125,935 reputation
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bio website math.ubc.ca/~israel
location Richmond, Canada
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visits member for 3 years, 5 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
9
answered Quadratic System of Equations
Jul
9
answered Help find hard integrals that evaluate to $59$?
Jul
8
answered Fastest way to compare fractions
Jul
8
answered Solving for unknown inside square root
Jul
8
comment Closed Form of Normal Distribution
For example, see Risch, "The problem of integration in finite terms", Trans. Amer. Math. Soc. 139 (1969), 167-189: dx.doi.org/10.2307%2F1995313
Jul
8
answered Finding the critical points of $\sin(x)/x$ and $\cosh(x^2)$
Jul
8
comment Finding the critical points of $\sin(x)/x$ and $\cosh(x^2)$
In fact, if $z \cos(z) - \sin(z) = 0$, $f(x) = \sin(z x)$ satisfies the differential equation $f'' = - z^2 f$ with boundary conditions $f(0)=0$ and $f'(1) - f(1) = 0$. It follows from the theory of Sturm-Liouville equations that $z^2$ must be real. The case where $z^2 < 0$ (i.e. $z$ purely imaginary) is easy to dismiss.
Jul
8
comment Closed Form of Normal Distribution
By the way, that Wikipedia entry is wrong: it's not just $n$'th roots, it's roots of polynomials. For example, $z$ satisfying $z^5 + x z + 1$ is an elementary function, although it can't be expressed in terms of radicals.
Jul
8
revised Closed Form of Normal Distribution
added 395 characters in body
Jul
8
answered Closed Form of Normal Distribution
Jul
8
comment Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$
@KasunFernando: Yes, you can replace "$f$ integrable on $[0,1]$ with "$x^n f$ integrable on $[0,1]$ for some $n$". And "integrable" can be Lebesgue rather than Riemann.
Jul
8
revised Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$
added 345 characters in body
Jul
8
answered Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$
Jul
8
comment $p$ is a $3$-digit prime, then there always exist $p$ consecutive composite numbers.
A more difficult question is whether there exist $p$ consecutive composite numbers preceded and followed by primes. Without the "3 digit" restriction this is not known. With it, it is a matter for explicit computation (see trnicely.net/gaps/gaplist.html#MainTable)
Jul
7
answered Limit of $a_n$ is $0$ iff Limit of $a_n \sin(n t)$ is $0$ for all $t\in[0,1]$
Jul
7
revised Fat Tail / Large Kurtosis Discrete Distributions?
added 471 characters in body
Jul
6
answered Fat Tail / Large Kurtosis Discrete Distributions?
Jul
6
comment Prove that an odd Collatz sequence will at some point have two consecutive even numbers
What do you mean by an "odd Collatz sequence"?
Jul
6
comment Approximating a probability
Unfortunately, when $D$ is small you really can't say anything about $Q$. In fact you could have $D = 0$ if $f$ and $g$ are supported on disjoint sets (e.g. $f=0$ on the intervals $[n,n+1]$ where $n$ is odd and $g=0$ when $n$ is even), and $Q$ could be anywhere in $[0,1)$.
Jul
5
comment Why does $(10^4 - 10^2) \cdot 0.0012121212\dots = 12$?
It's not that either. $10000 - 100 = 9900$