Re: [math-fun] NeilB's latest eye pickle
First time I recall the canceling-out Fred speaks of is noticing that a new refrigerator motor in my family home was quite audible at first, but over time became imperceptible. Until the thermostat would turn it off — at which moment someone might say, sincerely, "What was that?!" When I was away at college my parents moved to another town nearby. My first time home it was quite noticeable that a plane would fly overhead every 6 minutes. Pretty soon, whenever I was there, the planes became imperceptible — unless I thought about them, in which case they were suddenly there. ((( A different perceptual phenomenon is demonstrated by the short film De Duva (The Dove), at < https://www.youtube.com/watch?v=8X2QmLWWxq4 >, which is based on Ingmar Bergman films. I recommend not reading anything at that URL but just starting the video right away at full screen size. ))) Okay, so this frequency-canceling works for visual and auditory stimuli. What about taste, smell, and touch? ——Dan Fred wrote: ----- A quick search for "moving bar illusion" failed to find anything obviously relevant to RWG's impressively effective post. So I'll point out that it demonstrates very neatly the way our perceptive mechanisms filter out constant background frequencies, allowing through only changes in the signal spectrum --- visual, auditory, tactile, olfactory(?!). In this case, the subsequent apparent stretching of the text is patently opposite to that induced by the video, as our internal DPU continues to cancel out the assumed previous motion. I am reminded of an incident late one night when standing beside the Victorian clock in the hall of my home: completely unable to hear its normal loud tick, I inspected the pendulum to check whether it had stopped. As I watched in some bafflement, the tick gradually became perceptible, accompanied by the realisation that it had actually been present throughout. -----
I recently read that every sufficiently large integer in Z+ is representable as a linear combination — with coefficients in N_0 := {0,1,2,...} — of the first n primes P_n := {p_1,...,p_n} . (Actually this holds for relatively prime integers.) Let f(n) denote the largest integer *not* expressible as an N_0 combination of the primes in P_n with all coefficients nonnegative. I've also read there is no known expression for f(n). So: Is there a simple asymptotic expression for f(n) as n -> oo ??? ——Dan
f(1) ? If p_1 = 2, then f(2+) = 1 [2(k+1)+3*0, 2k+3*1] If p_1 = 3, then f(2) = 7, [3(k+3)+5*0, 3(k+1)+5*1, 3k+5*2] f(3+) = 4, [above, 7*1, 3*2, 5*1] Am I confused, or did I completely misunderstand what you've written? --ms On 14-May-15 23:57, Dan Asimov wrote:
I recently read that every sufficiently large integer in Z+ is representable as a linear combination — with coefficients in N_0 := {0,1,2,...} — of the first n primes
P_n := {p_1,...,p_n} .
(Actually this holds for relatively prime integers.)
Let f(n) denote the largest integer *not* expressible as an N_0 combination of the primes in P_n with all coefficients nonnegative.
I've also read there is no known expression for f(n).
So: Is there a simple asymptotic expression for f(n) as n -> oo ???
——Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Besides Mike and Andy, a third kind soul also pointed out to me the triviality of my question. It goes without saying that . (I had just run into the question of what is the largest number that is not a natural combination of two relatively prime natural numbers A and B, and the rest is history stupid.) ——Dan P.S. Hmm, I guess it might be interesting to ask the same question after replacing P_n := {p_1,...,p_n} by the infinite set of all primes but the first n of them: Q_n := {p_(n+1),...,p_(n+k),...}. P.P.S. My mathematical thinking has been heedless, headlong, and willy-nilly at least since 6th grade, when after my pleading, my elementary school teacher promised to teach me algebra if I got 3 consecutive perfect scores on our weekly arithmetic tests. I never managed to do so, solely because of careless error.
On May 15, 2015, at 7:09 AM, Mike Speciner <ms@alum.mit.edu> wrote:
f(1) ?
If p_1 = 2, then f(2+) = 1 [2(k+1)+3*0, 2k+3*1]
If p_1 = 3, then f(2) = 7, [3(k+3)+5*0, 3(k+1)+5*1, 3k+5*2] f(3+) = 4, [above, 7*1, 3*2, 5*1]
Am I confused, or did I completely misunderstand what you've written?
--ms
On 14-May-15 23:57, Dan Asimov wrote:
I recently read that every sufficiently large integer in Z+ is representable as a linear combination — with coefficients in N_0 := {0,1,2,...} — of the first n primes
P_n := {p_1,...,p_n} .
(Actually this holds for relatively prime integers.)
Let f(n) denote the largest integer *not* expressible as an N_0 combination of the primes in P_n with all coefficients nonnegative.
I've also read there is no known expression for f(n).
So: Is there a simple asymptotic expression for f(n) as n -> oo ???
——Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Habituation, or adaptation. Seen in humans, prairie dogs, and amoebas. I had a third-grade friend whose parents raised chinchillas. They had a strong smell, but my friend and his family had habituated and could not notice the smell, even after I pointed it out. They, in turn, may have remarked on the funny smell at my house. I chose to believe they were reacting defensively. There was no funny smell at my house. On Thu, May 14, 2015 at 6:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
First time I recall the canceling-out Fred speaks of is noticing that a new refrigerator motor in my family home was quite audible at first, but over time became imperceptible. Until the thermostat would turn it off — at which moment someone might say, sincerely, "What was that?!"
When I was away at college my parents moved to another town nearby. My first time home it was quite noticeable that a plane would fly overhead every 6 minutes. Pretty soon, whenever I was there, the planes became imperceptible — unless I thought about them, in which case they were suddenly there.
((( A different perceptual phenomenon is demonstrated by the short film De Duva (The Dove), at < https://www.youtube.com/watch?v=8X2QmLWWxq4 >, which is based on Ingmar Bergman films. I recommend not reading anything at that URL but just starting the video right away at full screen size. )))
Okay, so this frequency-canceling works for visual and auditory stimuli. What about taste, smell, and touch?
——Dan
Fred wrote:
----- A quick search for "moving bar illusion" failed to find anything obviously relevant to RWG's impressively effective post.
So I'll point out that it demonstrates very neatly the way our perceptive mechanisms filter out constant background frequencies, allowing through only changes in the signal spectrum --- visual, auditory, tactile, olfactory(?!).
In this case, the subsequent apparent stretching of the text is patently opposite to that induced by the video, as our internal DPU continues to cancel out the assumed previous motion.
I am reminded of an incident late one night when standing beside the Victorian clock in the hall of my home: completely unable to hear its normal loud tick, I inspected the pendulum to check whether it had stopped. As I watched in some bafflement, the tick gradually became perceptible, accompanied by the realisation that it had actually been present throughout. ----- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Tinnitus is a supposedly rather common condition which was devastating to me when I developed it. Yes, one can and does learn to ignore it but it’s unquestionably the elephant in the room.
On May 14, 2015, at 9:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
First time I recall the canceling-out Fred speaks of is noticing that a new refrigerator motor in my family home was quite audible at first, but over time became imperceptible. Until the thermostat would turn it off — at which moment someone might say, sincerely, "What was that?!"
When I was away at college my parents moved to another town nearby. My first time home it was quite noticeable that a plane would fly overhead every 6 minutes. Pretty soon, whenever I was there, the planes became imperceptible — unless I thought about them, in which case they were suddenly there.
I have never studied this topic in any formal fashion. My own (mild, high-frequency) tinnitus becomes noticeable only when I listen to music live, or on high-quality audio. This and further observation suggests that (in particular) your auditory channel is equipped with a sophisticated filter / Fourier analyser which detects when a particular frequency band is missing from the input, and increases its sensitivity to compensate. Paradoxically then, if for some reason you have developed narrow-band deafness to that frequency, the sensitivity may become cranked up to the point where it is reproducing random noise --- maybe even (positive) feedback, in some sense? Fred Lunnon On 5/15/15, Hans Havermann <gladhobo@teksavvy.com> wrote:
Tinnitus is a supposedly rather common condition which was devastating to me when I developed it. Yes, one can and does learn to ignore it but it’s unquestionably the elephant in the room.
On May 14, 2015, at 9:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
First time I recall the canceling-out Fred speaks of is noticing that a new refrigerator motor in my family home was quite audible at first, but over time became imperceptible. Until the thermostat would turn it off — at which moment someone might say, sincerely, "What was that?!"
When I was away at college my parents moved to another town nearby. My first time home it was quite noticeable that a plane would fly overhead every 6 minutes. Pretty soon, whenever I was there, the planes became imperceptible — unless I thought about them, in which case they were suddenly there.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I can hear mine anytime I am reminded of it, but, like Hans, I have learned to ignore it much of the time. I recall an abstract confirming Fred's final paragraph, that when hair cells in the cochlea are damaged, the corresponding brain circuits "go into a loop" "searching" for the non-existent signal. I found an abstract relating tinnitus to phantom pain in missing limbs (http://www.pnas.org/content/95/17/10340.short). I have read about "the mirror cure" for phantom limb pain (e.g. http://www.scientificamerican.com/article/the-mirror-cure/). Can anyone devise a corresponding "cure" for tinnitus, assuming that tinnitus and phantom limb pain have similar causes and assuming that the mirror cure for phantom pain works? On Fri, May 15, 2015 at 9:48 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I have never studied this topic in any formal fashion. My own (mild, high-frequency) tinnitus becomes noticeable only when I listen to music live, or on high-quality audio.
This and further observation suggests that (in particular) your auditory channel is equipped with a sophisticated filter / Fourier analyser which detects when a particular frequency band is missing from the input, and increases its sensitivity to compensate.
Paradoxically then, if for some reason you have developed narrow-band deafness to that frequency, the sensitivity may become cranked up to the point where it is reproducing random noise --- maybe even (positive) feedback, in some sense?
Fred Lunnon
On 5/15/15, Hans Havermann <gladhobo@teksavvy.com> wrote:
Tinnitus is a supposedly rather common condition which was devastating to me when I developed it. Yes, one can and does learn to ignore it but it’s unquestionably the elephant in the room.
On May 14, 2015, at 9:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
First time I recall the canceling-out Fred speaks of is noticing that a new refrigerator motor in my family home was quite audible at first, but over time became imperceptible. Until the thermostat would turn it off — at which moment someone might say, sincerely, "What was that?!"
When I was away at college my parents moved to another town nearby. My first time home it was quite noticeable that a plane would fly overhead every 6 minutes. Pretty soon, whenever I was there, the planes became imperceptible — unless I thought about them, in which case they were suddenly there.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I've had tinnitus in one ear for almost 25 years, alas. But while it began as a low whistling sound that would come and go, it soon morphed into a quiet "whoosh" that I'm usually unaware of. A funny thing about this whitish noise when I pay attention to it: With any tune I can more or less recall mentally, I can "will" it to play on my white noise system — and then I hear it. It's like having a free iPod. ——Dan
On May 15, 2015, at 3:38 PM, Jeff Caldwell <jeffrey.d.caldwell@gmail.com> wrote:
I can hear mine anytime I am reminded of it, but, like Hans, I have learned to ignore it much of the time. I recall an abstract confirming Fred's final paragraph, that when hair cells in the cochlea are damaged, the corresponding brain circuits "go into a loop" "searching" for the non-existent signal. I found an abstract relating tinnitus to phantom pain in missing limbs (http://www.pnas.org/content/95/17/10340.short).
I have read about "the mirror cure" for phantom limb pain (e.g. http://www.scientificamerican.com/article/the-mirror-cure/). Can anyone devise a corresponding "cure" for tinnitus, assuming that tinnitus and phantom limb pain have similar causes and assuming that the mirror cure for phantom pain works?
On Fri, May 15, 2015 at 9:48 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I have never studied this topic in any formal fashion. My own (mild, high-frequency) tinnitus becomes noticeable only when I listen to music live, or on high-quality audio.
This and further observation suggests that (in particular) your auditory channel is equipped with a sophisticated filter / Fourier analyser which detects when a particular frequency band is missing from the input, and increases its sensitivity to compensate.
Paradoxically then, if for some reason you have developed narrow-band deafness to that frequency, the sensitivity may become cranked up to the point where it is reproducing random noise --- maybe even (positive) feedback, in some sense?
Fred Lunnon
On 5/15/15, Hans Havermann <gladhobo@teksavvy.com> wrote:
Tinnitus is a supposedly rather common condition which was devastating to me when I developed it. Yes, one can and does learn to ignore it but it’s unquestionably the elephant in the room.
On May 14, 2015, at 9:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
First time I recall the canceling-out Fred speaks of is noticing that a new refrigerator motor in my family home was quite audible at first, but over time became imperceptible. Until the thermostat would turn it off — at which moment someone might say, sincerely, "What was that?!"
When I was away at college my parents moved to another town nearby. My first time home it was quite noticeable that a plane would fly overhead every 6 minutes. Pretty soon, whenever I was there, the planes became imperceptible — unless I thought about them, in which case they were suddenly there.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (6)
-
Dan Asimov -
Dan Asimov -
Fred Lunnon -
Hans Havermann -
Jeff Caldwell -
Mike Speciner