allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0311032343.312d19a7@posting.google.com>...> The dispute goes > over a "simple proof of Fermat's last theorem" (what else...). Harald > Hanche-Olsen, who made the page referenced above and disputes the > existence of such a proof,Ouch! That last sentence can be misunderstood. Hanche-Olsen does not dispute Fermat's last theorem being proved, he disputes the existence of a *simple* proof. As far as I know, Wiley, who proved the theorem, used just about every known aspect of mathemathics to get every piece in place. Which probably explains why it took 350 years to prove the theorem. Rune
Minimum Phase Impulse Response
Started by ●October 29, 2003
Reply by ●November 4, 20032003-11-04
Reply by ●November 4, 20032003-11-04
Rune Allnor wrote:> Jerry Avins <jya@ieee.org> wrote in message news:<bo6q8i$f3k$1@bob.news.rcn.net>... > >>The new poles and zeros yield the same magnitude response as the >>original, but the phase is minimum. I don't remember why, but I have >>it on reliable authority (O&S?). > > > With all due respect, Jerry, I get "bad itches" by that sort of argument. > Please don't misunderstand! I think you are right and I'm not capable > of doing better myself. >Since I gave it the trappings of a proof with "Q.E.D.!", I sympathize with your itch. I should have written that visualizing my old Spirule, I can see that zeros in the right-hand s plane introduce less phase shift than those reflected into the left, and that in the z plane right becomes outside and left becomes inside. reflections about the s-plane vertical axis become reciprocal along a radius in the z plane. So much for the geometry. I used the knowledge that that reflecting a zero about the jw axis leaves the magnitude response unchanged. I cited O&S as the source, but it may be Guillemin. I am happy to use those guys' results without re-deriving them. I know how to calculate the section modulus of a beam given its shape, but there are tables for that and I use them. Jerry P.S. Digital calculators are valuable tools, but those who have never become proficient with slide rule or Spirule lack a powerful way to visualize simple solutions to otherwise complicated problems. P.P.S. I know a simple way to trisect an angle with ruler, compass, and pencil. I sometimes use it. It works well. I'm not (for this) a nut. -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 4, 20032003-11-04
Jerry Avins <jya@ieee.org> wrote in message news:<bo8mp4$3vi$1@bob.news.rcn.net>...> Rune Allnor wrote: > > > Jerry Avins <jya@ieee.org> wrote in message news:<bo6q8i$f3k$1@bob.news.rcn.net>... > > > >>The new poles and zeros yield the same magnitude response as the > >>original, but the phase is minimum. I don't remember why, but I have > >>it on reliable authority (O&S?). > > > > > > With all due respect, Jerry, I get "bad itches" by that sort of argument. > > Please don't misunderstand! I think you are right and I'm not capable > > of doing better myself. > > > > Since I gave it the trappings of a proof with "Q.E.D.!", I sympathize > with your itch. I should have written that visualizing my old Spirule, I > can see that zeros in the right-hand s plane introduce less phase shift > than those reflected into the left, and that in the z plane right becomes > outside and left becomes inside.Now that you mention it, the 2nd edition of P&M I used when I took that class did show some figures with some vectors(?) from the zeros to the unit circle. I never understood why those "vectors" had to be moved counterclockwise around the zero. (My memory may be failing me now, I'm not sure if what I remember makes sense at all). If somebody tells me it had something to do with moving in the complex plane and staying on some particular sheet of a Riemann surface, things may not be as total voodoo as they appear at the moment.> reflections about the s-plane vertical > axis become reciprocal along a radius in the z plane. So much for the > geometry. I used the knowledge that that reflecting a zero about the jw > axis leaves the magnitude response unchanged. I cited O&S as the source, > but it may be Guillemin.It was probably O&S. I haven't seen that book, but from what I hear, those guys did things the mathemathical way.> I am happy to use those guys' results without re-deriving them. I know > how to calculate the section modulus of a beam given its shape, but there > are tables for that and I use them.Of course I agree with you. My problem is that I like to understand what's going on. Sure, there are plenty of people out there who do the maths way better than I will ever be able to. There are thousands of people who code up those numerical routines faster and more efficient, in any sense of the word, than me. Still, I like to understand how the stuff works. By doing that I have a chance of finding out what's easy and what's not. Which means I can anticipate problems with either coding the routines or using the routines. And I find out who possess actual knowledge and skills, and who are merely "politicians" or "imposters", which would be very useful knowledge whenever I or my projects get under pressure.> Jerry > > P.S. Digital calculators are valuable tools, but those who have never > become proficient with slide rule or Spirule lack a powerful way to > visualize simple solutions to otherwise complicated problems.I have no knowledge of either (I never found out what a spirule is), but I still trust you on this. Not because of any authority you might have, but because of the knowledge and skills you consistently demonstrate.> P.P.S. I know a simple way to trisect an angle with ruler, compass, and > pencil. I sometimes use it. It works well. I'm not (for this) a nut.Yeah, right. I suppose the margin of your post was just too small to describe your trick?[*] Anyway, if the occation ever arises where you can show me your method face to face, I'll by you a beer. Rune [*] Yep, you caught me. According to "authorities" (some maths book, I can't remember which one) it has been proven that trisecting the angle by means of the mentioned instruments is impossible. I can't give the proof myself, I can only refer to "authority". Somehow I suspect you knew I would respond like that...
Reply by ●November 5, 20032003-11-05
Jerry Avins wrote:> P.P.S. I know a simple way to trisect an angle with ruler, compass, and > pencil. I sometimes use it. It works well. I'm not (for this) a nut.(sideways glance) Using the ruler's scale, or just the straight edge?
Reply by ●November 5, 20032003-11-05
"Eric C. Weaver" <weav@sigma.net> wrote in message news:<3fa88fcb$1@news.announcetech.com>...> Jerry Avins wrote: > > > P.P.S. I know a simple way to trisect an angle with ruler, compass, and > > pencil. I sometimes use it. It works well. I'm not (for this) a nut. > > (sideways glance) > > Using the ruler's scale, or just the straight edge?Right... it took me a couple of hours, but if I can use the scale I too can trisect the angle. With only the straight edge things can become difficult. Rune
Reply by ●November 5, 20032003-11-05
Dangerously vocal young programmer wrote:> > > P.P.S. I know a simple way to trisect an angle with ruler, compass, and > > > pencil. I sometimes use it. It works well. I'm not (for this) a nut. > > > > (sideways glance) > > > > Using the ruler's scale, or just the straight edge? > > Right... it took me a couple of hours, but if I can use the scale > I too can trisect the angle. With only the straight edge things can > become difficult. > > RuneHey Rune, Archimedes knew how to do this already, with just a straight edge and (two I think) pencilmarks. Is that hint enough? Regards, Andor
Reply by ●November 5, 20032003-11-05
Rune Allnor wrote:> Jerry Avins <jya@ieee.org> wrote in message news:<bo8mp4$3vi$1@bob.news.rcn.net>... >...>>P.P.S. I know a simple way to trisect an angle with ruler, compass, and >>pencil. I sometimes use it. It works well. I'm not (for this) a nut. > > > Yeah, right. I suppose the margin of your post was just too small to > describe your trick?[*] Anyway, if the occation ever arises where you can > show me your method face to face, I'll by you a beer. > > Rune > > [*] Yep, you caught me. According to "authorities" (some maths book, I > can't remember which one) it has been proven that trisecting the angle > by means of the mentioned instruments is impossible. I can't give the > proof myself, I can only refer to "authority". Somehow I suspect you > knew I would respond like that...Rune, I wasn't trying to catch you. We needn't be face to face; I'll describe the method so you can try it yourself. [All geometry problems that are isomorphic to quadratic or linear equations can be solved with compass and straightedge. In general, those isomorphic to cubic and higher equations can not be so solved.]* Trisection is cubic*, so more powerful tools are needed. Instead of a straightedge, I need a ruler, as I wrote. A ruler has parallel sides and an end square to them. These extra- Euclidian features play a part in the construction. I'll give you time to play with the idea before I thrust it on you. I do not withhold it now to tease. The method came to my attention in the late 1940s when it was published on the front page of the second section of the then two-section New York Times. It illustrated someone's clam to have solved the trisection problem; The Times appeared to endorse that claim. Subsequent discussion was interesting! Jerry _______________________________ * I haven't proved that, but accept it from Authority. -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 5, 20032003-11-05
Andor wrote:> Dangerously vocal young programmer wrote: > >>>>P.P.S. I know a simple way to trisect an angle with ruler, compass, and >>>>pencil. I sometimes use it. It works well. I'm not (for this) a nut. >>> >>>(sideways glance) >>> >>>Using the ruler's scale, or just the straight edge? >> >>Right... it took me a couple of hours, but if I can use the scale >>I too can trisect the angle. With only the straight edge things can >>become difficult. >> >>Rune > > > Hey Rune, > > Archimedes knew how to do this already, with just a straight edge and > (two I think) pencilmarks. Is that hint enough? > > > Regards, > AndorThat -- and Rune's way -- sounds simpler than my way. Do those ways work for large angles, say, 90 degrees +/- a little? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 5, 20032003-11-05
an2or@mailcircuit.com (Andor) wrote in message news:<ce45f9ed.0311050704.1fe1b0af@posting.google.com>... :> Dangerously vocal young programmer wrote:Yep, that's what I was getting at... ;)> > > > P.P.S. I know a simple way to trisect an angle with ruler, compass, and > > > > pencil. I sometimes use it. It works well. I'm not (for this) a nut. > > > > > > (sideways glance) > > > > > > Using the ruler's scale, or just the straight edge? > > > > Right... it took me a couple of hours, but if I can use the scale > > I too can trisect the angle. With only the straight edge things can > > become difficult. > > > > Rune > > Hey Rune, > > Archimedes knew how to do this already, with just a straight edge and > (two I think) pencilmarks. Is that hint enough?I am sure he did. Using the scale of a rule or some other length reference, it's not difficult at all. However, some problems appear to have haunted maths throughout history: - Trisecting the angle using only a straight-edge (with no scale or length refernce) and compass - Constructing a square of the same area as a circle with given diameter (or was it vice versa?) using only straight-edge and compass - Proving one of Euclid's postulates on geometry. Proving that the first two were impossible was apparently among the first main contributions of abstract algebra. The disproof of Euclid's 5th(?) postulate spawned what we now know as "non-Euclidian geometry", and with it, modern mathematics. My mistake when I (too sarcastically) flamed Jerry's post was that I didn't check the basic assumtions of his claim. Rune
Reply by ●November 5, 20032003-11-05
Rune Allnor wrote:> ... Using the scale of a rule or some other length reference, > it's not difficult at all.I don't know that one, unless trial and error is alowed. The usual way using trial and error -- called by some "successive refinement" -- is with dividers. Draftsmen do that regularly.> However, some problems appear to have haunted maths throughout history: > > - Trisecting the angle using only a straight-edge (with no scale or length > refernce) and compass > - Constructing a square of the same area as a circle with given diameter > (or was it vice versa?) using only straight-edge and compass > - Proving one of Euclid's postulates on geometry. > > Proving that the first two were impossible was apparently among the first > main contributions of abstract algebra. The disproof of Euclid's 5th(?) > postulate spawned what we now know as "non-Euclidian geometry", and with > it, modern mathematics. > > My mistake when I (too sarcastically) flamed Jerry's post was that I > didn't check the basic assumtions of his claim.Now _that_ was a trap I did lay!> > RuneJerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
