122,996 reputation
478206
bio website math.ubc.ca/~israel
location Richmond, Canada
age
visits member for 3 years, 4 months
seen 1 hour 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


7h
comment Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?
Maple gave me a parametric solution which simplifies to $$\left\{ a={{\it \_Z1}}^{4}+2\,{{\it \_Z1}}^{3}+4\,{{\it \_Z1}}^{2}+3 \,{\it \_Z1}+3,b={{\it \_Z1}}^{2}+{\it \_Z1}+2,c=1+{\it \_Z1},d={\it \_Z1} \right\} $$
1d
comment Optimum set partitioning with constraint
The question is nontrivial if $\sum_{i \in A} i \ge m$. For example with $m=5$ and $A = \{1,2,3,4\}$, an optimal solution is $\{1,3\},\{2\},\{4\}$.
1d
comment Choosing random marbles until one is divisible by $X$
The way you stated the problem, $X$ is always $1$: choose one marble and its number will certainly be divisible by $1$. If that's not what you mean, please explain what you do mean.
1d
comment Solution of the Legendre's ODE using Frobenius Method
No. You can take $a_1$ to be anything. The recurrence then tells you want $a_3$ is in terms of $a_1$, what $a_5$ is in terms of $a_3$, etc.
2d
comment Radius of convergence of power series
@Libertron That is indeed what it means. The radius of convergence $\ge r$ because the function is analytic in $\{z: |z - z_0| < r\}$, and $\le r$ because $|f(z)| \to \infty$ as $z$ approaches the pole, so you conclude it is exactly $r$.
Jul
19
comment Find all positive integers $n$ such that sum of digits of $2^n$ is equal to $n$.
Yes, but the effect on most of the digits is very small.
Jul
18
comment Find all positive integers $n$ such that sum of digits of $2^n$ is equal to $n$.
BTW you might also mention some reasons the digits are not independent: their sum mod 9 is known, as is their alternating sum mod 11.
Jul
18
comment Find all positive integers $n$ such that sum of digits of $2^n$ is equal to $n$.
We don't. The most we can do is estimate the expected number of additional cases.
Jul
17
comment $n$-to-$1$ near zero of holomorphic function
Do you mean, a holomorphic function that has a zero of order $n$ is $n$-to-one near that zero? That follows from the argument principle.
Jul
17
comment A definite integral involving $\exp(-a\cosh x)$
You should be able to take the first derivative wrt $b$, divide by $b$, and then take the limit as $b \to 0$ to get the case $n=1$.
Jul
17
comment How to prove that some set is a Borel set
It's not quite true that "it is not possible to characterize a Borel set". See en.wikipedia.org/wiki/Borel_hierarchy
Jul
16
comment Find all positive integers $n$ such that sum of digits of $2^n$ is equal to $n$.
The other part of the Borel-Cantelli lemma says that if $E_n$ are independent events with $\sum_n P(E_n) = \infty$, then almost surely infinitely many of them occur.
Jul
15
comment Does the series $\sum_{n=1}^\infty \frac{n!}{n^n}e^n$ converge?
Not using the root test. The terms don't go to $0$, so ...
Jul
14
comment How to find all possible values of Ɵ for the equation cos4Ɵ=0
Not only there...
Jul
13
comment What are the properties of the set of the Real Numbers without the Integers?
Any finite ant can step over the missing number without even noticing it.
Jul
13
comment How to find all possible values of Ɵ for the equation cos4Ɵ=0
Do you know where $\cos(t) = 0$?
Jul
13
comment What's this number called and what are its properties?
OK, so let's give it a name: "user132181's number".
Jul
11
comment Why is $\sum_{i=0}^{n}1=(n+1)$?
You aren't "adding nothing", you are adding $1$ for each value of $i$ from $0$ to $n$.
Jul
11
comment Matrices and algebra
The Maple command is: product((x^3+1)/(x^3-1), x=2..infinity);
Jul
9
comment Matrices and algebra
According to Maple, $$\prod_{x=2}^\infty \dfrac{x^3+1}{x^3-1} = \dfrac{3}{2}$$