Mike Dubbeld

Look, these guys just aren't telling the straight story. Simply put you are not going to read this book without an extensive background in mathematics far beyond calculus and ordineary differential equations which by itself removes the vast majority of people right off the bat. Although he claims like just about every other author he will walk you through it - it is basically a lie. Don't get me wrong I love Penrose books but you better have at least vector analysis, complex analysis, linear algebra and partial differential equations if you hope to cope with this book. You can get a lot out of it of course without the above but I always find it very dissappointing if I find out too late I am not going to get the math used in a book. Even though most people won't be able to keep up with him, for the exact same reason people that do know the math greatly appreciate reading Penrose and his twistors etc. I find what holds a lot of people back from Penrose is complex analysis which from my perspective you almost have to go out of your way to get unless you are a math major. So don't be fooled or discouraged. It IS a fairly difficult book. I suggest this book inspire people to learn these subjects and go back to this book so they can enjoy it/not get stuck over and over again.

If you can do a little calculus see Who is Fourier - a book that goes into great detail on FFT Spectrum analyzers which is very easy to read with lots of simple illustrations piecing it together with the equations used to produce the a graph of any sound. The size of the bars are the Fourier coefficients for that component of sine wave frequencies associated with spectrum of frequencies.

This is specifically covered (Ahhhhh and Ohhhh) in the book Who is Fourier on p143 and elsewhere. The whole book is about Fourier analysis of waves and how a spectrum FFT analyzer works. When you hear the note C on a piano it is clearly distinguishable from the same note C on a flute. The difference is called the Quality or Timbre. The 2 instruments have the same fundamental frequency but the sounds are different because of the different amounts of the various harmonics. ANY form (geometry) can be represented/constructed by a Fourier synthesis of sine waves and likewise any form can be deconstructed into it a number of simple waves of various amplitudes and frequencies (Fourier analysis). The larynx in the throat and voice box we use to create rarefactions and compressions of air molecules (longitudinal waves) which can be represented mathematically as sinewaves. Intstead of Quality or Timbre I like to call the complex waves produced on a FFT (Fast Fourier Transform) Analyzer their Fourier Fingerprint. Every 'form' in the universe/conglomeration of atoms and molecules having a rigid structure has a unique Fourier fingerprint which I also call its 'real' name. In other words, every 'solid' shape has a form. Every form (geometry) has a corresponding set of Fourier component waves into which it can be broken down into. The form can be considered simply a complex wave made up of a large number of simple sinewaves (each of which having their own amplitude/coefficients and frequencies) such that when added together they produce the form/geometry. This entire subject goes much further than anyone on this forum might even likely to suspect. It can be taken into quantum mechanics as well using spherical waves (Nick Herbert Quantum Reality). As 3-dimenisonal creatures with our sole experience from the senses being in 3-d, we cannot visualize more than 3 dimensions. Yet everyone knows mathematically there is no reason we cannot go beyond 3 dimensions. As strange as it may seem, every frequency can be considered a dimension in Fourier series that is orthogonal to other dimensions. (orthogonal being perpendicular to) Thus, a vector with 8 dimensions can be represented as 8 waves each with a different frequency and the Fourier coefficients represtative of the amplitude of the vector in that dimension. We can visualize n dimensions as n waves to think about intuitively. For example: Suppose I have tomato juice as having A amount of tomato; B amount of celery; C amount of lettuce; and D amount of cabbage. Now if I wanted to plot this f(tomato,celery,lettuce,cabbage) I would need 4 dimensions. Leave off the cabbage (dimension). Use the coefficients A,B,C as the amplitude of a vector in the 3 component directions having a resultant vector in 3-d space along the x,y and z axes whose 'shadow' on each of the planes represents its component. The resultant vector would have a taste unique from another brand that also uses tomato,celery and carrot but in the amounts D,E and F. D, E and F being again the vector components in each of the cardinal directions on a right-handed Cartesian coordinate system. Since it has different component amounts of the same ingredients it tastes different than the other brand. Its resultant vector points in a different direction and magnitude also. But if you want to plot more than 3 variables (vegtables) you can't do it. But you can assign a frequency to each vegtable for any number of vegtables with A,B,C,D....n Fourier coefficients and add the waves together and visualize that! The above example is shown in great detail in the book Who is Fourier. If you have had calculus and done some vector things, the whole book reads as easy as a comic book (complete with cartoon characters!). They demonstrate what I said above much more clearly. They show Ahhh and Ohhh and other sounds as produced on a FFT specturm analyzer. They also show how this leads to the QFT Heisenberg Uncertainty Principle Heisenberg matrix vs Schrodinger wave - a mapping showing the 2 to be identical in much the same way as vectors and waves above.

I think my answer is a good answer. I see the poster as likely complaining about having to learn DE's as a computer science major - what are they good for? I didn't see the post until recently and since no one else responded that makes my post about 100 percent better than the rest...... I don't really care if I answer homework questions either. A lot of times people think they can just get an answer on the web but for things in math and science on tests chances are you have to show your work and the person getting a homework answer will likely see the answer is so complicated that even if they do copy it they could not explain it in class and it will not help them on a test anyway. It is for good reason there are so few people in school doing science and mathematics. I have a ton of math books but most of them totally suck unless you already know the subject. The arrogance in most math books is beyond belief --- "and so it is obvious that---" No, many times nothing about it is obvious in any way shape or form. Skipping 10 or 20 steps is not elegance it is lazy egoism. I have a bad attitude toward the priesthood of mathematics. (Another thing that irks me no end is how many mistakes authors make. It is very difficult to prove an author wrong when you are just learning something new.) Its simply not good enough in schools to teach without explicitly showing what good it is for which again is what this thread is about. I can take my knowledge of partial differential equations down to Borders Books and with about 2 dollars get a cup of coffee with it..... Curiouser and Curiouser, cried Alice.

Not much turnover on these forums. Computer programs are sometimes used to solve DE's because they have no exact analytical solution they have to be approximated by iteration using numerical methods. As someone working in a corporation you might be asked to do this but if you are not familiar with DE's you would have a lot of trouble. Formulating DE's is more of an engineering thing. Not only that but when you say 'computer science' you no doubt are pertaining to a Boolean-Church-Turing machine/digital beast. But there are such things as analog computers and that puts you squarely into the realm of differential equations. Continuous analog functions/block diagrams/transfer functions and feedback control systems. Electrical circuit analogs of billion dollar bridges and sky scrappers that have similar differential equations. It costs only a few dollars to build and test an electrical circuit but no one goes around building billion dollar buildings to see if they are structurally sound when built this way or that way. Analog computers are far faster than digital computers and one example of such a beast is an air speed indicator and other pilot instruments. By learning systems engineering with block diagrams/transfer functions in the form of Laplace transforms to represent governing DE's of some system, given a set of initial conditions and boundary conditions you can brainlessly construct an algorithm for solution on a computer by numerical iteration/approximation of solution. Any 6 year old can be a programmer. The same can't be said about formulating and solving DE's. Calculators use analog computers. They take things like the discharge rate on a capacitor as output and feed it into an analog to digital interface where it is digitized for LCD or LQD display digits. Also there is convolution and Fourier series - representing square waves by analog smooth waves - how do you modulate a set of pulses of digital output into a modem to be multiplexed with a few thousand other modulated signals and shot up to a satellite and sent to the other side of the planet and demux'ed and de-modulated on the other end? Wave superposition. Analog. DE's again. How does you CPU design interface with the outside world?

No black holes will be created at CERN LHC unless they can come up with 10^15 beyond the energy of LHC. Creating microscopic black holes at LHC is only possible if gravity is leaking into other dimensions per Scientific American. Even if they do create one it will last on the order of trillionths of a second and be vastly smaller than the size of a proton. They can create energy in LHC to come up with a 10^-23 KG but the smallest black hole possible theoretically is 10^-8 kg. Out of time. Mike D.

The above is true except consider that you can add enough waves to make it as square as you like in appearance - any number of decimal places. Something called Gibbs phenomena happens at the junction of the vertical and horizontal intersections where there is a spiked over shoot that cannot be avoided but still you can make it as small as you like. Since there is no such thing as a perfect anything in the universe to begin with and the most accurate thing measured in science is about 10^-13 meters who really cares? There is no such thing as a straight line or perfect circle or any other geometric shape (in the universe) because you can always go smaller and smaller to find deviations. Everything there is - is what it is to some number of decimal places for things in the universe. I used to know where on the web you could plug in numbers and create waveforms and there is one for spherical waves also.

Hi, I only posted once before a few months ago. I am playing around with DE and Fourier Series and FFT also. You can create shapes with any sort of waves you like. Sine waves are popular because no matter how many times you differentiate them they still come back as sinewaves with a phase shift. But you could use tuba waves or piano waves if you like. I believe spherical waves/harmonics are particularly important in understanding electron orbits. Fourier transforms are a powerful means of solving differential equations and nowhere will you find more analysis using Fourier than in Electrical Engineering. Fourier analysis can take any form and deconstruct it into a set of harmonic waveforms and its converse - Fourier synthesis can take a set of waves and use wave superposition which is just wave addition to add them. You just add their amplitudes together. Waves are linear in this way and are therefore a popular means of analysis. Before discussing waves in any sort of detail an understanding of the level of mathematics someone is comfortable with is necessary. Fourier Series is easiest involving the average value (a0) which is 1/T int from 0 to T of y(t)dt + an = 2/T int from 0 to T of f(t) cos(nwt)dt (the cosine waves) + bn = 2/T int from 0 to T of f(t) sin(nwt)dt where f(t) is your particular function and I am using int to mean the calculus integral of. If you have a square wave for instance you have y(t) = 2 on the interval [0,1) and y(t) = 0 on the interval (1,2] there is obviously a discontinuous jump at x = 1 from 2 to 0. Without going into the details of the integrations of this wave, the square wave function can be represented by the Fourier Series 1 + 4/pi(cos(pit) - 1/3 cos(3pit) + 1/5 cos(pit)..... the more terms you add to the series the more this combination of cosine waves looks like a square wave when added together. All the above may sound like a lot of work but thats life in the big city when you need to represent a discontinuous function by waves - for instance when you have an input of square waves from a signal generator and you have an electrical network function and the product of the input wave x network function = the output function things like Fourier analysis become necessary. And there are tables of waveforms you can select from and get a Fourier Series representation from/the Fourier Series equation at the same time and there are standard computer algorithms for calculating any such Fourier Series out as far as you like along with the resulting waveform plot. Associated with any Fourier Series comes the truncation error but for a series that converges rapidly the error is small with only a few terms. The minimum requirements for representing a complicated waveform including one that is discontinuous like a square wave or sawtooth wave is the Direchlet Conditions where the waveform has to have a finite number of discontinuities, have a finite number of maxima and minima and be absolutely convergent. (calculus subjects) One of the most fastinating things about Fourier Series is that they are analogous to vector analysis to some extent. When you want to construct a Fourier Series for a function you need to determine the coefficients or amplitudes of the harmonics. You could consider these amplitudes the vector components in each direction. But the really nice thing about using waves is that the frequencies of the waves are equivalent to the dimensions in a vector space. While you cannot visualize more than 3 dimensions you can visualize any number of frequencies....... But see, the thing is, you could construct a 'mathematical universe' where each dimension/frequency is some aspect of that 'universe' like say gravity or the electric force (or quantum-weird entity) as represented by vector component x with dot product coefficient A = a Fourier harmonic frequency x and amplitude A and adding up the vector components gives a final 'entity' consisting of the components and in the same way the harmonic Fourier frequency 'components' represent the directions while the Fourier coefficients represent the magnitude of the vector components in each direction!! The book 'Who Is Fourier?' points this out comparing orthogonality in vectors with orthogonality in Fourier series in terms of Euler's Formula and they are equivalent. (just construct the Maclaurin Series for each). There they use the analogy of vegetable juice to illustrate the same thing where the amount of carrot is the vector magnitude and carrot would be in the x direction for example and celery say in the y direction with B magnitude and there being a resultant vegtable juice when they are added. But in Fourier terms it would be carrot is one frequency with its coefficient and celery another frequency with its coefficient representing vegtable juice in terms of waves instead of vectors. Another very interesting thing about Fourier is you can extend it to include waves of infinite periods using the Fast Fourier Transform. You can take a snapshot of a wave of some length and infer/predict what the rest of the wave will look like based on that snapshot where the longer the snapshot wave - the more closely it will resemble the original wave. And this is where the Heisenberg Uncertainty Principle comes in - from the uncertainty found in waves. Same principle.

I think this whole discussion is off course. Replicator might be a star trek term I don't know but 'transporter' is a treky word. You don't have to start with something macroscopic here. As I see it the problem is to take a fermion and convert it into a boson, transport it at the speed of light and then have it convert back into what it was at its destination. An electron is a femion so it doesn't take much energy. As I see it if you could do that you could easily do 'replicator' things. When an electron and positron come together you get gamma rays. More than that I don't know where to go with it or if there is anywhere to go. Just a thought. Its likely that nature has built in obstacles to disallow it unless you do have some kind of fusion reactor you can tap into for energy.

Jay Leno cracked a joke not long ago - What do you call an American tourist in Iran? Answer: A kidnap victim...... Kiana? Mike Dubbeld