127,205 reputation
584218
bio website math.ubc.ca/~israel
location Richmond, Canada
age
visits member for 3 years, 5 months
seen 1 min 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


17m
comment Initial value problems with known solutions?
You could look at e.g. google.ca/… and references there.
28m
comment Tiling squares with L-Trominoes
... and of the $9 \times 6$ rectangle.
8h
comment Continuity in finding eigenvectors
But it will have complex eigenvalues and eigenvectors. Who asked for real ones?
9h
comment Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?
Yes, sorry, I forgot some $t$'s: $$e^{t\lambda} e^{tN} = e^{t\lambda} \left(I + t \dfrac{N}{1!} + t^2 \dfrac{N^2}{2!} + \ldots + t^{n-1} \dfrac{N^{n-1}}{(n-1)!}\right)$$
17h
comment Prove that there are infinitely many primes $P_i\equiv1\pmod6$
If there are only finitely many primes $p_1, \ldots, p_n \equiv 5 \mod 6$, consider $K = p_1 \ldots p_n + 6$ (if $n$ is odd) or $p_1^2 \ldots p_n + 6$ (if $n$ is even). Then $K \equiv 5 \mod 6$, and is not divisible by any of $p_1, \ldots, p_n$.
17h
comment Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?
What's the problem with it?
18h
comment Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?
An $n \times n$ Jordan block is of the form $B = \lambda I + N$, where $N^n = 0$. Then $$e^{tB} = e^{t\lambda} e^N = e^{t\lambda} \left(I + \dfrac{N}{1!} + \dfrac{N^2}{2!} + \ldots + \dfrac{N^{n-1}}{(n-1)!}\right)$$
19h
comment Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?
Yes, if $A$ is diagonalizable.
19h
comment Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?
That should be $\sum_{n=0}^\infty$.
23h
comment What's the sum of this series?
Which do you want, to estimate the sum or find the exact value?
1d
comment Math issue implementing an invoice API
Not at all clear what calculations are taking place here. In any case it's not a mathematical issue; it's a question of how to deal with particular software, or maybe a question of tax law.
1d
comment Does this PDE Have Two Different Solutions?
Yes, there seems to be an error in your transformation of the equation. If you do it right you should get the equation in my answer.
1d
comment What is the bound on the error that is given by Taylor's inequality?
There are several different forms for the error in Taylor's theorem (Lagrange form, Cauchy's form, integral form). All are correct. Which one are you using?
1d
comment A reference for a simple lemma on positive solutions of ODE
... and the reference is to Gronwall's inequality: see en.wikipedia.org/wiki/Gronwall%27s_inequality which lists the source as Gronwall, Thomas H. (1919), "Note on the derivatives with respect to a parameter of the solutions of a system of differential equations", Ann. of Math. 20 (2): 292–296
1d
comment Sum of random variable
And yes, of course Fubini is used to make a double integral into an iterated integral.
1d
comment Sum of random variable
Of course if it is not absolutely continuous, there is no such formula involving densities.
1d
comment Sum of random variable
The event $X + Y \le z$ is the same as $(X,Y) \in \{(x,y): x + y \le z\}$. The probability measure corresponds to the joint distribution of $X$ and $Y$. That it is absolutely continuous with respect to two-dimensional Lebesgue measure is exactly (thanks to Radon-Nikodym) the statement that $X$ and $Y$ have a joint density $f_{XY}$ (which I am assuming because you mentioned $f_{XY}$).
2d
comment What happens to a function when it is undefined?
@Jona: $\sqrt{2}$ is not "infinite". It is a perfectly good real number. The fact that it can't be exactly represented in your calculator is irrelevant to questions of definition. On the other hand, it is true that in numerical calculations you have to be very careful with expressions in which you take the difference of two numbers that are very close to each other, causing a large relative error. That is an entirely separate question.
2d
comment finite sum of cosines
Watch out for the case where $k$ divides $n$.
2d
comment polynomials and minimality
You could say that $1 + x + x^4$ is the minimal polynomial of a certain root $r$ of $x^{15}-1$ in some extension field of ${\mathbb Z}_2$. Such an $r$ would satisfy $1 + r + r^4 = 0$. It would not satisfy $1 + r^3 + r^4 = 0$, so $1+x^3+x^4$ would not be a minimal polynomial of $r$.