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bio website math.ubc.ca/~israel
location Richmond, Canada
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visits member for 3 years, 10 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


1h
comment Newton's Method: Found a case where it doesn't work. Why?
... also univariate, but in the complex plane.
2h
comment Newton's Method: Found a case where it doesn't work. Why?
Is Matlab's fzero function not robust?
6h
comment This may be odd to some of you (soln to 1st ODE with NO constant of integration)
Actually you lost a factor of $1/4$. Note that $y = (x + C)^2/4 + 2x - 3$ is a solution for $x \ge -C$. For $x < -C$ you get the wrong square root.
17h
comment Fixed Point Iteration Method
Because $r^n \to 0$ as $n \to \infty$ if $|r| < 1$.
18h
comment Fixed Point Iteration Method
Which behavior?
20h
comment Mutually exclusive AND independent event (help with examples)
It depends on whom you want to talk about.
1d
comment Algebraic relations between trigonometric numbers
If $d = 1$ it says $T_n(\cos(\pi/n))=T_m(\cos(\pi/m)$.
1d
comment Algebraic relations between trigonometric numbers
No, $T_n(\cos(x)) = \cos(nx)$. See e.g. mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
1d
comment Algebraic relations between trigonometric numbers
As to the new question, in general $\cos(\pi/(n+1))$ will not be in the field generated by $\cos(\pi/(n-1))$ over the rationals, so there is no such construction.
1d
comment Algebraic relations between trigonometric numbers
More generally, if $d = \gcd(m,n)$, then $T_{n/d}(\cos(\pi/n)) = T_{m/d}(\cos(\pi/m))$.
1d
comment Algebraic relations between trigonometric numbers
If $m \mid n$ then $T_{n/m}(\cos(\pi/n)) = \cos(\pi/m)$.
1d
comment Does this operator equation have solutions?
I don't know if it's because of a "linear bias", but I don't think anti-linear operators are studied very much, certainly a lot less than linear operators. Of course, they are in one-to-one correspondence with the linear operators, in the sense that given an anti-linear involution $J$, $A$ is anti-linear iff $JA$ is linear. But the algebra is somewhat different. I don't know what an anti-linear Spectral Theorem would look like.
2d
comment Finding Eigenvalues of given linear operator
... and solve a differential equation.
2d
comment Understanding Overdetermined System
Because one $(x_1,x_2)$ value satisfies all the equations.
2d
comment about a sequence of isometries' convergency.
The $x_m$ could all be the same. If you just need the convergence to be pointwise rather than uniform, you can take them all to be the same.
2d
comment Find the $n$ for which $σ(n) = 15$
When you eliminate all other possibilities, the one remaining is unique.
2d
comment Find the $n$ for which $σ(n) = 15$
@Sasha you mean $\sigma(n) \ge n + 1$.
Jan
27
comment 'Rational' solutions of sine
$2 \sin(\pi r) = -i \exp(i\pi r) + i \exp(-i \pi r)$. $\pm i$ and $\exp(\pm i \pi r)$ are algebraic integers, being roots of unity. The sum or product of algebraic integers is an algebraic integer.
Jan
27
comment Solve for matrix that is hidden inside a scalar
Sorry, typo: I'll edit.
Jan
27
comment Eigenspaces and jordan normal form
That's the point. Once you have $A$ diagonalized, you have its eigenspaces (corresponding to blocks of indices where $D$ has each eigenvalue of $A$ as diagonal element); $TBT^{-1}$ consists of blocks on the diagonal corresponding to these, and you put each of these into Jordan form.