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Feb
9
comment What general mobius transformation maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$.
Thank you, that is helpful. Another question: how do you know to choose $D$ and $G$ as you have chosen? It seems a little arbitrary to me. For example, how do you know not to to choose the interior of the two circles in the domain? And why is $G$ chosen to be the space between the two circles in the range?
Feb
9
comment What general mobius transformation maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$.
Thanks for this. The mathematics makes sense to me but how do you motivate such an approach? For example, how did you know we needed to consider a transformation $h$ from $E$ to $F$?
Feb
8
comment If $T$ is an orthogonally diagonalizable linear operator in an inner product space, show that $T^*$ is also orthogonally diagonalizable.
@Omnomnomnom But aren't the $PDP^*$ meant to be understood as matrices in such a representation? Or as the composition of linear operators? In which case, it could also be written $P \circ D \circ P^*$?
Feb
7
comment Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space. Prove $\exists \{f_1,\ldots,f_n\}: (e_i,f_j)=\delta_{ij}$
Does this really show they exist or just that $(A^T)^{-1}$ would have the $f_i$ as columns if they did exist?
Feb
1
comment Let $V$ be the space of complex polynomials on $[0,1]$. Is the differentiation operator self-adjoint?
@Qidi, I have revised my question to clearly define $D$. I don't think this aligns with your suggestion, does it? Where can I find that definition of the differential operator, which you say is the usual convention?
Feb
1
comment Let $V$ be the space of complex polynomials on $[0,1]$. Is the differentiation operator self-adjoint?
Integration by parts gives me $\langle Df, g \rangle = \sum\sum a_i \overline b_j - a_0 \overline b_0 - \int_0^1 f \frac{\partial \overline g}{\partial t} dt$. Does that give us something useful?
Jan
31
comment Suppose $T$ is diagonalizable in $\mathbb{C}$. Show $e^T = \sum_{\lambda \in sp(T)} e^\lambda P_\lambda$ is the matrix exponential series.
Thanks for the hint. Is there a reason to do this over the other suggested answers which use the series representation of the exponential function?
Jan
13
comment Let $f(x) = (x^n-1)/(x-1)$. Why does $f(1)=n$?
See my edit to the original post. $f$ is defined as a polynomial so I think you are right in that in order for $f$ to be a polynomial, it must be defined that $f(1)=n$.
Jan
3
comment Let $f(z) = \frac{z^{-2}}{\sin( \pi z )}$. What is the residue for $z \neq 0$?
Thank you for the clarification. Understood
Jan
2
comment Let $f(z) = \frac{z^{-2}}{\sin( \pi z )}$. What is the residue for $z \neq 0$?
Could you expand on how you know the value of the coefficient in your second to last sentence from the right side of the equality? I don't know how to perform a Laurent expansion on your result.
Dec
18
comment What is the statistical steady state of this poisson process?
Thanks, that would be helpful so that I can review your answer again in more detail.
Dec
18
comment What is the statistical steady state of this poisson process?
Also, I'm a little unsure of how you derived the balance equation?
Dec
18
comment What is the statistical steady state of this poisson process?
how did you arrive at the mass functions as functions of time "i", $\lambda / (\lambda + i \mu)$ and $i \mu / (\lambda + i \mu)$? I don't have this in my answers.
Dec
18
comment What is the statistical steady state of this poisson process?
@A.S. I'd like to understand both your approaches. How do you arrive at a rate of change of $n$ people of $\lambda - n \mu$? On your first comment, what is the equation that signifies zero change of probability density? If this is the density of the number of people on the island, how do you find that density from the given processes?
Dec
16
comment Suppose $X_{n+1} = 2X_n - X_{n-1} + Z_n$. What is the value function and backward equation to calculate $P(X_T > 1 \mid X_0 = X_1 = 0)$.
@A.S. that makes a lot of sense. Thanks for the remark. Although does that get us anywhere in terms of understanding how to use a value function and backward equation to solve the question? I'm doing this as practice and would like to understand the question using the method as stated. Appreciate your remarks though.
Dec
13
comment Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? (heat equation)
I notice your answer also looks like a convolution. I have another idea where if we write $B_t = B_0 + (B_t - B_0)$ where $B_0 \sim u_0$, we can use the independent increments property and the fact that the distribution of the sum of independent random variables is equal to the convolution of their densities. This gives us another probabilistic approach, though yours is much more general/flexible.
Dec
13
comment Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? (heat equation)
Thanks for the clarification. This makes sense to me now. The probabilistic approach is helpful to see and much more informative than just solving the heat equation using standard techniques from PDE.
Dec
13
comment Suppose $dX_t = a(X_t) dt + b(X_t) dW_t$ and $Y_s=X_t$ where $s=t^2$. What SDE does $Y_s$ satisfy in the weak sense?
@zhoraster you could be right. I will edit the question accordingly.
Dec
13
comment Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? (heat equation)
I know what the dirac delta function is but how does one interpret the notation $\delta_x(dz)$ in the integrand? And can you explain further what you mean by "mix the densities $(p_t^x)_{x \in \mathbb{R}^d}$"?
Dec
6
comment Is the determinant of $A$ is equal to the product of its eigenvalues for vector spaces over any field?
@Baloown. Thank you for the answer. What is a spectra? Additionally, in the comments to my question, Servaes says "It holds for vector spaces over any algebraically closed field". Yet you said it only holds in $\mathbb{C}$. These seem to be at odds. Which one is true?