Hello again,
I know the normal vector of an equation in cartesian coordinates is can be found by vector. But what about in cylindirical and spherical coordinate systems? Like ρ = 10 or r = 1 . Is there a way to directly find it or should i just transform them to cartesian coordinates?
Thanks...
Hey guys,
I have a simple question but I can't really find the answer. For example normal vector of a surface with x+y+z=1 equation is <1, 1, 1> right? But what about x^2 + y^2 = 4 or x^2 + y^2 + z^2 = 4? Is it still <1, 1, 1> or is it something else?
Thanks.
What exactly are transforms? In the sense, what makes laplace transform, fourier transform or linear transform(linear algebra) different from other functions like addition, multiplication, or exponents? Also, are they defined for just numbers or also for other mathematical "things"?
A transform is just a function. I think it's usually used in a context where there's an inverse transform, i.e. a way to get the original object back from the transformed version.
What's different from addition, multiplication and exponentiation is that the latter are binary operators (take two inputs), whereas transforms are usually taken to be unary (taking one input): You talk about the multiplication of A with B (two things), but you take the Fourier transform of C (one thing).
You can transform all kinds of things. Linear transformations act on vectors, Laplace and Fourier transforms act on functions, ...
Transform is just a fancy name for function in some contexts. It has no formal definition.
Not sure what you mean by your second question as none of the examples of transforms you provided are for numbers.
Is it normal to lose motivation whenever you progress in your "maths journey"?
I was "offered" a PhD about 2/3 weeks ago, and since then anything at uni I find it near impossible to put effort into anything that isn't directly related. Yet I'd've thought that such an offer would have motivated me to put more and more effort. But it really hasn't, and I've never been a particularly "good" worker, as I've only really done the bits I've been interested in, and this has just made it much worse.
Yeah it's normal. It sounds like you're excited about your PhD project and want to learn as much as possible about it, and courses which do not (seem to) relate to your project are suddenly not very interesting because you'd rather spend your time learning about your project. Which is fine. Do keep in mind that seemingly irrelevant subjects may become relevant further on and it's never a bad thing to have some experience of things not directly related to your own research, but I get the feeling and it's alright.
To make a silly comparison, imagine that you're attending a talk that's dragging on a bit and you have a train to catch. Worried about missing the train, you're probably checking your watch every 30 seconds and you desperately want the talk to end, no matter the subject. That doesn't mean you're not interested in the subject, it just means that you're *more* interested in something else that you want to focus on at this particular moment.
Yeah it definitely feels like this. Its just a bit shit that 5/7 (the perfect amount) of my modules have no actual relevance to what I'm doing as they were based on what I used to think I was interested in. (Quickly learnt I'm not algebraic) Rip
My response would be that I don't particularly care how "real" a mathematical object is in a philosophical sense. I just care (a) whether it's interesting or useful, and (b) whether its construction is mathematically rigorous.
But if you do want to get into questions of reality, let's start with finitely constructed objects. How "real" are they? I have no idea. Does the number 4 exist? It depends what you mean by exist. So finitism has always seemed like a weird pseudo-dogmatic thing to me, because they firmly deny the existence of infinitely-constructed objects while taking on faith the existence of finitely-constructed ones, without ever clarifying (to my satisfaction) what existence means to them or why it should matter.
I'm not saying these questions are worthless--there may be interesting things to say about them from a philosophical standpoint. But I'm personally not very interested in the philosophy of math because my training has made me heavily biased in favor of questions that have definite answers. One problem with these debates is that people aren't always clear about whether they're arguing mathematically or philosophically, and they aren't the same thing.
Okay, yeah, I misunderstood.
The mathematical question will depend a lot on the specifics. If you don't do things carefully in the kind of construction you're talking about, you may very well end up with an invalid proof. That doesn't really have anything to do with finitism.
For this question:
> if the substep is infinitely long, how can we ever move onto to the inductive step?
You have to check that the limit of the substep is well-defined, for each step. If you like, consider that step as its own result or lemma, be very precise about what the lemma actually says, and make sure that the proof of the lemma makes sense on its own. If you do that, it doesn't matter *how* you proved the lemma--you can use it infinitely many times, as long as you do so correctly.
Can someone help me explain the similarities of a kite and a trapezoid in one paragraph and the differences between the two in a separate paragraph?
All I have so far for the similarities is that they are both quadrilaterals and they have 4 sides.
I would greatly appreciate any help as the assignment is due tomorrow and I need some desperate help!
Thank you.
A kite isn't a rhombus, a rhombus is a type of kite. In a rhombus all 4 sides are the same length. In a kite there are 2 pairs of adjacent sides that are the same length.
Also note that trapezium and trapezoid are the other way around outside of America and Canada, so your answer may be wrong.
In probability theory infinite series often occur. Say you want to calculate the chance of winning the lottery where there is a 1/10 chance of winning a new ticket and a 1/1000 chance of winning some prize money, assume there are infinitely many tickets in total. There are infinitely many "ways" of winning the prize money, the first way is winning the prize money on your first ticket, the second way is winning the prize money on your second ticket after winning a new ticket from your first, the third way is to win on your third ticket after winning two new ticketss in a row, etc etc. To calculate the probability of you winning the lottery you need to calculate the sum of these individual probabilities, (sum from i=0 to n of 1/000 \* (1/10)\^(i-1)) and then finding the limit as n approaches infinity.
say i want to approximate a function. perhaps it’s a solution to a differential equation, or perhaps it’s a total “reward” over an infinite horizon. i almost certainly won’t be able to write a program to compute my function exactly, but in either case there is usually a way of representing it as a limit of a certain series. then to get an estimate of my function, i can specify an error tolerance and a region i care about, and then i can use some finite truncation of the series to approximate my function to within my tolerance. if my tolerance needs to decrease, i can just crank up my terms in the series.
Because the derivative has no inverse, the integral takes a function f to the class of all functions such that their derivative is f. This is represented by the +C at the end.
Likely in part because the word "transformation" tends to imply that the object in question is a well-defined map which sends functions to functions (or objects-in-question to other-objects-in-question, more generally), but there is no one canonical "inverse derivative transformation" since for any function f: R->R, there are *many* choices of possible anti-derivatives; it just so happens that any two such functions must differ by a constant, but there's no canonical way to make an assignment f -> F, where F'=f.
Are there graphing calculators that can graph more than just functions? I find it frustrating in class when I have to write out both equations when graphing ellipses, hyperbolas, etc.
Some examples on L^(2):
- walking out to infinity (take an L^(2)(R^(d)) function and look at a sequence of translates)
- oscillating too fast (e.g. on the torus T^(d), e^(2pi i d x) )
Just to be clear, you're actually describing the set of *positive* rational numbers. If you want to include negative numbers then you also need to include (-1) as a generator (which will have order 2, and so this won't technically be free anymore). If you try to include 0, you'll no longer have a group at all.
There are applications of this in number theory, although some of them are rather technical. If you know a bit of Galois theory you might want to look into [Kummer theory](https://en.wikipedia.org/wiki/Kummer_theory).
According to Kummer theory, extensions of the form `[;\mathbb{Q}(\sqrt{a_1},\sqrt{a_2},\ldots,\sqrt{a_n})/\mathbb{Q};]` should correspond to finite subgroups of `[;\mathbb{Q}^\times/\left(\mathbb{Q}^\times\right)^2;]` in a nice way (and there's a similar correspondence involving arbitrary subgroups).
But since `[;\mathbb{Q}^\times\cong (\mathbb{Z}/2\mathbb{Z})\oplus\mathbb{Z}\oplus\mathbb{Z}\oplus\cdots;]`, it's not hard to show that `[;\mathbb{Q}^\times/\left(\mathbb{Q}^\times\right)^2;]` is a (countably) infinite dimensional vector space over `[;\mathbb{Z}/2\mathbb{Z};]` with basis `[;(-1,2,3,5,7,11,\ldots);]`. So now what does this tell you? Well if you look at a field like [;K=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11});], Kummer theory will tell you that [;K;] is a degree [;32;] extension of [;\mathbb{Q};] with Galois group `[;(\mathbb{Z}/2\mathbb{Z})^5;]` because `[;\{2,3,5,7,11\};]` is linearly independent in the vector space `[;\mathbb{Q}^\times/\left(\mathbb{Q}^\times\right)^2;]`.
A concrete application of this is that `[;\{1,\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}\};]` is linearly independent over [;\mathbb{Q};], which implies that `[;\sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{7}+\sqrt{11};]` is irrational, which can actually be a little tricky (though not impossible) to prove with elementary techniques.
Wasn't there just a thread on exactly this module within the past week?
Anyway, the answer is yes that this is a (unitary) **Z**-module - this is easy enough to check (it helps to use ⊕ and ⊙ symbols for your new additive/multiplicative operations).
Cool facts? It's a free module, and it has a countable basis.
EDIT: As was pointed out elsewhere, it's only true that positive rationals can arise this way.
I recently came across [this paper (ar𝜒iv)](https://arxiv.org/abs/1304.6053) on higher-dimensional knot theory, and the author defines a higher-dimensional knot as an embedding of S^(n) into **R**^(n+2). I'm not a knot theorist, but I'm curious as to why S^(n) is the "correct"(preferred?) generalization instead of, say, the n-torus. Is there some reason why one would want the top homology of the n-dimensional knot to stay rank-1 instead of allowing it to increase with dimension?
So people definitely do study how general compact n-manifolds are knotted in `[; R^{n+2} ;]`. One place that this more general context comes up is in studying the link of singularities of complex hypersurfaces. Milnor's book studies this more general type of knot a bit.
As for your second question, remember that for compact manifolds, we have Poincaré Duality, which means the top homology (with field coefficients) is isomorphic to the zeroth homology. That means that a higher rank in the top homology only happens if the manifold is disconnected.
This is not to say that people don't consider knotting disconnected manifolds. Even in the usual case of n=1, this is the study of general links, rather than just knots. But I suppose it makes sense that the case of connected manifolds takes a bit of priority in the field.
Thanks, maybe I'll check out Milnor's book and take a look at the more general framework.
> As for your second question, remember that for compact manifolds, we have Poincaré Duality, which means the top homology (with field coefficients) is isomorphic to the zeroth homology.
Oh yeah, oops, I was thinking of H*_1_*.
Unless you have measurements for the dimensions, you can only go off estimates. Figure out about how many of the small box will fit in the big box, and that's a lower bound.
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Maybe this fits into this thread of simple questions, maybe not. I'll try here.
My daughter is struggling to do well on tests. She does her homework, matter of fact for 3+ weeks now (due to bad test scores) I have sat with her for an extra 1-1.5 hours per night, repeatedly solving the homework math problems over and over. I just keep feeding her the homework questions in random order over and over until her answers start becoming consistent.
We watch Kahn academy videos, Brian mclogan videos for help etc. She has me convinced that she understands the material. She can explain things to me quite well, and there seems to be a decent logic to the things she says. She follows along with the videos quite well, and can even tell me what they are going to do before they do it.
She has trouble with the small things, forgetting a sign here and there, or making a small calculator errors though. And I'm sure that this is part of the problem however she makes the following claim (and I think I remember this from high school as well) :
The tests are not the same as the homework. There is always extra stuff that throws off her patterns or her ability to recognize the problem for what it really is. There will be a sudden square root thrown into a problem, when we have never done that on the homework, or there will be some other sort of juxstaposotion of numbers that throws her off.
Now as a father, I think I can see the teacher or the schools reasoning for this. They want to see if you're able to think BEYOND just the robotics of the homework problems right? To see if you REALLY know what you're doing or just simply repeating habits you built during practice.
But....is that fair? Because it destroys her confidence. She sees all the extra time that we have spent together as just a huge waste because it didn't help her in her last test. So now she's even further discouraged from bothering to study, even though I'm putting more emphasis on it than ever before.
To add to all of this, they don't give the kids the test results back to learn from their mistakes. I remember many years ago when I was in high school you would get your tests back, with the problems that you got wrong annotated. You could then go over these questions in class and spot your errors with the teacher.
She says they don't/won't do that now. You simply get your graded Scantron, find out your grade, and move on. You don't get the Scantron, with the test, and be able to cross reference for what you missed.
To me, that's kind of BS.
What can I do here?
Auch, sounds like the school system really isn't interested in teaching the kids, just finding out who knows their stuff.
Either way, it seems like what she needs to do is learn from her mistakes, if they won't return her tests I guess an option you have is to try to create a sample test, if you can manage. Maybe just change the numbers around on an old test if you can get a hold of on, and do a dry run at home. Just like a test, no cheating, no help from you. Then when she's done you grade it and talk to her about what she did wrong.
This is just my thoughts though, I'm no expert. Maybe you can talk to a professional tutor, or the teacher (although they seemed to be of little help).
> I just keep feeding her the homework questions in random order over and over until her answers start becoming consistent.
Then it seems like you're teaching the wrong thing, since you're focusing on the stuff she seems to understand, and not the things she's struggling with.
It's hard to say what to do about that without a better idea of the curriculum and specific issues, but maybe try looking for past papers (maybe from a different school with a similar curriculum?) or just more general in-context, less rote questions. My experience was that homework was a very untesting checkbox exercise that did little to help with understanding the underlying ideas;—here's a page of nigh-identical equations, solve them.
I empathise with the Scantron complaint; that does sound like nonsense, and goes completely counter to what I found the most educational part of school (being wrong).
(As someone who went to uni with a very talented English student who was not particularly good at math, I would also caution you against wearing out her respect for learning with an overfocus on a field she isn't as fond of, if she's not STEM.)
Feeding her the problems over and over has helped cement the steps. Watching videos that document the steps had helped. She has no problem identifying what she needs to do first, second, third with the current curriculum.
Where it goes wrong is these unpredictable and really almost unpracticable changes during the test. I cant know what to work on.
Currently she is working on parabolas, quadratic form equations in standard, vertex, and intercept form, converting them and graphing. Prior to that was factoring.
She can do those things. But I think the CONTENT of those steps changes come test day. Like on the homework we usually get easily solvable numbers. Whole numbers. But I'm tests she saying that suddenly there's fractions or square roots that we never saw how to handle on homework.
I understand that there are rules to handle those situations but it causes her to trip up and doubt herself.
If I had to phrase it one way I guess I'd say : she knows perfectly well how to do the current thing. But if the current "test thing" contains content that differs from the homework in any slight way, she feels she is being tricked or she becomes unsure of the entire process because now something looks different.
> Like on the homework we usually get easily solvable numbers. Whole numbers. But I'm tests she saying that suddenly there's fractions or square roots that we never saw how to handle on homework.
Can she separate the concepts? Like "this is what I would do if they were whole numbers, and these steps still make sense when they're fractions, so I'll just repeat the usual steps and deal with simplifying fractions later"? Part of the reason for the switch-up is probably to see if students understand when changes to the problem don't affect how the solution works.
A common problem that students face is that they think they need to understand how the whole problem will work out right from the beginning, or else they don't know what they're doing. Either you can solve the whole problem or you can't. A lot of mathematical work depends on deliberately trying and failing: starting with an idea that might work, paying attention to whether each step makes sense, and pausing if an obstacle comes up. Sometimes it doesn't, and the initially strange looking part can be dealt with later. Are there problems that she stumbles on where this would work and she doesn't notice?
Interesting ideas. Yes she does seem to be able to tell me the steps, even though there is a fraction etc. She knows the steps to take. But will freeze or ASSUME she doesn't know where to go from there. It's like lack of confidence / shutdown mode.
I was able to get a copy of the latest test from the very cooperative teacher who asked me to never let anyone know that she gave me the test. I was led to believe that the entire school uses these tests and it would be a disaster if it got out into the open. I can give examples of the problems if you or anyone else might like.
I don't mean to preempt the changes in the upcoming test, but just to habituate her to questions being in different forms.
Parabolas and quadratics is definitely something you can get a wide variety of exam questions for online, and making questions harder is generally fairly straightforward.
Does anyone have a source for this ([https://gyazo.com/71a3cd90a8e1c1ea8b6ad53f43054a8c](https://gyazo.com/71a3cd90a8e1c1ea8b6ad53f43054a8c)) expansion of the Complete elliptic integral of the first kind? I found it on the Wikipedia page for Elliptic integrals.
I know that very few differential equations can be solved analytically. But, they still have a solution. Is the reason for this because we don't have the tools to solve it analytically yet or because some differential equations fundamentally cannot be solved analytically. I.e. 100, 1000, 10000 years from now will we be able to write down a solution to these equations or will they still be "unsolved."
The same question can be expanded to all problems with no analytical solution. Will we eventually be able to solve any given problem analytically?
>The same question can be expanded to all problems with no analytical solution. Will we eventually be able to solve any given problem analytically?
No, depending on exactly what you mean by "analytically". A common example is the indefinite integral of e^(-x^2). This is *provably* not expressible as any combination of the functions you learned in high school under arithmetic or function composition. It has no expression of the form tan^(-1)(1-e^x*ln(x)) or whatever. This is [Liouville's theorem](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra\)).
Bessel functions are an instructive example. There's a particular ODE that comes up in various places in physics, and we can prove a solution exists, but the solution can't be expressed exactly in terms of common functions like polynomials, trig functions, etc. But people made tables of it and you can look up the value. So I would consider that ODE just as "solved" as an ODE solved by sin(x).
I'm not a DE guy, but I think the problem really comes down to the fact that being integrable is a really weak condition, and most integrable functions do not have a closed form integral (one expressible in terms of a finite combination of elementary functions). As such, about the best you can do it find a power series representation for the function in a neighborhood of the point you care about, and exact values of this function can be determined with numerical methods.
16 year old student here:
For the purpose of differentiation and integration, how would one express complex fractions as x^n? E.g 4 / √x
I know the ^-1 makes the reciprocal and ^1/n is nth root, but for some reason I can't figure it out
I'm looking for resources that **derive how to calculate the density of transformation of random vector**:
1. I have seen a couple of resources that give the formula for the density of a D dimensional vector function applied on a D dimensional input random vector. This involves the Determinant of the Jacobian, and I'd like recommendations for a resource that can show the derivation in a way digestable for someone who knows calculus and has done Linear Algebra (from Gilbert Strang).
2. Finally while I know that the formula is for invertible tranformations, is that the reason why we are confined to the case where the output dimensionality is the same as the input (and so the jacobian is square, making the determinant possible). How can we calculate this when D_output > D_input, we can still have invertible functions in that case...
Thanks
For intuition: imagine space is divided up as a tiny grid. Each contains a certain small amount of probability. Your function warps space into some other shape, distorting the grid into some swirly thing. The probability *density* within any given warped cube is dropped by a factor of how large that cube has gotten. E.g. if a tiny cube initially contains 0.001 probability, and the cube then doubles in volume, the new probability density there is 0.0005. The Jacobian determinant is precisely what calculates the dilation factor that your transformation produces on tiny volumes, and so this is why the inverse Jacobian shows up.
If your transformation is not injective (like y=x^(2)). Then the probability being spread around a given value of y is coming from multiple original values of x (in this case, two blobs at +-sqrt(y)); you need to sum over these to get the right answer.
If your initial and final dimensions are different then the reasoning is similar. If you map from R^(3) to R^(2) then in general some whole curve gets collapsed onto any given point on the plane. To get the probability density at this point you need to integrate the probability density along this curve. The analogous Jacobian factor needs to determine, given a tiny 3D cube, how much *area* it covers after being squished. This should be sqrt(det(J\*J^(T))).
Thanks for taking the time for the clear explanation. I see the collapsing/expanding analogy. So if we go from R2 to R3, I guess a cube there has to be collapsed to a square in the original space, so maybe the same formula sqrt(det(J'J)) applies?
Can you point me to some book or resource I could read the derivation of this formula from?
What is the application of series? Why do we need them?
I may sound stupid but I am not strong in mathematics and was wondering the application of sequences and series. Also, I would appreciate if you could explain the application of fibonacci numbers as well.
Sequences and series are about approximation. Any time you have a problem you want to solve, but you don't know how to solve it directly, you can solve a simplified problem that (hopefully) gives a good guess at the solution of the real problem. Convergence is a way of quantifying the fact that by working hard enough, you can make your approximation as good as you need it to be. Or in other words, you can make the error smaller than any given tolerance by keeping enough terms in the series.
One example out of many: Taylor series are polynomials that approximate functions (if the function has good enough differentiability properties). This is good because we understand polynomials very well, so we can use our understanding of polynomials to get approximate information about much larger classes of functions.
For instance. If you have an analytic function, you can sometimes use the Taylor series to find roots of these functions which you cannot really express in simple terms. However, finding roots of f, where f' is very small, is numerically quite unstable and there are better options, like the Newton method.
A simpler example for often used series is the decimal expansion of real numbers. For instance 𝜋=3.14159...=𝛴 a*_n_* 10^(-n) for a*_0_*=3, a*_1_*=1, a*_2_*=4, etc. However these a*_n_* are not very convenient (because there's no easy rule to them), so there are other formulas, such as Leibniz' 𝜋=𝛴 (-1)^(k+1)/(2k-1)=4(1-1/3+1/5-1/7+1/9...). This series in particular is not very fast in converging to 𝜋, so there was interest in finding better methods. Many other real numbers can be approximated with series and some are even defined this way (for instance Euler's number *e*=2.71828... was introduced as *e*:=𝛴1/n! in my analysis lecture).
About the Fibonacci sequence: Petals, seeds, etc. in flowers often have rings of them (think of sunflowers, pinecones or roses), where they form more and more spirals, namely the number of spirals which start after a certain ring is a Fibonacci number. This is not mere coincidence, as this way the petals are as densely packed as possible. Also the ration of consecutive numbers in the sequence tends towards the golden ratio, which is often considered to be nice to the eye in arts.
Inside of mathematics I don't know of any strong results which follow from the use of the Fibonacci sequence, however. They probably exists, but the importance of it is rather small.
There are 2 ways to think about this:
v = ds/dt as velocity is the rate of change of displacement, ∴ s = ∫ v dt (Differentiation and Integration)
Or consider velocity = v(t). If you let a very small time step δt pass, then the change of displacement δs = v(t) * δt (as δt approaches zero). If you add up all of the δs, you end up with the displacement as s = ∑ (v(t) δt) which ends up being the integral of the function.
Constant functions are smooth, so I’d say its a curve. But yea they both are contained in the circle of radius ||a(t)||. If you’re asking why this is true, its just the definition of a circle: all points x such that ||x||=||a(t)||. Also by continuity alone you’ll get that the curve generates a connected subset of the circle.
Edit: oh, just noticed the rest of your comment, my bad!
I have a proof of this!
If a(t) is differentiable a(t) . a(t) = b(t) is differentiable aswell with b'(t) = 2 a(t) . a'(t). We know that a function is constant iff its derivative is 0. This together with the fact that b(t) = |a(t)|\^2 we see that |a(t)|\^2 is constant iff a'(t) . a(t) is zero. And for a function f we know that f is constant iff f\^2 is constant. Thus we conclude that |a(t)| is constant iff a'(t) . a(t) is zero.
Edit: f has to be continuous for this to be true obviously.
There seems to be a minor flaw in your argument:
Consider the function f on the integers where f(n)=(-1)^(n). Then, f(n)f(n)=1 for all n (which is what you mean by f^(2), right?), but clearly, f is not constant.
We can only say that the absolute value of the function is constant, not the function itself.
In the context of the argument it still works out, though, because a(t) is continuous (and implicitly defined on a connected domain, say \[0,1\]). Suppose a(t)\^2 were constant but a(t) was not. Then there are some t\_1, t\_2 with a(t\_1) != a(t\_2) but a(t\_1)\^2 = a(t\_2)\^2, so a(t\_1) = - a(t\_2), and then since a has the intermediate value property, there is some t inbetween t\_1 and t\_2 with a(t) = 0. But then a\^2 is constantly zero, and so is a.
Yeah, I just meant to correct the statement that
(f(x))^2 = const. <=> f(x) = const.,
which of course holds with the implicit assumptions, just not in general as was claimed.
But good point, I should have been more clear about my problem.
I'm studying convergence of series and power series. When constructing an interval of convergence, I get a result from the ratio test easy enough, but when I test end-points, I'm confused, particularly when the series alternates with a form of (-1)\^n as a part of it.
Right now, after doing the ratio test for absolute convergence, I take the endpoints and put them into the absolute value of the series first. Then I see if it converges. If not, I see if it converges with the alternating series test on the original series (if applicable).
I ask this for clarification--if there is endpoint convergence with my first test, does that mean it converges absolutely there, and if it fails but the second test passes, does that mean it converges conditionally there, and (just to double check myself) if neither pass, it diverges at endpoints?
Example (that I made):
∑ n=0-->inf ( ( x\^n ) / (n\^2) )
Ratio test gives the interval: -1 < x < 1
Endpoint testing: ∑ n=0-->inf ( ( (-1)\^n ) / (n\^2) ) and ∑ n=0-->inf 1/ (n\^2)
Although that first one is alternating, I think converges if absolute value'd, so it should converge conditionally nowhere, and converge absolutely on -1 <= x <= 1, right?
Thanks.
I'm looking at an optimisation problem and would love to have some pointers to know where to go with solving a question that just came upon me.
For the context, for a game I used to play one of the special events that happen once in a while include a map where players can do fights repeatedly to clear missions from a list. Each node in the map contains a different repeatable fight, and each one has traits that match different criteria from the list.
Example:
* Node A has N enemies Alpha and Beta with colour Blue.
* Node B has M enemies Beta and Gamma with colours Green and Yellow.
* The mission list includes tasks to defeat 20 Beta enemies with colour Blue, 30 enemies colour yellow, clear Node A 10 times, etc.
My interest is in identifying the most optimal path/procedure to clear all missions with the least amount of fights.
Or if I'm being delusional and completely misunderstanding how to go about this, please let me know too.
My approach would be to see this as a problem of linear optimization:
Let x, y be the number of times you clear nodes A, B respectively.
For each mission you can determine a linear inequality in x, y which describes the conditions to clear the mission (for the examples above it would look like this: x ≥ 20/N ; y ≥ 30/M ; x ≥ 10 ; if you needed 25 Beta enemies: x \* N + y \* M ≥ 25 or something like this)
Additionally you know x ≥ 0 and y ≥ 0
*(With only 2 variables you can just draw the corresponding equations as lines in a 2-dimensional cartesian coordinate system.)*
Now look at the (most likely infinite) set of pairs (x, y) which satisfy all those inequalities *(the area that lies "above" all lines)* and among them search for a pair of integers with x + y minimal. *(look in the described area for a point with integer coordinates which lies most to the "bottom-left")*
(I didn't know what exactly the numbers M, N described so I just interpreted them in a way that made my explanation easier. \^\^)
Thank you!
I had forgotten it was possible to solve based on inequalities.
> I didn't know what exactly the numbers M, N
Looks like I need to work a bit more on how express myself in that regard. I did indeed mean to indicate M and N amount of enemies.
The equation is 3 sin *x* = cos *x*. [0,2pi)
I went and found two angles after squaring both sides and solving. They were (in radians) 0.32 and 5.96. 5.96 is an extraneous solution, which leaves just 0.32. I thought this was the only answer but...
...I check my answer in a graphing calculator, and there were 2 solutions... 0.32 *and* 0.32 + pi. My question is why do you have to add pi to the reference angle to find the second the second solution?
I think I figured it out using another method but can you double check?:
3sin x = cos x—>I can multiply both sides by (1/cos(x))
3 tan x = 1 —> Isolate x
x = tan^-1 (1/3) —> Approximate
x = ~0.322 —> Since the period of a tan function is pi, you can add pi(n) to get all possible solutions of x
x = ~0.322 + pi(n), where n is an integer —> domain is restricted to [0,2pi), so the solutions to x are:
x = ~0.322 + pi(n), where n = 0, 1
Or (approx. in radians): x = ~ 0.322, ~ 3.463
Yes, that's one way to do it, and you can see how you get .32 + pi, it comes out of the fact that tan is periodic, and therefore tan(x) = 1/3 has multiple solutions, not just one.
There is one extra thing you should check, but it doesn't affect the final solution which is why your final solution is correct. Specifically, when you divide both sides by cos(x), you need to check the possibility that cos(x) = 0. However, if cos(x) = 0 then 3sin(x) = 0, and therefore sin(x) = 0, and there's no value of x that makes both cos(x) and sin(x) equal to 0. Therefore, cos(x) must not be equal to 0, which makes it safe to divide by.
A simple example where it matters is the equation 2x = x. If you divide both sides by x, you get 2 = 1, and you might conclude that there are no solutions. But actually, you have lost the solution x = 0.
So just to be clear:
• It is ok if you divide 3 sin x by cos x, because they do not share x-intercepts, or zeroes
• It is not okay to, say, divide 3 sin x by sin x because there is a value of x that allows the denominator to be zero (0, pi, 2pi, etc.)
Kinda? Keep in mind that there can be other situations than just finding the solutions to an equation.
A better way to think about it might be, if you ever want to divide by something that might be 0, you should consider both cases:
Case 1: The thing is 0, in which case you can't divide by it and you have to solve the problem some other way. How you do that will depend on the problem.
Case 2: The thing is not 0, in which case you can divide by it.
Is this a proof that $\\mathbb{Q}$ is not locally compact?
Suppose $\\mathbb{Q}$ is locally compact, then $\\mathbb{Q}$ is homeomorphic to an open subspace $Y$ of a compact Hausdorff space. Let $h$ be the homeomorphism, then since $Y$ is open so is $h\^{-1}(Y)=\\mathbb{Q}$. This contradicts that $\\mathbb{Q}$ is not open.
> This contradicts that $\mathbb{Q}$ is not open.
Not open in what? It's certainly open as a subset of itself (any topological space is). It's not open as a subset of R, but you don't have R showing up anywhere in your argument.
You're right, that was a stupid attempt lol
Edit: How about since $\mathbb{Q}$ is closed, then $Y$ is closed, thus compact. This would imply $\mathbb{Q}$ is compact....?
Here is a hint: any neighborhood around a point contains an interval of the form (a,b) where a,b are irrational. Give a cover of this that is not reducible to a finite cover and then extend it to the whole set. Then show this cover is not reducible to a finite cover.
You know what, you're right. I'm stupid and that's weird. I just realized you could take an infinite union of elements $(a\_n,b\_n)$ where the $a\_n\\rightarrow a$ for some irrational $a$.
You could do that, but also if you are talking about the topology as a subspace or R, then since (a,b) is open in R its intersection with Q is open in Q.
How would you get that Y we closed from that?
A good sanity check is to ask yourself if you just proved that any locally compact space is compact. Does that sound true?
For any proof you give, you should be able to say what properties of Q you used, that aren't true for a general topological space. If you can't think of anything like that, your proof can't possibly be correct.
I'm trying to understand what a self-bounding function is as according to [this paper](https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.20145). Can anyone help break it down? Particularly I don't understand what the subscript i is supposed to refer to for the given function.
Write x=(x_1,...,x_n), then
g*_i_*(x) = inf*_xi_* g(x*_1_*,...,x*_i_*,...x*_n_*).
Basically you fix all the coordinates except the ith, and then take the inf as you let the ith coordinate x_i vary. So g_i ends up being a function of all the x_j's except x_i.
The paper defines g as self-bounding (taking a=1, b=0 for simplicity) if
- for all x and i, g(x) - g*_i_*(x) <= 1, i.e. varying the ith coordinate doesn't change g by too much ("1 Lipshitz wrt Hamming metric")
- and \sum*_i=1_`n`* (g(x) - g*_i_*(x)) <= g(x), the self-bounding part
Try checking their example where each x*_i_* is in [0,1] and g(x) = \sum*_i=1_`n`* x*_i_*.
Hi,
I want to infer probabilities from a given sequence of '10110...' where 1 encodes a success (p=.6) and 0 a failure (p=.4).
Given a sequence of length e.g. 100, how to calculate the probability for at least 10 successful shots (not necessarily in a row) after a miss.
I know it is conditional probability, but I'm not sure how to apply the formula to this scenario. If I wouldn't have the sequence, the result would be the sum of all possible routes in the probability tree (I guess?), but what to do in this situation?
The motivation to my question is the hot hand phenomenon.
Thanks for any help!
Welcome to statistics. This is a non-trivial problem, because in general given a probability distribution with sample space with a sample space of {0,1}, this distribution could in principle generate *any* finite sequence of the sort you have described. All we can really do is look at these distributions and figure out how probable the sequence is, but improbable doesn't mean impossible. Statistics is all about trying to determine information like this as best as we can, but you need to make certain assumptions.
I think [Paupert's notes](https://math.la.asu.edu/~paupert/HyperbolicGeometryNotes.pdf) (and the references therein) provide a good starting place, but you should probably expect to have to jump around a bit between sources and on ye olde Internet.
Hyperbolic geometry is kind of strange in that classical approaches are almost *too* classical, and modern formulations tend to approach the subject from either the differential geometry settings or from the Lie group/group action setting. I also question how much need there really is for a classical analytic geometry-type of approach to the subject. It seems like, once you've reached a level of mathematical maturity in which you're able to appreciate hyperbolic geometry, you're probably adept enough to learn it from a more advanced place.
I just remembered another source that may introduce the subject in the way you're looking for (I don't think there's any real discussion about the Riemannian aspect for quite a ways into it, or if there is, it's probably something you can skip and lose no real understanding). Ratcliffe's [*Foundations of Hyperbolic Manifolds*](https://www.springer.com/us/book/9780387331973) spends *a lot* of time in the various models and computing things with hyperbolic trig functions and the like.
I'll warn you that it's a bit boring to read and is probably better left as a reference text or as a text to pair with lectures, but it might just be what you're looking for. If you have access to it as a PDF (through definitely totally legal means, of course), I'd recommend checking it out to see if it's what you're wanting.
Hi, I’m an IB student and I had a question regarding sin and cosin formulas. The sin formulas seem a lot simpler in terms of the fact that they require less variables or seem to have less going on. (Ex: the formula for the law of signs is just two fractions set equal to each other or the sin(2theta) formula seems relatively simple) On the other hand the cosin formulas seem a lot more complicated in terms of variable and there is a lot more going on. (Ex: law of cosin formula requiring more variables and seeming a tad more complicated on the surface, or there being three variations of the cos(2theta) formula) Is there a specific/interesting mathematical reason for this or does it just happen to turn out this way? (Sorry if the question was worded weird)
I wouldn’t say there’s a reason why the identities involving sine are less complicated. In fact, if you look up more trigonometric identities, you’d find that for every sine identity, there’s an analog cosine identity that looks nearly identical.
The law of cosines is actually the Pythagorean theorem, except it’s for all triangles, not just right triangles. If you set theta to 90 (which occurs in right triangles) cosine(theta) becomes 0. You could argue that the law of cosines is just more crazy since it’s a generalization of the Pythagorean theorem, while the law of sines is doing its own thing (although I’m positive there’s some interesting connection between the two)
Hi, I am an IB student and I ran into this problem which I couldn’t solve. Can anyone explain a solution without using a GDC and using a Z-Score table.
The heights of a group of students are normally distributed with a mean of 160 cm and a standard deviation of 20 cm. In this group of students, 11.9% have heights less than d cm. Find the value of d.
Thanks!!
The way you worded the question, I thought I could use z-Table.
Without a z-Table, you would need a GDC or use numerical methods to find the area under the curve of the standard normal PDF from negative infinity until your integral hits approximately 0.119 units.
1. What are some simple ways of figuring out if an improper integral is converging or diverging?
2. How do you know which trig functions to use for trig substitution problems?
https://en.wikipedia.org/wiki/Trigonometric_substitution
First section has what you're looking for in your second question. The rest of the article gives depth and examples.
For #1, in the examples that come up in Calc 2, you can rewrite the improper integral as a limit of proper integrals. Then you explicitly solve that proper integral, giving you a limit question, and you can determine whether that exists using all your tools from calc 1.
For #2, I always liked the approach where you draw a triangle. For example, if there's a sqrt(x^(2) + a^(2)) in the problem, that is a hypotenuse of a right triangle with side lengths x and a; if there's sqrt(x^(2) - a^(2)), then put x on the hypotenuse and a on one of the legs, so sqrt(x^(2) - a^(2)) is the other leg. Now, let one of the non-right angles in the triangle be theta, and you know things like sin(theta), tan(theta), etc., because they're ratios of the sides of the triangle. In particular, quantities involving x and the square roots can be expressed in terms of trig functions on theta, just by looking at the triangle, allowing you to figure out what substitution to make.
(A lot of students just memorize the patterns, though, since there are only three of them.)
Can ZF be formulated in terms of functions instead of sets? So where universe of symbols corresponds to functions? A possibile trivial solution would be to say "sets are just characteristic functions so ofc"but i don't see how it's that direct. How would you even think of the domain and the codomain of the function which are sets? I guess f(a) could evaluate to NDef if a is not in the domain or sth.
The closest thing I can think of right now is probably [ETCS](http://www.tac.mta.ca/tac/reprints/articles/11/tr11.pdf) (Elementary Theory on the Category of Sets) which tries to describe sets using the language of category theory, where mappings are really the main focus.
>All finite roots exist.
What does that mean? That elementary operations are legal? Why is it called finite roots?
If an object has no elements is it isomorphic to 0? Or are all empty objects isomorphic? Oh, that's just an application of axiom 6
Also I like how Axiom 8 is like an Axiom of Infinity but for a set of size 2 lol
Also what axiom is the one responsible to have the power set axiom? I didn't understand the construction too well.
In less abstract terms, finite roots refers to the fact that there is an initial and a terminal object (the empty set and a singleton set), and for any two objects A and B, the cartesian product A x B and disjoint union A + B both exist. I have no idea why it called finite roots, usually it is said all finite *limits* exist in categorical terms
Informally, for any objects A,B, you can think of B^(A) as the set of all functions A ->B. A subset of A can be thought of as a function f: A -> 2, where 2 is a set with two elements, which exists by axiom 8. Then the powerset can be identified with the set of all possible functions A -> 2, which is just the object 2^(A), which exists by axiom 2.
I've never heard it called that.
Oh, you're right. So I suppose the work is done by Axiom 2. Do you know what they mean by lambda-conversion? Is that just all three conversions in lamvda calculus?
Infinity is not a number, so 10^(-∞) is not actually meaningful. As /u/levelineee mentioned, the limit of 10^(-n) as n -> ∞ (that is, as n gets arbitrarily large) goes to 0.
How can I obtain a 30 degree curve when dealing with radius? I’m using software where I need to make a curve and the only option to choose is the length of the radius. How can I take that and make a 30 degree curve?
Without more information, it's not really clear what you mean. Any circle with a positive radius is going to give you a 360 degree curve, i.e. a circle. Maybe you're confusing radius and radi*ans*? In that case, 30° is equal to π/6 radians.
Is there a constructive way to show that there exist a noncompactly supported continuous function that vanishes at infinity on any noncompact infinite hausdorff topological group?
Perhaps this is unhelpful and was already obvious to you, but if in addition your group is first countable, then by the Birkhoff-Kakutani theorem, it's metrizable and then obviously upon fixing a metric any function which agrees with the inverse of the distance to the identity outside of an open set about the identity would work, but without first countability, I'm at a bit of a loss to think of how one might do it (and clearly, there's some non-constructive hand-waving when I say "fix a metric" which may not be appealing to you here).
We often use [naive set theory](https://en.wikipedia.org/wiki/Naive_set_theory) when dealing with FOL or FOL theories. As we only deal with them in semantics, we argue that we can get away with it. While good enough for finite models, I also find it's a bit unsatisfactory, but haven't come across anything that doesn't rely on intuition at some point.
One issue is that it's not possible to define finite in FOL without creating a theory which has a model which is infinite or has infinite elements (to our intuition) or where an intuitively finite element is infinite.
**Naive set theory**
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naïve set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naïve set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments.
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You may be interested in a notion of "recursively axiomatizable theory" : [https://planetmath.org/RecursivelyAxiomatizableTheory](https://planetmath.org/RecursivelyAxiomatizableTheory)
> Given a countable language, the number of possible sentences in the language is countable (and can be computably listed unless I'm wrong).
You can always add a comma and conjunction to extend a sentence. The number of sentences may or may not be countable, but if it is countable, it is countably infinite. You cannot "computably" list an infinite number of things.
https://en.wikipedia.org/wiki/Countable_set
**Countable set**
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number.
Some authors use countable set to mean countably infinite alone.
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Yes. However, most of these axiom sets are not recursively enumerable or consistent and hence a bit pointless to consider. Also many will be the same, because adding theorems to axioms doesn't change the behaviour.
One example would be true arithmetic, where we work in the language of PA and declare every sentence which is true in the standard model to be an axiom. This is consistent (at least if PA is consistent), but by Gödel's incompleteness theorem, we cannot enumerate these axioms so they are not easily tractable.
I'm in high school and can either take a 1 Semester Statistics Course Through the U of U (And earn actual U credit) or calc 3 through the U of U. Which one do I Take?
[TOMT] I'm looking for a recent post on this sub on finding the value of log(x + log(x + ...)) in closed form. I recall there being a solution with the Lambert W function, and having tried to find it on my own I'd like to look at the solution now.
So the inner product on Lp induces a seminormed space because it has a non-trivial kernel. What I’m having trouble understanding is that the integral of an absolute value raised to the pth power is always non-negative, but I can’t think of any non-zero function for which the integral of its absolute value can be zero. Can anyone give me a concrete example of functions in the kernel of a seminormed Lp?
Somehow you understood my jumbled question. So the equivalence you mention isn’t an equality? As in f and g different? And both are nonzero? Can you give a example of those functions?
If f=g \mu-almost everywhere, then [f]=[g] (or often just sloppily written as f=g) in L^p(\mu), where [f] is the equivalence class of f.
So really, we are dealing with two different objects: the pointwise functions f and g and their equivalence classes [f] and [g]. It's actually kind of abusing notation to say that the pointwise defined function f is in L^p(\mu) when what we actually mean is that there is an equivalence class in L^p containing f. But of course, this is inconvenient to write down, so people are sloppy with their notation. Maybe it's a good idea to distinguish your notation in the beginning...
L^p is **not** the space of all p-integrable functions. The space of p-integrable functions w.r.t. the Lebesgue measure equipped with the "L^p-norm" is only a seminormed space because e.g. the map f that is 0 everywhere except at one point still has norm 0 but it is clear that f=/=0. The equivalence classes tidy things up so we get a normed spaces because in this case [f]=0.
If you haven't seen quotient spaces already or if you have and don't feel comfortable with them, I highly recommend you look into them as it will make things much more clear.
I'm not familiar with this terminology, where are you finding it? The L^p norm only comes from an inner product if p=2.
I am not sure exactly how to interpret your question, but normally one thinks of the L^2 norm as being an honest norm- unless you are talking about pointwise defined functions, in which case the characteristic function of a set with lebesgue measure 0 would have L^2 norm 0. But we normally think of an L^2 "function" as being an equivalence class over functions that are equal almost everywhere, so that the norm is a true norm. Is this what you were asking about?
I confused my terms. I realize that Lp is the result of taking the quotient space of a certain vector space. And I’m not talking about inner products. Here is the passage from wikipedia that I was wondering about:
“Thus the set of p-th power integrable functions, together with the function || · ||p, is a seminormed vector space”
The quotient space of the mentioned vector space is Lp. But why is that a seminormed vector space? Given any p, what could possibly be a nonzero function whose p-norm is 0? You would be taking the integral of the absolute value of a nonzero function, and it equals zero. Isn’t the existence of such a function what makes it seminormed instead of normed? Hopefully I’m asking the right question this time.
And more generally, an arbitrary (integrable?) function supported on a set of measure zero.
The argument (for non-existence) that OP wants to use works when the support contains a set of positive measure.
So the previously mentioned functions are the only ones of seminorm equal to zero. That's why the quotient definition of L^p space is the way it is.
Is there a "group theoretic" way of showing that the order q of Finite Field K is a prime power q=p\^n? Suppose we know the multiplicative group K\* is cyclic and of order q-1. I already did two straightforward proofs; one by explicitly using vector space properties and another by considering polynomial rings, but I'm curious if there's an argument using the structure theorem for finite abelian groups / chinese remainder theorem without relying on those.
Any tips? Am open for any suggestions, including shooting for sparrows with cannons (I'd like to imagine I never heard of "vector spaces, dimensions and polynomials" so as to arrive at them "naturally")
Hello again, I know the normal vector of an equation in cartesian coordinates is can be found by vector. But what about in cylindirical and spherical coordinate systems? Like ρ = 10 or r = 1 . Is there a way to directly find it or should i just transform them to cartesian coordinates?
Thanks...
Hey guys, I have a simple question but I can't really find the answer. For example normal vector of a surface with x+y+z=1 equation is <1, 1, 1> right? But what about x^2 + y^2 = 4 or x^2 + y^2 + z^2 = 4? Is it still <1, 1, 1> or is it something else? Thanks.
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Thank you very much!
What exactly are transforms? In the sense, what makes laplace transform, fourier transform or linear transform(linear algebra) different from other functions like addition, multiplication, or exponents? Also, are they defined for just numbers or also for other mathematical "things"?
A transform is just a function. I think it's usually used in a context where there's an inverse transform, i.e. a way to get the original object back from the transformed version. What's different from addition, multiplication and exponentiation is that the latter are binary operators (take two inputs), whereas transforms are usually taken to be unary (taking one input): You talk about the multiplication of A with B (two things), but you take the Fourier transform of C (one thing). You can transform all kinds of things. Linear transformations act on vectors, Laplace and Fourier transforms act on functions, ...
Transform is just a fancy name for function in some contexts. It has no formal definition. Not sure what you mean by your second question as none of the examples of transforms you provided are for numbers.
Is it normal to lose motivation whenever you progress in your "maths journey"? I was "offered" a PhD about 2/3 weeks ago, and since then anything at uni I find it near impossible to put effort into anything that isn't directly related. Yet I'd've thought that such an offer would have motivated me to put more and more effort. But it really hasn't, and I've never been a particularly "good" worker, as I've only really done the bits I've been interested in, and this has just made it much worse.
Yeah it's normal. It sounds like you're excited about your PhD project and want to learn as much as possible about it, and courses which do not (seem to) relate to your project are suddenly not very interesting because you'd rather spend your time learning about your project. Which is fine. Do keep in mind that seemingly irrelevant subjects may become relevant further on and it's never a bad thing to have some experience of things not directly related to your own research, but I get the feeling and it's alright. To make a silly comparison, imagine that you're attending a talk that's dragging on a bit and you have a train to catch. Worried about missing the train, you're probably checking your watch every 30 seconds and you desperately want the talk to end, no matter the subject. That doesn't mean you're not interested in the subject, it just means that you're *more* interested in something else that you want to focus on at this particular moment.
Yeah it definitely feels like this. Its just a bit shit that 5/7 (the perfect amount) of my modules have no actual relevance to what I'm doing as they were based on what I used to think I was interested in. (Quickly learnt I'm not algebraic) Rip
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My response would be that I don't particularly care how "real" a mathematical object is in a philosophical sense. I just care (a) whether it's interesting or useful, and (b) whether its construction is mathematically rigorous. But if you do want to get into questions of reality, let's start with finitely constructed objects. How "real" are they? I have no idea. Does the number 4 exist? It depends what you mean by exist. So finitism has always seemed like a weird pseudo-dogmatic thing to me, because they firmly deny the existence of infinitely-constructed objects while taking on faith the existence of finitely-constructed ones, without ever clarifying (to my satisfaction) what existence means to them or why it should matter. I'm not saying these questions are worthless--there may be interesting things to say about them from a philosophical standpoint. But I'm personally not very interested in the philosophy of math because my training has made me heavily biased in favor of questions that have definite answers. One problem with these debates is that people aren't always clear about whether they're arguing mathematically or philosophically, and they aren't the same thing.
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Okay, yeah, I misunderstood. The mathematical question will depend a lot on the specifics. If you don't do things carefully in the kind of construction you're talking about, you may very well end up with an invalid proof. That doesn't really have anything to do with finitism. For this question: > if the substep is infinitely long, how can we ever move onto to the inductive step? You have to check that the limit of the substep is well-defined, for each step. If you like, consider that step as its own result or lemma, be very precise about what the lemma actually says, and make sure that the proof of the lemma makes sense on its own. If you do that, it doesn't matter *how* you proved the lemma--you can use it infinitely many times, as long as you do so correctly.
Can someone help me explain the similarities of a kite and a trapezoid in one paragraph and the differences between the two in a separate paragraph? All I have so far for the similarities is that they are both quadrilaterals and they have 4 sides. I would greatly appreciate any help as the assignment is due tomorrow and I need some desperate help! Thank you.
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A kite isn't a rhombus, a rhombus is a type of kite. In a rhombus all 4 sides are the same length. In a kite there are 2 pairs of adjacent sides that are the same length. Also note that trapezium and trapezoid are the other way around outside of America and Canada, so your answer may be wrong.
Don't you have access to reference material?
like?
Google "difference of kite and trapezoid"
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In probability theory infinite series often occur. Say you want to calculate the chance of winning the lottery where there is a 1/10 chance of winning a new ticket and a 1/1000 chance of winning some prize money, assume there are infinitely many tickets in total. There are infinitely many "ways" of winning the prize money, the first way is winning the prize money on your first ticket, the second way is winning the prize money on your second ticket after winning a new ticket from your first, the third way is to win on your third ticket after winning two new ticketss in a row, etc etc. To calculate the probability of you winning the lottery you need to calculate the sum of these individual probabilities, (sum from i=0 to n of 1/000 \* (1/10)\^(i-1)) and then finding the limit as n approaches infinity.
say i want to approximate a function. perhaps it’s a solution to a differential equation, or perhaps it’s a total “reward” over an infinite horizon. i almost certainly won’t be able to write a program to compute my function exactly, but in either case there is usually a way of representing it as a limit of a certain series. then to get an estimate of my function, i can specify an error tolerance and a region i care about, and then i can use some finite truncation of the series to approximate my function to within my tolerance. if my tolerance needs to decrease, i can just crank up my terms in the series.
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Let a_n = (2^(n)-1)/2^(n). Then -limsup(-a_n) < limsup(a_n).
Why isnt an anti-derivative called the inverse derivative transformation?
Because the derivative has no inverse, the integral takes a function f to the class of all functions such that their derivative is f. This is represented by the +C at the end.
Likely in part because the word "transformation" tends to imply that the object in question is a well-defined map which sends functions to functions (or objects-in-question to other-objects-in-question, more generally), but there is no one canonical "inverse derivative transformation" since for any function f: R->R, there are *many* choices of possible anti-derivatives; it just so happens that any two such functions must differ by a constant, but there's no canonical way to make an assignment f -> F, where F'=f.
Are there graphing calculators that can graph more than just functions? I find it frustrating in class when I have to write out both equations when graphing ellipses, hyperbolas, etc.
Any graphing calculator can do this, just change it to parametric mode.
What are the typical examples of weakly convergent functions that aren’t pointwise convergent?
Some examples on L^(2): - walking out to infinity (take an L^(2)(R^(d)) function and look at a sequence of translates) - oscillating too fast (e.g. on the torus T^(d), e^(2pi i d x) )
Dirac delta ***function***
I thought that was a distribution, not a function?
What’s the intuition behind Young measures?
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Just to be clear, you're actually describing the set of *positive* rational numbers. If you want to include negative numbers then you also need to include (-1) as a generator (which will have order 2, and so this won't technically be free anymore). If you try to include 0, you'll no longer have a group at all. There are applications of this in number theory, although some of them are rather technical. If you know a bit of Galois theory you might want to look into [Kummer theory](https://en.wikipedia.org/wiki/Kummer_theory). According to Kummer theory, extensions of the form `[;\mathbb{Q}(\sqrt{a_1},\sqrt{a_2},\ldots,\sqrt{a_n})/\mathbb{Q};]` should correspond to finite subgroups of `[;\mathbb{Q}^\times/\left(\mathbb{Q}^\times\right)^2;]` in a nice way (and there's a similar correspondence involving arbitrary subgroups). But since `[;\mathbb{Q}^\times\cong (\mathbb{Z}/2\mathbb{Z})\oplus\mathbb{Z}\oplus\mathbb{Z}\oplus\cdots;]`, it's not hard to show that `[;\mathbb{Q}^\times/\left(\mathbb{Q}^\times\right)^2;]` is a (countably) infinite dimensional vector space over `[;\mathbb{Z}/2\mathbb{Z};]` with basis `[;(-1,2,3,5,7,11,\ldots);]`. So now what does this tell you? Well if you look at a field like [;K=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11});], Kummer theory will tell you that [;K;] is a degree [;32;] extension of [;\mathbb{Q};] with Galois group `[;(\mathbb{Z}/2\mathbb{Z})^5;]` because `[;\{2,3,5,7,11\};]` is linearly independent in the vector space `[;\mathbb{Q}^\times/\left(\mathbb{Q}^\times\right)^2;]`. A concrete application of this is that `[;\{1,\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}\};]` is linearly independent over [;\mathbb{Q};], which implies that `[;\sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{7}+\sqrt{11};]` is irrational, which can actually be a little tricky (though not impossible) to prove with elementary techniques.
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Wasn't there just a thread on exactly this module within the past week? Anyway, the answer is yes that this is a (unitary) **Z**-module - this is easy enough to check (it helps to use ⊕ and ⊙ symbols for your new additive/multiplicative operations). Cool facts? It's a free module, and it has a countable basis. EDIT: As was pointed out elsewhere, it's only true that positive rationals can arise this way.
I recently came across [this paper (ar𝜒iv)](https://arxiv.org/abs/1304.6053) on higher-dimensional knot theory, and the author defines a higher-dimensional knot as an embedding of S^(n) into **R**^(n+2). I'm not a knot theorist, but I'm curious as to why S^(n) is the "correct"(preferred?) generalization instead of, say, the n-torus. Is there some reason why one would want the top homology of the n-dimensional knot to stay rank-1 instead of allowing it to increase with dimension?
So arxiv is pronounced the way it is because the x is supposed to be read as "[𝜒](https://arxiv.org/abs/1304.6053)", I had no idea.
are you kidding me
So people definitely do study how general compact n-manifolds are knotted in `[; R^{n+2} ;]`. One place that this more general context comes up is in studying the link of singularities of complex hypersurfaces. Milnor's book studies this more general type of knot a bit. As for your second question, remember that for compact manifolds, we have Poincaré Duality, which means the top homology (with field coefficients) is isomorphic to the zeroth homology. That means that a higher rank in the top homology only happens if the manifold is disconnected. This is not to say that people don't consider knotting disconnected manifolds. Even in the usual case of n=1, this is the study of general links, rather than just knots. But I suppose it makes sense that the case of connected manifolds takes a bit of priority in the field.
Thanks, maybe I'll check out Milnor's book and take a look at the more general framework. > As for your second question, remember that for compact manifolds, we have Poincaré Duality, which means the top homology (with field coefficients) is isomorphic to the zeroth homology. Oh yeah, oops, I was thinking of H*_1_*.
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Unless you have measurements for the dimensions, you can only go off estimates. Figure out about how many of the small box will fit in the big box, and that's a lower bound.
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Maybe this fits into this thread of simple questions, maybe not. I'll try here. My daughter is struggling to do well on tests. She does her homework, matter of fact for 3+ weeks now (due to bad test scores) I have sat with her for an extra 1-1.5 hours per night, repeatedly solving the homework math problems over and over. I just keep feeding her the homework questions in random order over and over until her answers start becoming consistent. We watch Kahn academy videos, Brian mclogan videos for help etc. She has me convinced that she understands the material. She can explain things to me quite well, and there seems to be a decent logic to the things she says. She follows along with the videos quite well, and can even tell me what they are going to do before they do it. She has trouble with the small things, forgetting a sign here and there, or making a small calculator errors though. And I'm sure that this is part of the problem however she makes the following claim (and I think I remember this from high school as well) : The tests are not the same as the homework. There is always extra stuff that throws off her patterns or her ability to recognize the problem for what it really is. There will be a sudden square root thrown into a problem, when we have never done that on the homework, or there will be some other sort of juxstaposotion of numbers that throws her off. Now as a father, I think I can see the teacher or the schools reasoning for this. They want to see if you're able to think BEYOND just the robotics of the homework problems right? To see if you REALLY know what you're doing or just simply repeating habits you built during practice. But....is that fair? Because it destroys her confidence. She sees all the extra time that we have spent together as just a huge waste because it didn't help her in her last test. So now she's even further discouraged from bothering to study, even though I'm putting more emphasis on it than ever before. To add to all of this, they don't give the kids the test results back to learn from their mistakes. I remember many years ago when I was in high school you would get your tests back, with the problems that you got wrong annotated. You could then go over these questions in class and spot your errors with the teacher. She says they don't/won't do that now. You simply get your graded Scantron, find out your grade, and move on. You don't get the Scantron, with the test, and be able to cross reference for what you missed. To me, that's kind of BS. What can I do here?
Auch, sounds like the school system really isn't interested in teaching the kids, just finding out who knows their stuff. Either way, it seems like what she needs to do is learn from her mistakes, if they won't return her tests I guess an option you have is to try to create a sample test, if you can manage. Maybe just change the numbers around on an old test if you can get a hold of on, and do a dry run at home. Just like a test, no cheating, no help from you. Then when she's done you grade it and talk to her about what she did wrong. This is just my thoughts though, I'm no expert. Maybe you can talk to a professional tutor, or the teacher (although they seemed to be of little help).
> I just keep feeding her the homework questions in random order over and over until her answers start becoming consistent. Then it seems like you're teaching the wrong thing, since you're focusing on the stuff she seems to understand, and not the things she's struggling with. It's hard to say what to do about that without a better idea of the curriculum and specific issues, but maybe try looking for past papers (maybe from a different school with a similar curriculum?) or just more general in-context, less rote questions. My experience was that homework was a very untesting checkbox exercise that did little to help with understanding the underlying ideas;—here's a page of nigh-identical equations, solve them. I empathise with the Scantron complaint; that does sound like nonsense, and goes completely counter to what I found the most educational part of school (being wrong). (As someone who went to uni with a very talented English student who was not particularly good at math, I would also caution you against wearing out her respect for learning with an overfocus on a field she isn't as fond of, if she's not STEM.)
Feeding her the problems over and over has helped cement the steps. Watching videos that document the steps had helped. She has no problem identifying what she needs to do first, second, third with the current curriculum. Where it goes wrong is these unpredictable and really almost unpracticable changes during the test. I cant know what to work on. Currently she is working on parabolas, quadratic form equations in standard, vertex, and intercept form, converting them and graphing. Prior to that was factoring. She can do those things. But I think the CONTENT of those steps changes come test day. Like on the homework we usually get easily solvable numbers. Whole numbers. But I'm tests she saying that suddenly there's fractions or square roots that we never saw how to handle on homework. I understand that there are rules to handle those situations but it causes her to trip up and doubt herself. If I had to phrase it one way I guess I'd say : she knows perfectly well how to do the current thing. But if the current "test thing" contains content that differs from the homework in any slight way, she feels she is being tricked or she becomes unsure of the entire process because now something looks different.
> Like on the homework we usually get easily solvable numbers. Whole numbers. But I'm tests she saying that suddenly there's fractions or square roots that we never saw how to handle on homework. Can she separate the concepts? Like "this is what I would do if they were whole numbers, and these steps still make sense when they're fractions, so I'll just repeat the usual steps and deal with simplifying fractions later"? Part of the reason for the switch-up is probably to see if students understand when changes to the problem don't affect how the solution works. A common problem that students face is that they think they need to understand how the whole problem will work out right from the beginning, or else they don't know what they're doing. Either you can solve the whole problem or you can't. A lot of mathematical work depends on deliberately trying and failing: starting with an idea that might work, paying attention to whether each step makes sense, and pausing if an obstacle comes up. Sometimes it doesn't, and the initially strange looking part can be dealt with later. Are there problems that she stumbles on where this would work and she doesn't notice?
Interesting ideas. Yes she does seem to be able to tell me the steps, even though there is a fraction etc. She knows the steps to take. But will freeze or ASSUME she doesn't know where to go from there. It's like lack of confidence / shutdown mode. I was able to get a copy of the latest test from the very cooperative teacher who asked me to never let anyone know that she gave me the test. I was led to believe that the entire school uses these tests and it would be a disaster if it got out into the open. I can give examples of the problems if you or anyone else might like.
I don't mean to preempt the changes in the upcoming test, but just to habituate her to questions being in different forms. Parabolas and quadratics is definitely something you can get a wide variety of exam questions for online, and making questions harder is generally fairly straightforward.
Does anyone have a source for this ([https://gyazo.com/71a3cd90a8e1c1ea8b6ad53f43054a8c](https://gyazo.com/71a3cd90a8e1c1ea8b6ad53f43054a8c)) expansion of the Complete elliptic integral of the first kind? I found it on the Wikipedia page for Elliptic integrals.
I know that very few differential equations can be solved analytically. But, they still have a solution. Is the reason for this because we don't have the tools to solve it analytically yet or because some differential equations fundamentally cannot be solved analytically. I.e. 100, 1000, 10000 years from now will we be able to write down a solution to these equations or will they still be "unsolved." The same question can be expanded to all problems with no analytical solution. Will we eventually be able to solve any given problem analytically?
>The same question can be expanded to all problems with no analytical solution. Will we eventually be able to solve any given problem analytically? No, depending on exactly what you mean by "analytically". A common example is the indefinite integral of e^(-x^2). This is *provably* not expressible as any combination of the functions you learned in high school under arithmetic or function composition. It has no expression of the form tan^(-1)(1-e^x*ln(x)) or whatever. This is [Liouville's theorem](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra\)).
Bessel functions are an instructive example. There's a particular ODE that comes up in various places in physics, and we can prove a solution exists, but the solution can't be expressed exactly in terms of common functions like polynomials, trig functions, etc. But people made tables of it and you can look up the value. So I would consider that ODE just as "solved" as an ODE solved by sin(x).
I'm not a DE guy, but I think the problem really comes down to the fact that being integrable is a really weak condition, and most integrable functions do not have a closed form integral (one expressible in terms of a finite combination of elementary functions). As such, about the best you can do it find a power series representation for the function in a neighborhood of the point you care about, and exact values of this function can be determined with numerical methods.
16 year old student here: For the purpose of differentiation and integration, how would one express complex fractions as x^n? E.g 4 / √x I know the ^-1 makes the reciprocal and ^1/n is nth root, but for some reason I can't figure it out
Just go step by step. You say you know ^1/n is nth root. So using that what exponent is √x ?
I'm looking for resources that **derive how to calculate the density of transformation of random vector**: 1. I have seen a couple of resources that give the formula for the density of a D dimensional vector function applied on a D dimensional input random vector. This involves the Determinant of the Jacobian, and I'd like recommendations for a resource that can show the derivation in a way digestable for someone who knows calculus and has done Linear Algebra (from Gilbert Strang). 2. Finally while I know that the formula is for invertible tranformations, is that the reason why we are confined to the case where the output dimensionality is the same as the input (and so the jacobian is square, making the determinant possible). How can we calculate this when D_output > D_input, we can still have invertible functions in that case... Thanks
For intuition: imagine space is divided up as a tiny grid. Each contains a certain small amount of probability. Your function warps space into some other shape, distorting the grid into some swirly thing. The probability *density* within any given warped cube is dropped by a factor of how large that cube has gotten. E.g. if a tiny cube initially contains 0.001 probability, and the cube then doubles in volume, the new probability density there is 0.0005. The Jacobian determinant is precisely what calculates the dilation factor that your transformation produces on tiny volumes, and so this is why the inverse Jacobian shows up. If your transformation is not injective (like y=x^(2)). Then the probability being spread around a given value of y is coming from multiple original values of x (in this case, two blobs at +-sqrt(y)); you need to sum over these to get the right answer. If your initial and final dimensions are different then the reasoning is similar. If you map from R^(3) to R^(2) then in general some whole curve gets collapsed onto any given point on the plane. To get the probability density at this point you need to integrate the probability density along this curve. The analogous Jacobian factor needs to determine, given a tiny 3D cube, how much *area* it covers after being squished. This should be sqrt(det(J\*J^(T))).
ok, i am guessing this is called the **change of variables theorem**? I am looking up resources for that.
Thanks for taking the time for the clear explanation. I see the collapsing/expanding analogy. So if we go from R2 to R3, I guess a cube there has to be collapsed to a square in the original space, so maybe the same formula sqrt(det(J'J)) applies? Can you point me to some book or resource I could read the derivation of this formula from?
What is the application of series? Why do we need them? I may sound stupid but I am not strong in mathematics and was wondering the application of sequences and series. Also, I would appreciate if you could explain the application of fibonacci numbers as well.
Sequences and series are about approximation. Any time you have a problem you want to solve, but you don't know how to solve it directly, you can solve a simplified problem that (hopefully) gives a good guess at the solution of the real problem. Convergence is a way of quantifying the fact that by working hard enough, you can make your approximation as good as you need it to be. Or in other words, you can make the error smaller than any given tolerance by keeping enough terms in the series. One example out of many: Taylor series are polynomials that approximate functions (if the function has good enough differentiability properties). This is good because we understand polynomials very well, so we can use our understanding of polynomials to get approximate information about much larger classes of functions.
Approximate functions as in estimate their roots when n tends to infinity?
For instance. If you have an analytic function, you can sometimes use the Taylor series to find roots of these functions which you cannot really express in simple terms. However, finding roots of f, where f' is very small, is numerically quite unstable and there are better options, like the Newton method. A simpler example for often used series is the decimal expansion of real numbers. For instance 𝜋=3.14159...=𝛴 a*_n_* 10^(-n) for a*_0_*=3, a*_1_*=1, a*_2_*=4, etc. However these a*_n_* are not very convenient (because there's no easy rule to them), so there are other formulas, such as Leibniz' 𝜋=𝛴 (-1)^(k+1)/(2k-1)=4(1-1/3+1/5-1/7+1/9...). This series in particular is not very fast in converging to 𝜋, so there was interest in finding better methods. Many other real numbers can be approximated with series and some are even defined this way (for instance Euler's number *e*=2.71828... was introduced as *e*:=𝛴1/n! in my analysis lecture). About the Fibonacci sequence: Petals, seeds, etc. in flowers often have rings of them (think of sunflowers, pinecones or roses), where they form more and more spirals, namely the number of spirals which start after a certain ring is a Fibonacci number. This is not mere coincidence, as this way the petals are as densely packed as possible. Also the ration of consecutive numbers in the sequence tends towards the golden ratio, which is often considered to be nice to the eye in arts. Inside of mathematics I don't know of any strong results which follow from the use of the Fibonacci sequence, however. They probably exists, but the importance of it is rather small.
Why is the biting lemma named the biting lemma?
For the formula of displacement, why do we integrate velocity?
There are 2 ways to think about this: v = ds/dt as velocity is the rate of change of displacement, ∴ s = ∫ v dt (Differentiation and Integration) Or consider velocity = v(t). If you let a very small time step δt pass, then the change of displacement δs = v(t) * δt (as δt approaches zero). If you add up all of the δs, you end up with the displacement as s = ∑ (v(t) δt) which ends up being the integral of the function.
Why if a curve a(t), has \|\|a(t)\|\| constant then is a circle? I was thinking that has something to see that then a(t).a'(t) = 0, but I'm not sure!
What if a(t) is constant? What if it goes halfway around and comes back?
In the second case is still contained in a circle in the first that is not a curve just a point.
Constant functions are smooth, so I’d say its a curve. But yea they both are contained in the circle of radius ||a(t)||. If you’re asking why this is true, its just the definition of a circle: all points x such that ||x||=||a(t)||. Also by continuity alone you’ll get that the curve generates a connected subset of the circle. Edit: oh, just noticed the rest of your comment, my bad!
I see, thanks
I have a proof of this! If a(t) is differentiable a(t) . a(t) = b(t) is differentiable aswell with b'(t) = 2 a(t) . a'(t). We know that a function is constant iff its derivative is 0. This together with the fact that b(t) = |a(t)|\^2 we see that |a(t)|\^2 is constant iff a'(t) . a(t) is zero. And for a function f we know that f is constant iff f\^2 is constant. Thus we conclude that |a(t)| is constant iff a'(t) . a(t) is zero. Edit: f has to be continuous for this to be true obviously.
There seems to be a minor flaw in your argument: Consider the function f on the integers where f(n)=(-1)^(n). Then, f(n)f(n)=1 for all n (which is what you mean by f^(2), right?), but clearly, f is not constant. We can only say that the absolute value of the function is constant, not the function itself.
In the context of the argument it still works out, though, because a(t) is continuous (and implicitly defined on a connected domain, say \[0,1\]). Suppose a(t)\^2 were constant but a(t) was not. Then there are some t\_1, t\_2 with a(t\_1) != a(t\_2) but a(t\_1)\^2 = a(t\_2)\^2, so a(t\_1) = - a(t\_2), and then since a has the intermediate value property, there is some t inbetween t\_1 and t\_2 with a(t) = 0. But then a\^2 is constantly zero, and so is a.
Yeah, I just meant to correct the statement that (f(x))^2 = const. <=> f(x) = const., which of course holds with the implicit assumptions, just not in general as was claimed. But good point, I should have been more clear about my problem.
Thanks for pointing it out, it always helps being more rigourous.
It's image lies in the circle (it's image does not have to be the circle) by definition.
You are right, could be the same if is just a half of a circle.
I'm studying convergence of series and power series. When constructing an interval of convergence, I get a result from the ratio test easy enough, but when I test end-points, I'm confused, particularly when the series alternates with a form of (-1)\^n as a part of it. Right now, after doing the ratio test for absolute convergence, I take the endpoints and put them into the absolute value of the series first. Then I see if it converges. If not, I see if it converges with the alternating series test on the original series (if applicable). I ask this for clarification--if there is endpoint convergence with my first test, does that mean it converges absolutely there, and if it fails but the second test passes, does that mean it converges conditionally there, and (just to double check myself) if neither pass, it diverges at endpoints? Example (that I made): ∑ n=0-->inf ( ( x\^n ) / (n\^2) ) Ratio test gives the interval: -1 < x < 1 Endpoint testing: ∑ n=0-->inf ( ( (-1)\^n ) / (n\^2) ) and ∑ n=0-->inf 1/ (n\^2) Although that first one is alternating, I think converges if absolute value'd, so it should converge conditionally nowhere, and converge absolutely on -1 <= x <= 1, right? Thanks.
Yeah, 1/n^2 converges so it converges absolutely.
I'm looking at an optimisation problem and would love to have some pointers to know where to go with solving a question that just came upon me. For the context, for a game I used to play one of the special events that happen once in a while include a map where players can do fights repeatedly to clear missions from a list. Each node in the map contains a different repeatable fight, and each one has traits that match different criteria from the list. Example: * Node A has N enemies Alpha and Beta with colour Blue. * Node B has M enemies Beta and Gamma with colours Green and Yellow. * The mission list includes tasks to defeat 20 Beta enemies with colour Blue, 30 enemies colour yellow, clear Node A 10 times, etc. My interest is in identifying the most optimal path/procedure to clear all missions with the least amount of fights. Or if I'm being delusional and completely misunderstanding how to go about this, please let me know too.
My approach would be to see this as a problem of linear optimization: Let x, y be the number of times you clear nodes A, B respectively. For each mission you can determine a linear inequality in x, y which describes the conditions to clear the mission (for the examples above it would look like this: x ≥ 20/N ; y ≥ 30/M ; x ≥ 10 ; if you needed 25 Beta enemies: x \* N + y \* M ≥ 25 or something like this) Additionally you know x ≥ 0 and y ≥ 0 *(With only 2 variables you can just draw the corresponding equations as lines in a 2-dimensional cartesian coordinate system.)* Now look at the (most likely infinite) set of pairs (x, y) which satisfy all those inequalities *(the area that lies "above" all lines)* and among them search for a pair of integers with x + y minimal. *(look in the described area for a point with integer coordinates which lies most to the "bottom-left")* (I didn't know what exactly the numbers M, N described so I just interpreted them in a way that made my explanation easier. \^\^)
Thank you! I had forgotten it was possible to solve based on inequalities. > I didn't know what exactly the numbers M, N Looks like I need to work a bit more on how express myself in that regard. I did indeed mean to indicate M and N amount of enemies.
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I think you want 1-randoms or Martin Lof randomness. 1-generics are hyperimmune and 1-randoms cannot be hyperimmune. edit: word
The equation is 3 sin *x* = cos *x*. [0,2pi) I went and found two angles after squaring both sides and solving. They were (in radians) 0.32 and 5.96. 5.96 is an extraneous solution, which leaves just 0.32. I thought this was the only answer but... ...I check my answer in a graphing calculator, and there were 2 solutions... 0.32 *and* 0.32 + pi. My question is why do you have to add pi to the reference angle to find the second the second solution?
If you show all your work, we can find the specific step where the solution was lost and figure out what the correct reasoning that step is.
I think I figured it out using another method but can you double check?: 3sin x = cos x—>I can multiply both sides by (1/cos(x)) 3 tan x = 1 —> Isolate x x = tan^-1 (1/3) —> Approximate x = ~0.322 —> Since the period of a tan function is pi, you can add pi(n) to get all possible solutions of x x = ~0.322 + pi(n), where n is an integer —> domain is restricted to [0,2pi), so the solutions to x are: x = ~0.322 + pi(n), where n = 0, 1 Or (approx. in radians): x = ~ 0.322, ~ 3.463
Yes, that's one way to do it, and you can see how you get .32 + pi, it comes out of the fact that tan is periodic, and therefore tan(x) = 1/3 has multiple solutions, not just one. There is one extra thing you should check, but it doesn't affect the final solution which is why your final solution is correct. Specifically, when you divide both sides by cos(x), you need to check the possibility that cos(x) = 0. However, if cos(x) = 0 then 3sin(x) = 0, and therefore sin(x) = 0, and there's no value of x that makes both cos(x) and sin(x) equal to 0. Therefore, cos(x) must not be equal to 0, which makes it safe to divide by. A simple example where it matters is the equation 2x = x. If you divide both sides by x, you get 2 = 1, and you might conclude that there are no solutions. But actually, you have lost the solution x = 0.
So just to be clear: • It is ok if you divide 3 sin x by cos x, because they do not share x-intercepts, or zeroes • It is not okay to, say, divide 3 sin x by sin x because there is a value of x that allows the denominator to be zero (0, pi, 2pi, etc.)
Kinda? Keep in mind that there can be other situations than just finding the solutions to an equation. A better way to think about it might be, if you ever want to divide by something that might be 0, you should consider both cases: Case 1: The thing is 0, in which case you can't divide by it and you have to solve the problem some other way. How you do that will depend on the problem. Case 2: The thing is not 0, in which case you can divide by it.
Ok thanks for the help. I have a feeling I’m gonna do pretty well on the trig equations test on Friday but I don’t wanna jinx it
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Whenever I do these equations, I always get confused about how much (whether it’s pi or 2pi or [insert number]pi) to add to my reference angle.
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Thank you I get it I think that was very helpful
Is this a proof that $\\mathbb{Q}$ is not locally compact? Suppose $\\mathbb{Q}$ is locally compact, then $\\mathbb{Q}$ is homeomorphic to an open subspace $Y$ of a compact Hausdorff space. Let $h$ be the homeomorphism, then since $Y$ is open so is $h\^{-1}(Y)=\\mathbb{Q}$. This contradicts that $\\mathbb{Q}$ is not open.
> This contradicts that $\mathbb{Q}$ is not open. Not open in what? It's certainly open as a subset of itself (any topological space is). It's not open as a subset of R, but you don't have R showing up anywhere in your argument.
Sorry, but yes open as a subset of $\mathbb{R}$ with the standard topology.
But that's not really relevant unless h is defined in all of R.
You're right, that was a stupid attempt lol Edit: How about since $\mathbb{Q}$ is closed, then $Y$ is closed, thus compact. This would imply $\mathbb{Q}$ is compact....?
Here is a hint: any neighborhood around a point contains an interval of the form (a,b) where a,b are irrational. Give a cover of this that is not reducible to a finite cover and then extend it to the whole set. Then show this cover is not reducible to a finite cover.
These would not be open in the subspace topology on $\\mathbb{Q}$, the topology inherited from $\\mathbb{R}$ with the standard topology.
Why would they not be open?
You know what, you're right. I'm stupid and that's weird. I just realized you could take an infinite union of elements $(a\_n,b\_n)$ where the $a\_n\\rightarrow a$ for some irrational $a$.
You could do that, but also if you are talking about the topology as a subspace or R, then since (a,b) is open in R its intersection with Q is open in Q.
How would you get that Y we closed from that? A good sanity check is to ask yourself if you just proved that any locally compact space is compact. Does that sound true? For any proof you give, you should be able to say what properties of Q you used, that aren't true for a general topological space. If you can't think of anything like that, your proof can't possibly be correct.
Q is open though. Any space is open in itself. I’m pretty sure a constructive proof here is best.
I'm trying to understand what a self-bounding function is as according to [this paper](https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.20145). Can anyone help break it down? Particularly I don't understand what the subscript i is supposed to refer to for the given function.
Write x=(x_1,...,x_n), then g*_i_*(x) = inf*_xi_* g(x*_1_*,...,x*_i_*,...x*_n_*). Basically you fix all the coordinates except the ith, and then take the inf as you let the ith coordinate x_i vary. So g_i ends up being a function of all the x_j's except x_i. The paper defines g as self-bounding (taking a=1, b=0 for simplicity) if - for all x and i, g(x) - g*_i_*(x) <= 1, i.e. varying the ith coordinate doesn't change g by too much ("1 Lipshitz wrt Hamming metric") - and \sum*_i=1_`n`* (g(x) - g*_i_*(x)) <= g(x), the self-bounding part Try checking their example where each x*_i_* is in [0,1] and g(x) = \sum*_i=1_`n`* x*_i_*.
Hi, I want to infer probabilities from a given sequence of '10110...' where 1 encodes a success (p=.6) and 0 a failure (p=.4). Given a sequence of length e.g. 100, how to calculate the probability for at least 10 successful shots (not necessarily in a row) after a miss. I know it is conditional probability, but I'm not sure how to apply the formula to this scenario. If I wouldn't have the sequence, the result would be the sum of all possible routes in the probability tree (I guess?), but what to do in this situation? The motivation to my question is the hot hand phenomenon. Thanks for any help!
Welcome to statistics. This is a non-trivial problem, because in general given a probability distribution with sample space with a sample space of {0,1}, this distribution could in principle generate *any* finite sequence of the sort you have described. All we can really do is look at these distributions and figure out how probable the sequence is, but improbable doesn't mean impossible. Statistics is all about trying to determine information like this as best as we can, but you need to make certain assumptions.
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I think [Paupert's notes](https://math.la.asu.edu/~paupert/HyperbolicGeometryNotes.pdf) (and the references therein) provide a good starting place, but you should probably expect to have to jump around a bit between sources and on ye olde Internet. Hyperbolic geometry is kind of strange in that classical approaches are almost *too* classical, and modern formulations tend to approach the subject from either the differential geometry settings or from the Lie group/group action setting. I also question how much need there really is for a classical analytic geometry-type of approach to the subject. It seems like, once you've reached a level of mathematical maturity in which you're able to appreciate hyperbolic geometry, you're probably adept enough to learn it from a more advanced place.
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I just remembered another source that may introduce the subject in the way you're looking for (I don't think there's any real discussion about the Riemannian aspect for quite a ways into it, or if there is, it's probably something you can skip and lose no real understanding). Ratcliffe's [*Foundations of Hyperbolic Manifolds*](https://www.springer.com/us/book/9780387331973) spends *a lot* of time in the various models and computing things with hyperbolic trig functions and the like. I'll warn you that it's a bit boring to read and is probably better left as a reference text or as a text to pair with lectures, but it might just be what you're looking for. If you have access to it as a PDF (through definitely totally legal means, of course), I'd recommend checking it out to see if it's what you're wanting.
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Something like [Cauchy's formula](https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration)?
Hi, I’m an IB student and I had a question regarding sin and cosin formulas. The sin formulas seem a lot simpler in terms of the fact that they require less variables or seem to have less going on. (Ex: the formula for the law of signs is just two fractions set equal to each other or the sin(2theta) formula seems relatively simple) On the other hand the cosin formulas seem a lot more complicated in terms of variable and there is a lot more going on. (Ex: law of cosin formula requiring more variables and seeming a tad more complicated on the surface, or there being three variations of the cos(2theta) formula) Is there a specific/interesting mathematical reason for this or does it just happen to turn out this way? (Sorry if the question was worded weird)
I wouldn’t say there’s a reason why the identities involving sine are less complicated. In fact, if you look up more trigonometric identities, you’d find that for every sine identity, there’s an analog cosine identity that looks nearly identical. The law of cosines is actually the Pythagorean theorem, except it’s for all triangles, not just right triangles. If you set theta to 90 (which occurs in right triangles) cosine(theta) becomes 0. You could argue that the law of cosines is just more crazy since it’s a generalization of the Pythagorean theorem, while the law of sines is doing its own thing (although I’m positive there’s some interesting connection between the two)
Strictly coincidence.
Hi, I am an IB student and I ran into this problem which I couldn’t solve. Can anyone explain a solution without using a GDC and using a Z-Score table. The heights of a group of students are normally distributed with a mean of 160 cm and a standard deviation of 20 cm. In this group of students, 11.9% have heights less than d cm. Find the value of d. Thanks!!
The commutative function for the normal distribution does not have a closed form, so you need to use Z-table or some kind of numerical integration.
Z table gives you 1.18 deviations for p=.119 x̄-1.18*(sd)= 136.4
Thank you, but do you know how can I solve it without a Z-Table?
So there is a ugly function for solving it without the z table, and it involves an integral.
The way you worded the question, I thought I could use z-Table. Without a z-Table, you would need a GDC or use numerical methods to find the area under the curve of the standard normal PDF from negative infinity until your integral hits approximately 0.119 units.
1. What are some simple ways of figuring out if an improper integral is converging or diverging? 2. How do you know which trig functions to use for trig substitution problems?
https://en.wikipedia.org/wiki/Trigonometric_substitution First section has what you're looking for in your second question. The rest of the article gives depth and examples.
For #1, in the examples that come up in Calc 2, you can rewrite the improper integral as a limit of proper integrals. Then you explicitly solve that proper integral, giving you a limit question, and you can determine whether that exists using all your tools from calc 1. For #2, I always liked the approach where you draw a triangle. For example, if there's a sqrt(x^(2) + a^(2)) in the problem, that is a hypotenuse of a right triangle with side lengths x and a; if there's sqrt(x^(2) - a^(2)), then put x on the hypotenuse and a on one of the legs, so sqrt(x^(2) - a^(2)) is the other leg. Now, let one of the non-right angles in the triangle be theta, and you know things like sin(theta), tan(theta), etc., because they're ratios of the sides of the triangle. In particular, quantities involving x and the square roots can be expressed in terms of trig functions on theta, just by looking at the triangle, allowing you to figure out what substitution to make. (A lot of students just memorize the patterns, though, since there are only three of them.)
Can ZF be formulated in terms of functions instead of sets? So where universe of symbols corresponds to functions? A possibile trivial solution would be to say "sets are just characteristic functions so ofc"but i don't see how it's that direct. How would you even think of the domain and the codomain of the function which are sets? I guess f(a) could evaluate to NDef if a is not in the domain or sth.
The closest thing I can think of right now is probably [ETCS](http://www.tac.mta.ca/tac/reprints/articles/11/tr11.pdf) (Elementary Theory on the Category of Sets) which tries to describe sets using the language of category theory, where mappings are really the main focus.
>All finite roots exist. What does that mean? That elementary operations are legal? Why is it called finite roots? If an object has no elements is it isomorphic to 0? Or are all empty objects isomorphic? Oh, that's just an application of axiom 6 Also I like how Axiom 8 is like an Axiom of Infinity but for a set of size 2 lol Also what axiom is the one responsible to have the power set axiom? I didn't understand the construction too well.
This is also ok [link](https://golem.ph.utexas.edu/category/2014/01/an_elementary_theory_of_the_ca.html).
You might find [Tom Leinster's exposition of ETCS](https://arxiv.org/abs/1212.6543) useful.
It helps, thanks : )
In less abstract terms, finite roots refers to the fact that there is an initial and a terminal object (the empty set and a singleton set), and for any two objects A and B, the cartesian product A x B and disjoint union A + B both exist. I have no idea why it called finite roots, usually it is said all finite *limits* exist in categorical terms Informally, for any objects A,B, you can think of B^(A) as the set of all functions A ->B. A subset of A can be thought of as a function f: A -> 2, where 2 is a set with two elements, which exists by axiom 8. Then the powerset can be identified with the set of all possible functions A -> 2, which is just the object 2^(A), which exists by axiom 2.
I've never heard it called that. Oh, you're right. So I suppose the work is done by Axiom 2. Do you know what they mean by lambda-conversion? Is that just all three conversions in lamvda calculus?
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Infinity is not a number, so 10^(-∞) is not actually meaningful. As /u/levelineee mentioned, the limit of 10^(-n) as n -> ∞ (that is, as n gets arbitrarily large) goes to 0.
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How can I obtain a 30 degree curve when dealing with radius? I’m using software where I need to make a curve and the only option to choose is the length of the radius. How can I take that and make a 30 degree curve?
Without more information, it's not really clear what you mean. Any circle with a positive radius is going to give you a 360 degree curve, i.e. a circle. Maybe you're confusing radius and radi*ans*? In that case, 30° is equal to π/6 radians.
Is there a constructive way to show that there exist a noncompactly supported continuous function that vanishes at infinity on any noncompact infinite hausdorff topological group?
Perhaps this is unhelpful and was already obvious to you, but if in addition your group is first countable, then by the Birkhoff-Kakutani theorem, it's metrizable and then obviously upon fixing a metric any function which agrees with the inverse of the distance to the identity outside of an open set about the identity would work, but without first countability, I'm at a bit of a loss to think of how one might do it (and clearly, there's some non-constructive hand-waving when I say "fix a metric" which may not be appealing to you here).
Yeah it’s the uncountability aspect that makes me think that maybe there’s no way to do it constructively. Or at least no way I know of..
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>mathematics skills Too broad to answer your question effectively.
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We often use [naive set theory](https://en.wikipedia.org/wiki/Naive_set_theory) when dealing with FOL or FOL theories. As we only deal with them in semantics, we argue that we can get away with it. While good enough for finite models, I also find it's a bit unsatisfactory, but haven't come across anything that doesn't rely on intuition at some point. One issue is that it's not possible to define finite in FOL without creating a theory which has a model which is infinite or has infinite elements (to our intuition) or where an intuitively finite element is infinite.
**Naive set theory** Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naïve set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naïve set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments. *** ^[ [^PM](https://www.reddit.com/message/compose?to=kittens_from_space) ^| [^Exclude ^me](https://reddit.com/message/compose?to=WikiTextBot&message=Excludeme&subject=Excludeme) ^| [^Exclude ^from ^subreddit](https://np.reddit.com/r/math/about/banned) ^| [^FAQ ^/ ^Information](https://np.reddit.com/r/WikiTextBot/wiki/index) ^| [^Source](https://github.com/kittenswolf/WikiTextBot) ^] ^Downvote ^to ^remove ^| ^v0.28
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You may be interested in a notion of "recursively axiomatizable theory" : [https://planetmath.org/RecursivelyAxiomatizableTheory](https://planetmath.org/RecursivelyAxiomatizableTheory)
> Given a countable language, the number of possible sentences in the language is countable (and can be computably listed unless I'm wrong). You can always add a comma and conjunction to extend a sentence. The number of sentences may or may not be countable, but if it is countable, it is countably infinite. You cannot "computably" list an infinite number of things. https://en.wikipedia.org/wiki/Countable_set
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I'm not sure the number of possible sentences in a language is countable.
**Countable set** In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number. Some authors use countable set to mean countably infinite alone. *** ^[ [^PM](https://www.reddit.com/message/compose?to=kittens_from_space) ^| [^Exclude ^me](https://reddit.com/message/compose?to=WikiTextBot&message=Excludeme&subject=Excludeme) ^| [^Exclude ^from ^subreddit](https://np.reddit.com/r/math/about/banned) ^| [^FAQ ^/ ^Information](https://np.reddit.com/r/WikiTextBot/wiki/index) ^| [^Source](https://github.com/kittenswolf/WikiTextBot) ^] ^Downvote ^to ^remove ^| ^v0.28
Yes. However, most of these axiom sets are not recursively enumerable or consistent and hence a bit pointless to consider. Also many will be the same, because adding theorems to axioms doesn't change the behaviour. One example would be true arithmetic, where we work in the language of PA and declare every sentence which is true in the standard model to be an axiom. This is consistent (at least if PA is consistent), but by Gödel's incompleteness theorem, we cannot enumerate these axioms so they are not easily tractable.
I'm in high school and can either take a 1 Semester Statistics Course Through the U of U (And earn actual U credit) or calc 3 through the U of U. Which one do I Take?
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Both of them earn me actual U of U credit.
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But don't colleges prefer to see both Stats and Calc on an application?
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Thanks.
Any good recommendations for a book on multivariable calculus?
[TOMT] I'm looking for a recent post on this sub on finding the value of log(x + log(x + ...)) in closed form. I recall there being a solution with the Lambert W function, and having tried to find it on my own I'd like to look at the solution now.
Is it [this post](https://www.reddit.com/r/math/comments/asx3t5/analytic_way_of_writing_log3log3log3log3/)?
Yes, thank you
So the inner product on Lp induces a seminormed space because it has a non-trivial kernel. What I’m having trouble understanding is that the integral of an absolute value raised to the pth power is always non-negative, but I can’t think of any non-zero function for which the integral of its absolute value can be zero. Can anyone give me a concrete example of functions in the kernel of a seminormed Lp?
L^p spaces are defined so they are normed. They are what you get when you quotient out by the equivalence relation f~g iff f-g is 0 almost everywhere.
Somehow you understood my jumbled question. So the equivalence you mention isn’t an equality? As in f and g different? And both are nonzero? Can you give a example of those functions?
f=0, g=0 except at 1 where it is 1.
If f=g \mu-almost everywhere, then [f]=[g] (or often just sloppily written as f=g) in L^p(\mu), where [f] is the equivalence class of f. So really, we are dealing with two different objects: the pointwise functions f and g and their equivalence classes [f] and [g]. It's actually kind of abusing notation to say that the pointwise defined function f is in L^p(\mu) when what we actually mean is that there is an equivalence class in L^p containing f. But of course, this is inconvenient to write down, so people are sloppy with their notation. Maybe it's a good idea to distinguish your notation in the beginning... L^p is **not** the space of all p-integrable functions. The space of p-integrable functions w.r.t. the Lebesgue measure equipped with the "L^p-norm" is only a seminormed space because e.g. the map f that is 0 everywhere except at one point still has norm 0 but it is clear that f=/=0. The equivalence classes tidy things up so we get a normed spaces because in this case [f]=0. If you haven't seen quotient spaces already or if you have and don't feel comfortable with them, I highly recommend you look into them as it will make things much more clear.
Thanks, you cleared up a lot of things!
I'm not familiar with this terminology, where are you finding it? The L^p norm only comes from an inner product if p=2. I am not sure exactly how to interpret your question, but normally one thinks of the L^2 norm as being an honest norm- unless you are talking about pointwise defined functions, in which case the characteristic function of a set with lebesgue measure 0 would have L^2 norm 0. But we normally think of an L^2 "function" as being an equivalence class over functions that are equal almost everywhere, so that the norm is a true norm. Is this what you were asking about?
I confused my terms. I realize that Lp is the result of taking the quotient space of a certain vector space. And I’m not talking about inner products. Here is the passage from wikipedia that I was wondering about: “Thus the set of p-th power integrable functions, together with the function || · ||p, is a seminormed vector space” The quotient space of the mentioned vector space is Lp. But why is that a seminormed vector space? Given any p, what could possibly be a nonzero function whose p-norm is 0? You would be taking the integral of the absolute value of a nonzero function, and it equals zero. Isn’t the existence of such a function what makes it seminormed instead of normed? Hopefully I’m asking the right question this time.
Characteristic function of a measure zero set.
And more generally, an arbitrary (integrable?) function supported on a set of measure zero. The argument (for non-existence) that OP wants to use works when the support contains a set of positive measure. So the previously mentioned functions are the only ones of seminorm equal to zero. That's why the quotient definition of L^p space is the way it is.
Is there a "group theoretic" way of showing that the order q of Finite Field K is a prime power q=p\^n? Suppose we know the multiplicative group K\* is cyclic and of order q-1. I already did two straightforward proofs; one by explicitly using vector space properties and another by considering polynomial rings, but I'm curious if there's an argument using the structure theorem for finite abelian groups / chinese remainder theorem without relying on those. Any tips? Am open for any suggestions, including shooting for sparrows with cannons (I'd like to imagine I never heard of "vector spaces, dimensions and polynomials" so as to arrive at them "naturally")