Rick Lyons <R.Lyons@_bogus_ieee.org> wrote: (snip, I wrote)>>When you touch something, it is the wave function of the electrons >>in your finger interacting with the electrons in the object you >>are touching. Pauli exclusion keeps them from overlapping.(snip regarding complex numbers)> It's sounds like you may be a physicist, or at > least have a strong background in physics.> If so, I have a seemingly super-silly question > for you. Do modern physicists know what is > an electron?Hmm. Seems to me that instead of knowing what it is, physicists know what it does. Classically, an electron was a small ball of charge. Fairly often, that is enough of an explanation. That explains capacitors and, without knowing the mechanism of friction, resistors. For inductors, you need the magnetic field and its interaction with electrons, and you get that from special relativity. If you consider electrons (balls of charge) boiling off a hot cathode, you can explain vacuum tubes, and so electronics at the beginning of the last century.> For example, if I tell my brother that electrons > from his car battery flow through the starter motor > and make the car engine start turning, and if he > asks me, "What are electrons?", what should I say?That some atoms hold onto their electrons stronger than others is enough to explain batteries, and magnetic induction (see above) the starter motor. But you need quantum mechanics to explain solid state physics, and especially semiconductors. First you find that the classical "electrons run into atoms" model for resistance doesn't quite work, but then you have to explain holes. Inside metals (and semiconductors), you have to think about electrons as quantum mechanical waves. No more little balls of charge, just solutions to wave equations with a periodic potential. For one, see: http://physics.stackexchange.com/questions/10800/how-can-the-hall-effect-ever-show-positive-charge-carriers Even more, much of the same math used in DSP also works for wave equations in solids, especially Fourier transforms. An important difference is that in solids you have to do the transform in 3D. In quantum mechanics, frequency space (usually omega, saves having lots of 2*pi in the equations) multiplied by hbar (that 2*pi again) gives the energy. If you read the above linked explanation, you will find that group velocity, which corresponds to group delay in signal processing, is important. The dispersion function for electrons in a crystal is not linear, but is actually sinusoidal. http://en.wikipedia.org/wiki/Dispersion_relation Notice, though, that all that is left is a description of what an electron does. It satisfies the solutions to certain equations. That is, in the end, all you can say about one. -- glen
real valued impulse response
Started by ●August 15, 2014
Reply by ●August 26, 20142014-08-26
Reply by ●August 26, 20142014-08-26
On Tue, 26 Aug 2014 19:12:51 +0000 (UTC), glen herrmannsfeldt <gah@ugcs.caltech.edu> wrote:>Rick Lyons <R.Lyons@_bogus_ieee.org> wrote: > >(snip, I wrote) > >>>When you touch something, it is the wave function of the electrons >>>in your finger interacting with the electrons in the object you >>>are touching. Pauli exclusion keeps them from overlapping. > >(snip regarding complex numbers) > >> It's sounds like you may be a physicist, or at >> least have a strong background in physics. > >> If so, I have a seemingly super-silly question >> for you. Do modern physicists know what is >> an electron? > >Hmm. Seems to me that instead of knowing what it is, >physicists know what it does. > >Classically, an electron was a small ball of charge. Fairly >often, that is enough of an explanation. That explains capacitors >and, without knowing the mechanism of friction, resistors. > >For inductors, you need the magnetic field and its interaction >with electrons, and you get that from special relativity. If you >consider electrons (balls of charge) boiling off a hot cathode, >you can explain vacuum tubes, and so electronics at the beginning >of the last century. > >> For example, if I tell my brother that electrons >> from his car battery flow through the starter motor >> and make the car engine start turning, and if he >> asks me, "What are electrons?", what should I say? > >That some atoms hold onto their electrons stronger than others >is enough to explain batteries, and magnetic induction (see above) >the starter motor. > >But you need quantum mechanics to explain solid state physics, >and especially semiconductors. First you find that the classical >"electrons run into atoms" model for resistance doesn't quite work, >but then you have to explain holes. > >Inside metals (and semiconductors), you have to think about electrons >as quantum mechanical waves. No more little balls of charge, just >solutions to wave equations with a periodic potential. For one, see: > >http://physics.stackexchange.com/questions/10800/how-can-the-hall-effect-ever-show-positive-charge-carriers > >Even more, much of the same math used in DSP also works for wave >equations in solids, especially Fourier transforms. An important >difference is that in solids you have to do the transform in 3D. >In quantum mechanics, frequency space (usually omega, saves having >lots of 2*pi in the equations) multiplied by hbar (that 2*pi again) >gives the energy. > >If you read the above linked explanation, you will find that group >velocity, which corresponds to group delay in signal processing, >is important. The dispersion function for electrons in a crystal >is not linear, but is actually sinusoidal. > >http://en.wikipedia.org/wiki/Dispersion_relation > >Notice, though, that all that is left is a description of what >an electron does. It satisfies the solutions to certain equations. >That is, in the end, all you can say about one. > >-- glenHi glen, thanks a lot for your detailed and cogent explanation. It sounds like a meaningful description of an electron depends on what you're doing with that electron. I was afraid you'd reply with, "Rick. Electrons are like women. Nobody understands them." [-Rick-]
Reply by ●August 26, 20142014-08-26
On Tue, 26 Aug 2014 10:19:13 -0700 (PDT), makolber@yahoo.com wrote: [Snipped by Lyons]>regarding the physicality of complex numbers.... if you replace the word "complex" with "2 dimensional" then it doesn't seem so mysterious.. > >In fact I do have an oscope that will display a "complex" or 2 dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and put the ocsope in XY mode. > >MarkHi Mark, right you are. I've used those exact ideas in my writing about quadrature (I/Q) processing. The great mathematician and electrical engineer Charles Steinmetz, who pioneered electrical power generation and distribution for General Electric (GE) in the late 1800s, did NOT like the term "complex numbers." He preferred to call them "general numbers." [-Rick-]
Reply by ●August 26, 20142014-08-26
Rick Lyons <R.Lyons@_bogus_ieee.org> wrote: (snip)> thanks a lot for your detailed and cogent > explanation. It sounds like a meaningful > description of an electron depends on what > you're doing with that electron.> I was afraid you'd reply with, "Rick. Electrons > are like women. Nobody understands them."When I was in grad. school, we had weekly group seminars. There was one subject that we were specifically not allowed to talk about, as my advisor knew we didn't understand. -- glen
Reply by ●August 27, 20142014-08-27
On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com wrote:> On Tuesday, August 26, 2014 12:32:12 PM UTC-4, Rob Gaddi wrote: > > > On Mon, 25 Aug 2014 19:53:21 -0700 > > > > > > Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote: > > > > > > > > > > > > > On Mon, 25 Aug 2014 20:43:20 +0000 (UTC), glen herrmannsfeldt > > > > > > > <gah@ugcs.caltech.edu> wrote: > > > > > > > > > > > > > > [Snipped by Lyons] > > > > > > > > > > > > > > > >When you touch something, it is the wave function of the electrons > > > > > > > >in your finger interacting with the electrons in the object you > > > > > > > >are touching. Pauli exclusion keeps them from overlapping. > > > > > > > > > > > > > > > >Who believes that wave functions are complex, and who that it should > > > > > > > >be two separate real functions? Remember, unlike phasors and voltages, > > > > > > > >it is the absolute square of the wave function that gives the > > > > > > > >probability density. > > > > > > > > > > > > > > > >-- glen > > > > > > > > > > > > > > Hi glen, > > > > > > > It's sounds like you may be a physicist, or at > > > > > > > least have a strong background in physics. > > > > > > > > > > > > > > If so, I have a seemingly super-silly question > > > > > > > for you. Do modern physicists know what is > > > > > > > an electron? > > > > > > > > > > > > > > For example, if I tell my brother that electrons > > > > > > > from his car battery flow through the starter motor > > > > > > > and make the car engine start turning, and if he > > > > > > > asks me, "What are electrons?", what should I say? > > > > > > > > > > > > > > Thanks, > > > > > > > [-Rick-] > > > > > > > > > > > > > > > > > > > Little pieces of electricity. That is a statement that, while being > > > > > > almost entirely wrong, conveys everything that you actually wanted to > > > > > > say. > > > > > > > > regarding the physicality of complex numbers.... if you replace the word "complex" with "2 dimensional" then it doesn't seem so mysterious.. > > > > In fact I do have an oscope that will display a "complex" or 2 dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and put the ocsope in XY mode. > > > > MarkWhy stop at 2? Why not use quaternions http://en.wikipedia.org/wiki/Quaternion
Reply by ●August 28, 20142014-08-28
On 8/27/14 5:07 AM, gyansorova@gmail.com wrote:> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com wrote: >> >> regarding the physicality of complex numbers.... if you replace the word "complex" with "2 dimensional" then it doesn't seem so mysterious.. >>but complex numbers are *not* merely ordered pairs of real numbers. there is more to them than that.>> In fact I do have an oscope that will display a "complex" or 2 dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and put the ocsope in XY mode.same issue.> > Why stop at 2? Why not use quaternions > > http://en.wikipedia.org/wiki/Quaternionhey, why not N-dimensional "complex numbers". better define how operations like multiplication get done. and, if one of those 4 dimensions is the "real axis", then the rules of those operations should default to the existing real operations when the other 3 dimensions have a value of 0. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●August 28, 20142014-08-28
On Thu, 28 Aug 2014 12:24:56 -0400, robert bristow-johnson wrote:> On 8/27/14 5:07 AM, gyansorova@gmail.com wrote: >> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com >> wrote: >>> >>> regarding the physicality of complex numbers.... if you replace the >>> word "complex" with "2 dimensional" then it doesn't seem so >>> mysterious.. >>> >>> > but complex numbers are *not* merely ordered pairs of real numbers. > there is more to them than that. > >>> In fact I do have an oscope that will display a "complex" or 2 >>> dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and >>> put the ocsope in XY mode. > > same issue. > > >> Why stop at 2? Why not use quaternions >> >> http://en.wikipedia.org/wiki/Quaternion > > hey, why not N-dimensional "complex numbers". better define how > operations like multiplication get done. and, if one of those 4 > dimensions is the "real axis", then the rules of those operations should > default to the existing real operations when the other 3 dimensions have > a value of 0.I'm not sure what you're trying to get at, Robert. You seem to start by asking why not an arbitrary number of dimensions for "complex numbers", then you finish off the sentence talking about "those four dimensions". At any rate, as to your stipulation for the real part, real numbers, complex numbers, quaternions and octonions all follow that rule. As for N-dimensional "complex numbers", if you stipulate that they must be a certain kind of algebra or group (I can't remember which -- I think they must be a Lie algebra), then there are exactly four kinds possible: reals, complex, quaternions, and octonions. Hamilton, in fact, was stuck for years trying to extend the notion of complex numbers to three dimensions, and only got unstuck when he realized that he must represent the operations with four components, not three. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●August 28, 20142014-08-28
On 8/28/14 1:55 PM, Tim Wescott wrote:> On Thu, 28 Aug 2014 12:24:56 -0400, robert bristow-johnson wrote: > >> On 8/27/14 5:07 AM, gyansorova@gmail.com wrote: >>> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com >>> wrote: >>>> >>>> regarding the physicality of complex numbers.... if you replace the >>>> word "complex" with "2 dimensional" then it doesn't seem so >>>> mysterious.. >>>> >>>> >> but complex numbers are *not* merely ordered pairs of real numbers. >> there is more to them than that. >> >>>> In fact I do have an oscope that will display a "complex" or 2 >>>> dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and >>>> put the ocsope in XY mode. >> >> same issue. >> >> >>> Why stop at 2? Why not use quaternions >>> >>> http://en.wikipedia.org/wiki/Quaternion >> >> hey, why not N-dimensional "complex numbers". better define how >> operations like multiplication get done. and, if one of those 4 >> dimensions is the "real axis", then the rules of those operations should >> default to the existing real operations when the other 3 dimensions have >> a value of 0. > > I'm not sure what you're trying to get at, Robert. You seem to start by > asking why not an arbitrary number of dimensions for "complex numbers", > then you finish off the sentence talking about "those four dimensions". > > At any rate, as to your stipulation for the real part, real numbers, > complex numbers, quaternions and octonions all follow that rule. > > As for N-dimensional "complex numbers", if you stipulate that they must be > a certain kind of algebra or group (I can't remember which -- I think they > must be a Lie algebra), then there are exactly four kinds possible: reals, > complex, quaternions, and octonions. Hamilton, in fact, was stuck for > years trying to extend the notion of complex numbers to three dimensions, > and only got unstuck when he realized that he must represent the > operations with four components, not three. >Tim, i dunno much about quaternions (like what they're good for), but my main issue is that while a complex number is fully described as a pair of real numbers, the set of complex numbers and its algebra are more than that of 2-vectors. C is not the same as R^2 . -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●August 28, 20142014-08-28
On Thu, 28 Aug 2014 18:04:40 -0400, robert bristow-johnson wrote:> On 8/28/14 1:55 PM, Tim Wescott wrote: >> On Thu, 28 Aug 2014 12:24:56 -0400, robert bristow-johnson wrote: >> >>> On 8/27/14 5:07 AM, gyansorova@gmail.com wrote: >>>> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com >>>> wrote: >>>>> >>>>> regarding the physicality of complex numbers.... if you replace the >>>>> word "complex" with "2 dimensional" then it doesn't seem so >>>>> mysterious.. >>>>> >>>>> >>> but complex numbers are *not* merely ordered pairs of real numbers. >>> there is more to them than that. >>> >>>>> In fact I do have an oscope that will display a "complex" or 2 >>>>> dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and >>>>> put the ocsope in XY mode. >>> >>> same issue. >>> >>> >>>> Why stop at 2? Why not use quaternions >>>> >>>> http://en.wikipedia.org/wiki/Quaternion >>> >>> hey, why not N-dimensional "complex numbers". better define how >>> operations like multiplication get done. and, if one of those 4 >>> dimensions is the "real axis", then the rules of those operations >>> should default to the existing real operations when the other 3 >>> dimensions have a value of 0. >> >> I'm not sure what you're trying to get at, Robert. You seem to start >> by asking why not an arbitrary number of dimensions for "complex >> numbers", then you finish off the sentence talking about "those four >> dimensions". >> >> At any rate, as to your stipulation for the real part, real numbers, >> complex numbers, quaternions and octonions all follow that rule. >> >> As for N-dimensional "complex numbers", if you stipulate that they must >> be a certain kind of algebra or group (I can't remember which -- I >> think they must be a Lie algebra), then there are exactly four kinds >> possible: reals, >> complex, quaternions, and octonions. Hamilton, in fact, was stuck for >> years trying to extend the notion of complex numbers to three >> dimensions, and only got unstuck when he realized that he must >> represent the operations with four components, not three. >> >> > Tim, i dunno much about quaternions (like what they're good for), but my > main issue is that while a complex number is fully described as a pair > of real numbers, the set of complex numbers and its algebra are more > than that of 2-vectors. C is not the same as R^2 .Your confusion confuses me. Rather, I'm not quite sure where your confusion lies. Quaternions are, as close as you can come, the next higher dimensional "complex numbers". They have a complete algebra that is more than 4- vectors. They're like complex numbers in that they have imaginary parts, but unlike complex numbers in that they have three imaginary parts. You can add them, subtract them, and multiply them. Multiplication is not commutative -- a * b does not necessarily equal b * a, in quaternion-land. They're exceedingly useful as a compact way of representing rotations is 3D space, however there are so many rules and exceptions that you have to remember that they'll make your brain twitch. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●August 28, 20142014-08-28
Tim Wescott <seemywebsite@myfooter.really> writes:> [...] > Your confusion confuses me. Rather, I'm not quite sure where your > confusion lies. > > Quaternions are, as close as you can come, the next higher dimensional > "complex numbers". They have a complete algebra that is more than 4- > vectors. They're like complex numbers in that they have imaginary parts, > but unlike complex numbers in that they have three imaginary parts. You > can add them, subtract them, and multiply them. Multiplication is not > commutative -- a * b does not necessarily equal b * a, in quaternion-land. > > They're exceedingly useful as a compact way of representing rotations is > 3D space, however there are so many rules and exceptions that you have to > remember that they'll make your brain twitch.I did a little reading on quaternions and division rings. What a rich set of mathematics this algebra! I surely wish I had the time and money to study this area of mathematics intead of writing my umpteenth thousandth line of C/C++ code... -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
