James S. Cook
Reputation
9,633
Top tag
Next privilege 10,000 Rep.
Access moderator tools
 1m reviewed Approve About the boundary conditions of the Black-Scholes-Merton PDE 3m comment reoder basis vectors to get 'more diagonal' representation of NxM matrix I see, well, I guess you need to decide what is first then sort through your columns and select all the first columns and put them at the start of your new list. Then, look for what comes next according to the order as indicated by the pixelation scheme. Put all those second.... keep sorting until you have them all in a nice tidy list. 17h comment Eigenvalues of Matrix Product. $det(AB) = det(A)det(B)$ and $det(M) = \lambda_1 \cdots \lambda_n$ so there is a relationship. If $AB=BA$ then they share e-vectors, with possibly different e-values... there is much to say here. 19h comment reoder basis vectors to get 'more diagonal' representation of NxM matrix see pages 174-175 of supermath.info/LinearNotes2015.pdf essentially the idea is to choose the inverse image of the range basis to start your domain basis then stick the basis for the kernel at the end. This gives you an identity matrix padded with zeros. 23h reviewed Reject Is this a valid example of a non-euclidean Sierpinski attractor? 23h comment Does a mathematical construct exists which explains all theories? @sasha category theory is very general, if I understand correctly, it includes just about everything you can think of, or, at least gives a language in which those things can be reformulated. As to complexity theory, I don't know details... but, I'd be surprised if could not be fit... I'll leave the details to those who have studied these things properly :) 1d comment Does a mathematical construct exists which explains all theories? As the years go on, more and more physics is decided in computer simulations. Some even go so far as to say physics should be computer simulations (this is roughly what Wolfram says in the big book). This much we can say, physics is incomplete thus any existence theorem about "where" physics fits has to deal with the inevitable ignorance of physics. 1d comment Calculus of Variations transformation I usually assume this is because we are implicitly working on some jet-space where $y'$ is not really the derivative of $y$ with respect to $x$. This is relevant to variational calculus where we often take derivatives with respect to $y'$ and for example $f(x,y,y') = y$ would have partial derivative w.r.t. $y'$ of zero; $\frac{\partial f}{\partial y'} = 0$. Sort of like, but even more frustratingly in my experience, the independence of $z$ and $\bar{z}$ when those are used as a complex notation for real derivatives... 2d awarded Yearling 2d comment Sanity check: smooth structure of tangent bundle The set $TU$ described below is open in the bundle. I can't see using smaller sets than $T_pM$ since over a given point $p \in M$ you want to allow for all possible $v \in T_pM$ in $(p,v)$. Using an open set smaller than all of $T_pM$ would put some tangent vectors outside the chart of the tangent bundle. That would defeat much of the purpose of studying $TM$. So, we use adapted charts. Jul25 comment Sanity check: smooth structure of tangent bundle good news, you have not lost your mind. Jul25 answered Sanity check: smooth structure of tangent bundle Jul25 comment What are some applications of stochastic processes and advances probabilities in real world? maybe stochastic electrodynamics. If this is not stochastic then it's a really bad name for it. Jul25 comment Finding a Gradient Vector given only a derivative and direction @Vlad just the point in question, but, since we are given the direction and magnitude for $(\nabla f)(P)$ neither the formula for $f$ nor the details of $P$ particularly matter. That was my knee-jerk read of this. Jul25 comment Finding a Gradient Vector given only a derivative and direction the rate of change is largest in the direction of the gradient and that maximum value is the magnitude of the gradient. Thus, multiply your unit vector by $2 \sqrt{3}$ and that's the gradient. Jul24 comment What am I working with? [Inferring a theory in Category Theory using associativity of Cartesian Product] I don't know enough to properly answer your question, but, this much is easy $((a,b),c)$ corresponds to $(a,(b,c))$ and this gives us the bijective correspondence of $(A \times B) \times C$ and $A \times (B \times C)$. That said, I'm not sure how you define $A \times B \times C$. I usually think of it as both $(A \times B) \times C$ and $A \times (B \times C)$. Anyway, I'm sure someone will help with this interesting question. Jul24 comment Determine the null space of a linear map well, these are polynomials and a bunch of differentiation. Basically, use calculus and attack directly if I had to guess. Jul24 comment Question on vector fields along maps (need a quick sanity check) @a student often it is the case that if $U$ is small enough then $f$ restricted to $U$ is injective, but, if $f$ is constant then we'd have an insurmountable problem. Fortunately, the condition that the derivative of $f$ at $p$ be nonzero is sufficient to give a local inverse. It's funny this comes up again, just the other day I answered math.stackexchange.com/a/1367605/36530 and I think you'll find Jack Lee's answer useful there as well. Jul23 comment Are there identities for combinations of traces of products of four Gell-Mann matrices? I don't understand the formatting of your question. Maybe if you provided more details, even if just links to wikipedia articles with specifics, we might be able to say something... Jul23 reviewed Approve How can a = x (mod m) have multiple meanings in modular arithmetic?