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Low carb diets
OmegaZero2003 wrote:
"Wayne S. Hill" wrote... OmegaZero2003 wrote: My original point to the OP on the topic was a retort to the statement that *all* systems are nonlinear. That is not true. Nonsense. The OP never said any such thing. 8-p Here is the context; I see you were pointing to biological systems. I wasn't the OP. -- -Wayne |
#202
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Low carb diets
"Wayne S. Hill" wrote in message ... OmegaZero2003 wrote: "Wayne S. Hill" wrote... OmegaZero2003 wrote: My original point to the OP on the topic was a retort to the statement that *all* systems are nonlinear. That is not true. Nonsense. The OP never said any such thing. 8-p Here is the context; I see you were pointing to biological systems. I wasn't the OP. OK - nevermind. I thought it was you that said that about biological systems? -- -Wayne |
#203
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Low carb diets
"Ignoramus32303" wrote in message ... Some systems are linear, within some limits, and within a certain accuracy. There is no perfect linear system. What is "perfect"? Fist - ther is a rigorouus *mathematical* defintion of what it takes for a system to be characterized as linear by definition. You need to find out what that is (I gave a lot of info/refs for you to do so easily). Most systems that we talk about as linear, fail at being linear as you increase the input, the failure and non-linearity growing with the input. Many do; not all. Sometimes, to become sufficiently non-linear, the input has to grow to unreasonable levels, Yeah - like an (audio) amp being turned way up to saturation whenre you hear screeches in the output. reasonable defined as what we experience in real life. Well - some audio systems are inadvertently turned up too loud! I believe that I have given a fairly accurate summary that should be supported by consensus. Fairly accurate is not accurate enough, apparently. The devil is in the details and there are systems that fall within the mathematical definition that are not "llinearized", subclassed (as with the audio amp), or approximated. i |
#204
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Low carb diets
"Ignoramus32303" wrote in message ... In article m, OmegaZero2003 wrote: "Ignoramus32303" wrote in message ... Some systems are linear, within some limits, and within a certain accuracy. There is no perfect linear system. What is "perfect"? that gives a response exactly proportional to the output, to any output. Good! I should have mentioned an obvious one: optics! See: http://www.wkonline.com/a/Linear_Sys...712 92885.htm Fist - ther is a rigorouus *mathematical* defintion of what it takes for a system to be characterized as linear by definition. You need to find out what that is (I gave a lot of info/refs for you to do so easily). Most systems that we talk about as linear, fail at being linear as you increase the input, the failure and non-linearity growing with the input. Many do; not all. Sometimes, to become sufficiently non-linear, the input has to grow to unreasonable levels, Yeah - like an (audio) amp being turned way up to saturation whenre you hear screeches in the output. try getting the amp to amplify 1 megawatt of power... reasonable defined as what we experience in real life. Well - some audio systems are inadvertently turned up too loud! I believe that I have given a fairly accurate summary that should be supported by consensus. Fairly accurate is not accurate enough, apparently. The devil is in the details and there are systems that fall within the mathematical definition that are not "llinearized", subclassed (as with the audio amp), or approximated. Yes, details are important. i |
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Low carb diets
OmegaZero2003 wrote:
"Wayne S. Hill" wrote... I wasn't the OP. OK - nevermind. I thought it was you that said that about biological systems? Yeah, in a follow-up. -- -Wayne |
#206
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Low carb diets
OmegaZero2003 wrote:
"Wayne S. Hill" wrote... But a tiny NN is very simple, and it's quite clear how it stores information. No is isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!! A tiny artificial neural network (which is what I've been referring to all along)? Sure it is. I'm not talking about brains here (although a friend of mine has mapped the function of the sense of smell of a crayfish brain, and he thinks he knows how it works in detail). Clamping is functionally a sudden saturation. No! Not theoretically or practically. Control systems are clamped at a setpoint that assures continued, NORMAL operation, in the vast majority of cases. sigh -- -Wayne |
#207
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Low carb diets
"Wayne S. Hill" wrote in message ... OmegaZero2003 wrote: "Wayne S. Hill" wrote... But a tiny NN is very simple, and it's quite clear how it stores information. No is isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!! A tiny artificial neural network (which is what I've been referring to all along)? Sure it is. I'm not talking about brains here (although a friend of mine has mapped the function of the sense of smell of a crayfish brain, and he thinks he knows how it works in detail). Did you bother to see the papers on computing and representation I referenced? I do not care what the neurophsyiological response of a crayfish CNS is; there ain't any instruments yet that can tell just what processes and properties in *any* part of a NN exactly *represent* a "piece" of information - even given what the definition of "information" is (beyond a difference). Ask your friend how the smell is represented to the crayfish? What is it like to be a crayfish smelling a pizza for example? Do you know what I am talking about when I use the word: "representation"? It is key. Otherwise , your friend is seeing Neural Correlates Of Smelling. Those are correlates and not the representation of the qualia of the smell. Let me give a quick example. We know how the VC (visual cortex) works in detail. Trust me on this. Yet, we now next to nothing about what it is that the NNet (together with its associated environmental context - the neurochemicals, fields etc. - the soup the neurons are immersed in) does to *represent*, say, the image of me performing a 1000 pound squat in a meet. Or the image that Michael Jordan used to visualize himslef in any number of patterns in which he scored. The image is the representation and it is an image presented to a conscious process that is made aware, or is aware, of that *represenation* qua (as) the real thing! This is a key issue in cognitive neuroscience - the binding problem is an associated issue - how do all those things (smell, taste, thoughts etc.) come together to represent a *whole* to the conscious being? Much of that issue arises from the unknown manner in which a neural net in situ (as I qualified above - in the soup), represents that information and performs "computations" upon it. In terms of computational neuroscience and ANNs(artificial neural net), remember that the ANN is a couple orders of magnitude less sophisticated (at least) (using simple I/O transforms and connection schemes used to build multi-layer ANNS) than the real thing in situ. Clamping is functionally a sudden saturation. No! Not theoretically or practically. Control systems are clamped at a setpoint that assures continued, NORMAL operation, in the vast majority of cases. sigh -- -Wayne |
#208
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Low carb diets
"OmegaZero2003" wrote in message s.com... "Wayne S. Hill" wrote in message ... OmegaZero2003 wrote: "Wayne S. Hill" wrote... But a tiny NN is very simple, and it's quite clear how it stores information. No is isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!! A tiny artificial neural network (which is what I've been referring to all along)? Sure it is. I'm not talking about brains here (although a friend of mine has mapped the function of the sense of smell of a crayfish brain, and he thinks he knows how it works in detail). Did you bother to see the papers on computing and representation I referenced? Here is the page(s) with live links to the online papers. http://www.u.arizona.edu/%7Echalmers/online.html The Consciousness and Physics set and the Consciousness and Artificial Intellgence set are appropo. If you merely look through the titles, you will absorb a lot - that is - you will see that the characterization of any *real* live NN *in situ* involves a huge amount of parameters and processes, from the neurochemical and field distributions and desities (and how that afffects signalling ) to oscillatory effects on processing and on and on. Lots of progress, but the upshot is that neurosci has not come up with a theory of how the brain (or parts of the brain) represent "information" such that it can be cognized/processed/computed by some conscious entity (like a crayfish). And if you tell the story that knowing a simple I/O scheme (based on the go-no_go/thresholding/etc)of the black box model of a neuron (or group of neurons), then you are missing, oh - perhaps 99% of what is going on! Signaling, computation and representation are accomplished through a host of mechansims that no theory has provided adequate explanation for yet. I do not care what the neurophsyiological response of a crayfish CNS is; there ain't any instruments yet that can tell just what processes and properties in *any* part of a NN exactly *represent* a "piece" of information - even given what the definition of "information" is (beyond a difference). Ask your friend how the smell is represented to the crayfish? What is it like to be a crayfish smelling a pizza for example? Do you know what I am talking about when I use the word: "representation"? It is key. Otherwise , your friend is seeing Neural Correlates Of Smelling. Those are correlates and not the representation of the qualia of the smell. Let me give a quick example. We know how the VC (visual cortex) works in detail. Trust me on this. Yet, we now next to nothing about what it is that the NNet (together with its associated environmental context - the neurochemicals, fields etc. - the soup the neurons are immersed in) does to *represent*, say, the image of me performing a 1000 pound squat in a meet. Or the image that Michael Jordan used to visualize himslef in any number of patterns in which he scored. The image is the representation and it is an image presented to a conscious process that is made aware, or is aware, of that *represenation* qua (as) the real thing! This is a key issue in cognitive neuroscience - the binding problem is an associated issue - how do all those things (smell, taste, thoughts etc.) come together to represent a *whole* to the conscious being? Much of that issue arises from the unknown manner in which a neural net in situ (as I qualified above - in the soup), represents that information and performs "computations" upon it. In terms of computational neuroscience and ANNs(artificial neural net), remember that the ANN is a couple orders of magnitude less sophisticated (at least) (using simple I/O transforms and connection schemes used to build multi-layer ANNS) than the real thing in situ. Clamping is functionally a sudden saturation. No! Not theoretically or practically. Control systems are clamped at a setpoint that assures continued, NORMAL operation, in the vast majority of cases. sigh -- -Wayne |
#209
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Low carb diets
On Sun, 21 Dec 2003 21:35:47 GMT, "OmegaZero2003"
wrote: "Wayne S. Hill" wrote in message ... OmegaZero2003 wrote: BTW, as a PS to my other answer post, here are some linear systems. - those characterizable by linear algebra. there are lots of these! 'ang on there, when we refer to linearity in such systems, we're referring to linearity of the dynamics, i.e., x-dot = A * x + b 'ang on there yourself!!! Do you know WHAT characterizes a linear system mathematically? Do you know the difference between an additive versus a multiplicative factor? Well there is your answer! In case you have not understood - will will take it slowly at first. Atend: here is help for you: "The response of a [ linear ] system to a sum of inputs is the sum of the responses to each individual input separately. All engineers should know what superposition is. Are you trying to be condescending? These two nice properties allow a whole range of tools to be applicable in designing linear systems and predicting their behavior. Some more examples of linear systems in real life: a) Frequency filters -- circuits which only pass low frequencies and reject high, or vice-versa. They only work within a narrowly-bounded input-output range. Too much signal input and they will break (non-linear). b) Delays are linear. Echos from faraway canyons are linear. Shout twice as loud, get an echo twice as loud. Two people shouting at the same time comes back as two people echoing at the same time. Two much volume and you will heat the air, changing the transmission characteristics (speed) of your pressure wave (non-linear). c) Many different kinds of economic systems -- looking at the apple juice production (output) vs. the apple crop yield, for example. This too will fail linearity at the extremes of yield. A bumper crop one year could mean an inability to harvest all the fruit before it goes bad. d) Limiting cases of non-linear systems for small inputs: Even if the system's response may not satisfy the equation above exactly, it often will well enough for small enough inputs. In this case, even if the number of apples bought by consumers, say, is inversely proportional to the price of apples, you can still model small changes around a reference price with linear systems (but beware when the inputs get large!). (http://van.hep.uiuc.edu/van/qa/secti...20020319233837. htm) Note that the last item (d) talks about linearization/signal decomposition/approximation_techniques of/for non-linear systems, which I am NOT talking about. We are talking about bona fide LINEAR systems fulfilling all the mathematical/theoretical properties that decades of science has determined qualifies a system as "Linear" *by definition*!!!! Real systems are always limited on their inputs. Real systems break down at some point if too much energy enters the system. Linearity is only a trait of a system over its useful operating range. I am also not talking about trivial examples of what is a linear system in toto (including extreme), but is subclassed to non-interesting behavior (like an amp NOT in saturation mode). In reality, people who make things that work must take nonlinearity into account. It's the only way to make reliable systems. You may not be thinking in toto, but I am, because I have to. We know what linear is. We also know it doesn't really exist. We try to make things work as linear as possible, because many useful things exhibit a limited version of it (like amps and filters). The definition of linearity for that amp doesn't allow for limiting the input. If it were truly linear, you wouldn't have to place extra limits on it. So I think you are coming around to accept the assertion that linear systems don't really exist. I will admit, though, they are nice on paper. This omits simple algebraic behaviors (because x-dot = 0). - Hamiltonian oscillators and like systems. (the direction field specifically) Not of great interest in biological systems, except as a backdrop. We were not restricting the discussion to biological systems. My original retort to the OP's statement: "All systems are non-linear" was to say: That is not true. It is easy to falsify hypotheses of the form: "All...(x) are (y)" in the sense that only one counter-example need be provided. FYI, a special class of linear control systems known as singularly perturbed control systems, uses the Hamiltonian approach ( recursive) approach based on t exact pure-slow and pure-fast decoupling of optimal control problems. Another interesting special class of linear systems:: "Linear systems with non-rational transfer functions (In this project linear systems described by partial differential equations having non-rational transfer functions are studied. The aim of this project is to analyse the dynamic behaviour and properties of linear input-output systems with non-rational transfer functions (such as flexible robot arms and heat processes) in the frequensy domain. For some classes of such systems it is possible to develop the overall transfer function in analytical form by the transmission matrix method. The interaction between different parts of the system (including the way they are coupled to each other) can then be analysed. The transmission matrix method has been succesfully applied to multi-link flexible robot arms and to buckling of multi-segment columns. Another application concerns the stability of heat processes described by parabolic partial differential equations. Based on the transmission matrix method a Nyquist stability test was developed for sandwich-layered materials with linear inner heat source." (M. Vajta DISC Project 1999) Good grief. Robot arms are the last thing in the world that would be linear. There may be some use of linear mathematics in the control law, but that's it. Linear math in control algorithms is the norm. for a linear system characterized using N first-order linear homogeneous differential equations with constant coefficient can be found at: (with a little more detailed math): http://www.mathpages.com/home/kmath440/kmath440.htm I leave it to the reader to come up with only three biological systems that are so charaterized (this is an easy quiz to see if you understand what a linear system is). You can also look up sparse linear systems for funzies! - continuous-time systems like electrical networks, many mechanical systems Aside from the fact that linearity is an approximation in all such systems, they all have their nonlinear limits. The nonlinearity is whatever it is that keeps the systems operating in their linear range. - any discrete system with a transfer function whose input, response and output functions depend on one variable Huh? I think you have to learn the meaning of linear. Note that I am NOT talking about linearisation of non-linear systems (which seems to be talked about in this thread instead of the major contentions about what exists.)!!! I am not talking about signal decomposition to acheive a linear treatment (what you called piecewise). Here is some help. Note that he points out some examples in the text. "Signals, Linear Systems, and Convolution Professor David Heeger Characterizing the complete input-output properties of a system by exhaustive measurement is usually impossible. Instead, we must find some way of making a finite number of measurements that allow us to infer how the system will respond to other inputs that we have not yet measured. We can only do this for certain kinds of systems with certain properties. If we have the right kind of system, we can save a lot of time and energy by using the appropriate theory about the system's responsiveness. Linear systems theory is a good time-saving theory for linear systems which obey certain rules. Not all systems are linear, but many important ones are. When a system qualifies as a linear system, it is possible to use the responses to a small set of inputs to predict the response to any possible input. This can save the scientist enormous amounts of work, and makes it possible to characterize the system completely." Please note the statement: "Not all systems are linear, but many important ones are." And now for a little math: "Linear Systems A system or transform maps an input signal x(t) into an output signal y(t): y(t) = T[x(t)]; where T denotes the transform, a function from input signals to output signals. Systems come in a wide variety of types. One important class is known as linear systems. To see whether a system is linear, we need to test whether it obeys certain rules that all linear systems obey. The two basic tests of linearity are homogeneity and additivity. 4 Homogeneity. As we increase the strength of the input to a linear system, say we double it, then we predict that the output function will also be doubled. For example, if the current injected to a passive neural membrane is doubled, the resulting membrane potential fluctuations will double as well. This is called the scalar rule or sometimes the homogeneity of linear systems. Additivity. Suppose we we measure how the membrane potential fluctuates over time in response to a complicated time-series of injected current x1(t). Next, we present a second (different) complicated time-series x2(t). The second stimulus also generates fluctuations in the membrane potential which we measure and write down. Then, we present the sum of the two currents x1(t) + x2(t) and see what happens. Since the system is linear, the measured membrane potential fluctuations will be just the sum of the fluctuations to each of the two currents presented separately. Superposition. Systems that satisfy both homogeneity and additivity are considered to be linear systems. These two rules, taken together, are often referred to as the principle of superposition. Mathematically, the principle of superposition is expressed as: T( x1 + x2) = T(x1) + T(x2) (2) Homogeneity is a special case in which one of the signals is absent. Additivity is a special case in which = = 1. Shift-invariance. Suppose that we inject a pulse of current and measure the membrane potential fluctuations. Then we stimulate again with a similar pulse at a different point in time, and again we measure the membrane potential fluctuations. If we haven't damaged the membrane with the first impulse then we should expect that the response to the second pulse will be the same as the response to the first pulse. The only difference between them will be that the second pulse has occurred later in time, that is, it is shifted in time. When the responses to the identical stimulus presented shifted in time are the same, except for the corresponding shift in time, then we have a special kind of linear system called a shift-invariant linear system. Just as not all systems are linear, not all linear systems are shift-invariant. In mathematical language, a system T is shift-invariant if and only if: y(t) = T[x(t)] implies y(t " That is pretty much what I told the OP. Not ALL systems are non-linear. Period, End of story. QED and all that. Gabel, Robert A. and Roberts, Richard A., 1973, Signals and Linear Systems, New York: John Wiley & Sons, 415 pp. Gaskill, Jack D., 1978, Linear Systems, Fourier Transforms, and Optics, New York: John Wiley & Sons, 554 pp. Lathi, B. P., 1992, Linear Systems and Signals, Carmichael, California: Berkeley-Cambridge Press, 656 pp. Lewis, Laurel J., Reynolds, Donald K., Bergseth, F. Robert, Alexandro, Jr., Frank J., 1969, Linear Systems Analysis, New York: McGraw-Hill Book Company, 489 pp. Schwarz, Ralph J. and Friedland, Bernard, 1965, Linear Systems, New York: McGraw-Hill Book Company, 521 pp The most common form of discrete systems is the iteration (strobing based on time or state) of a continuous system. A single-variable system would be of a form x-dot = f(x) Here, f can be (generally is) a nonlinear function of x, so the system will show nonlinear behavior, both continuously and discretely. - any systems preserving homogeneity (output proportional to input) and superposition (a way of combining linear functions such that the result is a linear function) In other words, linear systems. And there are lots of those! Do yourself a favor and merely google "linear system", read what you care to, then come back. I do not believe you do not get this. I can give you a good start: Linear System Theory and Design by Chi-Tsong Chen Read Chapter 2! Even non-deterministic systems can be modeled using statistics for linear dynamics. Not in their dynamics. You do not know what you are talking about. This is the classic engineer's mistake of characterizing in statistical terms what is not understood. When you delve into the NLD of such systems, you gain true insight into what makes them work. I can give examples. I have more examples in my head than you can provide. I worked on this for 20 years. But the main point to not be belabored is that there are linear systems on nature and manmade (1) Most systems *are* non-linear but some of those are characterizable using linear methods to some degree of accuracy; you did make something like this point. That is an approximation that people sometimes find useful to make. That doesn't make it so, especially when you take the system outside of the limited context in which you placed it for convenience. There are systems that are approximated as linear; that is called *linearization* and there is a gamut of math to deal with how to do that properly! But there are many systems that are inherently and demonstrably linear. -- -Wayne --- Proton Soup "If I drink water I will have to go to the bathroom and how can I use the bathroom when my people are in bondage?" -Saddam Hussein |
#210
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Low carb diets
"Proton Soup" wrote in message ... On Sun, 21 Dec 2003 21:35:47 GMT, "OmegaZero2003" wrote: "Wayne S. Hill" wrote in message ... OmegaZero2003 wrote: BTW, as a PS to my other answer post, here are some linear systems. - those characterizable by linear algebra. there are lots of these! 'ang on there, when we refer to linearity in such systems, we're referring to linearity of the dynamics, i.e., x-dot = A * x + b 'ang on there yourself!!! Do you know WHAT characterizes a linear system mathematically? Do you know the difference between an additive versus a multiplicative factor? Well there is your answer! In case you have not understood - will will take it slowly at first. Atend: here is help for you: "The response of a [ linear ] system to a sum of inputs is the sum of the responses to each individual input separately. All engineers should know what superposition is. Are you trying to be condescending? Who? Me? Sorry. I was probably a little. It seemed elementary considering the responses seemed to be intelligent but missed major aspects of this. |
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