On 5/31/14, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, May 30, 2014 at 4:28 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
In answer to Andy: it shouldn't matter which compactification is used --- so long as we don't forget about the necessity. "Or some other ..." --- trick question? There aren't any others!
Not true; the Stone-Cech compactification is much, much, larger than RP2.
Or without using such heavy machinery, consider the sequences (0,n) and (1, n). In a compactification, they each have to have a convergent subsequence. But there's no reason that these two convergent subsequences have to converge to the same point, as they do in the two compactifications you mention.
Andy
Compactness is all very well, but practical applications are liable to involve "regions" extending to infinity. Having earlier casually postponed consideration on the grounds that any old completion should solve the problem, I must now admit that it's not so simple. For a start, as Andy pointed out, there are quite a few choices. A principle which I elucidated early in life --- I believe it was on the occasion I was first exposed to the serial trumpet abuse of Miles Davis --- is that the potential for an activity does not automatically imply a recommendation to engage in it. Numerous other instances since observed include: dropping a 5lb hammer on one's foot; credit default swaps; and Stone–Čech c*mp*ct*f*c*t**n. Indeed, it's not clear that any of these alternatives has much to offer from the standpoint of computational geometry, besides the classical duo that in my innocence I was relying on. However, Andy's suggestion that (0, n) and (1, n) might be assigned different limits proves prescient, as this example involving a rectangular hyperbola demonstrates (it was time to get away from unit-circle counterexamples) --- Consider region R = { (x, y) | x >= 0 & y >= 0 } and functions (A) f = x*y + 1 ; or (B) f = x*y - 1 . In case (A) there are no zeros in R , in case (B) there are plenty; however neither projective nor complex boundary points at infinity can distinguish them. [Other examples might involve some complicated singularity at infinity.] However, I have an elementary workaround which looks intuitively plausible, at least in simple non-compact situations. Apply the (relaxed) intrinsic criterion with respect to THREE different coordinate frames, say: (x, y) ; (y, x) ; (x+y, x-y) . Courtesy of its asymptotes, an interior hyperbola may evade detection in two frames; but it cannot then also hide in the third. What constraints on a non-compact region & boundary would stand this up? And just how does one go about proving something like this anyway? Fred Lunnon On 6/2/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Yesterday I proposed the intrinsic criterion for real 2-space zero-detection below. Nobody has yet shot it down --- which may well of course be because nobody has actually read it --- so I shall press on, making the optimistic assumption that it involves no grossly obvious errors.
<< Theorem: Given a continuously differentiable function f(x, y) , and a compact region R of the plane ( |R^2 ) with f nonzero on the boundary of R : f has a zero within R if and only if there is some (intrinsic critical) point (x, y) in R where *** df/dx = f = 0 ***.
Proof (offered tentatively): By the implicit function theorem, the constraint f(x, y) = 0 defines a function x(y) single-valued in any interval of y where df/dx <> 0 . Such an interval meeting R is finite since R is compact, and its endpoints lie in the interior of R since f <> 0 on the boundary; therefore df/dx = 0 at those endpoints. The converse is trivial. QED.
On reflection, "region" is used rather loosely above: in fact, it seems that all that is required is that R is compact. If so, the condition cannot be relaxed in the same way that Dan did for the extrinsic version --- to f being merely C^0 continuous --- since that depended on connectivity. Which does smell just a little bit fishy ...
Fred Lunnon
On 6/1/14, Dan Asimov <dasimov@earthlink.net> wrote:
[I sent this 1.5 hours ago, but through some glitch it still hasn't shown up in my Inbox.]
CORRECTION:
Where I wrote "In R^n, let W be a compact connected (n-1)-dimensional submanifold with connected boundary M = bd(W)," the correct dimension of W should be n (not n-1), as corrected below.
Also, the affiliation mentioned is no longer current.
--Dan
----- Hmm, Why wouldn't C^0 (just continuous) work.
In R^n, let W be a compact connected n-dimensional submanifold with connected boundary M = bd(W).
Claim: ------ If a continuous function f: W -> R^1 (= the reals) for which 0 lies in f(W) but not in f(M), then f must have an absolute extremum in int(W) = W - M.
Proof à la Latto: ----------------- Since f is continuous on the compact set W, it must have both an absolute minimum and an absolute maximum on W. Since f is nonzero on the connected set M, f must take M into either (0,oo) or (-oo,0). If the former, f must have a local minimum in int(W); if the latter f must have a local maximum in int(W). [ ]
Corollary: ---------- If f is also differentiable on int(W), then it has a critical point there (since any local extremum x of a function differentiable in a neighborhood of x must have a critical point at x). -----