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"I would never die for my beliefs because I might be wrong."

-- Bertrand Russell


3h
comment Question about solutions of $y''+(w^2+b(t))y=0$ .
What are the initial conditions for $\phi(t)$?
1d
comment Question about solutions of $y''+(w^2+b(t))y=0$ .
Shouldn't that be $\lim_{t\to \infty}$?
Nov
21
comment Probability and Quantum mechanics
Terence Tao's page on the subject is also quite nice.
Nov
21
comment Probability and Quantum mechanics
Now, it has to be said that QM is less radical than free probability. While it involves non-commuting variables, we usually still work with the ordinary notion of indepence. Therefore, you still get the usual statistical distributions like the normal distribution. Only when non-commuting variables are involved does one get stranger things.
Nov
21
comment Probability and Quantum mechanics
This is where the work of Dan Voiculescu, for instance, fits in. He studied what is called free probability in which the notion of independence of random variables is replaced by the notion of freeness, which leads to a new type of statistics. Maybe this intro of the subject can give you a feel for what it is like.
Nov
21
comment Probability and Quantum mechanics
The thing is, there are generalisations of probability theory in which we need no such thing as sample spaces, or $\sigma$-algebras. These do encompass the usual probabilistic descriptions, but they use "random variables" and "expectations" as the fundamental entities and not sample spaces, algebras and probability measures.
Nov
20
comment Exponential Function of Quaternion - Derivation
I think a natural way to extend the exponential function to quaternions would be to use the Taylor series of the exponential over the complex and just extend domain and range to include quaternions.
Nov
20
comment Why do we write proofs “forward?”
@Steve Jessop: Yeah, I also think we see "backward" and "forward" differently. I mean one works backwards w.r.t. to how the proof is written at the end.
Nov
20
comment Why do we write proofs “forward?”
@Steve Jessop: you say you do it "forward" twice, but that was precisely my point, the first time is not done forwards, but backwards, because you compute back from what you want to obtain (something must be smaller than any epsilon) and you look for a fitting delta. In the end, you write it in the forwards way: being smaller than this delta makes you smaller than that epsilon and you can find a delta for each epsilon.
Nov
18
comment How to solve an inverse of derivative ode
So we could alternatively write:$y=\phi'(y^{-c_1}+y^{-c_2})$? Or not?
Nov
16
awarded  Yearling
Oct
30
reviewed Approve suggested edit on How would one solve this system
Oct
24
reviewed Approve suggested edit on Why is it true that $∀x((-x)^2=x^2)$?
Oct
24
reviewed Reject suggested edit on Algebra. Modeling with one-variable equations and inequalities
Oct
9
awarded  Taxonomist
Oct
7
comment Do mathematicians, in the end, always agree?
Actually, scientists disagreeing about something is the tell-tale sign that you are at the frontier of research. It means that there's still something to be discovered and it does occur in mathematics too, it's just that you haven't reached those outer boundaries yet because mathematics is one of the vastest subjects there is.
Oct
4
comment how is the signum function neither continuous nor discontinuous at $x=0$.
The function is undefined at $x=0$, maybe your book requires the function to at least be defined at the point to be able to even speak about continuity/discontinuity?
Sep
30
awarded  Explainer
Sep
24
awarded  Autobiographer
Sep
20
comment When is the product $(1+1/3)\cdots(1+1/n)$ equal to an integer?
Nope, the correct answer is B. Hint, try to write every factor as one fraction.