This question was asked me in binomial theorem chapter. I don’t know if I am doing it right. Here are my steps
Let $(1+x)^n=C_0+C_1x+C_2x^2+\cdots C_n x^n$
Let $x=1,n=2013$
$2^{2013}=C_0+C_1+C_2+\cdots C_{2013}$
Let $2^{2013}=17k+r$ from here I am lost. Help.
ah this one, yeah, you used to post these 2 years ago which is where I am now as I tried to crawl the maths chat to find the very first maths chat post of mine
Btw I have been dreaming about a lot of stuff that is a mix of integrals, recursive functions and ordinals recently. however, with the exception of the most recent one, the others are not even mathematically consistent
Outside of these dreams, I also came up with some integrals which unfortunately I have not though about how to approach them yet, but some conjectures were being proposed which will be investigated later (one of these are the tetration integrals that I tagged you and simpleart in)
However, after some random pondering and free association about the dream itself after I woke up, I then came up with something that might be a little bit more sensible:
Let $I=\int f(x)dx$. There exists functions $f(x)$ such that the following holds for all $n \in \Bbb{N}$
$$\int I^n dx =I^{n+1}$$ Ok it turns out we can do something about this: Differentiate both sides to get $$I^n =(n+1)I^n f(x)$$ We can rearrange this equation to get $$I^n (1- (n+1) f(x))= 0$$ This give us two conditions: $$I^n=0\text{ or } f(x) = \frac{1}{n+1}$$ Now consider the base case: $$f(x)=0\text{ or } f(x) = 1$$
Therefore the only consistent solution is the zero function
@Secret Happily these days I manage to bring into reality what in other times I only dreamt like you. Particularly, I'm studying some results entirely special, pretty similar to some obtained by Ramanujan in his second Notebook.
As you know, there is always some difficulty about anything related to the last results one gets. Some mystery is always present to some extent, and this is not bad at all (it's part of the fantastic flavour of mathematics).
I see. Meanwhile, as it takes qutie a bit of time for me to get number theory background solid, which is why mostly the integral investigations I am currently dealing with are mostly symbolic manipulations.
I do have a preference about tetration due to my interest in ordinals, which is why you seemed to see quite a bit of my integrals often involving a tetration integrand. However to deal with these properly, one needs good Lambert W function manipulations
Most of the time, I and my dreams came up with a lot of ideas and I don't have any idea what they mean yet, thus I will often just store in notebooks or posts bits of it so that when other users react to them, they will indirectly inform me what I can do about them
For me, my preference is rather than asking them to solve the integrals, I often just post the integral and leave it as is. Most users tend to react with them in different ways, and from that, I gained some insights on what I can do about them
This way I handle mathematical obejcts is very similar to how fine art is presented: You just leave your artwork in the venue, and the audience will came up and connect with their own interpretations
@LeakyNun So you suggests there are nontrival solutions $f$ that will satisfy that integro recurence relation?
@LeakyNun because if you said it is invalid to split it up into these two cases analogous to polynomials, it means the equation has to be dealt with as a whole, which might provide gaps for a nontrival f to solve it
@Secret Yes, that's fine, but at the same time you might like to find a way to solve the problem, a way which defines you in a particular way, creating your mathematics there, not like in books, papers, or like suggested by others on chat. That way which defines you mathematically in an unique way.
@Daminark Suppose $F : C \to \mathrm{Set}$ is a functor from a small category $C$ (ie where the hom-sets are actually sets). Then for any fixed object $A$ of $C$, natural transformations between $F$ and $\hom(A, -)$ are in one-to-one correspondence with the elements of the set $F(A)$.
I have a bit of a question about a paper i'm reading.
they define a flux vector is a vector $\mathbf{u}\in\mathbb{R}^R_{\geq 0}$ where $R$ is a finite set of reactions over $\Lambda$ which is a finite set of species.
That sometimes happens, but it is currently rare in the maths domain for me. If I recall, the only truly original stuff I have done so far is proving theorems that showed that almost all algebraic systems that involve division by zero must be nonassociative (It is original because the maths community think it is crazy to mess with pathological things like division by zero, but I am kinda a reckless rule breaker in some way. One of the things that attract me most are precisely those pathological mathematical objects that most fear to work with, because they are weird and helps to flex the mind)
Hm, maybe it's useful to think about the contravariant version (you know what contravariant functors are, right?). That says natural transformations between $F$ and $\hom(-, A)$ are in one-to-one correspondence with $F(A)$, where $F$ is a contravariant functor, ie, a functor $C^{op} \to \mathrm{Set}$ instead.
@SohamChowdhury It's also super-easy to prove. You can prove it.
(this is a paper about continuous chemical reaction networks; in other words, a system of chemicals where reactions are used to get from one state to another as a way of solving problems.)
@Secret That's OK, a good amount of courage is needed to go there and try things that most wouldn't dare or think of in mathematics. That makes a huge difference.
it then says that the support of $\mathbf{u}$ is the set $\text{supp}(\mathbf{u}) = \{\rho\in R\mid\mathbf{u}(\rho)>0\}$ where $\rho$ is a reaction $\in R$ (of which I can give details if necessary).
it says that a flux vector is applicable at a state if every $\rho \in \text{supp}(\mathbf{u})$ is applicable at the state and if (the state is $\mathbf{c}$) $\mathbf{c}(s) + \sum_\limits{\rho\in R}\mathbf{u}(\rho)\Delta\rho(s)\geq 0$ for every $s\in\Lambda$ and finally that a flux vector sequence $\mathbf{U}$ is a tuple of flux vectors (sorry for so much detail; I am unsure what's important here).
the paper then says in the main algorithm to "compute the max support flux vector sequence $\mathbf{U}_{\mathbf{c},\epsilon}$
so first: what do the two subscripts mean here? second, what is the reference to "max support" seeing as it defines no such thing as a support flux vector sequence or a max flux vector sequence or anything, just a flux vector sequence?
@Daminark Maybe I should add one thing: Suppose $F = \hom(-, B)$ for some other object $B$ of $C$. Then natural transformations between $\hom(-, B)$ and $\hom(-, A)$ are in one-to-one correspondence with $\hom(A, B)$.
Applying this to our homotopy category of CW complexes: First, a fact: $H^n(\Sigma X; G)$ is naturally isomorphic to $H^{n-1}(X; G)$ for any $n$ (you will soon prove this). Using that, you get $[\Sigma X, K(G, n)]$ is naturally isomorphic to $[X, K(G, n-1)]$. Or, $[X, \Omega K(G, n)]$ is naturally isomorphic to $[X, K(G, n-1)]$.
This gives a natural transformation $[-, \Omega K(G, n)] \to [-, K(G, n-1)]$ which is also an isomorphism. By Yoneda this should correspond to an isomorphism $\Omega K(G, n) \cong K(G, n-1)$ in the homotopy category (which is precisely a homotopy equivalence), if I am not wrong.
@heather Without reading the paper in detail, I am guessing you have a sequence of flux vectors (which is stored in $\mathbf{U}$) for a given state $\mathbf{c}$ (thus that might explain the subscript $\mathbf{c}$). So the max support flux vector sequence probably means to maximise $\mathbf{U}$ given $\mathbf{c}$ under some parameters (given by the $\epsilon$ subscript). But I felt like we might need to read that paper in more detail to deduce what it is doing
Yeah, it definitely should. Yoneda should say $\text{Nat}(\hom(-, B), \hom(-, A)) \cong \hom(A, B)$ is not only a bijection but sends "natural isomorphisms" to isomorphisms.
Bashing out the proof explicitly should do this.
What I'm worried about is I'm not really working on a category, but a homotopy category, where "eveything is coherent upto homotopy". I guess I can handwave this off.
@Daminark yeh peter may is doing everything backwards man
@Secret Just to be aware of one more thing: one needs a huge (hard to measure) investment in mathematics to get some results, some sacrifice that needs good consideration, to be sure you want to do it. I mean not just going half a way, that means a lot of your time was wasted.
@Secret sometimes the results simply do not come, not that fast, and maybe you even need to crawl at some point.
It took me 5 months to prove those theorems I stated to you earlier, and I expected when I get my nonassociative algebra and cateogry theoric background solid, the rest would definitely took many years to prove as division by zero enters the nonassociative regime
@MikeMiller I am right in saying that in the homotopy category of CW complexes, $[Z, X] \cong [Z, Y]$ for all objects $Z$ implies $X$ is homotopy equivalent to $Y$, right? I guess I just want to know if the Yoneda lemma works in the "homotopy localization" or something.
The interesting thing about that division by zero algebra personal project of mine is its nature means it cannot fail (the only way for it to fail is when it is shown to be equivalent to a halting problem without oracles, which I highly doubt it will happen since the axiomatic systems are quite consistent)
Currently the direction suggests at least one nonasssociative division by zero algebra exists. If we can find that, everything we knew about division by zero will be revised
But of course, the alternate outcome is that we instead found a series of no-go theorems ruling out the existence…
My philosophy of answering any questions thrown by people or nature itself is to find the most perfect answer possible, an answer that once given, is already all that is need to know thus closing the question completely.
(btw, for those who are unfamilar with the terminologies I made up for my research, zero terms are basically things like x/0, 0x, x0, 0x0, x0y 0^n etc. that does not collapse to 0)
But anyway, without at least a solid abstract algebra background, I cannot continue my analysis on nonassociative systems, which is why the project is temporary put on hold
Yeah, ok, so I can construct a map $\Omega K(G, n) \to K(G, n-1)$ but that involves the representability too. Basically choosing the identity element from $H^{n-1}(\Omega K(G, n); G)$ and representing that. Then that's a homology isomorphism, which gives a weak homotopy equivalence, I think.
The only thing is that it's part-time, the lady said I applied a little late and that the program was full, but that I could be admitted for part-time. i said alright and she said okay you're in. :}
But I think it'll be good, I haven't been in a school day program in 4 years.
@Astyx so basically at first it was "sorry the program will be full by the time we review your application"
then "well you could apply to part time it will cost 50 bucks"
turns out it costs 145 bucks so i said "well i don't want to pay 145 bucks"
so then she said "i'll email you back tonight" she never emailed me. I called her today and she said "good news, if you want to do part-time I can admit you right now" and I said "alright!" and now I am a Western Mustang.
Hi, I'm trying to find a question I saw here a week or so ago. It was about three 3-dimensional points and showing whether or not they form a right-angled triangle
V: Finally get the standalone program to work, thus I can finally get the answers for a certain section of the paper after 2-3 weeks of lack of progress) X: The H2Se calculation still refuses to converge. Upon advice of my peers, I think I will file a error report to the help centre of the software soon
Meanwhile, I need to figure out how to reorganise my too maths saturated brain to back to chemistry, so I can read through the literature faster to prepare my proposal
Hi, could anyone help me with the calculation of the cohomology with compact support of R^n, as done in Hatcher's book on page 244? I dont understand why $H^i(\mathbb R^n, \mathbb R^n \setminus B)$ is zero for $i=0$. To me it seems that $H^i(\mathbb R^n, \mathbb R^n \setminus B) = H^i(S^n)$, so $H^0(\dots)=\mathbb Z$.
I need to find a factorization of both $f_{1}=2x^{2}+4x+6$ and $f_{2}=2x^{2}+4x-6$ into a product of irreducible elements of $\mathbb{Q}[x]$.
I already was able to do so in the case of $\mathbb{Z}[x]$:
$f_{1}=2(x^{2}+2x+3)$
$f_{2}=2(x-(-3))(x-1)$
Note that for $f_{1}$, $x^{2}+2x+3$ has no r...
Let $\varphi:A\to B$ be a ringhomomorphism. Show that the induced map $$^a\varphi:Spec(B)\to Spec(A),\ \mathfrak{p}\mapsto \varphi^{-1}(\mathfrak{p})$$
satisfies
$$\overline{^a\varphi(V(\mathfrak{b}))} = V(\varphi^{-1}(\mathfrak{b}))$$
for any ideal $\mathfrak{b}$ in $B$. Here, for a ring $A$ we ...
Sorry for bothering@BalarkaSen, but I wanted to know. My uni is giving category theory with just some basic intro to topology. So should I need to do a bit more topology before category theory?
Differential Geometry question here: If I want to follow the flow of two vector fields $X$ and $Y$ over a manifold, but $[X, Y] \neq 0$, for instance trying to reduce two different varieties of error, $\mathcal L_X Y$ can be used here to account for that, right?
nvm, I just broke my chatjax plugin at some point.
I'm still trying to wrap my head around exactly what the lie derivative $\mathcal L_X Y$ is in this case.
Actually, the example I gave of two different sources of error commutes, ignore that.
Or this question that was on the exam I had to supervise today: find $a$ such that the following integral vanishes: $$\int_0^\infty x^3 e^{-ax} \sin(x) dx $$
This question was asked me in binomial theorem chapter. I don’t know if I am doing it right. Here are my steps
Let $(1+x)^n=C_0+C_1x+C_2x^2+\cdots C_n x^n$
Let $x=1,n=2013$
$2^{2013}=C_0+C_1+C_2+\cdots C_{2013}$
Let $2^{2013}=17k+r$ from here I am lost. Help.
I need to find a factorization of both $f_{1}=2x^{2}+4x+6$ and $f_{2}=2x^{2}+4x-6$ into a product of irreducible elements of $\mathbb{Q}[x]$.
I already was able to do so in the case of $\mathbb{Z}[x]$:
$f_{1}=2(x^{2}+2x+3)$
$f_{2}=2(x-(-3))(x-1)$
Note that for $f_{1}$, $x^{2}+2x+3$ has no r...
