Does all math have to start in the middle? Let's say you had no mathematical literacy. Could you learn arithmetic formally? Like, would you start with formal logic that never depended on doing basic arithmetic and build up from there? (i.e. you can't say anything about a set's cardinality or do any proofs involving basic arithmetic in them without first defining that arithmetic).
I remember a while back reading a paper about the proof that you can divide by 3, and it made me seriously consider just how arduous such a task really would be.
@XanderHenderson I don't doubt that it would be an inefficient endeavor. Teaching children axiomatically that 3 comes after 2 has worked for centuries. However, it seems like there's spaces between knowing nothing about math and knowing the topics taught in elementary school that get lost.
If $\Delta$ABC is a triangle with sides $a$,$b$ and$c$ and satisfies $a^2+b^2=c^2$. then $\Delta$ABC is right triangle. I am not getting the counter. I think the statement is true. How to prove it? Please give hints.
Start mathematics from $\mathsf{ZFC}$ and just work your way up?
Or teach kindergarteners formal arithmetic form day one?
I don't see any reasonable way to start in kindergarten, as the students will (1) lack the cognitive tools to understand what you are doing and (2) lack the background and intuition to understand why you are doing it.
Now, suppose that you had a fully formed adult who had, somehow, grown up in a vacuum and never once been exposed to a single mathematical idea.
It seems possible that you could take that tabula rasa and work with it, but that is an entirely made-up situation.
Hi all! Given two complex valued polynomials $p(z) = a_n z^n + \ldots + a_0$ and $q(z) = b_n z^n + \ldots + b_0$, if $|a_k| \geq |b_k|$ for all $k$, then is it true that $|p(z)| \geq |q(z)|$ for all $z \in \C$? My gut feeling is that is that it is true, but I'm having difficulty proving it's true even for the $n=1$ case in which I'm trying to show that $|a_1 z + a_0 | \geq | b_1 z + b_0|$ using the triangle inequality. I'd like to use this result in a proof involving Rouche's theorem later...
For example, the typical career for most of the students that I work with is probably something like AP Calculus (which is really just a class where they learn recipes), College Calculus (the words "epsilon" and "delta" might appear, but are not vital), "Advanced" Calculus or Undergrad Real Analysis (finally, something approaching rigour), followed by graduate topology, measure theory, and various specialized analysis classes (bend over, guys; here comes the RIGOR!)
At each step of the process, we can build rigor onto a framework of intuition which has already been developed, hence it is possible to motivate the kinds of arguments that are going to be made.
@Poptart hmm, I wonder if that's actually true since you can always pick $z$ to be one of the roots of $p(z)$
in which case the only way for the inequality to be valid is if $q(z)$ vanishes as well...but if they have the same roots, they're the same polynomial up to an overall constant
In which case, I think the easy counterexample is just $p(z)=z-1$ and $q(z)=z$.
Can someone convince me why $Tp(\partial M)$ is of cxdimension one? (Tangent space of a boundary). All I read from Lee is that for any open set around a $p \in M$, its tangent space $T_p(U) \sim \mathbb{R}^n$. I don't see how this implies it has cxdimension one.
Let $X$ be a collection of $k$ double points, and $I_X(d)$ the subspace of all homogenous polynomial through $X$, that is with all first partial derivatives vanishing at the points of $X$. In general, do we know $\dim(I_X(d)$?
Yeah, there is a reason the only open problems which actually require some proper math to state and which attract cranks are the ones with big money prizes
I am not aware of even a single crank attempt at Lusztig's conjectures for example.
I wonder if the map $f:S^2\to\Bbb R^9$ given by $v\mapsto vv^\top$ (that is, $\begin{bmatrix}x\\y\\z\end{bmatrix}\mapsto \begin{bmatrix}x^2&xy&xz\\ xy&y^2&yz\\ xz&yz&z^2 \end{bmatrix}$) is anything approaching an isometry. (It looks like an embedding of $\rm\Bbb RP^2$ in (a five-dimensional subspaces of) $\Bbb R^9$.)
($x^2+y^2+z^2=1$)
It seems the image of a vector $w\in T_vS^2$ is $vw^\top+wv^\top\in T_{vv^\top}{\rm im}(f)$
@0celo7 huh. That's probably a math talk I'd attend
some of that stuff I can sorta understand right off the bat, like "the invariance of a Gibbs measure associated to an infinite dimension Hamiltonian system"
which basically sounds like it should be "yay for stat mech"
it's basically just the same as saying that, for a system at temperature $T$, the probability (density) of the system being in a state of energy $E$ should be proportional to $e^{-E/T}$
that works for finite systems, anyways. for infinite-dimensional systems it seems one needs to generalize that appropriately
@0celo7, is this reasoning of mine correct: for $g(x)=0$, the riemann stieltjis integral holds for each finction f and, it is 0. But we know that some functions are not riemann integrable, hence...
Is there some intuitive interpretation of multiplying a unit vector of the form $e^{i \theta}$ by another unit vector, say $(1, 0)^T$? I was reading about phasors and phase vectors, but I am not sure...
Yes, I suppose is the only way to interpret it as a vector. I initially was not thinking of it as a vector, but then I heard a person calling it a "phase vector"...
It's a vector in the complex plane
Effectively, it represents a vector, because of Euler's formula.
I'm reading through the opencv documentation and some questions in SO but it doesn't seem to provide this information.
I've an image $I(x,y)$ and I want to find a gaussian function $f_{\mu,\Sigma}(x,y)$ such that
$$
f_{\mu,\Sigma}(x,y) \approx I(x,y)
$$
using LSE or MLE estimation. $\mu$ is the...
@AkivaWeinberger Have you ever heard of Monge's theorem?
It says that if you have three circles on the plane, if you draw the three cones with the common (external) tangents of each of the three pairs of the circles, then the vertex of those three cones are colinear
Mind-blowing proof: Think of the picture as a section of the same scenario with three 2-spheres and cones tangential to three pairs of them.
Think of the two planes that are tangential to the three spheres from "above" and "below". These are itself tangential to the cones. That means the vertex points of the cones lie on the intersection of these two planes.
The professor showed us this proof while doing some projective geometry in my first year in uni, many minds (mine included) were blown in that lecture!
@BalarkaSen You can also think of the three circles in the plane as a perspective drawing of three unit spheres in space. (The smaller circles are just farther away spheres.) For each pair of spheres, create the cylinder tangent to them; this becomes the pair of tangent lines in the drawing. Then you can draw the plane through the spheres' center; the line in the drawing is the "horizon" of that plane.
@BalarkaSen 3D generalization of Monge's theorem: consider four spheres of distinct radii in space, and for each pair of spheres construct the cones tangent to them. Then the vertices of those six cones are coplanar.
If I can up my functional analysis a bit, I will expect myself to spam the chat with functional analysis because it is very interesting (especially for infinite operators)
but right now, I don't even have enough knwoledge to knew whether I am making sense
Well they are also interesting in an algebraic perspective, such as how the solution curves behave in method of (I forgot the name) used in solving quasilinear PDEs
The most interesting PDEs are of coruse the nonlinear ones, but they are hard to solve even numerically. We computational chemists deal with these on a daily basis
thus there is quite a bit of overlap between PDEs and numerical analysis
IMO, combinitorics is a lot more tedious than other fields of mathematics, cause even if you know what's going on, it still involve a bit tedious counting of some sorts
if you have an ellipse and you draw its tangents of slope i and -i, and look at their intersection points, can you guess what real points you obtain this way ?
Meanwhile, for number theory, well, it has some rules... they are just hard to master. I am starting to get that number theory is basically about decomposing numerical expressions
I don't know if there is an elegant way to approach that, but for me I just grab an random ellipse $x^2/a^2 + y^2 / b^2 = c^2$, plot the equation for all its intersection points of said tangents, and then solve for those where the imaginary part are zero, so at least 4 simuutaneous equations
Geometrically, assuming the ellipses are not tilted in some way, then there should be some relationships between the ratio of its major and minor radius with where the points are intersecting in the x axis or something, but I might need to think more about it
This is based on the intuition of a real ellipse, the intersection of two lines of slopes m and -m are going to form a parallelogram that circumscribe the ellipse if I recall
Well, in theory the tilted case can be handled by introducing a suitable sheer rotation mapping to the system of tangent lines and ellipse, but my brain is a bit tired to think of that scenario atm
@EricSilva There’s an old theorem/conjecture by Brian White that if you take a surface repping a nontrivial homology class, and flow it by MCF, you end up with a minimal surface. Something like that. A lot of MCF in GR is based on ideas like that.
that should in theory work if we made the identification $a+bi \to (a,b)$
So given an arbitrary warped ellipse, solve for the unique A that maps the standard ellipse to it, and then take $A^{-1}$ to find where the intersections would have been
Hey guys, could someone tell me if i have a relation between $||\widehat{u}||_{L^2}$ and $||u||_{L^2}$? i know about Plancherel, but Plancherel tells me the relation with the $l^2$ norm. Hope u can help me. $\widehat{u}$ is the fourier transform
the slope should in theory be modelled by a vector, but I have not figure out how it will be affected under the map A. I have an equation that relates m with a,b. I guess the hardest problem is to find the $(x_0, y_0)$ that gives the required slope $m$
o wait a sec.., it seems I can just plug $(x_0,y_0)$ and the given slope $m$ directly into the equation for $y'$ by $y' \to n$, $(x,y) \to (x_0,y_0)$ and then after rearranging it gives the equation of the tangent directly. I still need to figure out how to get $k$ though, let me think...
Wait, the slope is complex, then I cannot just treat that as an xy plane problem because I will need the complex multiplication structure in order for the slope to multiply correctly...
The geometric intuition I have is treating it as an $\Bbb{R}^2$ vector space problem, which need the slope to be real for it to work out (otherwise under that correspondance, you will have slope i mapped to the vector (0,1) which makes no sense when you multiply them
ok I give up, it's too late and my brain is not functioning
while I now have the explicit form of the y intercept and slope, i have yet to figure out the way the rotation matrix relates to the new rotated ellipse
@EricSilva So part of my summer project is to try to reprove some results of Brian White for this situation using estimates produced by Schoen-Yau and my professor in her JDG paper that I will send you when it gets published.
Hi, I have a metric space $X$ and the usual definition of convergence of sequences with the $\varepsilon, n$ definition. I understand why this is equivalent to saying that every $\varepsilon$-neighborhood contains all but finitely many terms of the sequence. But how can I argue, that this is true iff every neighborhood contains finitely many terms (not $\varepsilon$-neighborhood anymore)?
Problem: If $S,T : X \to \Bbb{R}$ are bounded linear functionals s on some normed linear space $X$ that agree on a dense subset $D \subseteq X$, then $S=T$ on $X$. Proof: Let $x \in X$ and $\epsilon > 0$, and let $M_T > 0$ and $M_S > 0$ be upper bounds for $T$ and $S$, respectively. Then there exist $d \in D$ such that $||x-d|| < \epsilon$. Hence $$|S(x)-T(x)| \le |T(x-d)| + |S(d-x)| \le M_S ||x-d|| + M_T ||x-d|| < \epsilon (M_S + M_T).$$ Letting $\epsilon \to 0^+$, we conclude that $S(x)=T(x)$.
@AkivaWeinberger hello, i want to study the continuity $f(x)=\begin{cases}0,~\text{if}~ x<0\\ x^2+1,~\text{if}~ x\geq0$ where $f:(\mathbb{R},\sigma)\to (\mathbb{R},|.|)$ $\sigma$ is the co-finite topology, i say let $x<0$ then $f(x)=0$, let $W=]-\varepsilon,+\varepsilon[$ how to find $f^{-1}(]-\varepsilon,+\varepsilon[)$ please ?
it's easy, you take your calculator and look at $(9+4\sqrt 5)^{1/3}+(9-4\sqrt 5)^{1/3}$, you find it's $3$ so it gives you the $3/2$ part and then the $\sqrt 5/2$ part is just a matter of making the cube root work out
I'm reading through the opencv documentation and some questions in SO but it doesn't seem to provide this information.
I've an image $I(x,y)$ and I want to find a gaussian function $f_{\mu,\Sigma}(x,y)$ such that
$$
f_{\mu,\Sigma}(x,y) \approx I(x,y)
$$
using LSE or MLE estimation. $\mu$ is the...
