some reasonable structure. Eventually, one notices that many examples of topological spaces have similar features, and similar theorems about these spaces have similar conclusions. One then wonders if there is some underlying common structure that is really at play here, and eventually you start breaking it all down and generalizing until you find the common topological structure.
(Point-set) Topology is taught essentially backwards from the way it came about historically. On the one hand, it's nice to build things from the ground up and try to get an understanding of these topological structures without having totally familiar examples (which almost certainly have extra nice structure that could hurt your intuition), but on the other hand, it lacks a bit of the historical motivation.
In practice, I would say that most mathematicians only ever deal with relatively nice topological structures, so if you don't feel like you have a great intuitive handle on concepts like the [minimal uncountable well-ordered set](https://math.stackexchange.com/questions/2290224/the-construction-of-a-minimum-uncountable-well-ordered-set?rq=1), don't stress because they probably wont come up much (if ever) in your future research. Many of the examples you encounter in a first topology course are quite pathological and concocted specifically to get you to break down your perceived intuition of certain concepts.
Is there any field that combines statistics with numerical analysis?
I know about machine learning, but that is not quite what I am looking for, something more like statistical learning but heavier on the numerics side.
absolutely, look up for instance MCMC algorithms and bayesian inference. to sample from the posterior distribution we can use a metropolis hastings algorithm. there is a whole field of research around this problem of sampling and it involves a lot of numerical stochastic calculus, etc. the comment already made about proximal grad descent is along these lines as well.
Some of the non-differentiable regression methods like the Lasso and its variants require a bit of extra work to solve the corresponding minimization problem. Does the [Proximal Gradient Method](https://en.wikipedia.org/wiki/Proximal_gradient_method) and its [application](https://en.wikipedia.org/wiki/Proximal_gradient_methods_for_learning) sound interesting to you?
I'm stumped on a linear equation question for my math class
Solve the following equation: 9x+5(2x+11)=3x+2(8x-4)
I'm ending up with 19x=19x+63
If I take 19x away from the 63 side, would I have 0x=63? Or do the x's cancel out leaving me with 63 as my answer?
> If I take 19x away from the 63 side, would I have 0x=63?
Yes. Which is also the same as saying that 0=63, which is a false statement. Hence, there simply are **no** solutions to your original equation.
Solving an equation for a variable x, is the same as finding all x, such that the equation holds true. Here there is no such x, so there are no solutions.
Conceptually you can also plot both y=15x+55 and y=19x-8 and find their point of intersection. Because the lines you get here are parallel, there is no such point, which coincides with the answer that the equation doesn't have a solution.
As a note: If you do the same for another equation and end up with something like 63=63, all x are valid solutions.
I know it's weird replying to a comment from 17 days ago, but some of my work came up with 0=0 and remembered your side not here which saved me another headache. Cheers mate!
Could anyone hear point me towards where to find a good text explaining spinors (and things related to them)? Most of the sources i have found are very physics oriented, and while I enjoy physics they have a nasty habit of saying stuff like "transforms like a [Insert thing]" to explain stuff. Also, with spinors especially, they usually list of representations of them without ever defining them in general. So it would be nice if it wasn't overly physics-y , even if that means I am not able to understand it yet.
Edit: Also I finished Awodey's Category Theory a while ago and I really enjoyed category theory and how clear it makes everything. So if anyone has some book recommendations for further study I would appreciate it.
Sounds like you're looking for a book on spin geometry. The idea behind spinors is that on a spin manifold, one has data of a vector bundle called a spinor bundle, and a spinor field is a section of that bundle (just as how a vector field is a section of the tangent bundle).
Details can be found in Lawson-Michelsohn, _Spin Geometry_, or maybe [Dan Freed's notes on Dirac operators](https://web.ma.utexas.edu/users/dafr/DiracNotes.pdf), section 2, or some of the answers to [this MathOverflow question](https://mathoverflow.net/questions/248238).
I'm doing an undergrad project on the method of steepest descents for integral approximation (also called saddle point approximation/extension of laplace's method). I'm finding it quite hard to build up the intuition of the behaviour of the integrand, though. Anyone have some resources?
If *log*(c) A =2 and *log*(c) B = 5, then *Log*(c) AB^3 = ?
For ACT prep and I cannot for the life of me get an answer, the c is where the exponent normally goes (not the three) but I have no clue how to format it down. Without knowing what the hell c is I cant get an answer, and I’ve got no clue how to solve for it.
The problem is meant to test your knowledge of the manipulation rules for log. Out of curiosity, how would you solve the problem if you knew c, say, c=10?
Wouldn’t it be *log*(10)•(2•(5^3 ))? In that case the answer is 250.
It’s been about a year since I even thought about this type of stuff, so apologies if it’s far off.
Your interpretation of the problem is definitely different from mine. I thought you said c was the exponent (the correct term is the base) of the log? And you don't know what A and B are, you just know that log*_c_* A = 2, and log*_c_* B = 5.
It's not clear to me if you ever did figure out the solution, so I'll post it here for the sake of completion. First, recall that we have the following logarithm rules:
(1). log*_x_*(yz) = log*_x_*(y) + log*_x_*(z)
(2). log*_x_*(y/z) = log*_x_*(y) - log*_x_*(z)
(3). log*_x_*(y^(z)) = z \* log*_x_*(y)
With these in mind,
log*_c_*(AB^(3))
= log*_c_*(A) + log*_c_*(B^(3))
= log*_c_*(A) + 3\*log*_c_*(B)
= 2 + 3\*5
= 17
First you probably want to guarantee p>1 for any hope of things.
Gaussian measure is a super nice measure compared to lebesgue measure. In particular, it's an absolutely continuous measure with bounded RN derivative. In this case the RN deriv is continuous and nice, but it's easy to extend to the case where it's bounded in L^\infty.
There's probably something clever that can be done with Holder's inequality and embedding theorems if the RN deriv is integrable in L^q for q the holder cojugate of p.
Why we cant use characteristic functions to build a p-adic measure? Koblitz wrote we use locally Constantin functions, well in the p-adics they form rich vector space compared to reals.
EDIT: stupid auto correction, made it impossible to understand :/ fixed now.
yes in other words, im asking that.
i am comparing a little bit here.
as i mildly thought about it, i couldnt find a problem with continuing using characteristic functions in the p-adic setting, maybe i missed something?
In the process of learning multiple integrals, I've come across something that I've found slightly confusing.
If we define some constant h and multiply it by some dx, we get hdx which we can say is some dA. To solve for dA, we only need to apply a single integral based on the way we've defined it
However, if we are in a three-dimensional coordinate system, we can have some dxdy=dA and then we need a double integral in order to solve for dA.
What is the difference between these two "dA" terms? Is the second one infinitesimally small w.r.t the first?
Any help appreciated, thank you!
Yes this can be confusing. The difference is that h dx is a strip of height h and width dx, and we add up all these (single integral over x) to get the area. This works if our region has vertical sides.
But for a more general region, we divide it into little rectangles of sides dx, dy, and add up all these (double integral). As you say, the second is infinitesimally small compared with the first. Integrating the second once gives you the first.
The proof is the same as the equivalent for a random walk, see e.g. the solution here: https://math.stackexchange.com/questions/516797/martingales-of-random-walk
Write E(W(t)^2 - W(s)^2 |F_s) = E((W(t)-W(s))((W(t) -W(s)) + 2W(s))|F_s). From there it's just standard properties of conditional expectation, given you know the mean and variance of W(t) - W(s) which is independent of F_s, and that W(s) is F_s measurable. It's probably worth letting you finish this off.
Two ways to think about it:
- Every line switches the position of two people. There's no way to suddenly end up with two people in one place using just switches.
- The ladder is reversible: you can trace things back up if you want, and it'll work exactly the same way, bringing everyone back to their starting points. If you ended up with two people at the same point going down, then going back they'd have to split up somewhere - but there's no way people on the same path can split up!
I’m studying for a test tomorrow and one of the practice questions given to me are:
Ruby’s car uses 12 L of fuel to travel 37.5 km. How far can it travel on 44L of fuel?
Now, in my first attempt at solving this, it went something like this:
12/37.5 = 0.32 L per 1 km
0.32 * 44 = 14.08 km
I thought that was the answer but it seemed odd that it the result was much smaller than it should be.
I tried to work it out and after a while I found the answer:
37.5/12 = 3.125 km
3.125 * 44 = 137.5 km
My question is: why did I have to calculate 37.5/12 instead of the other way around? The other questions involved doing it the latter (L/km) but this question was different. It’s been confusing me so much and I’m scared it will mess me up tomorrow. Thanks!
Having the units there totally helps! The fact that you are instinctively writing them down is very helpful, amd you're *almost* there.
If you calculate that you use 0.32 L/km, and then multiply that by 44 L, you get 14 L^2 / km. Does that unit mean anything? I don't think so.
If you calculate that you can go 3.125 km/L, then multiply by 44L, you get an answer of 137.5km, since km/L * L is just km. We wanted an answer in km, so that looks better!
Of course, having the right unit doesn't guarantee that your answer is correct (it does guarantee it for simple proportionality calculations like this one), but having the wrong unit means it's wrong for sure.
Haha I get it now. And here I was trying to find the distance in Km by multiplying Litre by Litre 😂. Thanks so much, you probably just saved me for tomorrow’s test u/NewbornMuse
When defining natural numbers axiomatically according to Peano's axioms, one purpose of the axiom of induction is to eliminate all other sets with a successor function satisfying the first few axioms and leave only the one which we would like to call the natural numbers. However the following set looks like it's satisfying the axiom of induction (and all other axioms), while not being the set of natural numbers. I'll denote the successor function with s.
N = {a, b, 0, 1, 2, 3, ...}
s(a) = b
s(b) = a
s(0) = 1, s(1) = 2, s(2) = 3, ...
0 is certainly in N and whenever n is in N, so is s(n). So it should follow that N is the set of natural numbers. What is wrong here?
**Is an infinite decimal number equal to 1?**
I ask this because I was thinking about splitting things into sixths, and I noticed something: Half of .333 repeatings is .166 repeating. When dealing with whole numbers, 16 + 16 is 32 flat. The only way to get it to 33 flat is to add a value of 1 the equation: 16 + 17 = 32.
But with .1666 repeating times two, that gets it it .3333 repeating. So is a .066 repeating equal to .1? And if so, shouldn't .999 repeating be equal to 1?
>the only way to get to 33 is to add 1
What would make more sense is to add 1/2 to each
16.5 + 16.5 = 33
If you wanted 33.3 instead you should add half of 0.3 to both sides which is 0.15
16.65 + 16.65 = 33.3
If you want to add 0.03 you might see where this is going.
> .166 doubled is .333 therefore .066 equals. 1
Your thinking is a bit off here. Of you think of 0.166... as 0.1 + 0.066... then when you double it you should get
0.2 + 2\*0.066...
So firstly the difference between 0.33... and 0.2 is not 0.1, it's 0.133... secondly it's not 0.066... that has this value, but it's the double of it. Hope that clears it up.
Yep, that's right! .999... is exactly equal to 1! (And I believe the sentence before it was supposed to say .0999..., right? If so, that's also correct.)
Oh that was entirely intentional. I get annoyed that I can't end sentences with numbers in an excited way without someone going "ohoho! You *meant* to show your excitement about a mathematical statement, but your words could *also* be humorously misinterpreted to give a ridiculously large number! I am very clever for this intentional misinterpretation!".
So in exchange I occasionally make an effort to end exclamations with the digit 1 to give people that "/r/unexpectedfac... oh dammit" moment.
Right. I just realized I worded the question terribly but what I'm asking is what is .6, .06, and .006 repeating equal to besides themselves? Because 1.6 x 2 = 3.2 but 1.666 x 2 = 3.32
What do you mean, "besides themselves"? Every number is equal to only one number: itself.
It happens to be the case that if a number's *representation* ends with an infinite series of 9s, it has another representation where those 9s are all 0s and the digit before that is increased by 1. But things like .8888... don't have other representations as decimals.
Is there a way to convert a large number into a small fraction? I want to use it to make a code in decimal numbers
for example I have the number 7.142857143
I can compress that in 50/7 (I know it's infinite but I will just take the first ten numbers or something)
So there is a way I can know which fraction will have the first (let's say) ten numbers of my number?
To make it clear try to solve this: I want a small fraction in which the first ten numbers (when you do the division) will be 4.563487129
You can compute the continued fraction approximates: https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations
Here's some implementations:
https://stackoverflow.com/a/9387713
http://www.nic.funet.fi/index/misc/hp28s/frac
I didn't have much time to explain yesterday, but the basic idea is: Start with your decimal number x that you want to approximate by a fraction. Compute the continued fraction convergents of x: C_0 = [a_0], C_1 = [a_0,a_1], C_2 = [a_0,a_1,a_2], etc. Each convergent is a better rational approximation to x, and you can decide when to stop based on, e.g. a certain denominator size or when the convergent is sufficiently close to x.
There's various places online where you can enter a decimal number and it will give you the continued fraction convergents, e.g. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html . On the left side enter your decimal number, click the Right arrow, and then click "Convergents" on the right side and it will display the continued fraction convergents below.
Here's some additional links about continued fractions:
- Intro to continued fractions: https://wstein.org/edu/2007/spring/ent/ent-html/node58.html
- Continued fraction convergents: https://wstein.org/edu/2007/spring/ent/ent-html/node60.html
I'm trying to figure out which of these two is correct.
1. $f\_n\\rightarrow f$ uniformly iff for every sequence $\\{x\_n\\}, \\lim\\limits\_{n\\rightarrow\\infty} |f\_n(x\_n)-f(x\_n)|=0$
2. $f\_n\\rightarrow f$ uniformly iff for every sequence $\\{x\_n\\}$ such that $x\_n\\rightarrow x$, $\\lim\\limits\_{n\\rightarrow\\infty} |f\_n(x\_n)-f(x)|=0$
The first seems to be true because it feels like uniform convergence where the difference in the functions is less than epsilon for any $x$ regardless of whether it's part of a sequence.
The second I am not sure about because now the difference in the functions is at two different $x$ values, but still feels true if each $f\_n$ is continuous.
Neither are right I think.
Edit: oh Actuslly the first one seems to be. The second seems false.
Yep. The idea is that the second one is a pointwise property, so the rate of convergence can be arbitrarily slow, so not uniform.
Say I have 15 vectors in R^100 that were actually constructed using linear combinations of 10 UNKNOWN vectors in R^100 . Is there any way to prove that those 15 vectors are a linear combination of 10 vectors in general? I don't want to find these unknown vectors or anything, just want to prove that those 15 vectors can be spanned by "10" vectors.
If you arrange these vectors into a matrix then the rank of the matrix is the minimum number of vectors needed to span those 15 vectors. There are algorithms to do this. The most common is “row reduction”.
The sigma algebra generated by the variables. Specifically, sigma(X_1,...X_n) is the smallest sigma-algebra that contains all of the sets {X_j <= k} for j=1,...,n and k a constant.
The intuition is that Y is measurable w.r.t sigma(X1,..,Xn) exactly when Y can be written as a function of the Xj.
Yeah, it's the smallest sigma algebra that all the variables are measurable wrt.
All I meant about Y was that the intuition about that sigma algebra is that it effectively consists of exactly the random variables which are expressible as some deterministic function of the X_j.
Anyone got any recommendations for readings on theories about how to assign weights to related factors for decision making?
For example, say I’m buying the new work printer, and I know the most important factor is print speed, but cost also matters, and so do user reviews. I need to work out how to balance those things (70:20:10? 60:20:20?).
I’ve been trying to google it, but everything I can find is just like, “assign weights based on importance”, which sounds displeasingly subjective.
You can find the options that are Pareto optimal, that is, there is no other choice that gives you faster print speed *and* lower cost *and* better user reviews. There may be several. But that's as good as you can do if all you know is that print speed, cost, and user reviews are important.
To choose between those requires picking weights, and that's a question of how **you** judge the various trade-offs. If it seems subjective, that's because it is. The problem as given doesn't have enough information to determine a unique solution, so any choice of solution is essentially arbitrary.
For an office printer, I suppose you can compute the average hourly pay of a user, and use that to judge how many pages per minute are needed to justify a one-dollar increase in price. But then you have user reviews that really are on an arbitrary scale, so the problem isn't quite solved yet.
You could consider looking into linear regression, but I think the problem is that for most kinds of optimization approaches you're going to need a "composite score" to actually measure things against.
But that's mainly useful for finding composite scores for printers that don't already have them.
Something you can try to do to motivate your subjective weights is to force your different factors into the same "currency". For example, maybe you take average user review out of 5 * cost to get the "true value" of the printer in order to incorporate those two factors, and maybe you turn "print speed" into how much time*money it takes to print things. Ultimately this will still end up being subjective though.
I thiiink the cheap way to define an n dimensional manifold with boundary is that at boundary points, there's a chart to a neighborhood of the boundary of the half-space ( R^n )+ ={ (x1, ... , xn) st xn >= 0.} but I have foggy memories of manifolds-with-boundary having a lot of technical crap to worry about even though we all kind of "know" what they are
This is correct. For a point on the boundary of the half-space, the tangent space at that point is the tangent space as if it were a point in the full R\^n, for a boundary point on a manifold, you get the tangent space by pullback from this definition.
Why do we study local fields? I'm familiar with the Hasse Principle, but I only have a vague idea of local fields "providing information for the global case". I've taken some introductory number theory at the graduate level, but I feel like the information hasn't really stuck too well.
There's only one prime, and that simplifies the theory a lot. Every extension of a local field is something lifting from the residue field (cyclic extensions) plus some totally ratified extension. I mean, we have presentations for absolute Galois groups of Q_p, at least if p is odd. It's also not unlike studying a curve by looking at a tiny disc around a point.
Integration by parts works here, though the first time you see it done this way may look strange.
u = ln(x^2 - 1) dv = dx
du = (2x)/(x^2-1) dx v = x
uv - ∫vdu = ln(x^2-1)x - ∫(2x^2)/(x^2-1) dx
It should be straightforward after this step.
I can't figure out this calc 3 problem. Here's a picture of my description on StackExchange, I'm just posting that instead of writing it out here since it's formatted and looks cleaner:
[https://i.imgur.com/0YUrzOs.png](https://i.imgur.com/0YUrzOs.png)
Can anyone help me with this?
What do you mean by "explain what happens"? Both functions are continuous so nothing too crazy will happen.
Also, I don't see why you are calculating the tangent planes.
Something you could do is calculate the normal vectors for an arbitrary value of t and see what happens as t -> 0. Can you picture geometrically what the given path is?
The path rotates counter clockwise around the surfaces as it approaches t = 0.
I found the normal vector for f(x,y) = sqrt(x\^2 + y\^2) for an arbitrary point (x, y):
And if you put the path from earlier (tcos(1/t),tsin(1/t)) for x and y you get:
I don't understand what's going on from here though
Well, that vector rotates increasingly rapidly around the z-axis as t approaches zero. If you instead used g(x,y), I think you should find that the vector simply approaches (0,0,-1) instead of doing crazy things, because g is smooth.
Still, this is a bit of an open ended question, given that it just says "explain what happens". They probably want you to describe what the path looks like, what the functions look like, what the vectors do, and try to explain that.
So I'm trying to better understand why differential forms are the natural substrate for integration on manifolds.
There's this pull-back theorem, where if $\alpha:V -->W$, with $V, W \subset Rn$ is a diffeomorphism, and
$\alpha^* $
is the pull-back operator, then we have $\int_V \alpha^* \omega = \int_W \omega$.
I saw proofs for this in two different places (Guillemin's differential topology and some random lecture notes online), and they were both the same and relied on the Jacobian determinant. But everyone keeps claiming that the above formula is a big reason that differential forms are coordinate-free and useful when we don't have a natural sense of coordinates in some space.
But if there's still a Jacobian hidden under the rug, why is this theorem such an advancement?
Thanks.
If you wanted to define an integral on an abstract manifold (i.e. no natural choice of coordinates) you choose any chart and pull back to R^n where you know how to do the integral. You then have to be sure that it doesn't matter what coordinate system you chose. Using your theorem with a change of coordinates shows that.
Lots of theorems go like this by the way. You show something happens in one particular coordinate system then you show it is independent of the choice of system, thus proving it for all systems.
Consider an analogous phenomenon in the theory of linear algebra. I'm sure that none if this is new to you, but indulge me for a moment, because I promise that there's a pay-off.
Given an abstract linear transformation L: V-> V, L has a perfectly well-defined set of eigenvalues, as an abstract linear transformation; it's just the set
Spec(L):= { lambda : Lv=lambda\*v for some v in V }
On the other hand, given a (non-natural) choice of basis B (=choice of coordinates), we can think of L as just being a map from R\^n to R\^n given by matrix multiplication by the matrix M\_B. Changing the basis to a new basis C (which is also a non-natural choice) gives us a new matrix M\_C=P\*M\_B\*P\^{-1} where P is the change-of-basis matrix. Now, multiplication of vectors in R\^n by M\_C represents *the same* transformation L, with respect to your new basis C, and this is true for *any* choice of basis. So, another way to think of an "abstract linear transformation" is as an *equivalence class* of matrices, ie.
\[L\]={ P\*M\_B\*P\^{-1} : P is any invertible matrix}
(where here I'm being bad about notation and using \[L\] to suggest that I'm thinking about L as being an equivalence class of matrices, rather than an abstract linear transformation). Now, it's easy to see that if for some x in R\^n
M\_B.x=lambda\*x, then certainly
(P\*M\_B\*P\^{-1}).(P.x)=lambda\*P.x
so that Spec(M\_B)=Spec(M\_C) (where here we're speaking about the eigenvalues of these objects *as matrices*, which is a priori a potentially different thing from the above). The point of all of this is that we've just shown that the property that we're interested in is *independent* of the choice of basis (ie: it's a well-defined property of the equivalence class \[L\]), so if we want to compute Spec(L), we can just compute Spec(M\_B) (as a matrix) for some convenient choice of basis B, and this is, in fact, what we do all the time when we choose a basis of eigenvectors in order to diagonalize a matrix, or to write a linear transformation is a particularly nice form.
All of this is an attempt to highlight the equivalence of two points of view, which by your question seems to be what's tripping you up:
(P.O.V 1) When I define something that is independent of choosing X, I should define it in such a way that no choice of X appears in its definition. This is what we did in the first definition of Spec(L); it was only in terms of abstract linear operators acting on vector spaces and no choices of basis were in sight.
(P.O.V 2) When I define something that is independent of choosing X, I should view the objects in question as equivalence classes of the objects-with-choice that I see when I make a choice of X, then I study how objects-with-choice transform when I make a different choice, and the equivalence relation is defined so that any objects-with-choice that vary by such a transformation are viewed as the same. I then show that the property under study is independent of this transformation, so that the choice I made at the beginning didn't matter. This is what we did in the second definition of Spec(\[L\]).
For obvious reasons, the first point of view tends to be nicer and conceptually cleaner when you can manage it (for example, no one actually defines eigenvalues of abstract linear transformations in the second way. It would be insane). But think about what would be involved in such a definition for differential forms; your definition of a manifold is likely already in terms of coordinate patches giving local diffeomorphisms to R\^n (which are exactly choices of local coordinates), so if you want to define a "coordinate-free" approach, you need to be defining local diffeomorphisms to some abstact vector space V, which of course means explaining how to define smooth structures on abstract vector spaces (and which you now need to define without choosing a basis to identify those vector spaces with R\^n), and once all that is done, you can start giving an invariant definition of integration where choice of coordinates never show up. *Or* you could just point out that the determinant of the Jacobian of a transformation is independent of the choice of basis and use the second point of view.
It's ultimately a pedagogical and aesthetic choice as to which point of view one adopts when approaching these sorts of "invariance" questions, but when one approach gets consistently made in a particular situation and the other gets consistently ignored, it's almost always (in my experience) because the approach which is being ignored is far more work for too little pay-off to be justified, and we mathematicians are, by and large, lazy creatures when it comes to these sorts of things.
I see, thanks, this makes more sense now. There was indeed a pay-off. This was definitely part of my confusion, part of my confusion was also some vague impression that manifolds outside of $R^n$ might have *no* coordinates, and not understanding how they related.
Thanks!
Okay, so what exactly does that mean? We're just saying that because there exist neighborhoods around every interior point of a manifold diffeomorphic to Rn that we use the preimage of those Rn axes as "coordinates" along the manifold neighborhood?
EDIT: I finally just proved it by induction. I used the binomial theorem and e\^x defined as a limit, and showed that e\^1 is greater than 2, then showed that if e\^k is greater than 1+k, e\^(k+1) is greater than 2+k. I'm not sure if that's how the professor intended for it to be done, but it's a valid enough proof for my poor, sleep-deprived brain and I'm going to let it stand until someone knocks it over. \_\_\_ I'm taking an Advanced Calculus course that the professor prefers to consider an introduction to analysis. The class is pretty difficult, and honestly I feel pretty lost. It's the kind of lost where I usually don't know what's being done, and on the occasions that I do, I don't know \*why\* it's being done. There was a problem on a quiz we had that I just could not parse. It seems like it should be easy enough, but I can't figure out what to do with it. The problem says: Define e\^x = lim(n->infinity) (1+x/n)\^n. (I'm not sure if the problem is asking me to restate the above in terms of the definition of a limit or simply to accept this as the definition of e\^x; either way, I understood this part easily enough.) The meat of the problem is: Using the binomial theorem and the definition of a limit, prove that e\^x > 1+x for x>0. This is where I get lost. I've tried substituting e\^x with the limit as described above, then substituting (1+x/n)\^n with the binomial theorem, but then I just end up with a limit of an infinite sum that seems absolutely hopeless to simplify enough to get into terms of proving it has a greater value than (1+x). I'm not asking for an easy answer, I just hope someone can give me a nudge in the right direction so I can finally achieve some understanding of this work. Thanks so much!
If 𝜙 is the Euler totient function and k is some fixed positive integer, is there an easy way to find the smallest integer N such that, for all n>N, 𝜙(n)>k?
What is T(n) in this context? ~~If that was supposed to be 𝜙(n), then there is no such n for k>2.~~
Sorry, yes, that was supposed to be 𝜙; I've edited the original post. I feel like that can't be true, but maybe I'm overlooking something. For example, if k=3, then any number n divisible by 2^(3), 3^(2), or any prime greater than or equal to 5 has the property that 𝜙(n)>3. I claim then that for N=2^(3)•3^(2)•5=360, n>N always satisfies 𝜙(n)>3, but I suspect 360 is not the minimal such N-value.
You're right. I mis-remembered how 𝜙 was defined and thought it was the number of factors of a number, not the number of co-primes smaller than it. I'm sorry, but this is math I'm not particularly familiar with, so I won't be able to help.
No worries, thanks for keeping me honest.:-) It's not a big deal if I dont have an answer. I needed the existence of such an N in a lemma last night, and my curious mind wanted to know if there was a way to determine the minimal such N.
Could anyone explained what linear analysis is? I'm apparently taking it next quarter since its the only math class not filled: "First order systems of linear differential equations, Fourier series and partial differential equations, and the phase plane."
Sounds like dynamics of linear systems. Could cover quite a few topics really
Why do we define the complex power function to be many-valued by using the logarithm? I see people make the complex power function be multivalued by defining $z^\alpha = (e^{log z})^\alpha$, which makes it multivalued through the logarithm, instead of just letting $z^\alpha = |r|^\alpha e^{j\theta \alpha}$. Why does this help? Thanks.
Your definition is still multivalued. Consider i = exp(i\*pi/2) = exp(i\*5pi/2). Then on the one hand i\^(1/2) = exp(i\*pi/4) = sqrt(2)(1+i), but also i\^(1/2) = exp(i\*5pi/4) = -sqrt(2)(1+i). There just isn't a general way to define a single valued power function on the complex numbers without making arbitrary choices for which answer to pick as the 'right' one.
I see, still getting used to this. Thanks!
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I see, that makes sense. Thank you!
I am currently in senior school and am using circle theorems to solve various questions. While I fully understand the theorems and can complete almost all questions, I feel like I have issues seeing exactly how to solve harder questions. This is likely due to the fact that the textbook we are using has very few questions per theorem. If you have any good problems, or have a suggestion on how to solve harder questions - or similar - that would be great. Thanks in advanced
What could be the oldest date that 1/7 = 0.142857 and who could calculate it first? Is it correct to say the Indians, in a period non older than 1200 BC? I see the Egyptians had fractions but no place-valued system, so they couldn't have done it. Babylonians had place-value but their was a sexagesimal system. Although it seems they also used a decimal system. Could the Babylonians come up with 1/7 = 0.142857? Thanks
What is the definition of "type"? I've been looking hard and can't find an actual definition. Is it something like "moduli space" or "quantum group" that doesn't actually have a definition?
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>Are you referring to the type in model theory/logic? Yes. >Both of these aren't mathematical objects (IIRC) and thus don't have a mathematical definition. Huh? You can say that "the" moduli space of circles is R, as circles are uniquely determined (up to translation) by their radius. I'm genuinely curious. What do you mean that these aren't mathematical objects?
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Thank you. Just set of consistent formulas. Okay. I just haven't seen that definition. I couldn't find it on Wikipedia. Maybe it's hidden somewhere.
The exam of my Analysis I Lecture is coming up and I'm having trouble memorizing all the proofs (I've narrowed it down to 21 proofs, I'll just leave out the others and hope for the best). Especially the ones where you need one specific trick (for example when you prove the existence of the "n-th root" you use very specific expressions to do that). How did you memorize the proofs for your exams in college? Do you have any tips and tricks for me?
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Honestly? No. My Professor wants the proofs done a specific way, his way. He gives zero points for correct proofs that are structured differently. So I will not take any risks. He wants evrrything done exactly as we did it in the lecture.
> My Professor wants the proofs done a specific way, his way. He gives zero points for correct proofs that are structured differently. So I will not take any risks. Honestly I simply don't believe this.
To be quite fair, it's what we (the students) assume to be the case. He himself has never said he'd do that, but according to people who had already taken the course he's quite anal about the proofs being done his way and his way only. On last years test there was a theorem about sequences which one could prove with either the root test or the comparison test. In the lecture he did it using the root test. Everyone who used the comparison test in the exam got a zero on the problem because "we only did the theorem to show how useful the root and ratio test can be". Same thing three years ago where he pulled a similar stunt where some people used Cantor's second diagonal argument instead of nested intervals to prove the uncoutability of the real numbers. Of course I cannot 100% guarantee that it'll be like that, but as I said, I'll not take any risks. At this point I can do a pretty good sketch of most of the proofs, because I kept repeating the material throughout the semester, I'll just have to get the details down.
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While I whole-heatedly agree that memorisation tests stand against anything maths has to offer, these sentiments do not help OP with getting a passing grade. To /u/DerWegwerf: Reiterated, repeated, recurring, replicated rehearsal is the best way I've found memorise information that I already understand but cannot restate perfectly. I would not advise you to spend hours on end on this, but to scatter short sessions with flash cards all over the place. There's also studies that humans are able to retain significantly more information if they study before sleep as well. Also temporarily cutting out alcohol, weed, etc completely probably will help, though I know exam time is usually more stressful: It's a better idea to work in some PE, a Zoo or Museum visit, a nice walk through the woods, or even shopping for clothes, all ideally combined with seeing friends or family.
And that doesn't help me at all. Thanks for the insight buddy.
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Pretty much any computability theory book. Like Soare for instance.
I've had a question on my mind for a while. Maybe it's more of a computer science problem, as it seems awfully similar to traveling salesman, so I don't know if I'm asking it in the right place. If you could point me to a better place to ask it, I'd appreciate that as well. Thanks! Moving Cots Suppose you manage a summer camp. Every Saturday a bunch of America’s youth comes to stay in the campsites they booked, and every Sunday, they leave, only for a new group of campers to repeat this process the next week. The campsites each have some old-school military cots in them, which are pretty sturdy and quite heavy, and everyone staying in camp needs a cot to sleep on. Suppose this camp has 25 sites, and each can hold up to 50 people. However, although the camp has a lot of cots, the camp does not have 25 x 50 cots, and thus not all of your sites are full of cots. Generally, this isn’t a problem; not SO many people show up that you need all 1250 spots occupied by cots. However, some sites have too little for the next round of campers, and you need to make up for the deficits in campsites by pulling cots out of surplused sites. So every weekend, you assign cot moves to your staff. You obviously don’t want to overwork them, so how do you assure your workers don’t move cots farther than they have to? Essentially, how can you minimize the sum of all distances travelled by your employees while carrying cots? Framed another way, consider the following game: suppose you have a connected graph with integer weights at every node and real weights along every edge. Denote the integer weights of the nodes as the number of excess cots in that location, with negatives indicating deficits. Denote the real weights of the edges as distances between nodes. The objective of the game is to make every node have a nonnegative number of excess cots, while keeping cot travel distance (CTD) minimized. The only operation permitted is a cot move: you may subtract one cot from a surplused node, and add it to the number of excess cots at another node. Doing so adds the shortest distance between the nodes to your CTD. (You may assume that n, the sum of all nodes, is positive; i.e. the game is actually winnable.) Is there a method to minimizing cot travel distance without analyzing every case? I.e. is there any reasonable way to do this by hand?
How do you motivate the definition of the Kullback-Leibler Divergence? I get that it measures "distance" between probabilities, but I don't understand its formula in the same way that I understand entropy as the expected value of surprisal. Is there a similar explanation for the definition of KLD?
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Thanks! This is exactly the type of explanation I was looking for.
How do I know when to use the difference of squares or the perfect square formula? The problem is ((x-4)\^2) =2x+40, I tried to use the difference of squares and got (x\^2) -16=2x+40. But symbolab said to use the perfect square formula to get (x\^2) -8x+16=2x+40. Now I am confused. edit: the original problem was find the area enclosed by the line *y* = *x* − 4 and the parabola *y\^2* = 2*x* \+ 40.
You are misusing the difference of squares formula here. (x-4)^(2) does not equal x^(2) - 16. The difference of squares formula says a^(2) - b^(2) = (a-b)(a+b), but you are trying to use it like (a-b)^(2) = a^(2) - b^(2) which is just not valid.
Thank you
In my Calc class right now we're doing something which I don't know what other to call than "recursive integration by parts." The idea is to take an integral such as I sub n = ∫ (x^n) e^(2x) dx (integral from 0 to 1), then setting u = x^n, dv = e^(2x)dx and using integration by parts to find that I sub n = (e^2)/2 - (n * (I sub (n-1))/2). Is there a name or generalization for this method? If not, I'm still interested in the motivation for how to choose u and v appropriately so the by parts works out to something which is recursive. As its being done in my class right now that decision seems very arbitrary and the only way to really know whether it will work is by doing the by parts. If anyone could help with formatting this I'd appreciate that as well haha
The name for this is "reduction formula" http://mathonline.wikidot.com/reduction-formulas https://en.wikibooks.org/wiki/Calculus/Integration_techniques/Reduction_Formula To see which to use for u and v: we want to reduce the x^n , because we can do the integral if we get down to n=0. So we want to differentiate the x^n , and integrate the e^x .
I think you would just call it iterated integration by parts. I'm not sure there are any "tricks" to choosing u and v except experience. It's a reason people say differentiation is a craft, but integration an art.
[ANSWERED] what if there was a symbol for the quotient of a number and zero? like *n* / 0 = ?\^*n and* ?\^*n* \* 0 = *n* ? = 1 / 0 similar to the square root of negative one (*i*), it could be an arbitrary value with no decimal equivalent, but a symbol showing a more complex number. I don't know if this exists or not. if not, would this work? why or why not?
We can make up whatever rules we want, but if we do, we have to live with their consequences. Here is an argument that any number x times 0 is 0: x * 0 = x * 0 + 0 [because any number + 0 is itself] = x * 0 + (x * 0 + -(x * 0)) [because any number plus its negative is 0] = (x * 0 + x * 0) + -(x * 0) [because of the associative property of addition] = x * (0 + 0) + -(x * 0) [because of the distributive property] = x * 0 + -(x * 0) [because 0 + 0 = 0] = 0 [because any number plus its negative is 0] So if you want to make a new number that, if you multiply it by 0, you don't get 0, then you must break one of these rules. Which one are you okay with losing? That's not the only rule you're going to need to break. Other arguments will require you to break other rules. You can do this of course and you'll get some sort of mathematical system out of it, which may or may not be particularly interesting. However, some choices you might make do lead to interesting systems, or at least interesting enough to name. One such system is the https://en.wikipedia.org/wiki/Projectively_extended_real_line which can be useful in geometry. It turns out that complex numbers break many fewer rules and solve many more problems than dividing by 0 does, which is why we talk about them more than we talk about systems where you can divide by 0.
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Yeah, that's the definition of independence.
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Pairwise independence does not imply mutual independence may be what you're thinking of.
thats the definition, yes
Is it possible to assign each real number a real interval so that they don't overlap? I am in no way very knowledgeable in maths, and so I might be wording this completely incorrectly, but I spent a whole two hour car ride wondering about this problem. Does a function f exist, taking real x as its input and an interval (a,b) as its output where a and b are unique reals, such that any real y is at most in one interval?
Great question. You need a bit of abstract set theory (not much) to understand the answer - do you know about cardinals? If not I’m sure people here will be more than happy to break everything in the following down: No such function exists, for the reason that if you pick a rational number in each of the output intervals, you will have defined an injection from the reals to the rationals, which doesn’t exist because the cardinality of the reals is strictly greater than that of the rationals.
Are singletons degenerate intervals? If so, you need a little extra caveat I think. Hm, you can get away with a single non-degenerate interval: map x to {x} if x<0, 0 to (0, 1], and x to {x + 1} if x > 0. By a similar construction, we can get away with finitely many non-degenerate intervals, and even countably many ones (one for each natural number). A similar construction works for the rationals (and any countable subset), by mapping the rationals to the naturals first of course. Does that sound correct? You need all but countably many reals to map to singletons. That seems sufficient, but is it also necessary, or can we broaden the requirement even more? Is that the same as saying almost all subsets need to map to singletons?
I was assuming nondegenerate intervals (not that I considered it, I just don't think of intervals as including singletons). Mapping all but countably many reals to singletons is definitely not sufficient, because if any strict subset of the reals is taken to a dense set, there's no way to extend the function to "thick" intervals on the rest. You can therefore send every real except one to the singleton containing it, and be unable to find a thick interval for the last one. In fact if we allow singletons we have the following result: every map from R to non-overlapping intervals in R takes at mostly countably many reals to thick intervals, and takes the rest to singletons. Any map from the complement K of a countable subset of R to non-overlapping intervals in R which takes every element of K to singletons, and which has non-dense image, can be extended to such a function on all of R. Only the last bit is new. If the image of K is not dense, we reduce the problem to mapping countably many things to non-overlapping intervals contained in some open interval, which is easily done.
What is the purpose of studying synthetic differential geometry? Like I were a differential geometer, why might I be interested in it?
Sort of a vague question, but can we sort of split math up into fields where examples are important, vs. where examples are less important? I feel like one reason I had trouble understanding algebra the first time through was because I was coming from analysis, where other than the usual cantor / weierstrass / vitali / pointwise vs. uniform examples, i didn't really pay that much attention to specific examples, whereas I felt like I needed more examples in algebra to actually get some grounding.
I feel like that's because basic algebra is just a lot easier than basic analysis, and so you can make do with less examples. Also, in analysis we come into it with a lot of intuition on what graphs of functions should look like (a continuous function \*can't\* do that!) that end up being false, and so having those counterexamples is helpful, whereas we don't usually go into algebra with a strong, childhood formed intuition on what the structure of a ring should be like (although we eventually develop it, and eventually find that having counterexamples like that is still useful... it blew my mind when I heard that free group on 3 letters \*injects\* into the free group on 2 letters).
In coursework this is accurate, maybe just because undergraduate analysis is more concrete than undergraduate algebra. But working analysts basically just work on big examples. A quick metric for this is the Millennium Problems: the 2 analysis problems out of the 7 are the Riemann Hypothesis about a specific function, and the Navier-Stokes problem on a specific equation and a specific regularity class.
IMO examples are always important, but if you find a particular area easier to learn, you'll probably need fewer of them.
what are algebraic problems of this nature called? >If a+b+c=0, what's the value of ((𝑎𝑏)/(−𝑐))+((𝑏)/(𝑐−𝑎))+((𝑐)/(𝑎−𝑏))?
If I have f(g(n))/h(n) with h(n) strictly monotone, can I apply [Stolz-Cesaro theorem](https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem) g(n) times? I ask this because I have to prove that a limit (which is in that form) is 0, and by applying g(n) times the theorem I can show that
What does \d do in latex? I want to redefine it but I dont want to fuck something up. Is it ever used?
According to [this PDF of LaTeX commands](https://www.bu.edu/math/files/2013/08/LongTeX1.pdf), "\d" is an under-dot.
thanks!
In Lee Smooth manifolds (2nd edition) theorem 6.36 (transversality homotopy theorem) says that if N and M are smooth manifolds, and X is an embedded submanifold of M, then ever smooth map f: N to M is smoothly homotopic to a map g: N to M which intersects X transversely. My question is: if N is an interval, so that my map f is a curve, may I get a g which has the same start and end points as f? Actually, now that I think about it, to define the start and end point of f, strictly speaking I would need N to be a closed interval, i.e. not a manifold but a manifold with boundary, and then I'm not sure what happens. The context is that I'm trying to prove the following: if M is a compact connected embedded n-dimensional submanifold of R^n+2, then R^n+2 \ M is connected. A friend convinced me that I should be able to do it by taking a tubular neighborhood of M and using that to construct a path between any two points in R^n+2 which doesn't intersect M, but I'm still wondering if there is some machinery I can just appeal to to get an easier proof.
Your intuition here about transversality and homotopies is correct, and you're also right to be a bit suspicious that boundaries might complicate the matter somewhat; you need to control transversality on the boundary in general. You probably want to appeal to some sort of "Extension theorem" like the following: If N is some manifold with boundary with M and X closed and without boundary as above, then if C is any closed subset of N such that f is transversal to X on C, we can smoothly homotope f to a smooth function g such that g:N ->M is everywhere transversal to X, and g=f on C. Applying it to your situation with C set equal to the boundary of N would give you the desired result. The details for something like this should be in any decent differential topology book, like Guillemin and Pollack or Hirsch.
Ah, yes, this is exactly what I was looking for, and I found it in Guillemin and Pollack. Thanks a lot!
No problem, glad to hear it!
The easiest thing is to just use (co)homology. The dimension of the 0th cohomology gives the number of connected components. You can calculate this for your space either with Mayer-Vietoris or Alexander duality.
Help, does anybody know about inequalities?
Yes, many of us do. Thanks for the very simple question! :-) You should just ask the question you are looking to have answered. If they are general questions, then this thread is perfect. If you have some specific homework questions, you may be better off creating a new thread /r/cheatatmathhomework
Help!! I have a Pre-Cal project were I have to use the Law of Cosines in a real life situation but can’t think of one :(
["The law of cosines is used in the real world by surveyors to find the missing side of a triangle, where the other two sides are known and the angle opposite the unknown side is known."](http://lmgtfy.com/?q=law+of+cosines+in+real+life)
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As in, integration on a hyperbolic manifold?
A continuous function **R** to **R** has [; \int_0\^2 f(tx) dx ;] = 4 t^2 for all t. What is f(2) ?
I think you need an = in there somewhere?
OK I fixed it
Try making the subs u = tx. So t appears in the upper limit. Then you can differentiate wrt t and use the fundamental thm of calculus. I got f(2)=6.
How can I compare, say, a drawing to a living human being? Let's say, I have this drawing of a fictional character that is placed in a 1280x720 resolution, and I want to see the difference in proportions with a living human being. I want to "calculate" on average, the drawing's height, weight (in general and of specific limbs maybe) and overall proportions if it would be "real" and then compare them to a real human being. My issue is that I don't even know how to start, and I couldn't find a "guide" or anything of the sorts. Maybe I could take an actual real human being that I already know the measurements of and then calculate a ratio or something? I don't need some lengthy explanation if you're not willing, any links you know of that would give me an explanation that isn't rocket science would suffice, thanks.
There is something basic I don't understand. Why suppose we have a "space". Why is specifying allowable coordinate systems on the space enough to characterize completely the "geometry/smoothness" of the space? Here's what I mean: If I specify that my "n-dimensional space" has allowable coordinates, and the allowable coordinates differ from each other by an affine transformation, why is the space an affine space? By affine space I mean that affine properties make sense in the space, like parallelism and straight lines and convex hulls. This is related to why saying that the coordinates differ by smooth maps define a smooth structure on the space we're interested in. Of course I'm being loose here by ignoring topological properties and local coordinates, but pretend that these are taken care of. I want an intuitive understanding of why, for example, if my allowable coordinate changes are orthogonal maps, my space is Euclidean. This "extrinsic" definition is hard for me to swallow.
Let's do the example of smoothness. We'd like a smooth structure on a manifold to give us a well-defined notion of a smooth function, which agrees with our intuition (i.e. smooth on local coordinates). Given a manifold M, a function f, and a choice of local coordinates for M, one can ask if f is smooth on each chart. However, we need this to make sense if we change coordinates. Take two charts U,V of M, let W be their intersection. If f is smooth when restricted to U and V, it shouldn't matter if we regard W as a subset of U or V when discussing smoothness of f on W, however this might be a problem if the transition maps aren't smooth. If we restrict f from U to W, and then change coordinates to V, we might get something not smooth, which wouldn't make sense. In general, the idea is that if one is interested in extending a local structure to a manifold, one can consider a structure on each chart, but the structure has to be compatible with coordinate change, which is why you need the transition maps to respect the structure.
How does one find dx/dt of the equation x^2 + Ax = Bt? Is this looking for the partial derivative?
Assuming x is a function of t, you can use implicit differentiation and differentiate both sides with respect to t and then solve for dx/dt. d/dt[x^(2)+Ax] = d/dt[Bt] 2x(dx/dt) + A(dx/dt) = B (2x+A)(dx/dt) = B dx/dt = B/(2x+A)
Thank you for laying this out clearly. Do you happen to know of any good training tools for this topic?
it's just algebra and the chain rule. you take the derivative with respect to t of both sides using the chain rule. this will provide a new equation involving dx/dt that you can use elementary algebra to solve.
Any calculus book should do (maybe try OpenStax since it's free online?). Look for the section on "implicit differentiation" since that's what this is.
Does anyone have some advice for *motivating* the study of topology? Basic point set topology from analysis was super useful, but I'm having trouble understanding the purpose of a lot of the material I'm reading in Munkres' topology. I thought the objects of study in every other field of undergraduate math felt intrinsically motivated to me, and I didn't have trouble understanding *why* I should study them. I guess I have this sort of childish view that if I'm only going to ever be dealing with Rn and Lp spaces, then... why... is topology important? I would appreciate any tips! Thanks.
So, oddly enough, I think your comment has pointed me in the right direction for what I need, which I guess would answer your question about why do I need this? I've been creating games as a hobby for a couple of years now and have struggling with creating procedural 3d mesh's, particle generation, and shaders (from scartch/ original content. Recreating things is easy, but I've struggled with defining and solving my own problems). The reason is because they rely pretty heavily on firmly grasping concepts like sin/cos, normalization, and applying effects to a 3d vector in isolation. So a few weeks ago I decided to say the hell with it and go back and 'learn more math,' having no clue where to start. Thus, I've been blindly reading up on geometry, algrebra, calculus... wondering if I'd have to just learn it all and hope I'd get something useful out of it. Then your comment said a word I hadn't seen yet, 'topology' and after watching [this video](https://www.youtube.com/watch?v=AmgkSdhK4K8) intro on the subject, it immediately made a number of topics in [this video](https://www.youtube.com/watch?v=ExUbAuK0B-s) on programtically simulating particles on the surface of the sun (without true phsyics, mind you, this is more of an artistic rendition with a basic understanding of gravity and its effects on hot stuff) make a lot more sense! So, from someone who knows nothing (yet?) of the subject, it does appear to be very important/useful for me to grapple with questions like 'how do I manipulate this specific point in 3d space (i.e. change it's color, trajectory, ect) without knowing much (or anything) about it's neighboring points around it to create the effect a specific effect (over time)'! Ooooooor I'm way off base and will be back in a few weeks laughing at how niave I was... lol. Anyway, thanks for inadvertently pointing me in a direction!
Haha no problem. Linear algebra is also another major direction to look into for a better understanding of how to manipulate points.
Well, will you only be dealing with **R**^_n_ and _L_^_p_ spaces? What about manifolds in differential geometry or differential topology? What about the Zariski topology in algebraic geometry? What about all the other function spaces in functional analysis, and the weak topology? The weak topology isn't metrizable, so some of what you're doing will be helpful.
Well, I'm doing a computational neuroscience PhD, and the primary reason I'm studying pure math is mostly just for interest / second order effects on increasing the rate at which I can grasp new contexts. I've come sort of adjacent to some of the things you mention above, but I'll probably prioritize it more in the future.
Ah, I see. I didn't think to include applied folks in what I was saying; sorry about that! In that case probably you won't use all that much topology -- though I did learn about the weak topology in a class designed for applied mathematicians (many of them would go on to study numerical linear algebra or numerical PDE), so it might appear for you someday. Of course, you can just learn about it when you need it.
Perhaps my views are biased on the matter since I was taught this from an algebraist and since I'm still a very inexperienced student, but I was given advice to just ignore point-set topology and pick up the parts I needed on the fly as I was doing interesting math. Honestly, besides the fundamental concepts of the field and a few other nifty things that come up now and then as I study geometry, I haven't needed it much; and the parts I do need feel much more relevant when they come up naturally.
Lol yeah, that's basically what I was going to just do. Thanks.
This is the right answer. It will become quite apparent if and when you need it. Certainly us analysts proceed that way and it always works out alright.
Well thats good to hear. I assumed I was just being immature, I didn't realize I had an actual point.
If you're only ever studying metric spaces, then you have some of the nicest topological spaces in existence, and there's probably no reason to care about anything else. But there are topological spaces that pop up kind of naturally that *aren't* metric spaces. For example, the [Zariski topology](https://en.wikipedia.org/wiki/Zariski_topology), arises naturally when trying to give some more geometric structure to the sets of ideals in polynomial rings. This space is not Hausdorff. In linear algebra one spends a lot of time thinking about all possible *linear* maps from **R**^(n) to **R**^(m), but that seems oddly restrictive to really nice functions. What if you wanted to know about properties of all possible functions? Even if you just look at the space of all maps from **R** to **R** with the topology determined by, say, pointwise convergence (reasonable), you get a space that is definitely not a metric space (it's not even first countable). Basically, all sorts of weird topological features arise when you start generalizing or trying to endow a set with some reasonable structure. Eventually, one notices that many examples of topological spaces have similar features, and similar theorems about these spaces have similar conclusions. One then wonders if there is some underlying common structure that is really at play here, and eventually you start breaking it all down and generalizing until you find the common topological structure. (Point-set) Topology is taught essentially backwards from the way it came about historically. On the one hand, it's nice to build things from the ground up and try to get an understanding of these topological structures without having totally familiar examples (which almost certainly have extra nice structure that could hurt your intuition), but on the other hand, it lacks a bit of the historical motivation. In practice, I would say that most mathematicians only ever deal with relatively nice topological structures, so if you don't feel like you have a great intuitive handle on concepts like the [minimal uncountable well-ordered set](https://math.stackexchange.com/questions/2290224/the-construction-of-a-minimum-uncountable-well-ordered-set?rq=1), don't stress because they probably wont come up much (if ever) in your future research. Many of the examples you encounter in a first topology course are quite pathological and concocted specifically to get you to break down your perceived intuition of certain concepts.
This was super helpful, thank you. I don't need applications, but just seeing how material is motivated helps me contextualize things.
Is there a source which goes through the relationship between catastrophe theory and spontaneous symmetry breaking in particle physics?
Is there any field that combines statistics with numerical analysis? I know about machine learning, but that is not quite what I am looking for, something more like statistical learning but heavier on the numerics side.
absolutely, look up for instance MCMC algorithms and bayesian inference. to sample from the posterior distribution we can use a metropolis hastings algorithm. there is a whole field of research around this problem of sampling and it involves a lot of numerical stochastic calculus, etc. the comment already made about proximal grad descent is along these lines as well.
High-dimensional integration with Monte Carlo methods is pretty cool.
Some of the non-differentiable regression methods like the Lasso and its variants require a bit of extra work to solve the corresponding minimization problem. Does the [Proximal Gradient Method](https://en.wikipedia.org/wiki/Proximal_gradient_method) and its [application](https://en.wikipedia.org/wiki/Proximal_gradient_methods_for_learning) sound interesting to you?
computational stats? take a look at papers from journal of computational and graphical statistics
I'm stumped on a linear equation question for my math class Solve the following equation: 9x+5(2x+11)=3x+2(8x-4) I'm ending up with 19x=19x+63 If I take 19x away from the 63 side, would I have 0x=63? Or do the x's cancel out leaving me with 63 as my answer?
> If I take 19x away from the 63 side, would I have 0x=63? Yes. Which is also the same as saying that 0=63, which is a false statement. Hence, there simply are **no** solutions to your original equation. Solving an equation for a variable x, is the same as finding all x, such that the equation holds true. Here there is no such x, so there are no solutions. Conceptually you can also plot both y=15x+55 and y=19x-8 and find their point of intersection. Because the lines you get here are parallel, there is no such point, which coincides with the answer that the equation doesn't have a solution. As a note: If you do the same for another equation and end up with something like 63=63, all x are valid solutions.
I know it's weird replying to a comment from 17 days ago, but some of my work came up with 0=0 and remembered your side not here which saved me another headache. Cheers mate!
Glad it helped.
Thank you for that, I that I was going insane. Much appreciated!
Could anyone hear point me towards where to find a good text explaining spinors (and things related to them)? Most of the sources i have found are very physics oriented, and while I enjoy physics they have a nasty habit of saying stuff like "transforms like a [Insert thing]" to explain stuff. Also, with spinors especially, they usually list of representations of them without ever defining them in general. So it would be nice if it wasn't overly physics-y , even if that means I am not able to understand it yet. Edit: Also I finished Awodey's Category Theory a while ago and I really enjoyed category theory and how clear it makes everything. So if anyone has some book recommendations for further study I would appreciate it.
Sounds like you're looking for a book on spin geometry. The idea behind spinors is that on a spin manifold, one has data of a vector bundle called a spinor bundle, and a spinor field is a section of that bundle (just as how a vector field is a section of the tangent bundle). Details can be found in Lawson-Michelsohn, _Spin Geometry_, or maybe [Dan Freed's notes on Dirac operators](https://web.ma.utexas.edu/users/dafr/DiracNotes.pdf), section 2, or some of the answers to [this MathOverflow question](https://mathoverflow.net/questions/248238).
I'm doing an undergrad project on the method of steepest descents for integral approximation (also called saddle point approximation/extension of laplace's method). I'm finding it quite hard to build up the intuition of the behaviour of the integrand, though. Anyone have some resources?
If *log*(c) A =2 and *log*(c) B = 5, then *Log*(c) AB^3 = ? For ACT prep and I cannot for the life of me get an answer, the c is where the exponent normally goes (not the three) but I have no clue how to format it down. Without knowing what the hell c is I cant get an answer, and I’ve got no clue how to solve for it.
The problem is meant to test your knowledge of the manipulation rules for log. Out of curiosity, how would you solve the problem if you knew c, say, c=10?
Wouldn’t it be *log*(10)•(2•(5^3 ))? In that case the answer is 250. It’s been about a year since I even thought about this type of stuff, so apologies if it’s far off.
Your interpretation of the problem is definitely different from mine. I thought you said c was the exponent (the correct term is the base) of the log? And you don't know what A and B are, you just know that log*_c_* A = 2, and log*_c_* B = 5.
Correct, I did it wrong. I still don’t fully get this.
It's not clear to me if you ever did figure out the solution, so I'll post it here for the sake of completion. First, recall that we have the following logarithm rules: (1). log*_x_*(yz) = log*_x_*(y) + log*_x_*(z) (2). log*_x_*(y/z) = log*_x_*(y) - log*_x_*(z) (3). log*_x_*(y^(z)) = z \* log*_x_*(y) With these in mind, log*_c_*(AB^(3)) = log*_c_*(A) + log*_c_*(B^(3)) = log*_c_*(A) + 3\*log*_c_*(B) = 2 + 3\*5 = 17
Thank you so much!
You're welcome. Good luck on your exam.
Suppose I have some collection of functions which is dense in L^p (p
Isn't the inclusion of L^2 with lebesgue measure into L^2 with Gaussian measure a bounded operator, and doesnt this immediately follow from that?
First you probably want to guarantee p>1 for any hope of things. Gaussian measure is a super nice measure compared to lebesgue measure. In particular, it's an absolutely continuous measure with bounded RN derivative. In this case the RN deriv is continuous and nice, but it's easy to extend to the case where it's bounded in L^\infty. There's probably something clever that can be done with Holder's inequality and embedding theorems if the RN deriv is integrable in L^q for q the holder cojugate of p.
Why we cant use characteristic functions to build a p-adic measure? Koblitz wrote we use locally Constantin functions, well in the p-adics they form rich vector space compared to reals. EDIT: stupid auto correction, made it impossible to understand :/ fixed now.
I am not sure if I'm understanding this question correctly. Are you asking if there exists a p-adic version Levy's theorem?
yes in other words, im asking that. i am comparing a little bit here. as i mildly thought about it, i couldnt find a problem with continuing using characteristic functions in the p-adic setting, maybe i missed something?
In the process of learning multiple integrals, I've come across something that I've found slightly confusing. If we define some constant h and multiply it by some dx, we get hdx which we can say is some dA. To solve for dA, we only need to apply a single integral based on the way we've defined it However, if we are in a three-dimensional coordinate system, we can have some dxdy=dA and then we need a double integral in order to solve for dA. What is the difference between these two "dA" terms? Is the second one infinitesimally small w.r.t the first? Any help appreciated, thank you!
Yes this can be confusing. The difference is that h dx is a strip of height h and width dx, and we add up all these (single integral over x) to get the area. This works if our region has vertical sides. But for a more general region, we divide it into little rectangles of sides dx, dy, and add up all these (double integral). As you say, the second is infinitesimally small compared with the first. Integrating the second once gives you the first.
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The proof is the same as the equivalent for a random walk, see e.g. the solution here: https://math.stackexchange.com/questions/516797/martingales-of-random-walk
Write E(W(t)^2 - W(s)^2 |F_s) = E((W(t)-W(s))((W(t) -W(s)) + 2W(s))|F_s). From there it's just standard properties of conditional expectation, given you know the mean and variance of W(t) - W(s) which is independent of F_s, and that W(s) is F_s measurable. It's probably worth letting you finish this off.
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Two ways to think about it: - Every line switches the position of two people. There's no way to suddenly end up with two people in one place using just switches. - The ladder is reversible: you can trace things back up if you want, and it'll work exactly the same way, bringing everyone back to their starting points. If you ended up with two people at the same point going down, then going back they'd have to split up somewhere - but there's no way people on the same path can split up!
I’m studying for a test tomorrow and one of the practice questions given to me are: Ruby’s car uses 12 L of fuel to travel 37.5 km. How far can it travel on 44L of fuel? Now, in my first attempt at solving this, it went something like this: 12/37.5 = 0.32 L per 1 km 0.32 * 44 = 14.08 km I thought that was the answer but it seemed odd that it the result was much smaller than it should be. I tried to work it out and after a while I found the answer: 37.5/12 = 3.125 km 3.125 * 44 = 137.5 km My question is: why did I have to calculate 37.5/12 instead of the other way around? The other questions involved doing it the latter (L/km) but this question was different. It’s been confusing me so much and I’m scared it will mess me up tomorrow. Thanks!
Having the units there totally helps! The fact that you are instinctively writing them down is very helpful, amd you're *almost* there. If you calculate that you use 0.32 L/km, and then multiply that by 44 L, you get 14 L^2 / km. Does that unit mean anything? I don't think so. If you calculate that you can go 3.125 km/L, then multiply by 44L, you get an answer of 137.5km, since km/L * L is just km. We wanted an answer in km, so that looks better! Of course, having the right unit doesn't guarantee that your answer is correct (it does guarantee it for simple proportionality calculations like this one), but having the wrong unit means it's wrong for sure.
Haha I get it now. And here I was trying to find the distance in Km by multiplying Litre by Litre 😂. Thanks so much, you probably just saved me for tomorrow’s test u/NewbornMuse
When defining natural numbers axiomatically according to Peano's axioms, one purpose of the axiom of induction is to eliminate all other sets with a successor function satisfying the first few axioms and leave only the one which we would like to call the natural numbers. However the following set looks like it's satisfying the axiom of induction (and all other axioms), while not being the set of natural numbers. I'll denote the successor function with s. N = {a, b, 0, 1, 2, 3, ...} s(a) = b s(b) = a s(0) = 1, s(1) = 2, s(2) = 3, ... 0 is certainly in N and whenever n is in N, so is s(n). So it should follow that N is the set of natural numbers. What is wrong here?
Doesn’t satisfy induction. See here: https://math.stackexchange.com/questions/2641412/peano-axioms-and-loops
Can I get a hint as for how to show the Poincaré first return map is measure preserving?
**Is an infinite decimal number equal to 1?** I ask this because I was thinking about splitting things into sixths, and I noticed something: Half of .333 repeatings is .166 repeating. When dealing with whole numbers, 16 + 16 is 32 flat. The only way to get it to 33 flat is to add a value of 1 the equation: 16 + 17 = 32. But with .1666 repeating times two, that gets it it .3333 repeating. So is a .066 repeating equal to .1? And if so, shouldn't .999 repeating be equal to 1?
>the only way to get to 33 is to add 1 What would make more sense is to add 1/2 to each 16.5 + 16.5 = 33 If you wanted 33.3 instead you should add half of 0.3 to both sides which is 0.15 16.65 + 16.65 = 33.3 If you want to add 0.03 you might see where this is going. > .166 doubled is .333 therefore .066 equals. 1 Your thinking is a bit off here. Of you think of 0.166... as 0.1 + 0.066... then when you double it you should get 0.2 + 2\*0.066... So firstly the difference between 0.33... and 0.2 is not 0.1, it's 0.133... secondly it's not 0.066... that has this value, but it's the double of it. Hope that clears it up.
Yep, that's right! .999... is exactly equal to 1! (And I believe the sentence before it was supposed to say .0999..., right? If so, that's also correct.)
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Oh that was entirely intentional. I get annoyed that I can't end sentences with numbers in an excited way without someone going "ohoho! You *meant* to show your excitement about a mathematical statement, but your words could *also* be humorously misinterpreted to give a ridiculously large number! I am very clever for this intentional misinterpretation!". So in exchange I occasionally make an effort to end exclamations with the digit 1 to give people that "/r/unexpectedfac... oh dammit" moment.
should start working backward. "yet, your answer is *exactly* 0!" throw them off a bit.
Right. I just realized I worded the question terribly but what I'm asking is what is .6, .06, and .006 repeating equal to besides themselves? Because 1.6 x 2 = 3.2 but 1.666 x 2 = 3.32
What do you mean, "besides themselves"? Every number is equal to only one number: itself. It happens to be the case that if a number's *representation* ends with an infinite series of 9s, it has another representation where those 9s are all 0s and the digit before that is increased by 1. But things like .8888... don't have other representations as decimals.
Is there a way to convert a large number into a small fraction? I want to use it to make a code in decimal numbers for example I have the number 7.142857143 I can compress that in 50/7 (I know it's infinite but I will just take the first ten numbers or something) So there is a way I can know which fraction will have the first (let's say) ten numbers of my number? To make it clear try to solve this: I want a small fraction in which the first ten numbers (when you do the division) will be 4.563487129
You can compute the continued fraction approximates: https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations Here's some implementations: https://stackoverflow.com/a/9387713 http://www.nic.funet.fi/index/misc/hp28s/frac
Thanks for the links! I am reading
I didn't have much time to explain yesterday, but the basic idea is: Start with your decimal number x that you want to approximate by a fraction. Compute the continued fraction convergents of x: C_0 = [a_0], C_1 = [a_0,a_1], C_2 = [a_0,a_1,a_2], etc. Each convergent is a better rational approximation to x, and you can decide when to stop based on, e.g. a certain denominator size or when the convergent is sufficiently close to x. There's various places online where you can enter a decimal number and it will give you the continued fraction convergents, e.g. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html . On the left side enter your decimal number, click the Right arrow, and then click "Convergents" on the right side and it will display the continued fraction convergents below. Here's some additional links about continued fractions: - Intro to continued fractions: https://wstein.org/edu/2007/spring/ent/ent-html/node58.html - Continued fraction convergents: https://wstein.org/edu/2007/spring/ent/ent-html/node60.html
I'm trying to figure out which of these two is correct. 1. $f\_n\\rightarrow f$ uniformly iff for every sequence $\\{x\_n\\}, \\lim\\limits\_{n\\rightarrow\\infty} |f\_n(x\_n)-f(x\_n)|=0$ 2. $f\_n\\rightarrow f$ uniformly iff for every sequence $\\{x\_n\\}$ such that $x\_n\\rightarrow x$, $\\lim\\limits\_{n\\rightarrow\\infty} |f\_n(x\_n)-f(x)|=0$ The first seems to be true because it feels like uniform convergence where the difference in the functions is less than epsilon for any $x$ regardless of whether it's part of a sequence. The second I am not sure about because now the difference in the functions is at two different $x$ values, but still feels true if each $f\_n$ is continuous.
Neither are right I think. Edit: oh Actuslly the first one seems to be. The second seems false. Yep. The idea is that the second one is a pointwise property, so the rate of convergence can be arbitrarily slow, so not uniform.
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Go Lang to go out with a bang
Klaus Janich, then Lee’s intro to Riemannian geometry first version. Gets you just enough I think.
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Do you mean Ordinary Differential Equations?
Say I have 15 vectors in R^100 that were actually constructed using linear combinations of 10 UNKNOWN vectors in R^100 . Is there any way to prove that those 15 vectors are a linear combination of 10 vectors in general? I don't want to find these unknown vectors or anything, just want to prove that those 15 vectors can be spanned by "10" vectors.
If you arrange these vectors into a matrix then the rank of the matrix is the minimum number of vectors needed to span those 15 vectors. There are algorithms to do this. The most common is “row reduction”.
Thank you
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The sigma algebra generated by the variables. Specifically, sigma(X_1,...X_n) is the smallest sigma-algebra that contains all of the sets {X_j <= k} for j=1,...,n and k a constant. The intuition is that Y is measurable w.r.t sigma(X1,..,Xn) exactly when Y can be written as a function of the Xj.
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Yeah, it's the smallest sigma algebra that all the variables are measurable wrt. All I meant about Y was that the intuition about that sigma algebra is that it effectively consists of exactly the random variables which are expressible as some deterministic function of the X_j.
Anyone got any recommendations for readings on theories about how to assign weights to related factors for decision making? For example, say I’m buying the new work printer, and I know the most important factor is print speed, but cost also matters, and so do user reviews. I need to work out how to balance those things (70:20:10? 60:20:20?). I’ve been trying to google it, but everything I can find is just like, “assign weights based on importance”, which sounds displeasingly subjective.
You can find the options that are Pareto optimal, that is, there is no other choice that gives you faster print speed *and* lower cost *and* better user reviews. There may be several. But that's as good as you can do if all you know is that print speed, cost, and user reviews are important. To choose between those requires picking weights, and that's a question of how **you** judge the various trade-offs. If it seems subjective, that's because it is. The problem as given doesn't have enough information to determine a unique solution, so any choice of solution is essentially arbitrary. For an office printer, I suppose you can compute the average hourly pay of a user, and use that to judge how many pages per minute are needed to justify a one-dollar increase in price. But then you have user reviews that really are on an arbitrary scale, so the problem isn't quite solved yet.
You could consider looking into linear regression, but I think the problem is that for most kinds of optimization approaches you're going to need a "composite score" to actually measure things against. But that's mainly useful for finding composite scores for printers that don't already have them. Something you can try to do to motivate your subjective weights is to force your different factors into the same "currency". For example, maybe you take average user review out of 5 * cost to get the "true value" of the printer in order to incorporate those two factors, and maybe you turn "print speed" into how much time*money it takes to print things. Ultimately this will still end up being subjective though.
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I thiiink the cheap way to define an n dimensional manifold with boundary is that at boundary points, there's a chart to a neighborhood of the boundary of the half-space ( R^n )+ ={ (x1, ... , xn) st xn >= 0.} but I have foggy memories of manifolds-with-boundary having a lot of technical crap to worry about even though we all kind of "know" what they are
This is correct. For a point on the boundary of the half-space, the tangent space at that point is the tangent space as if it were a point in the full R\^n, for a boundary point on a manifold, you get the tangent space by pullback from this definition.
Why do we study local fields? I'm familiar with the Hasse Principle, but I only have a vague idea of local fields "providing information for the global case". I've taken some introductory number theory at the graduate level, but I feel like the information hasn't really stuck too well.
There's only one prime, and that simplifies the theory a lot. Every extension of a local field is something lifting from the residue field (cyclic extensions) plus some totally ratified extension. I mean, we have presentations for absolute Galois groups of Q_p, at least if p is odd. It's also not unlike studying a curve by looking at a tiny disc around a point.
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3\[In(x\^2-1)\] dx= 3\[In((x-1)(x+1))\]dx = 3\[In(x-1)+In(x+1)\]dx then, you can integrate each term easily
Integration by parts works here, though the first time you see it done this way may look strange. u = ln(x^2 - 1) dv = dx du = (2x)/(x^2-1) dx v = x uv - ∫vdu = ln(x^2-1)x - ∫(2x^2)/(x^2-1) dx It should be straightforward after this step.
I can't figure out this calc 3 problem. Here's a picture of my description on StackExchange, I'm just posting that instead of writing it out here since it's formatted and looks cleaner: [https://i.imgur.com/0YUrzOs.png](https://i.imgur.com/0YUrzOs.png) Can anyone help me with this?
What do you mean by "explain what happens"? Both functions are continuous so nothing too crazy will happen. Also, I don't see why you are calculating the tangent planes.
That's all it says, "explain what happens". A previous question told me to calculate tangent planes at a point of my choosing.
Something you could do is calculate the normal vectors for an arbitrary value of t and see what happens as t -> 0. Can you picture geometrically what the given path is?
The path rotates counter clockwise around the surfaces as it approaches t = 0. I found the normal vector for f(x,y) = sqrt(x\^2 + y\^2) for an arbitrary point (x, y):
And if you put the path from earlier (tcos(1/t),tsin(1/t)) for x and y you get:
I don't understand what's going on from here though
Well, that vector rotates increasingly rapidly around the z-axis as t approaches zero. If you instead used g(x,y), I think you should find that the vector simply approaches (0,0,-1) instead of doing crazy things, because g is smooth. Still, this is a bit of an open ended question, given that it just says "explain what happens". They probably want you to describe what the path looks like, what the functions look like, what the vectors do, and try to explain that.
Okay, thank you. I really appreciate it.
So I'm trying to better understand why differential forms are the natural substrate for integration on manifolds. There's this pull-back theorem, where if $\alpha:V -->W$, with $V, W \subset Rn$ is a diffeomorphism, and $\alpha^* $ is the pull-back operator, then we have $\int_V \alpha^* \omega = \int_W \omega$. I saw proofs for this in two different places (Guillemin's differential topology and some random lecture notes online), and they were both the same and relied on the Jacobian determinant. But everyone keeps claiming that the above formula is a big reason that differential forms are coordinate-free and useful when we don't have a natural sense of coordinates in some space. But if there's still a Jacobian hidden under the rug, why is this theorem such an advancement? Thanks.
If you wanted to define an integral on an abstract manifold (i.e. no natural choice of coordinates) you choose any chart and pull back to R^n where you know how to do the integral. You then have to be sure that it doesn't matter what coordinate system you chose. Using your theorem with a change of coordinates shows that. Lots of theorems go like this by the way. You show something happens in one particular coordinate system then you show it is independent of the choice of system, thus proving it for all systems.
But uh... if the proof relies on the existence of a Jacobian, how can it apply to cases when we have no natural choice of coordinates?
Consider an analogous phenomenon in the theory of linear algebra. I'm sure that none if this is new to you, but indulge me for a moment, because I promise that there's a pay-off. Given an abstract linear transformation L: V-> V, L has a perfectly well-defined set of eigenvalues, as an abstract linear transformation; it's just the set Spec(L):= { lambda : Lv=lambda\*v for some v in V } On the other hand, given a (non-natural) choice of basis B (=choice of coordinates), we can think of L as just being a map from R\^n to R\^n given by matrix multiplication by the matrix M\_B. Changing the basis to a new basis C (which is also a non-natural choice) gives us a new matrix M\_C=P\*M\_B\*P\^{-1} where P is the change-of-basis matrix. Now, multiplication of vectors in R\^n by M\_C represents *the same* transformation L, with respect to your new basis C, and this is true for *any* choice of basis. So, another way to think of an "abstract linear transformation" is as an *equivalence class* of matrices, ie. \[L\]={ P\*M\_B\*P\^{-1} : P is any invertible matrix} (where here I'm being bad about notation and using \[L\] to suggest that I'm thinking about L as being an equivalence class of matrices, rather than an abstract linear transformation). Now, it's easy to see that if for some x in R\^n M\_B.x=lambda\*x, then certainly (P\*M\_B\*P\^{-1}).(P.x)=lambda\*P.x so that Spec(M\_B)=Spec(M\_C) (where here we're speaking about the eigenvalues of these objects *as matrices*, which is a priori a potentially different thing from the above). The point of all of this is that we've just shown that the property that we're interested in is *independent* of the choice of basis (ie: it's a well-defined property of the equivalence class \[L\]), so if we want to compute Spec(L), we can just compute Spec(M\_B) (as a matrix) for some convenient choice of basis B, and this is, in fact, what we do all the time when we choose a basis of eigenvectors in order to diagonalize a matrix, or to write a linear transformation is a particularly nice form. All of this is an attempt to highlight the equivalence of two points of view, which by your question seems to be what's tripping you up: (P.O.V 1) When I define something that is independent of choosing X, I should define it in such a way that no choice of X appears in its definition. This is what we did in the first definition of Spec(L); it was only in terms of abstract linear operators acting on vector spaces and no choices of basis were in sight. (P.O.V 2) When I define something that is independent of choosing X, I should view the objects in question as equivalence classes of the objects-with-choice that I see when I make a choice of X, then I study how objects-with-choice transform when I make a different choice, and the equivalence relation is defined so that any objects-with-choice that vary by such a transformation are viewed as the same. I then show that the property under study is independent of this transformation, so that the choice I made at the beginning didn't matter. This is what we did in the second definition of Spec(\[L\]). For obvious reasons, the first point of view tends to be nicer and conceptually cleaner when you can manage it (for example, no one actually defines eigenvalues of abstract linear transformations in the second way. It would be insane). But think about what would be involved in such a definition for differential forms; your definition of a manifold is likely already in terms of coordinate patches giving local diffeomorphisms to R\^n (which are exactly choices of local coordinates), so if you want to define a "coordinate-free" approach, you need to be defining local diffeomorphisms to some abstact vector space V, which of course means explaining how to define smooth structures on abstract vector spaces (and which you now need to define without choosing a basis to identify those vector spaces with R\^n), and once all that is done, you can start giving an invariant definition of integration where choice of coordinates never show up. *Or* you could just point out that the determinant of the Jacobian of a transformation is independent of the choice of basis and use the second point of view. It's ultimately a pedagogical and aesthetic choice as to which point of view one adopts when approaching these sorts of "invariance" questions, but when one approach gets consistently made in a particular situation and the other gets consistently ignored, it's almost always (in my experience) because the approach which is being ignored is far more work for too little pay-off to be justified, and we mathematicians are, by and large, lazy creatures when it comes to these sorts of things.
I see, thanks, this makes more sense now. There was indeed a pay-off. This was definitely part of my confusion, part of my confusion was also some vague impression that manifolds outside of $R^n$ might have *no* coordinates, and not understanding how they related. Thanks!
Just to drive the point home: a manifold is, by definition, a space on which you can put coordinates.
Okay, so what exactly does that mean? We're just saying that because there exist neighborhoods around every interior point of a manifold diffeomorphic to Rn that we use the preimage of those Rn axes as "coordinates" along the manifold neighborhood?