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bio website math.ubc.ca/~israel
location Richmond, Canada
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visits member for 3 years, 1 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


2d
comment Is there any geometry where ratio of circle's circumference to its diameter is rational?
In non-Euclidean geometries it's misleading to talk about the circumference-diameter ratio, because it changes from circle to circle.
2d
comment Number of integer solutions by generating functions method
... and then use partial fractions.
2d
comment Let p be an odd prime, q the smallest quadratic non residue (mod p). Prove q is prime.
Because if $a$ or $b$ is a nonresidue, $q$ wasn't the smallest nonresidue.
2d
comment Number of integer solutions by generating functions method
Presumably $x,y,z$ are nonnegative integers?
2d
comment Is it possible to reach 50k in the game 2048 with only having a highest tile of 512?
Some explanation might help. I have no idea what you're talking about in this question.
2d
comment PDF of Sum of Two Random Variables
Further hint: rotate by 45 degrees.
2d
comment Simplification of Double Integral with Independent Parameters
If you're integrating over a rectangular domain, you can write your integral in the form $\int \ldots \; dp_1 \int \ldots\; dp_2$
Apr
20
comment Does weak convergence of $\nu_{n}$ imply convergence of $\int{f_{n}(x)d\nu_{n}(x)}$?
Presumably $\nu_n$ and $\nu$ are in $\mathcal A$?
Apr
20
comment Prove that the convergence of the sequence (sn) implies the convergence of (s3n)
It might help if you told us what the question was. And what is OH?
Apr
20
comment Why do you have to begin with the largest angle or side when using law of cosines
Perhaps it means the problem I mentioned, that $\sin(x)$ doesn't determine $x$ completely if $x$ could be more or less than 90 degrees. Or maybe it's just that having every student do the problem the same way makes grading homework and tests easier for her.
Apr
19
comment Direct product norm
In what way does $\|(v_1, v_2)\| = \|v_1\| + \|v_2\|$ not work for you?
Apr
19
comment problem with inequality of modulus
$f(t) = t/(1+t) = 1 - 1/(1+t)$ is an increasing function of $t \ge 0$.
Apr
18
comment Convergence of $\int_{0}^{+\infty}\ln(1+\frac{1}{t^2})$
As Zarrax noted, $\ln(1 + 1/t^2) = \ln(t^2+1) - 2 \ln(t)$. You could use l'Hospital on $t \ln t$ as $t \to 0+$.
Apr
18
comment Minimal polynomial: is $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$?
It depends on which cube root of the negative number $2-\sqrt{5}$ you take. Mathematica uses the principal branch. If you use the real cube root $-(-2+\sqrt{5})^{1/3}$ you do get $1$.
Apr
18
comment Why do you have to begin with the largest angle or side when using law of cosines
When you have all sides, the Law of Cosines can be used for all angles. There is no ambiguity, because $\cos(\gamma)$ is a one-to-one function on $[0,180^o]$. But if (as jnh suggested) you use the Law of Cosines for one of the angles and Law of Sines to get the others from that, you can run into ambiguity there because $90^o+x$ and $90^o - x$ have the same sine.
Apr
18
comment Erdos-Renyi Model Intuition
What exactly are you asking about? In the $G(n,p)$ model, there are $n$ nodes in the graph, and $n \choose 2$ pairs, each of which could be an edge. Each of these is an edge with probability $p$, independently.
Apr
18
comment Explanation for the uniformity of the distance between a Gaussian variable to its nearest integer?
Your integral formula is not correct: $|X - R(X)| < \delta$ is not the same as $0 < X < \delta$.
Apr
18
comment Expected distance for a gaussian variable to its nearest integer.
Any text that develops the elementary theory of Fourier series, or en.wikipedia.org/wiki/Triangle_wave
Apr
18
comment Laplace’s equation in the Polar Coordinate System
Because both give you the value of $u$ at the same point, $(a,0)$ in cartesian (rectangular) coordinates.
Apr
18
comment How to find an irrational number in this case?
$m(n)$ is the index $j$ such that $a_j = \sum_{i=1}^n 2^{-k_i}$.