21,555 reputation
23382
bio website
location
age 29
visits member for 4 years, 4 months
seen 7 hours ago

May
18
comment Why can't you simulate isotropic fluid flow on a square lattice?
I'm afraid isotropy of fourth degree tensors is beyond my knowledge. I can make a conjecture: a second-order tensor being invariant under rotations of $\pi$ does not imply that it is necessarily a multiple of the identity, but invariance under rotations of $\pi/2$ does. Perhaps an analogous fact with $\pi/2$ and $\pi/3$ holds for fourth-order tensors. But you've probably thought of this already.
May
18
comment Why is convexity more important than quasi-convexity in optimization?
I wrote that poorly. It should be "use sufficient decrease criteria to ensure that the value at the next iterate $f(x_n−\alpha\nabla f(x_n))$ is (sufficiently) less than the current value $f(x_n)$."
May
18
comment Why is convexity more important than quasi-convexity in optimization?
Fromthe Wikipedia article on quasiconvex functions one can infer the advantages and disadvantages of this class of functions relative to convex ones. One I find particularly striking is that the sum of two quasiconvex functions need not be quasiconvex.
May
18
comment Why is convexity more important than quasi-convexity in optimization?
-1: This is not how gradient descent works. In short, you perform successive line searches along the ray $x_n - \alpha \nabla f(x_n)$ with $\alpha$ positive, so you never go up the gradient, and use sufficient decrease criteria to ensure that $f(x_n-\alpha\nabla f(x_n))$ is (sufficiently) less than $f(x_n)$. So you do approach the minimum even on the nonconvex function in the question.
May
18
revised Initial value problem, $dy/dt=1/y$, $y(0)=0$
corrected minor typos
May
17
comment Logarithms - How to handle factors within a difference or sum term of logarithms?
You know that $(x+1)\ln 16 = x\ln 16 + 1\ln 16$, right?
May
17
comment Curves on a circle
Why don't I have to bend them into the shape of a smiley face either?
May
17
comment Curves on a circle
Take the segments of radial lines and wiggle them a bit.
May
17
comment Curves on a circle
"[Segments of lines] aren't continuous, because they have endpoints" -- Do you mean closed curves rather than continuous curves? Curves are necessarily continuous according to the usual definition.
May
16
comment How can I show that a given system of differential equations is stiff?
Since your system is two-dimensional, you could visualize $(\dot C, \dot R)$ as a vector field over the $(C, R)$ space, which will give you a feel of the trajectories of your ODE. It looks like there is rapid descent in $C$ to a stable manifold near $\dot C = 0$, and then presumably much slower dynamics within that manifold; this is a common characteristic of stiff systems. (Also, you have a nasty singularity all along $R = 0$.)
May
16
awarded  Nice Answer
May
16
comment maximization and minimization
I came across this question when it was recently bumped to the front page, so I wrapped your chain of equivalences in an align environment. Hope that's OK.
May
16
revised maximization and minimization
formatting
May
16
comment Chameleons Riddle
Does this strategy only work when the number of variables and the number of possible actions is equal? Or, what would you do if, unluckily, your matrix was rectangular?
May
15
answered Why are there isosceles triangles?
May
15
comment Why are there isosceles triangles?
If you're ever doing a geometrical construction involving two radii of a circle with their endpoints joined by a line, you'll probably need some facts about isosceles triangles.
May
15
comment Why can't you simulate isotropic fluid flow on a square lattice?
Ah, this is not a limitation of general grid-based methods, but rather of the specific method of lattice gas automata which attempts to simulate fluids through cellular automata. As far as I am aware, the more commonly used grid-based fluid simulation methods that directly discretize the Navier-Stokes equations do not have this problem.
May
15
revised Logarithm of a Convex Function
rolled back to a previous revision
May
12
comment Interesting Recurrence Relation $T(n) = T(\sqrt{n}) + T(n-\sqrt{n}) + n$
It's really the asker's responsibility to clarify the question, but given the context of analysis of quicksort, I would guess that the actual recurrence ought to be $T(0) = T(1) = 1$ and $T(n) = T(\lfloor\sqrt n\rfloor) + T(n-\lfloor\sqrt n\rfloor) + n$ for $n \ge 2$.
May
12
comment To prove $f(x)\rightarrow \infty$ with a “home made” strategy
I don't see where in your induction step you go from $N$ to $N+1$ or something like that.