Are linear equations and linear maps linear in the same way? I get what a linear map is. I am confused whether we can show an equation is linear in the same way we show a linear map is linear.
@Stan: Linear equations come from linear maps. Suppose it's one equation, $a_1 x_1 + \dots + a_n x_n = 0$. Bake this into a map: $f(x) = a_1 x_1 + \dots + a_n x_n$. Then solutions to your equation are elements of the zero set of $f$.
And linear equations come from/give rise to linear maps.
One of my favorite examples comes from Seiberg-Witten theory. Normally you write down the SW equations, count the solutions, and that's your invariant. Furuta had the brilliant idea to instead consider the Seiberg-Witten map. By doing so and some reasonably simple analysis, he was able to prove the 10/8 theorem.
The same idea leads one to the Bauer-Furuta invariants: the SW map becomes your actual invariant, and lives in a "co homotopy group" instead of, well, the intwgers
@SemiC: Think of it as a wacky infinite-dimensional version of this. I can consider maps $\Sigma_2 \to T^2$. They all have an associated number called the degree (cardinality of generic preimage of a point, essentially, counted with a sign).
Or I could think of the homotopy class of the map as, uh, an invariant of the map. This is annoyingly tautological but sometimes there's value in all that information.
In this setting I have a 4-manifold, and then I have a nasty map between infinite-dimensional spaces, and then I roughly say "Take its homotopy class!"
That's a lot better than just a number associated to it. Maybe a lot harder to compute, but more theoretically powerful, since I throw away less data.
The relationship to degree is that "cardinality of ppreimage of point" is here replaced by "Cardinality of zero set" - the usual SW invariant
is that when one is using the residue theorem in complex analysis, one way to look at it is that one just counts the winding numbers and figures out how residues contribute from that
but one could also think of the integration cycle as an element of the first homology group. or if one wanted to hold on to even more information, as an element of the fundamental group
that doesn't make a difference if one is just doing residues, but it does if one has branch cuts (though now i have to be careful about whether I'm thinking of said cycles as in the complex plane, or in an appropriate branched cover of it)
Quick meta-math question. If I have a sequence $(\pi_1,\pi_2,\dots,\pi_n,\tau_1,\tau_2,\dots,\tau_n)$ that qualifies as a formal proof, am I correct in arguing that $(\pi_1,\tau_1,\pi_2,\tau_2,\dots,\pi_n,\tau_n)$ does NOT qualify as a formal proof (since each step in the sequence is no longer immediately derivable from a preceding axiom)?
I have a theory for a highly effective study system:
1) Find something in a field you enjoy, that is far beyond your understanding 2) Break down into a chain of things you need to learn in order 3) Attack from the start.
@JoshuaA I am not sure if I study like that, but note that one drawback of picking up arbitrary things you know nothing about and try learning about it is that your learning order is most often bound to get wrong, if you don't have a good textbook.
For example, sheaves and ringed spaces come after regular maps and algebraic varieties, not before :D
@robjohn how is it possible that my account on db disappeared?
@robjohn oh, it's still there, but I did something wrong. I was scared not for the having an account deleted but I thought my address was compromised by some hackers.
@Evinda Most of my results are not known. And I suppose there are very few persons that know to deal with them and probably no one with the stuff from my top research.
@Evinda Many are included in my book, but there are also proposed problems and articles with them.
@Evinda I don't plan to publish all I get, but just a part because some of the results can be used in many places and I prefer to first explore all their potantial. Some of my tools are extremely powerful.
@Evinda About techniques of calculating integrals, series and limits, most of them coming from personal research. It's far from being a usual book, at least if referring to the problems in it.
@Evinda I think we talked in the past a couple of times about it.
@Evinda It's what I call the art of mathematics. I mean I have nothing to do with solving in the usual sense, but I attend the art of mathematics which is far different.
While one is happy or very happy for solving a problem, I'm happy or very happy onlywhn I manage to bring the art in my solutions, to turn the problems and solutions into masterpieces.
oh, thats more than in norway, i think. if i recall correctly you have 6 courses for msc, then one year for msc thesis, then if accepted to a phd program here you need 2 courses in addition to your thesis.
(where the phd lasts for 3 years, 4 if you are hired with a year of teaching.)
@BalarkaSen Fun problem for you: Let $X$ be a connected CW complex and $Y$ be a $K(G,1)$. Then any group homomorphism $\pi_1(X)\to \pi_1(Y)$ is induced by a map $X\to Y$.
@MikeMiller So far we have essentially only covered simplicial objects and the simplicial identities, however some time in spent on analogues (i.e. do this and that and you can recover this and that which you know well.)
@BalarkaSen $G$ be a group, $\varphi:G\to G$ an automorphism. Then $\varphi$ is induced by a homotopy equivalence $f:K(G,1)\to K(G,1)$. Prove that the mapping torus $T(f)$ is a $K(G\rtimes_{\varphi} \Bbb Z,1)$.
@iwriteonbananas: The long exact sequence of a fibration is just an algebraic repackaging of the homotopy lifting property. You might as well take the LES as an exercise.
By the way, @BalarkaSen, from that exercise it follows that the group $G\rtimes_{\varphi}\Bbb Z$ has vanishing $\ell^2$-Betti numbers (those are defined as the ones of the classifying space). All $\ell^2$-Betti numbers of mapping tori are zero.
Right, sorry. The classifying space of that group is a mapping torus. It's a theorem which was proved in my course that mapping tori have vanishing l^2 betti numbers.
@MikeMiller question. how much (elementary) index theory do you know? i thought i remembered you bringing up some of it at one point, but i could be wrong
@MikeMiller I'll phrase it like this, stealing some phrases from Wikipedia's page on Fredholm operators
Let $\psi$ be a complex continuous function on the complex unit circle $\mathbb{T}$ and let $T_\psi$ denote the Toeplitz operator with symbol $\psi$, equal to multiplication by $\psi$ followed by the orthogonal projection $P$ from $L^2(\mathbb{T})$ to the Hardy space $H^2(\mathbb{T})$. Then $T_\psi$ is Fredholm on $H^2(\mathbb{T})$ iff $\psi$ has a well-defined winding number i.e. $\psi\neq 0$ on $\mathbb{T}$
Sure, I see what you're asking. I don't know a coherent description of "wall-crossing" in this context (what happens when you pass between components of the Fredholm operators).
Such a crossing can be perturbed so that you cross the other walls transversely, and the effect on the index is given by doing the others one at a time. There's not really much of a mathematical reason (if you're only interested in 1-parameter families) to try to cross there.
actually, i don't start until 1:25. so i have a few more minutes to ramble :P
i guess maybe all i'm looking for is that the 'non-Fredholm' subsets of that parameter space are 'usually' codimension 1, but it's not impossible to have sets of codimension 2
whether they matter is a different question, but they certainly can occur
Suppose I have a function $g \geq 0$ defined by
$$g(x) = \int_{-\infty}^{x}f(t)\text{ d}t \geq 0\text{, }x \in \mathbb{R}\text{. }$$
I know for a fact that $g$ is right-continuous and nondecreasing.
Is this enough to show that $f \geq 0$? If show, I'm not sure how to prove it.
[For those of ...
I have a theoretical meta-math question that I would like to ask. Since Gödels first THM states that there is no formal proof system to satisfy all of the following; 1. there is an algorithm to test if a sequence of sentences is a formal proof, 2. every true sentence is formally provable, 3. every formally provable sentence is true. How would it satisfy all three if there were only FINITELY many sentences unprovable in a system such as Peano Arithmetic?
I think your background is enough. If we denote this interval with $[a,b]$ (where $f<0$) and $λ(a,b)>0$, where $λ$ denotes the Lebesgue measure then we have $g(b)=\int_{-\infty}^b f(t)dt=\int_{-\infty}^af(t)dt+\int_{a}^b f(t)dt<\int_{-\infty}^af(t)dt+λ(a,b)max_{x\in (a,b)}f(x)<\int_{-\infty}^af(t)dt+0=g(a)$ which violates the assumption of nondecreasing. It was necessary in the above calculation that $λ(a,b)>0$. Otherwise there would be no problem. — Jimmy R.9 mins ago
Make it simpler: $$\int_{a}^b f(t)dt < \int_{a}^bmax f(t)dt=\max f(t)\int_{a}^bdt=(b-a)\max f(t)$$ So, if $b-a>0$ and $f(t)<0$ for all $t$ in $(a,b)$ this is negative. — Jimmy R.1 min ago
I wouldn't have thought to do a proof by contradiction lol
I discovered a new family of integrals and I wanted to ask you all if you know how to compute such integrals or it is time to write an article about them. Are they known with higher powers of log?
The solutions are optionalas usual.
$$\int_0^1 \frac{\text{Li}_2(x) \log ^8(1-x)}{x} \, dx$$
Do we...
Je résule tu prends une application $f$ de $R^2\times R^2$ tel que $f(x,y)=abscisse de x fois ordonnée de y. Donc pour f(x+z,y)=f((a+e,b+f),(c,d))=(a+e)d ?
Suppose I have a function $g \geq 0$ defined by
$$g(x) = \int_{-\infty}^{x}f(t)\text{ d}t \geq 0\text{, }x \in \mathbb{R}\text{. }$$
I know for a fact that $g$ is right-continuous and nondecreasing.
Is this enough to show that $f \geq 0$? If show, I'm not sure how to prove it.
[For those of ...
I discovered a new family of integrals and I wanted to ask you all if you know how to compute such integrals or it is time to write an article about them. Are they known with higher powers of log?
The solutions are optionalas usual.
$$\int_0^1 \frac{\text{Li}_2(x) \log ^8(1-x)}{x} \, dx$$
Do we...
