134,018 reputation
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bio website math.ubc.ca/~israel
location Richmond, Canada
age
visits member for 3 years, 7 months
seen 2 hours 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


2h
comment A strange 3rd order ODE
As far as numerical solutions are concerned, be careful because the solution is likely to go to $\infty$ at finite values of $x$. For example, with $c=0$ and $y(0)=1$, $y'(0) = y''(0)=0$, you get some very large values of $y$ by $x=8$ or so.
2h
comment A strange 3rd order ODE
Take as long as you want.
3h
comment Closed form of a difficult definite integral
This is the mixed algebraic-transcendental case for the Risch algorithm, which I don't think any of the usual CAS's have implemented fully. However, it's probably true that there is no closed-form antiderivative. Yes, the fact that it's a definite integral does give some hope, but I don't immediately see any way to do it using residues etc.
8h
comment Integral of a product with any continuous function which has integral 0 is equal to 0
Yes, and one of these terms is $0$...
8h
comment $X,Y$ are Banach spaces, $T$ is linear, $x_n\to 0$ and $Tx_n\to y$, then $y=0$ and $T$ is continuous.
$T(x_n - x) \to y - T(x)$. And the "suppose that..." tells us what?
17h
comment Integral of a product with any continuous function which has integral 0 is equal to 0
No, $c = \int_0^1 f(x)\; dx$. Why would that be $0$ if $f$ is continuous? And yes, it does have useful information: it identifies what linear functional on $C[0,1]$ corresponds to $g$.
18h
comment Integral of a product with any continuous function which has integral 0 is equal to 0
No, that's not not what I mean. With $c = \int_0^1 f(x)\; dx$ you would have $\int_0^1 f(x) g(x)\; dx = \int_0^1 (f(x) - c) g(x)\; dx + \int_0^1 c g(x)\; dx = ...$
18h
comment $X,Y$ are Banach spaces, $T$ is linear, $x_n\to 0$ and $Tx_n\to y$, then $y=0$ and $T$ is continuous.
Hint: if $x_n \to x$ and $T x_n \to y$, what do you know about $x_n - x$ and $T(x_n - x)$?
18h
comment Integral of a product with any continuous function which has integral 0 is equal to 0
Yes, it would. So what does that say about $\int_0^1 f(x) g(x)\; dx$?
1d
comment Divergence of series
Also $\ln(\cos(x)) = -1/2 x^2 + o(x^2)$.
1d
comment Integral of a product with any continuous function which has integral 0 is equal to 0
No. $\int f = 0$ only for the $f$ in the hypothesis.
1d
comment Integral of a product with any continuous function which has integral 0 is equal to 0
Further hint: find $c$ so that $\int_0^1 (f(x) -c)\; dx = 0$.
1d
comment Solving Simultaneous Equations - Hill Cipher
Use it in the first equation to find $c$ in terms of $a$. Then substitute into the second equation.
1d
comment Infinite series and the ratio test
... and since you mentioned the value of $\sum_{n=1}^\infty 1/n^2$, I might also note that $\sum_{n=1}^\infty \sin^2(n)/n^2 = (\pi - 1)/2$.
1d
comment Solve quadric equation system
Hm, well, $1$ or $0$. So there are also the cases $y_1^2 = y_2^2 + y_3^2$ and $y_1^2 = y_2^2$.
2d
comment Solve quadric equation system
You can also rescale so $a_1, \ldots a_4$ all have absolute value $1$. There are two nontrivial cases: $y_1^2 + y_2^2 = y_3^2 + y_4^2$ and $y_1^2 = y_2^2 + y_3^2 + y_4^2$.
2d
comment Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$
For example, $(1+1/k)^a < 2$ for $k > 1/(2^{1/a}-1)$, and $b/(k+1) < r/2$ for $k \ge 2b/r$.
Oct
15
comment Log integrals III
This is the Maple dilog.
Oct
15
comment Solve quadric equation system
OK, any nontrivial solutions.
Oct
15
comment Nested recurrence sequence with interesting properties
OEIS sequence A003605: oeis.org/A003605