For example, my one professor always shows his first year students the proof for sqrt(2) being irrational and tells them that if they don't think it is beautiful, it is not likely they will enjoy mathematics (jokingly of course).
I particularly have liked this one question about a game of tictactoe where players take turns filling a 3x3 matrix with 1's and 0's (first and second palyer resp.) until it is full. Then they take the determinant and player 1 wins if it is non-zero, otherwise player 2 wins. The question is which player has a winning strategy and what is it.
Another interesting question I have not yet been able to solve, but which a professor always gives as a fave problem is: "Find all positive rational solutions to a^b=b^a."
Note that since P2 has a winning strategy (so the determinant can be forced to be 0), you can just find a winning strategy for P1 playing in a specific spot first.
Hello fellow math lovers, I have a quick question. I have been laboriously looking for similar problems on the internet and I am using this place as a last resort. Does anyone here know where I can find a problem set covering questions about finding equation of line tangent to a curve? This type of problem appears in Chapter 3 Section 1 of Stewart Calculus (Single Variable) but I am not satisfied with my performance. I am primarily concerned with finding values of unknown variables.
That is a basic one. I have the basics down, no problem there. I am stuck at more advanced ones. For example: find the value of c such that the line y = 3x/2 + 6 is tangent to the curve y = c*sqrt(x). This problem is not so straight forward, and there are many like it that require finding value of missing variable. I am good to go once I have the equations and I can then use matrix algebra to get the values but getting all the equations correctly is where I get stumped at times.
@anakhronizein Quite right. In general, an orientation of a rank n vector bundle is equivalent to the data of a trivialization of $\Lambda^n E$, which is determined by a non vanishing section of $\Lambda^n E^*$. An n-form on a manifold restricts to an n-form in this sense on any subbundle of TM.
(Note that this is not the same as what I would normally call an E-valued form, but rather just an appropriately skew-symmetric and multilinear functional on E.)
I just realized that this gives a sorta pleasant fact: (4n-1)-dimensional contact manifolds are orientable, and (4n+1)-dimensional contact manifolds have orientable contact hyperplane field, then a contact manifold has trivial normal bundle to the hyper plane field (is co-orientable) if and only if it’s both orientable and has orientable contact field. So all of those orientations are given if and only if we have a global defining 1-form.
@MikeMiller I had heard a quote that I think was attributed to Arnold that said something along the lines of "all geometry is contact geometry". Do you know what was meant by this?
@anakhronizein Well I have already solved this one. But When I first solved it, I found out my answer was incorrect then I spent half an hour finding the correct one. First thing I do is write down what is required of me, what I need to get there and I start plugging away. But in general, my go to steps are, find the slope, find x value, find value...not necessarily in that order.
Geiges states in his preface: "One of the most eloquent of modern panegyrists of contact geometry is Vladimir Arnold, who proclaimed on several occasions since 1989 that ‘contact geometry is all geometry’"
I want to announce to all the US-based grad students that the current version of the tax bill in the US congress would make tuition waivers taxable income. For almost all of us, I imagine, that would substantially increase our tax burden, even after accounting for the larger standard deduction. So just a heads up to everyone to be aware, and prepare if necessary.
1) Kreck, “differential algebraic topology”, reframes homology as bordism of a kind of singular smooth manifold (the fun to say stratifold), thereby allowing one to use smooth techniques from transversalitt theory in a pleasantly direct way, and reinterpret a lot of standard algebra in this way
It’s often how I like to think about these things (also useful is the brief paper of Lipyanskiy, “geometric homology”, which uses manifolds with corners instead)
I was an undergraduate when I took that algebraic topology course. Also it was a category theorist so I feel like there was a lot of unnecessary categorical background that took up time. Hence the need for a refresher.
@TedShifrin Yeah, I'm salty about it. There doesn't seem to be any logic to raising taxes of grad students to pay for an estate tax cut other than they have to placate their donors. If there's a bright center to the tax-reform universe, this bill is on the planet furthest from
Somehow 'salty' seems the wrong word for that @KevinDriscoll
When you lose a video game and get frustrated afterwards, you're salty. When the GOP in Washington is planning to change the tax code so that the rich pay less and you, personally, pay more...
I think unironic contempt/anger are quite appropriate.
Hey has anybody ever heard that the rigidity of a graph is equivalent to the definability of equality on vertices of the graph in first order logic? I just proved it and I keep looking it up online and cannot find it shown anywhere else
I don't know the right terminology for this, so it will look naive. If I have a function y = f(x) (y and x are typically vectors), and I have some set of "smallest" basis vectors {a_i} such that f(x+a_i) = f(x) for any x (from which a whole vector space of such terms could be built), is there a standard way to refer to an individual a_i, and to the smallest set of them that could generate all possible cases where f is unchanged?
This is some kind of symmetry, so I presume this is related to Group theory (of which I have none) -- I don't really know what I am doing here.
I've been calling these minimal a_i things as "atoms" (they're essentially vectors that are almost all 0, but with a few elements that are +1 or -1 in particular patterns) but it's better if I find out what they're more conventionally called.
There's nothing with the same property that can have fewer non-zero elements than they have, so they're the simplest units to work with.
The underlying point of all this is to demonstrate to practioners in a particular application area who are constructing predictions using a particular f that their favorite f is completely blind to particular kinds of structure in x (i.e. f(x')=f(x) where x'-x is not something they would wish to be blind to). Being able to talk about how to generate a collection of interesting (i.e. surprising to them) x' more conventionally would be handy.
I can do the required calculations readily enough but better notation and terminology will help (not just in describing it but in looking up stuff in order to understand more about what I am doing rather than just building it all from scratch)
I'm not even sure where to start looking really.
Any suggestions of terms to use for an a_i or the smallest collection of them? Theorems relating to how to show I really do have the smallest set that generates all x' with the same output? Useful things to read about this stuff?
@Glen_b Sorry I'm not sure I understand. You have a vector-valued function $f$. And then what you find is that there is a single set of basis vectors $a_i$ such that $f( x + \sum_i c_i a_i) = f(x)$ for any $x$
It's right, but there's no unique set of basis vectors - you could make a different basis easily; my collection of a's just happen to be made up of vectors with the fewest possible number of non-zero terms in them (and by restricting them to only have +1 or -1 they'll then be unique up to an overall change of sign)
No. only that the 0-vector is another possible x, I guess. Oh, actually, ... I have to be careful, f(a_i) might strictly be undefined, more correctly for any case where a linear combination of a's is defined, the value would be zero
even "invariant on some linear subspace" is helpful at the moment.
So one thing to figure out is exactly what linear subspace your $a_i$ span
Ah right. And so you can think of $x + c_i a_i$ as being $x + v$ for some vector $v$. And then think of $v$ as being $A w$ for some matrix $A$ and vector $w$. That vector will be arbitrary. But the matrix will have to have a very particular structure
related to the $a_i$ and where they ahve 0s and where they don't
And knowing that matrix $A$ should give you a better idea of what kind of symmetry this is
Okay; these vectors are stacked up from a rectangular array that is cut off on the lower right (it most often will be triangular) in some convenient order. The ai's are such that all rows and columns add to 0.
Suppose $x \in \mathbb{R}^3$ and $a_1 = \{1, 0, 0 \}$ and $a_2 = \{0, -1, 0 \}$. Then we can say $f(x + A w) = f(x)$ for arbitrary $w$ provided that we make $A$ the matrix $$\begin{array}{lcr} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{array}$$
My contention is firstly that every such set of values (v in your notation) where all rows and columns sum to 0 are buildable from a's that in the original array have a 2x2 matrix +1,-1 and -1,+1
Hang on, while I read the link on how to see the LaTeX markup in chat.
Okay. So the particular A's then denote the group -- or is there some characteristic of A that does? This is probably the point where I should stop using up your valuable time and go read something about exactly how to do what you're saying to do.
The $A$s should tell you what group is hidden in there. They aren't themselves the group, they're actually something like a sum of representations of elements of the group
Since vocab is one thing I was seeking -- how to talk about things.
The stuff I have tried to read on group theory doesn't seem to relate directly to what I am trying to do but everything you have said makes sense. This suggests that I am just looking in the wrong places
Ah. I see even from the wikipedia page that that is quite probably the kind of thing I need, but I will need something like a set of notes with some exercises or a reasonable text on it.
@Glen_b So check out this lecture by William Harter and particularly the first 15 minutes or so. There's a lot of physics mumbo-jumbo in there that isn't important for you. But the thing I think is important is that he shows this matrix $H$
and that matrix doesn't have arbitrary entires. for exmaple, all its diagonal entries have to be the same.
and there are similar relations among the other entries
And what he shows here is the very basics of how you break this $6 \times 6$ matrix that he has into 6 pieces. And each piece corresponds to the element of a group. In this case its $C_6$. The matrix $A$ that you will find should play the same role as his matrix $H$. And of course who knows what, if anything, the group you'll have will be.
@Glen_b Indeed. And if all this works out, at the end of the day you'll be able to say something like. This $f$ is invariant under the transformation generated by the group $G$, so if you have 2 inputs that are related to each other by some $g \in G$ you're gonna get the same result. And so if you don't want to treat those 2 inputs the same, then you gotta come up with a different $f$
I can show all manner of interesting properties but I don't currently have the tools I need to do a proper job of saying everything important there is to say; I didn't even have the words to go looking.
Although the easiest way to go in this group theory direction is to somehow wrangle someone who does this for a living, and sadly Im not at that level yet
Yes, although I'm not exactly sure what the math people who do this actually do all day. Like what their specific expertise is. You might actually have more luck in a physics department.
For example let me give you the analogy I have in mind
Prof. HArter in that lecture looks as the case of 6 atoms or balls or whatever arranged in the shape of a hexagon and each one is connected by a spring to its nearest neighbor
Possible. However, I'd want to go in with enough background that I come off as serious-but-ignorant rather than a crank. Physicists see even more of them than mathematicians.
and then you look at a different problem which has little particles arranged in the shape of a 12-gon but htis time there are 2 kinds of spring. And 1 spring goes between each pair of balls and you alternate which one you use
So its like A-type, B-type, A-type, B-type etc. til you get all teh wya round
And again you do this matrix thing
And its some $ 12 \times 12$ mess, so not at all teh $6 \times 6$ thing you had earlier
but hten you find out by analyzing it that they both contain representation of the same group
... Looking with hindsight that doesn't sound helpful to you. But there's some connection there about guys of different sizes nevertheless hiding the same structure
Okay. Let $A \subseteq \Bbb{R}^\omega$ be the set of all bounded sequences in $\Bbb{R}$. The problem I am working on is trying to show that $x \in \Bbb{R}^\omega$ is lies in the same component of $0$ if and only if $x$ is a bounded sequence, where $\Bbb{R}^\omega$ is endowed with the uniform topo...
@MathematicsAminPhysics A room for experiments with MathJax was unfrozen. Just in case it's useful for you. (I've seen that you've occasionally do some experiments with formatting in various rooms. And in the past, you have similar room of your own.)
(With Riemann sums and real integrals, $dz$ represents the width of each of those tiny rectangles. But here $dz$ can be complex, so that picture gets confused.)
I need to calculate $\int_{\gamma} z \ ^ 3 dz$ where $\gamma = \gamma_1 *\gamma_2*\gamma_3*\gamma_4$ .
$\gamma_1(t) = -2t +1+i$
$\gamma_2(t) = -2it +i-1$
$\gamma_3(t) = 2t -i-1$
$\gamma_4(t) = 2it +1-i$
$t \in [0,1]$
That is, $\gamma$ is a square that starts at 1+i , with positive directio...
I need to calculate $\int_{\gamma} z \ ^ 3 dz$ where $\gamma = \gamma_1 *\gamma_2*\gamma_3*\gamma_4$ .
$\gamma_1(t) = -2t +1+i$
$\gamma_2(t) = -2it +i-1$
$\gamma_3(t) = 2t -i-1$
$\gamma_4(t) = 2it +1-i$
$t \in [0,1]$
That is, $\gamma$ is a square that starts at 1+i , with positive directio...
