130,825 reputation
587223
bio website math.ubc.ca/~israel
location Richmond, Canada
age
visits member for 3 years, 6 months
seen 31 secs ago

I'm an Emeritus Associate Professor of Mathematics at University of British Columbia and an Optimization Algorithms Researcher at D-Wave Systems in Burnaby BC


8h
comment For all $\xi \in \mathbb{C}$ we have $e^{-\pi\xi^2}=\int_{-\infty}^\infty \! e^{-\pi x^2}e^{2\pi ix\xi}\ \mathrm{d}x.$
... using the Cauchy Integral Theorem with estimates on integrals over two line segments to justify this complex change of variables.
8h
comment 2nd order nonlinear ODE question
A lot of this sort of thing goes on in trying to solve differential equations: write down a general form and see what has to be true to make it work.
10h
comment Calculating Expected Value and Variance Given Random Variable Distributions.
Your calculation of $E(X^2)$ is wrong: $(-2)^2 = 2^2$, not $-(2^2)$. Also notice that $(100 X)^2 = 100^2 X^2$, not $100 X^2$.
11h
comment Difference between identity and diagonal covariance matrices
The identity matrix is a diagonal matrix where all the diagonal entries happen to be equal. In this case it means the random variables (predictors) all have the same variance.
12h
comment Effect of the nature of noise on the spectrum of a random matrix
I'll assume this means the largest in absolute value, since the eigenvalues are complex in general. This still doesn't specify which one, though, since in the case of $X$ when the eigenvalues are not real they are complex conjugates of each other, and both members of a complex conjugate pair have the same absolute value.
14h
comment Effect of the nature of noise on the spectrum of a random matrix
An $n \times n$ matrix has $n^2$ entries. If each entry is random, the covariance matrix should be $n^2 \times n^2$, not $n \times n$. Do you mean that the matrix entries of $\eta_1$ and $\eta_2$ are independent and Gaussian with mean $0$ and covariance $\sigma^2$?
14h
comment Are there particular techniques to find the general formula for an arithmetic function, neither multiplicative nor additive?
Is your function supposed to be multiplicative? Then concentrate on the prime powers. These "regularities" don't tell you anything about most values of the function.
18h
comment Matrix norm square properties.
As I commented in response to 5xum's answer, you need quantifiers. My guess is that the statement is "For every $A$ there exist $\alpha$ and $\beta$ such that for all $x$, ...". With $\alpha = 0$ the first inequality is true, yes.
19h
comment Limit Point of Zero Points Implying a Zero Function
It is a very well-known theorem, found in any decent complex analysis text. Follows easily from considering the Taylor series of $f$ about $c$.
1d
comment Matrix norm square properties.
Obviously you can't take just any $\alpha$ and $\beta$. Quantifiers are important!
1d
comment Matrix norm square properties.
Hint: It wouldn't be true if you required $0 < \alpha$.
1d
comment Solve Identity about Combination
Just expand these out and solve two equations for the coefficients of $n$ and $n^2$.
1d
comment Strategy In auction bidding
You still haven't addressed the problem. Let's take a simple situation. Suppose you know for sure that a bid of $x$ will get you the last lot, and nothing lower will. You can either take it or leave it. What do you do? Presumably there is some maximum $x$ that you would be willing to spend. But you haven't told us what this is.
2d
comment Strategy In auction bidding
The two parts of your objective are in conflict. At the lowest possible price, the probability of acquiring a lot is nearly $0$. If you bid more, you will increase the probability of acquiring one, but it will not be at the lowest possible price. There is a tradeoff. You have not given us any information that might allow us to resolve this.
2d
comment Are primitive row stochastic matrices diagonalizable?
Try $$\pmatrix{1/2 & 1/2 & 0 & 0\cr 0 & 0 & 1/2 & 1/2\cr 1/2 & 0 & 0 & 1/2\cr 0 & 1/2 & 1/2 & 0\cr}$$ where eigenvalue $0$ has algebraic multiplicity $2$ and geometric multiplicity $1$.
2d
comment Are primitive row stochastic matrices diagonalizable?
The basic methodology was to select the basic form of the matrix subject to parameters, and adjust the parameters to make the characteristic polynomial have discriminant $0$.
Sep
19
comment If A is a matrix, what does A' mean?
Transpose, not transposition.
Sep
18
comment What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\rm J_{0}}\left(2\,n\right)} ^{2}}{{n}^{2}}}$?
That's not a closed form, it's a numerical approximation.
Sep
18
comment What does the Fundamental Theorem of Algebra say about…
That's what I said (counting multiplicity). If you count only distinct zeros, then no. For example, $x^2$ has degree $2$ and only one zero.
Sep
18
comment Random variable $X^2$ determined by moments
Thanks, I'll edit