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bio website math.ubc.ca/~israel
location Richmond, Canada
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visits member for 3 years, 6 months
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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


Jul
31
comment For which angles we know the $\sin$ value algebraically (exact)?
It's really not so bad if you don't try to put it in one expression. Here's how Maple can optimize it. $$\eqalign{t_1 &= 5^{1/2}\cr t_4 &= (30+6 t_1)^{1/2}\cr t_9 &= (2 t_4(1-t_1)+28-4 t_1)^{1/2}\cr t_{11} &= (-1+t_1+t_4+i t_9)^{1/3}\cr t_{13} &= t_{11}^2\cr t_{22} &= (128+32 t_{11}+2 t_4 t_{13}+2 t_1 t_{13}-2 t_{13}- 2 i t_9 t_{13})^{1/2}\cr t_{25} &= (32-2 t_{22})^{1/2}\cr \text{result} &= t_{25}/8 \cr}$$
Jul
31
awarded  Nice Answer
Jul
31
comment continued fraction to rational polynomial in maple?
Use > normal(...);
Jul
31
answered Solving $5^n > 4,000,000$ without a calculator
Jul
30
revised Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?
added 26 characters in body
Jul
30
comment Sampling a combination randomly
@EmreA: in the calculation of $Z$, you need to count the $M$-element sets $A$ that contain a given element $i$. Such a set is the union of $\{i\}$ and an arbitrary set of $M-1$ elements chosen from the $N-1$ elements that are not $i$.
Jul
30
revised Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?
added 119 characters in body
Jul
30
answered Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?
Jul
30
answered Show the result of the following infinite sum, based on a binomial random variable conditioned on a Poisson random variable
Jul
30
comment Distribution of the sum of independent r.v.
Of course, there's no guarantee that the integral can be found in closed form, even in the case of normal distributions. Thus if $X_1$ and $X_2$ have standard normal distributions, the probability is $$ \frac{\sqrt{2}}{4 \sqrt{\pi}} \int_{c-a}^b e^{-y^2/2} \left( \text{erf}(a/\sqrt{2}) - \text{erf}((c-y)/\sqrt{2})\right)\ dy$$ which, as far as I know, does not have a closed form in general.
Jul
30
comment For which angles we know the $\sin$ value algebraically (exact)?
If $x/\pi$ is an algebraic irrational, $\sin(x)$ is transcendental by the Gelfond-Schneider theorem.
Jul
30
comment For which angles we know the $\sin$ value algebraically (exact)?
Actually there are others besides $m \pi/n$, e.g. $\sin(\arctan(1/2))=1/\sqrt{5}$. You may find this a bit of a cheat though, since the angle is specified using inverse trig functions.
Jul
30
comment For which angles we know the $\sin$ value algebraically (exact)?
@ja72 : $\sin(3 \pi/8) = \sqrt{2+\sqrt{2}}/2$
Jul
30
answered Prove trigonometry identity for $\cos A+\cos B+\cos C$
Jul
30
awarded  Cleanup
Jul
30
revised Scalar product equals weighted sum of projection of the vectors onto the edges of a simplex
added 11 characters in body
Jul
30
revised Scalar product equals weighted sum of projection of the vectors onto the edges of a simplex
rolled back to a previous revision
Jul
30
comment LP relaxation for ILP\IP (integer linear programming)
Maybe I should try again: it may be too confusing to have the objective appearing in the constraint. Minimize $x+y$ subject to $x+2y \ge 3$, $x \ge 0$, $y\ge 0$. The optimal solution of the LP is $x=0$, $y=1.5$, for an objective value of $1.5$. Therefore there can't be a solution of the ILP with $x+y < 1.5$, so $1.5$ is a lower bound on the objective value for the ILP. In fact the optimal solution of the ILP is $x=0,y=2$ for objective value $2$.
Jul
30
comment LP relaxation for ILP\IP (integer linear programming)
@user36774: The objective is $2x$, so $x=2$ has an objective value of $4$.
Jul
30
answered How to conceptualize conditional expectation inductively?