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bio website math.ubc.ca/~israel
location Richmond, Canada
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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


May
9
comment Measurable Functions as a Limit of Continuous Functions
I see no problems with the use of Tietze here, though you might be able to get away without using it.
May
9
comment Measurable Functions as a Limit of Continuous Functions
@JTian: For each $n$, take a continuous function $g_n$ with $|g_n - f_n| < 1/n$ on $E_n \subset A \cap B_n$ where $m(A \cap B_n - E_n) < 1/n$ ...
May
9
revised Continuity of Parametric Integral
added 169 characters in body
May
9
answered Continuity of Parametric Integral
May
9
comment Is it true that a fourier transform of $f$ never vanishes if the translates of $f$ is $L^1(\mathbb{R})$
The point is that $L^1$ contains functions $g$ such that $\hat{g}(\xi_0) \ne 0$, and so those can't be in $V_f$.
May
9
comment Measurable Functions as a Limit of Continuous Functions
Hint: if $m(A) = \infty$, write $A = \bigcup_{n} (A \cap B_n)$ where $B_n$ is a ball of radius $n$. If $|f| = \infty$ on a set of positive measure, consider "cutoff" functions that take the values $\pm n$ where $f = \pm \infty$.
May
9
comment Finding $\int x^xdx$
These identities for $\int_0^1 x^{-x}\ dx$ and $\int_0^1 x^x\ dx$ are sometimes called the "sophomore's dream". Look that up on Wikipedia.
May
9
comment Finding $\int x^xdx$
And slightly more generally, $\int_0^1 x^{r-zx}\ dx = \sum_{n=1}^\infty \dfrac{z^{n-1}}{(r+n)^n}$ for $r > -1$.
May
9
answered Checking diagonalizability of a given $2\times 2$ matrix
May
9
comment Power Series Definition
Note that when you get to Laurent series centred at $a$ (i.e. series in powers of $z-a$ allowing both positive and negative integer powers), the region of convergence might be everything outside a disk centred at $a$, or between two concentric disks (an annulus).
May
9
answered Rotation $x \to x+a \pmod 1$ of the circle is Ergodic if and only if $a$ is irrational
May
9
answered This second order Equation
May
9
comment Sequence generation
Also note that e.g. if $k$ and $j$ are odd, $\lfloor \dfrac{jk-1}{2} \rfloor = \dfrac{jk-1}{2}$, otherwise $\lfloor \dfrac{jk-1}{2} \rfloor = \dfrac{jk-2}{2}$.
May
9
comment Sequence generation
Your code seems to be for producing a single element of the table, rather than a whole row. Which do you really want? Also you're using the same index variable i in two nested loops. Do you really want to do that?
May
9
comment expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$
Of course, that the series is asymptotic to, rather than convergent to, the integral is separate from the question of whether the series has a nonzero radius of convergence. I would think that the series for $V'(u)$ has a finite radius of convergence, perhaps around $4$ or $5$ from looking at the first $20$ coefficients, and then the factor $\Gamma(j+1/2)$ would make your series have radius of convergence $0$.
May
8
comment expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$
Note that if $p$ is a half-integer, the integral will have a nonzero imaginary part due to the intervals where $\sin(t)$ is negative. The sum of a convergent series with real coefficients would not...
May
8
answered expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$
May
8
comment eigenvector computation
@JohnSmith: You still think that? e.g. $C = \pmatrix{a & 0\cr 0 & b\cr}$ with $a, b \ne 0$, $X = \pmatrix{1/a & 1/a\cr 0 & 1/b\cr}$ (diagonalizable with eigenvalues $1/a$ and $1/b$), $CX = \pmatrix{1 & 1\cr 0 & 1\cr}$ (not diagonalizable).
May
8
answered What are some examples of a mathematical result being counterintuitive?
May
8
comment eigenvector computation
Just take your favourite non-diagonalizable matrix for $C X$, and multiply it on the left by a diagonal full-rank $C^{-1}$ to get $X$; for most $C$ the result will be diagonalizable.