19,250 reputation
1746
bio website math.ubc.ca/~israel
location Vancouver BC
age 64
visits member for 4 years
seen 7 hours ago
Associate Professor Emeritus, University of British Columbia, and Optimization Algorithms Researcher, D-Wave Systems, Burnaby BC

12h
comment “Epicycles” (Ptolemy style) in math theory?
For a discussion of epicycles as trigonometric series, see Shlomo Sternberg, "Celestial Mechanics vol. 1", W. A. Benjamin, 1969.
15h
comment “Epicycles” (Ptolemy style) in math theory?
You can see Stokes's original formulation (due to Kelvin) at <books.google.ca/…;. It precedes vector calculus (as formulated by Gibbs et al) by about 50 years.
19h
comment Consecutive numbers with mutually distinct exponents in their canonical prime factorization
Another $5$: $1431124, 1431125, 1431126, 1431127, 1431128$.
19h
awarded  Good Answer
20h
comment Consecutive numbers with mutually distinct exponents in their canonical prime factorization
You might also mention $1,2,3,4,5$ and $241, 242, 243, 244, 245$. Those and the ones you listed are all the $5$'s less than $10^6$, and there are no sequences of length $>5$ in this range.
1d
answered Proof that no differentiable space-filling curve exists
Mar
28
comment How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
That's a series in powers of $x/(x-1)$. And what do you get when you express it as a series in powers of $x$?
Mar
27
comment Variance under time-scaling of sample measurements
Of course, a normal random variable doesn't really make sense here, since $X$ is supposed to be nonnegative integer-valued. The question is what model the normal distribution is supposed to be approximating. A more usual model for this sort of thing would be a Poisson process, but that would have mean and variance equal (and proportional to the duration of the interval). It should be up to the authors to specify what model is being used here.
Mar
27
comment How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
Note that this series is not for the hypergeometric itself, but for $h(x)$ which is $(1-x)^K$ times the hypergeometric.
Mar
27
comment How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
I'm pretty sure these formulas, which were obtained with the help of Maple, do involve the correct use of Pochhammer symbols.
Mar
26
revised How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
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Mar
26
revised How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
added 530 characters in body
Mar
26
revised How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
added 201 characters in body
Mar
26
answered How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
Mar
24
revised Certain inverse problem related to moments
added 790 characters in body
Mar
24
answered Certain inverse problem related to moments
Mar
24
comment Is there any simpler form of this function
$F(2n+1) = c_n + d_n \sqrt{2}$ where $$c_{{m}}={\frac {- \left( 4\,{m}^{2}+4\,m \right) c_{{m-2}}- \left( -6 \,{m}^{2}-5\,m+1 \right) c_{{m-1}}}{2\,{m}^{2}+m}} $$ and $$d_n = \dfrac{n!(n+1)!}{(2n+1)!} 2^{3n+1}$$
Mar
24
comment Is there any simpler form of this function
It appears that $F(2n) = a_n + b_n \pi$ where $$a_n ={\frac {- \left( -2\,{n}^{2}+7\,n-6 \right) a _{ n-3} - \left( 9\,{n}^{2}-14\,n+7 \right) a _{ n-2 } - \left( -10\,{n}^{2}+10\,n \right) a_{ n-1} }{3 \,{n}^{2}-3\,n}} $$ while $$b_n = \dfrac{(2n+1)!}{(n!)^2 2^{n+2}}$$
Mar
20
comment quadratic matrix equation
(Correction of earlier comment) If $D$ is nonsingular, $\det(X) = \pm i$ in odd dimension, $\pm 1$ in even dimension.
Mar
20
comment quadratic matrix equation
How do you know $X$ is diagonalizable?