[math-fun] f(0,0)-f(0,1)-f(1,0)+f(1,1)
Is there an easy way to show that if f(x,y) is twice-differentiable on [0,1]x[0,1] with continuous second-order partial derivatives, then there must exist a point at which (d/dx)(d/dy) f(x,y) equals f(0,0)-f(0,1)-f(1,0)+f(1,1)? (I wonder if forums like math-fun, and sci.math in its heyday, are bad for a mathematician's brain. The above problem is probably a fairly standard exercise in advanced calculus, and in the old days, I would've had to wait until coffee hour to ask a colleague, by which time my own subconscious would have dredged up the needed facts and supplied the answer. Nowadays, I can just post the question and wait for someone in the 24-hour-a-day virtual coffee hour to post an answer. So my brain doesn't get as much exercise. Then again, my subconscious has more chew-toys to play with, thanks to *other* people's questions!) Jim Propp
Is there an easy way to show that if f(x,y) is twice-differentiable on [0,1]x[0,1] with continuous second-order partial derivatives, then there must exist a point at which (d/dx)(d/dy) f(x,y) equals f(0,0)-f(0,1)-f(1,0)+f(1,1)?
Let g = (d/dx)(d/dy) f(x,y).
From the Fundamental Theorem of Calculus,
df/dy (x,y) = int_0^x g dx + df/dy (0,y) Again, from the Fundamental Theorem of Calculus, f(x,y) = int_0^y df/dy dx + f(x,0) = =int_0^y (int_0^x g dx) dy + f(0,y) - f (0,0) + f(x,0) .
From here,
f(1,1)-f(0,1)-f(1,0)+f(0,0) = int_0^1 (int_0^1 g dx) dy Now, if g < the l.h.s. on the unit square, the double integral would be less than the l.h.s. Similarly, if g > the l.h.s on the unit square, the double integral would be greater than the l.h.s. Thus, from the continuity of g, there is a point at which g = the l.h.s. Alec Mihailovs
Hi all, I see that on 1 July 2008, analytic number theorist Xian-Jin Li (Math, Brigham Young University) announced a proof of RH. Interestingly enough, Li is a former Ph.D. student (Purdue, 1993) of Louis de Branges, who has been somewhat notorious of late for announcing his own "proof" of RH back in 2004, which as I understand it was never accepted by experts. See http://genealogy.math.ndsu.nodak.edu/id.php?id=16641 This initially gave me pause, but looking at Li's papers (via Google Scholar) I see that they have been published and cited by others. Most relevant, perhaps, is a paper by Bombieri and Lagarias which as I understand it generalizes an RH criterion due to Li. This isn't my field at all, but I have the initial impression that Li's work appears to fit into a community effort by many of the best mathematicians now working. Does anyone have information or comments? (I think I've seen Lagarias post here, so I'd be particularly interested comments from him.) How about some interesting consequences of RH?
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