mm-1479 === Subject: Re: Two basic questions regarding random processes and systems > Randy, i'm not sure what you mean by total power. > I mean the average power in the signal over all time: > lim_{tau rightarrow infty} 1/tau int_{-tau/2}^{+tau/2} |x(t)|^2 dt > Contrast this to the power in the signal at some specific point in time t_0 > and over some averaging time T, > 1/T int_{t_0-T/2}^{t_0+T/2} |x(t)|^2 dt. > Is there something wrong with this language? not too much. total power means that i am summing up a bunch of different powers somewhere and there is a total sum. average or mean is exactly correct. total energy has more meaning to me in the present context than total power. r b-j === Subject: Re: Two basic questions regarding random processes and systems > [...] > Randy, i'm not sure what you mean by total power. > I mean the average power in the signal over all time: > lim_{tau rightarrow infty} 1/tau int_{-tau/2}^{+tau/2} > |x(t)|^2 dt > Contrast this to the power in the signal at some specific point in > time t_0 > and over some averaging time T, > 1/T int_{t_0-T/2}^{t_0+T/2} |x(t)|^2 dt. > Is there something wrong with this language? > not too much. total power means that i am summing up a bunch of > different powers somewhere and there is a total sum. Yes, I see how the phrase could be taken that way. However, it doesn't necessarily mean that. Webster's (http://www.webster.com/) defines total in definition 2 as not lacking any part or member that properly belongs to it It is this sense that I mean. > average or > mean is exactly correct. True, but one can have an average or mean over a finite extent at a specific place in the signal as well. We need a term that does not impart the wrong connotation but also distinguishes the infinite-extent case from this case. > total energy has more meaning to me in > the present context than total power. I see your point, but you don't have to choose that connotation of total, as I stated above. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124 === Subject: Re: What Lester is on about ... <1114545353.dc6565966f539572e370810b160ba62e@teranews> <42766ea5.42450668@netnews.att.net> >> Let me give a brief idea of what I think Lester's differences >between >> differences is on about. He can tell me if I get it wrong, but if I >am I >> still think this is interesting and so would likely still talk about >this >> idea anyway. >(It oughtn't to be necessary to say this, but I'd like to thank you for >making the effort to post on Lester's side, without resorting to >juvenile insults or smart-alec evasion.) > I concur. >> What Lester wants to do is reduce objects and experiences to their >essential >> qualities: the things that allow me to distinguish one object or >experience >> from another object or experience. Therefore, these qualities are >the >> essential qualities of the experiences/objects that we have, and so >they are >> in fact always true about the object. If you change one of these >qualities, >> then the object is something else, and not what we've identified. >And so >> these are the basic units of knowledge, and hopefully we'd be able to >build >> from these starting principles to a solid understanding of other >principles >> and ultimately justify all of our qualities. Something that >epistemology >> desperately wants. >Hmm. Well, it says philosophy in the headers, so perhaps this isn't >meant to be maths. OK, it seems to me it's easy to dismiss the idea >that things (what, just Every Thing) have Essential Qualities, at least >in any context in which you are going to be able to include maths. But >dismissal isn't very productive, so let's suppose we embark on this >quest for absolute qualities of things. > The original context of my essays was science in general. It was > turned aside into the narrow confines of math for some reason right > off the bat. Perhaps because you posted to sci.math? People do manage to post messages that strictly are physics not maths in sci.math, without causing confusion, but if you talk about 'space' and 'dimensions' in sci.math without very carefully making explicit that you are not talking about the normal mathematical meanings of those words, you can't really expect sci.mathites to follow you. >Since Lester _seems_ to be talking about what mathematicians naturally >generalise as a vector space (or linear space), people ask reasonable, >basic questions, trying to understand the sense in which he means this, >and naturally getting frustrated by the absence of any clear answers. > I think people are equally frustrated by the mathematical answers they > get. Can you give actual examples of this? (You don't mean, I hope, that when people come along to sci.math and say I feel that if there are an infinite number of integers some of them just must be infinite, that they feel frustrated because mathematicians try to explain to them that they are wrong?) >Does he mean 2-dimensional space? 3-dimensional space? Actually, a line >embedded in a 2-dimensional space does divide it into two parts, but a >line in 3d space doesn't: well any attempt to elucidate the intended >meaning is bound to fail, since as Lester demonstrated a long while ago understanding >the abstraction that mathematicians mean by (vector) space, and quite >unable to answer any simple question involving the word 'dimension'. > More pejoration. I doubt many mathematicians understand what they > mean by space or dimension either. They by and large just draft a list > of properties and assume that whatever has such properties is space or > is a number and then complain when others claim there are universally > regressible properties that don't subsume all their properties. Well, your doubt is ill-founded, though I suppose I can't expect you to accept that. Making your sort of post on sci.math, and reading the responses isn't a very good strategy for learning what mathematicians _do_ mean by space or dimension, because you are not asking an elementary question, you are writing as though you understand the standard terms, and being surprised by responses and refutations on wildly differing levels. Have you tried reading an elementary exposition on vector spaces? I recommended that little, old book A concrete approach to abstract algebra by W W Sawyer to you a long time ago: have you got a copy and read Chapter 7? Incidentally, are you claiming that there are universally regressible properties of vector spaces in mathematics, which could somehow be used to prove that particular axiom sets are wrong? Any chance of a concrete example, that W W Sawyer might have understood? >My >guess is that he has some intuitive notion of space, being 'where we >live', or 'the universe of discourse'. So I suspect that if his claim >were rewritten to avoid abusing mathematical terminology, it would be >something like Entities promulgate differentiation in the universe - >something with which not many could violently disagree, but which says >not very much. > In other words if I vague out definitions so they have no mathematical > or scientific implications, mathematicians and scientists wouldn't > mind because they could safely ignore the implications of what I say. > In other words as long as I make no claims to universal significance > and truth, mathematicians and scientists wouldn't care Sorry. No deal. >Then Lester goes on to some sort of treatment of Znumbers, which he >puts in three classes: Zrationals, which actually do appear to coincide >with the rationals, Zirrationals, which are an ill-defined subset of >algebraic numbers, somewhat resembling the constructible numbers of >rule and compass construction, and Ztranscendentals, which are >described only by a vague expression (involving pointing out and >curves). I think most people have assumed that these three classes >are supposed to be mutually exclusive, and to encompass all of the >Znumbers (which might be most of the reals). But Lester vigorously >avoids explaining any of the obvious borderline test cases (e.g., if >the Zirrational/Ztranscendental distinction has something to do with >the lengths of curved lines, what about catenaries, which are curves >which precisely have rational lengths, despite having complicated >(non-algebraic) equations to describe them). > Scarcely true. I've more than gone out of my way to explain exactly > how the three classes are interpreted. I don't know anything about > catenaries and I seriously doubt there are curves which can be pointed > out anywhere on straight lines. Well, which is it? Are these categories of number (whatever you mean by that, exactly), or categories of curve (where a straight line is a special case of a curve)? If you claim that a curve cannot be pointed out on a straight line, I'm not going to look for a counterexample. But you've only identified two sorts of line that I can see - straight ones and curved ones. Are you saying that there is a third sort of line identified with Zirrationals? If two numbers (curves?) are pointed out on different curves, do you have any sort of formally specified procedure for identifying when these numbers (curves?) are the same? I imagine that pi could be pointed out on any number of circles (is that right?) Is there any sort of algebra: I mean if x is a Zirrational and y is a Ztranscendental, can we know anything about x+y or x*y? Why did you choose this seemingly rather arbitrary classification for numbers, and perhaps more importantly, if you want to get on with your work unpestered, why did you choose the immensely confusing terms you did? Is it too late to rename your numbers rational, squarable (for this Pythagoras thing: don't think 'squarable' has an existing meaning), and curvable, for example? Or are these names somehow an intrinsic part of this regressible universal truth? In which case, why not call them yuri, muri, and choetsu numbers, which would only confuse a tiny proportion of sci.math's readership, while you would have the satisfaction of knowing they are really the correct names, albeit in Japanese? >So what is this Znumber stuff? Maths, or philosophy? Can philosophers >understand it in any genuine sense? It's clear that mathematicians >can't. If it has some validity in philosophy, could it not be rephrased >in terms totally apart from maths? > What I have said and have to say isn't philosophy. It's regressible > tautologically to self contradictory alternatives. It's universally > true and not merely true in modern math and empiricist contexts. Not merely implies that you are indeed claiming to make substantive statements about mathematics; yet it seems to me you have an inordinately long task ahead of you to put any of this in a form that it can convince anyone who already knows anything about mathematics. Good luck, anyway! Brian Chandler http://imaginatorium.org === Subject: Re: What Lester is on about ... > Why did you choose this seemingly rather arbitrary classification for > numbers, and perhaps more importantly, if you want to get on with your > work unpestered, why did you choose the immensely confusing terms you > did? Because Leseter either: 1. A troll. 2. A psycopath 3. An ignoramus 4. A mathematical incompetent. The or implied above is inclusive, not exlclusive. Bob Kolker === Subject: Re: What Lester is on about ... >> Why did you choose this seemingly rather arbitrary classification for >> numbers, and perhaps more importantly, if you want to get on with your >> work unpestered, why did you choose the immensely confusing terms you >> did? >Because Leseter either: >1. A troll. >2. A psycopath >3. An ignoramus >4. A mathematical incompetent. >The or implied above is inclusive, not exlclusive. Very good, Bob, I've missed your coniptions fits the last few days. === Subject: Re: What Lester is on about ... >> > Let me give a brief idea of what I think Lester's differences >>between > differences is on about. He can tell me if I get it wrong, but >if I >>am I > still think this is interesting and so would likely still talk >about >>this > idea anyway. >> >>(It oughtn't to be necessary to say this, but I'd like to thank you >for >>making the effort to post on Lester's side, without resorting to >>juvenile insults or smart-alec evasion.) >> I concur. > What Lester wants to do is reduce objects and experiences to their >>essential > qualities: the things that allow me to distinguish one object or >>experience > from another object or experience. Therefore, these qualities are >>the > essential qualities of the experiences/objects that we have, and >>they are > in fact always true about the object. If you change one of these >>qualities, > then the object is something else, and not what we've identified. >>And so > these are the basic units of knowledge, and hopefully we'd be able >>build > from these starting principles to a solid understanding of other >>principles > and ultimately justify all of our qualities. Something that >>epistemology > desperately wants. >> >>Hmm. Well, it says philosophy in the headers, so perhaps this >isn't >>meant to be maths. OK, it seems to me it's easy to dismiss the idea >>that things (what, just Every Thing) have Essential Qualities, at >least >>in any context in which you are going to be able to include maths. >But >>dismissal isn't very productive, so let's suppose we embark on this >>quest for absolute qualities of things. >> The original context of my essays was science in general. It was >> turned aside into the narrow confines of math for some reason right >> off the bat. >Perhaps because you posted to sci.math? People do manage to post >messages that strictly are physics not maths in sci.math, without >causing confusion, but if you talk about 'space' and 'dimensions' in >sci.math without very carefully making explicit that you are not >talking about the normal mathematical meanings of those words, you >can't really expect sci.mathites to follow you. I posted to the groups I thought might have an abiding interest in universal truth. I didn't post to sci.math for mathematicians' input on conventional wisdom in mathematics nor did I post to sci.physics for conventional wisdom in physics. > >>Since Lester _seems_ to be talking about what mathematicians >naturally >>generalise as a vector space (or linear space), people ask >reasonable, >>basic questions, trying to understand the sense in which he means >this, >>and naturally getting frustrated by the absence of any clear >answers. >> I think people are equally frustrated by the mathematical answers >they >> get. >Can you give actual examples of this? (You don't mean, I hope, that >when people come along to sci.math and say I feel that if there are an >infinite number of integers some of them just must be infinite, that >they feel frustrated because mathematicians try to explain to them that >they are wrong?) Circles are the set of all points equidistant from any point? Lines are the integrals of points? etc. etc. etc. >>Does he mean 2-dimensional space? 3-dimensional space? Actually, a >line >>embedded in a 2-dimensional space does divide it into two parts, but >>line in 3d space doesn't: well any attempt to elucidate the intended >>meaning is bound to fail, since as Lester demonstrated a long while >ago >understanding >>the abstraction that mathematicians mean by (vector) space, and >quite >>unable to answer any simple question involving the word 'dimension'. >> More pejoration. I doubt many mathematicians understand what they >> mean by space or dimension either. They by and large just draft a >list >> of properties and assume that whatever has such properties is space >> is a number and then complain when others claim there are universally >> regressible properties that don't subsume all their properties. >Well, your doubt is ill-founded, though I suppose I can't expect you to >accept that. Making your sort of post on sci.math, and reading the >responses isn't a very good strategy for learning what mathematicians >_do_ mean by space or dimension, because you are not asking an >elementary question, you are writing as though you understand the >standard terms, and being surprised by responses and refutations on >wildly differing levels. Have you tried reading an elementary >exposition on vector spaces? I recommended that little, old book A >concrete approach to abstract algebra by W W Sawyer to you a long time >ago: have you got a copy and read Chapter 7? Are you saying mathematics' regressions for dimensionality and space are universally true or just axiomatically true? >Incidentally, are you claiming that there are universally regressible >properties of vector spaces in mathematics, which could somehow be used >to prove that particular axiom sets are wrong? Any chance of a >concrete example, that W W Sawyer might have understood? I don't deal with vector spaces. Sets of all points are highly suspect. I can't say what axioms they're based on though. >>My >>guess is that he has some intuitive notion of space, being 'where >>live', or 'the universe of discourse'. So I suspect that if his >claim >>were rewritten to avoid abusing mathematical terminology, it would >>something like Entities promulgate differentiation in the universe >>something with which not many could violently disagree, but which >says >>not very much. >> In other words if I vague out definitions so they have no >mathematical >> or scientific implications, mathematicians and scientists wouldn't >> mind because they could safely ignore the implications of what I say. >> In other words as long as I make no claims to universal significance >> and truth, mathematicians and scientists wouldn't care Sorry. No >deal. >>Then Lester goes on to some sort of treatment of Znumbers, which he >>puts in three classes: Zrationals, which actually do appear to >coincide >>with the rationals, Zirrationals, which are an ill-defined subset of >>algebraic numbers, somewhat resembling the constructible numbers of >>rule and compass construction, and Ztranscendentals, which are >>described only by a vague expression (involving pointing out and >>curves). I think most people have assumed that these three classes >>are supposed to be mutually exclusive, and to encompass all of the >>Znumbers (which might be most of the reals). But Lester vigorously >>avoids explaining any of the obvious borderline test cases (e.g., if >>the Zirrational/Ztranscendental distinction has something to do with >>the lengths of curved lines, what about catenaries, which are curves >>which precisely have rational lengths, despite having complicated >>(non-algebraic) equations to describe them). >> Scarcely true. I've more than gone out of my way to explain exactly >> how the three classes are interpreted. I don't know anything about >> catenaries and I seriously doubt there are curves which can be >pointed >> out anywhere on straight lines. >Well, which is it? Are these categories of number (whatever you mean by >that, exactly), or categories of curve (where a straight line is a >special case of a curve)? Straight lines aren't categories of curves and curves aren't categories of straight lines. The two are completely different concepts. The only thing they have in common is the idea of lines. Straight line segments are defined between points and curves aren't. > If you claim that a curve cannot be pointed >out on a straight line, I'm not going to look for a counterexample. I do claim that and there are no counterexamples I'm aware of. >But you've only identified two sorts of line that I can see - straight >ones and curved ones. Are you saying that there is a third sort of line >identified with Zirrationals? No. Rationals and irrationals are identified with straight line segments. Transcendentals are identified with curves. > If two numbers (curves?) are pointed >out on different curves, do you have any sort of formally specified >procedure for identifying when these numbers (curves?) are the same? No but I imagine the straight line radius of curvature in a plane would suffice for plane curves. > I >imagine that pi could be pointed out on any number of circles (is >that right?) Yes. > Is there any sort of algebra: I mean if x is a Zirrational >and y is a Ztranscendental, can we know anything about x+y or x*y? If you mean sums and multiples of irrationals and transcendentals, I don't see why they wouldn't follow the usual rules of algebra. >Why did you choose this seemingly rather arbitrary classification for >numbers, and perhaps more importantly, if you want to get on with your >work unpestered, why did you choose the immensely confusing terms you >did? The reason for seemingly arbitrary classification for numbers is just to eliminate logical inconsistencies in conventional interpretations of such terms as irrational and transcendental. My work at the moment is marketing so I don't need or expect to get on with it unpestered. > Is it too late to rename your numbers rational, squarable (for >this Pythagoras thing: don't think 'squarable' has an existing >meaning), and curvable, for example? Or are these names somehow an >intrinsic part of this regressible universal truth? The names are irrelevant. They refer to the same concepts conventional mathematics uses incorrectly. Whatever names are used simply reflect universally regressible usage. > In which case, why >not call them yuri, muri, and choetsu numbers, which would only confuse >a tiny proportion of sci.math's readership, while you would have the >satisfaction of knowing they are really the correct names, albeit in >Japanese? Sure as long as the underlying concepts are corrected in conventional mathematics: meainings such as irrational, transcendental, and empirical in physics. >>So what is this Znumber stuff? Maths, or philosophy? Can >philosophers >>understand it in any genuine sense? It's clear that mathematicians >>can't. If it has some validity in philosophy, could it not be >rephrased >>in terms totally apart from maths? >> What I have said and have to say isn't philosophy. It's regressible >> tautologically to self contradictory alternatives. It's universally >> true and not merely true in modern math and empiricist contexts. >Not merely implies that you are indeed claiming to make substantive >statements about mathematics; Of course I am. I wouldn't be spending the inordinate amount of time arguing over these concepts if I weren't. > yet it seems to me you have an >inordinately long task ahead of you to put any of this in a form that >it can convince anyone who already knows anything about mathematics. Not an inordinately long task at all once basic revisions are defined. General acceptance is another matter. Luck I don't need. Finite tautological regressions to self contradictory alternatives I need. === Subject: Re: What Lester is on about ... <1114545353.dc6565966f539572e370810b160ba62e@teranews> <42766ea5.42450668@netnews.att.net> <4276d003.56664926@netnews.att.net> >>Since Lester _seems_ to be talking about what mathematicians >naturally >>generalise as a vector space (or linear space), people ask >reasonable, >>basic questions, trying to understand the sense in which he means >this, >>and naturally getting frustrated by the absence of any clear >answers. >> >> I think people are equally frustrated by the mathematical answers >they >> get. >Can you give actual examples of this? (You don't mean, I hope, that >when people come along to sci.math and say I feel that if there are an >infinite number of integers some of them just must be infinite, that >they feel frustrated because mathematicians try to explain to them that >they are wrong?) > Circles are the set of all points equidistant from any point? Sounds good, and unfrustrating. (I.e. testable on a candidate circle) > Lines are the integrals of points? etc. etc. etc. Who said that? No mathematician, I would think. >>Does he mean 2-dimensional space? 3-dimensional space? Actually, a >line >>embedded in a 2-dimensional space does divide it into two parts, but >a >>line in 3d space doesn't: well any attempt to elucidate the intended >>meaning is bound to fail, since as Lester demonstrated a long while >ago >understanding >>the abstraction that mathematicians mean by (vector) space, and >quite >>unable to answer any simple question involving the word 'dimension'. >> >> More pejoration. I doubt many mathematicians understand what they >> mean by space or dimension either. They by and large just draft a >list >> of properties and assume that whatever has such properties is space >or >> is a number and then complain when others claim there are universally >> regressible properties that don't subsume all their properties. >Well, your doubt is ill-founded, though I suppose I can't expect you to >accept that. Making your sort of post on sci.math, and reading the >responses isn't a very good strategy for learning what mathematicians >_do_ mean by space or dimension, because you are not asking an >elementary question, you are writing as though you understand the >standard terms, and being surprised by responses and refutations on >wildly differing levels. Have you tried reading an elementary >exposition on vector spaces? I recommended that little, old book A >concrete approach to abstract algebra by W W Sawyer to you a long time >ago: have you got a copy and read Chapter 7? > Are you saying mathematics' regressions for dimensionality and space > are universally true or just axiomatically true? Mathematicians don't in general go in for regression, and I still don't understand what it means. I'm saying that in practice the (axiomatic) notions of dimensionality of a space, or of a manifold are very clear. I've never understood what you mean by dimension, but you seem to use it as a verb in a quite different way from mathematicians. >Incidentally, are you claiming that there are universally regressible >properties of vector spaces in mathematics, which could somehow be used >to prove that particular axiom sets are wrong? Any chance of a >concrete example, that W W Sawyer might have understood? > I don't deal with vector spaces. Sets of all points are highly > suspect. I can't say what axioms they're based on though. ... >>Then Lester goes on to some sort of treatment of Znumbers, which he >>puts in three classes: Zrationals, which actually do appear to >coincide >>with the rationals, Zirrationals, which are an ill-defined subset of >>algebraic numbers, somewhat resembling the constructible numbers of >>rule and compass construction, and Ztranscendentals, which are >>described only by a vague expression (involving pointing out and >>curves). I think most people have assumed that these three classes >>are supposed to be mutually exclusive, and to encompass all of the >>Znumbers (which might be most of the reals). But Lester vigorously >>avoids explaining any of the obvious borderline test cases (e.g., if >>the Zirrational/Ztranscendental distinction has something to do with >>the lengths of curved lines, what about catenaries, which are curves >>which precisely have rational lengths, despite having complicated >>(non-algebraic) equations to describe them). >> >> Scarcely true. I've more than gone out of my way to explain exactly >> how the three classes are interpreted. I don't know anything about >> catenaries and I seriously doubt there are curves which can be >pointed >> out anywhere on straight lines. >Well, which is it? Are these categories of number (whatever you mean by >that, exactly), or categories of curve (where a straight line is a >special case of a curve)? > Straight lines aren't categories of curves and curves aren't > categories of straight lines. The two are completely different > concepts. The only thing they have in common is the idea of lines. > Straight line segments are defined between points and curves aren't. > If you claim that a curve cannot be pointed >out on a straight line, I'm not going to look for a counterexample. > I do claim that and there are no counterexamples I'm aware of. No, I don't suppose so. But if I claim that no splunge can be trinculated with a terminating algorithm, could you find a counterexample? >But you've only identified two sorts of line that I can see - straight >ones and curved ones. Are you saying that there is a third sort of line >identified with Zirrationals? > No. Rationals and irrationals are identified with straight line > segments. Transcendentals are identified with curves. OK. Identified I can understand. > If two numbers (curves?) are pointed >out on different curves, do you have any sort of formally specified >procedure for identifying when these numbers (curves?) are the same? > No but I imagine the straight line radius of curvature in a plane > would suffice for plane curves. What is the straight line radius of curvature of a general curve? A sine wave, for example (y = sin x in the x-y plane). (I am understanding correctly that Zcurves are not necessarily arcs of circles, yes?) > Is there any sort of algebra: I mean if x is a Zirrational >and y is a Ztranscendental, can we know anything about x+y or x*y? > If you mean sums and multiples of irrationals and transcendentals, I > don't see why they wouldn't follow the usual rules of algebra. >Why did you choose this seemingly rather arbitrary classification for >numbers, and perhaps more importantly, if you want to get on with your >work unpestered, why did you choose the immensely confusing terms you >did? > The reason for seemingly arbitrary classification for numbers is just > to eliminate logical inconsistencies in conventional interpretations > of such terms as irrational and transcendental. My work at the moment > is marketing so I don't need or expect to get on with it unpestered. I didn't mean a day job, I meant developing your theories. OK, well here it seems to me we have a Big Problem. You claim there are logical inconsistencies in conventional interpretations of such terms as irrational and transcendental - I don't suppose you could actually show us any, in a way that can be understood by someone whose speciality is, for example, algebra? You wish to define your own categories, which (in my Z-ified version) are called Zirrationals and Ztranscendentals. I suppose you really hope that mathematicians will come to use your definitions? But suppose they do - what happens to all of the work done on what I'm still calling (un-Z-ed) irrationals and transcendentals. In nonZ-maths, non-transcendental numbers are called algebraic, and they are defined (algebraically!) as numbers which are roots of polynomials with integer coefficients. (So transcendentals are not such roots.) There is a whole subject which many of us have learnt about by watching James Harris blundering around, and getting free instruction concerning a subset of the algebraic numbers called algebraic integers. Do you know what they are? Are you claiming that somehow this work is all wrong? Surely even if you succeed in getting the names applied to your definitions, the theorems about algebraic numbers will still be true; will they just have to be renamed? (It is obvious that your Zirrationals are a proper [you _do_ know what that means, don't you?] subset of the algebraic numbers, so the theorems will not apply to them. > Is it too late to rename your numbers rational, squarable (for >this Pythagoras thing: don't think 'squarable' has an existing >meaning), and curvable, for example? Or are these names somehow an >intrinsic part of this regressible universal truth? > The names are irrelevant. They refer to the same concepts conventional > mathematics uses incorrectly. Whatever names are used simply reflect > universally regressible usage. Huh? Conventional mathematics uses the *concepts* incorrectly?? No: mathematics is not talking about _your_ concepts, it's talking about something else. To be honest, I don't think you have more than the vaguest idea what a mathematician means by transcendental number (I just gave you a definition, but I bet you skipped over it), so your claim that somehow mathematicians are using this concept incorrectly is a pretty bizarre category error. >>So what is this Znumber stuff? Maths, or philosophy? Can >philosophers >>understand it in any genuine sense? It's clear that mathematicians >>can't. If it has some validity in philosophy, could it not be >rephrased >>in terms totally apart from maths? >> >> What I have said and have to say isn't philosophy. It's regressible >> tautologically to self contradictory alternatives. It's universally >> true and not merely true in modern math and empiricist contexts. >Not merely implies that you are indeed claiming to make substantive >statements about mathematics; > Of course I am. I wouldn't be spending the inordinate amount of time > arguing over these concepts if I weren't. > yet it seems to me you have an >inordinately long task ahead of you to put any of this in a form that >it can convince anyone who already knows anything about mathematics. > Not an inordinately long task at all once basic revisions are defined. > General acceptance is another matter. Luck I don't need. Finite > tautological regressions to self contradictory alternatives I need. Is part of your project a hope that mathematicians will stop using sets of axioms for purely formal derivations of cute and interesting bits of abstract structure, and will start pronouncing absolute truth? Seems rather unlikely: you might as well start campaigning that mathematicians should all do English literary criticism, or Belgian lacework theory, instead of maths. Brian Chandler http://imaginatorium.org === Subject: Re: What Lester is on about ... > >Since Lester _seems_ to be talking about what mathematicians >>naturally >generalise as a vector space (or linear space), people ask >>reasonable, >basic questions, trying to understand the sense in which he means >>this, >and naturally getting frustrated by the absence of any clear >>answers. > > I think people are equally frustrated by the mathematical answers >>they > get. >> >>Can you give actual examples of this? (You don't mean, I hope, that >>when people come along to sci.math and say I feel that if there are >>infinite number of integers some of them just must be infinite, >that >>they feel frustrated because mathematicians try to explain to them >that >>they are wrong?) >> Circles are the set of all points equidistant from any point? >Sounds good, and unfrustrating. (I.e. testable on a candidate circle) That's the problem with empiricism. It doesn't show what's wrong with an idea. This definition defines a sphere and not a circle. It only defines a circle if you assume the points are on a plane. But in set theory alone there is no way to keep the points equidistant on a plane. Nor is there any way to assume equidistance without assuming Euclidean geometric concepts to begin with. >> Lines are the integrals of points? etc. etc. etc. >Who said that? No mathematician, I would think. Bob Kolker said that. As to mathematician I dunno. He seems to think he's enough of a mathematician to insult me routinely as not one. >Does he mean 2-dimensional space? 3-dimensional space? Actually, >>line >embedded in a 2-dimensional space does divide it into two parts, >but >>a >line in 3d space doesn't: well any attempt to elucidate the >intended >meaning is bound to fail, since as Lester demonstrated a long >while >>ago >>understanding >the abstraction that mathematicians mean by (vector) space, and >>quite >unable to answer any simple question involving the word >'dimension'. > > More pejoration. I doubt many mathematicians understand what they > mean by space or dimension either. They by and large just draft a >>list > of properties and assume that whatever has such properties is >space >>or > is a number and then complain when others claim there are >universally > regressible properties that don't subsume all their properties. >> >>Well, your doubt is ill-founded, though I suppose I can't expect you >>accept that. Making your sort of post on sci.math, and reading the >>responses isn't a very good strategy for learning what >mathematicians >>_do_ mean by space or dimension, because you are not asking an >>elementary question, you are writing as though you understand the >>standard terms, and being surprised by responses and refutations on >>wildly differing levels. Have you tried reading an elementary >>exposition on vector spaces? I recommended that little, old book A >>concrete approach to abstract algebra by W W Sawyer to you a long >time >>ago: have you got a copy and read Chapter 7? >> Are you saying mathematics' regressions for dimensionality and space >> are universally true or just axiomatically true? >Mathematicians don't in general go in for regression, and I still >don't understand what it means. Tautological regression? And yet math is often described as a tautological discipline. Go figure. > I'm saying that in practice the >(axiomatic) notions of dimensionality of a space, or of a manifold are >very clear. I've never understood what you mean by dimension, but you >seem to use it as a verb in a quite different way from mathematicians. I do use dimension as a verb to describe the dimensioning of space by the intersection of straight lines. I admit I haven't gone into exhautive detail on the subject because I've been too busy with more basic concepts. >>Incidentally, are you claiming that there are universally >regressible >>properties of vector spaces in mathematics, which could somehow be >used >>to prove that particular axiom sets are wrong? Any chance of a >>concrete example, that W W Sawyer might have understood? >> I don't deal with vector spaces. Sets of all points are highly >> suspect. I can't say what axioms they're based on though. >... >Then Lester goes on to some sort of treatment of Znumbers, which >puts in three classes: Zrationals, which actually do appear to >>coincide >with the rationals, Zirrationals, which are an ill-defined subset >algebraic numbers, somewhat resembling the constructible numbers >rule and compass construction, and Ztranscendentals, which are >described only by a vague expression (involving pointing out >and >curves). I think most people have assumed that these three >classes >are supposed to be mutually exclusive, and to encompass all of >the >Znumbers (which might be most of the reals). But Lester >vigorously >avoids explaining any of the obvious borderline test cases (e.g., >the Zirrational/Ztranscendental distinction has something to do >with >the lengths of curved lines, what about catenaries, which are >curves >which precisely have rational lengths, despite having complicated >(non-algebraic) equations to describe them). > > Scarcely true. I've more than gone out of my way to explain >exactly > how the three classes are interpreted. I don't know anything about > catenaries and I seriously doubt there are curves which can be >>pointed > out anywhere on straight lines. >> >>Well, which is it? Are these categories of number (whatever you mean >>that, exactly), or categories of curve (where a straight line is a >>special case of a curve)? >> Straight lines aren't categories of curves and curves aren't >> categories of straight lines. The two are completely different >> concepts. The only thing they have in common is the idea of lines. >> Straight line segments are defined between points and curves aren't. >> If you claim that a curve >cannot be pointed >>out on a straight line, I'm not going to look for a counterexample. >> I do claim that and there are no counterexamples I'm aware of. >No, I don't suppose so. But if I claim that no splunge can be >trinculated with a terminating algorithm, could you find a >counterexample? I hardly think straight lines and curves are quite in the same category. I've defined straight lines as existing between points and curves as not existing between points. So I don't see there is any lack of definition or precision of definition involved in finding counter examples if they are to be found. >>But you've only identified two sorts of line that I can see - >straight >>ones and curved ones. Are you saying that there is a third sort of >line >>identified with Zirrationals? >> No. Rationals and irrationals are identified with straight line >> segments. Transcendentals are identified with curves. >OK. Identified I can understand. OK, good. >> If two numbers (curves?) are pointed >>out on different curves, do you have any sort of formally specified >>procedure for identifying when these numbers (curves?) are the same? >> No but I imagine the straight line radius of curvature in a plane >> would suffice for plane curves. >What is the straight line radius of curvature of a general curve? A >sine wave, for example (y = sin x in the x-y plane). (I am >understanding correctly that Zcurves are not necessarily arcs of >circles, yes?) I'm not familiar with Zcurves but, no, curves in general are not necessarily arcs of circles. I have no idea what the straight line radius of curvature means exactly but I have seen the phrase and it strikes me that most curves would have one or they wouldn't be curved and we couldn't define the curvature of the curves that are curved. >> Is there any sort of algebra: I mean if x is a >Zirrational >>and y is a Ztranscendental, can we know anything about x+y or x*y? >> If you mean sums and multiples of irrationals and transcendentals, I >> don't see why they wouldn't follow the usual rules of algebra. >>Why did you choose this seemingly rather arbitrary classification >for >>numbers, and perhaps more importantly, if you want to get on with >your >>work unpestered, why did you choose the immensely confusing terms >you >>did? >> The reason for seemingly arbitrary classification for numbers is just >> to eliminate logical inconsistencies in conventional interpretations >> of such terms as irrational and transcendental. My work at the moment >> is marketing so I don't need or expect to get on with it unpestered. >I didn't mean a day job, I meant developing your theories. OK, well >here it seems to me we have a Big Problem. I didn't mean day job either. When I said marketing I meant marketing the ideas I've developed. >You claim there are logical inconsistencies in conventional >interpretations of such terms as irrational and transcendental - I >don't suppose you could actually show us any, in a way that can be >understood by someone whose speciality is, for example, algebra? Sure. Both straight line segments and curves are called irrationals. I don't understand why they are confused this way but it represents a logical inconsistency. >You wish to define your own categories, which (in my Z-ified version) >are called Zirrationals and Ztranscendentals. I suppose you really hope >that mathematicians will come to use your definitions? But suppose they >do - what happens to all of the work done on what I'm still calling >(un-Z-ed) irrationals and transcendentals. In nonZ-maths, >non-transcendental numbers are called algebraic, and they are defined >(algebraically!) as numbers which are roots of polynomials with integer >coefficients. (So transcendentals are not such roots.) There is a whole >subject which many of us have learnt about by watching James Harris >blundering around, and getting free instruction concerning a subset of >the algebraic numbers called algebraic integers. Do you know what they >are? Here I expect you're just ranting and raving to relieve tension. >Are you claiming that somehow this work is all wrong? Did I say it was all wrong or just entailed certain logical inconsistencies. > Surely even if >you succeed in getting the names applied to your definitions, the >theorems about algebraic numbers will still be true; will they just >have to be renamed? (It is obvious that your Zirrationals are a proper >[you _do_ know what that means, don't you?] subset of the algebraic >numbers, so the theorems will not apply to them. If the names are just changed to protect the innocent I will certainly be satisfied on this count. >> Is it too late to rename your numbers rational, squarable >(for >>this Pythagoras thing: don't think 'squarable' has an existing >>meaning), and curvable, for example? Or are these names somehow an >>intrinsic part of this regressible universal truth? >> The names are irrelevant. They refer to the same concepts >conventional >> mathematics uses incorrectly. Whatever names are used simply reflect >> universally regressible usage. >Huh? Conventional mathematics uses the *concepts* incorrectly?? No: >mathematics is not talking about _your_ concepts, it's talking about >something else. To be honest, I don't think you have more than the >vaguest idea what a mathematician means by transcendental number (I >just gave you a definition, but I bet you skipped over it), so your >claim that somehow mathematicians are using this concept incorrectly >is a pretty bizarre category error. I skipped over your definition because it's kinda irrelevant if transcendentals are defined on curves and rationals and irrationals are defined on straight line segments. Besides Bob Kolker has given this definition many times both before and after he said lines were the integration of points. >So what is this Znumber stuff? Maths, or philosophy? Can >>philosophers >understand it in any genuine sense? It's clear that >mathematicians >can't. If it has some validity in philosophy, could it not be >>rephrased >in terms totally apart from maths? > > What I have said and have to say isn't philosophy. It's >regressible > tautologically to self contradictory alternatives. It's >universally > true and not merely true in modern math and empiricist contexts. >> >>Not merely implies that you are indeed claiming to make >substantive >>statements about mathematics; >> Of course I am. I wouldn't be spending the inordinate amount of time >> arguing over these concepts if I weren't. >> yet it seems to >me you have an >>inordinately long task ahead of you to put any of this in a form >that >>it can convince anyone who already knows anything about mathematics. >> Not an inordinately long task at all once basic revisions are >defined. >> General acceptance is another matter. Luck I don't need. Finite >> tautological regressions to self contradictory alternatives I need. >Is part of your project a hope that mathematicians will stop using sets >of axioms for purely formal derivations of cute and interesting bits of >abstract structure, and will start pronouncing absolute truth? If not why are they mathematicians and why aren't they running around the universe instead looking for exceptions to 2+2=4? > Seems >rather unlikely: you might as well start campaigning that >mathematicians should all do English literary criticism, or Belgian >lacework theory, instead of maths. Yeah, well, they might definitely consider universal truth instead of problematic pronunciamenti on a universal truth they don't understand. === Subject: Re: What Lester is on about ... <1114545353.dc6565966f539572e370810b160ba62e@teranews> <42766ea5.42450668@netnews.att.net> <4276d003.56664926@netnews.att.net> <4277e550.73038770@netnews.att.net> Me: >> If two numbers (curves?) are pointed >>out on different curves, do you have any sort of formally specified >>procedure for identifying when these numbers (curves?) are the same? >> >> No but I imagine the straight line radius of curvature in a plane >> would suffice for plane curves. >What is the straight line radius of curvature of a general curve? A >sine wave, for example (y = sin x in the x-y plane). (I am >understanding correctly that Zcurves are not necessarily arcs of >circles, yes?) > I'm not familiar with Zcurves but, no, curves in general are not > necessarily arcs of circles. I have no idea what the straight line > radius of curvature means exactly but I have seen the phrase and it > strikes me that most curves would have one or they wouldn't be curved > and we couldn't define the curvature of the curves that are curved. Lester, Lester: look, I add a Z to a word to indicate your usage of it. If you really have any real contribution you make, independent of mellifluity, you could help a lot by not using mathematical terms to mean something different. Oh, but this seems to be an explicit description of your modus operandi: I have seen the phrase indeed! Try going and learning something about what it means: only a planar curve of constant curvature has a single radius of curvature. So what you imagined would suffice above is totally hopeless. > I didn't mean day job either. When I said marketing I meant marketing > the ideas I've developed. OK, please let me give you a bit of advice. If you hope ever to get anywhere with mathematicians, drop marketing. Play the word-association game - what comes after marketing? Lies. But if I can do so without contradicting myself, a word of marketing advice: you let drop somewhere that you have a simple proof of F3rm@t's 1@st theorem [see, I've protected you from searches]. Out there, Lester, are lots and lots of mathematicians, many of them with brains unimaginably more powerful, nimble, and probably young than either yours or mine, and you'd better take it from me that if you let them know that you claim that |= |_ T [hope you can read that] follows from an absolute Truth that space can't have more than three dimensions, well you can kiss your chances goodbye. >You claim there are logical inconsistencies in conventional >interpretations of such terms as irrational and transcendental - I >don't suppose you could actually show us any, in a way that can be >understood by someone whose speciality is, for example, algebra? > Sure. Both straight line segments and curves are called irrationals. I > don't understand why they are confused this way but it represents a > logical inconsistency. In real mathematics neither straight line segments nor curves are called irrationals. What ever gave you this odd idea? It certainly explains why you can claim that you have proved there is no single real line; problem is that this just means your misconception of what is meant by real line doesn't exist, which probably surprises no-one. > Here I expect you're just ranting and raving to relieve tension. >Are you claiming that somehow this work is all wrong? > Did I say it was all wrong or just entailed certain logical > inconsistencies. By wrong I intend to include entailing certain logical inconsistencies. Doesn't exactly sound right, does it? >Huh? Conventional mathematics uses the *concepts* incorrectly?? No: >mathematics is not talking about _your_ concepts, it's talking about >something else. To be honest, I don't think you have more than the >vaguest idea what a mathematician means by transcendental number (I >just gave you a definition, but I bet you skipped over it), so your >claim that somehow mathematicians are using this concept incorrectly >is a pretty bizarre category error. > I skipped over your definition because it's kinda irrelevant if > transcendentals are defined on curves and rationals and irrationals > are defined on straight line segments. So you claim that theorems proved from definitions you have skipped over entail logical inconsistencies? Hmm. Quite bold, at least. > ... Besides Bob Kolker has given > this definition many times both before and after he said lines were > the integration of points. Which definition did BK give? Brian Chandler http://imaginatorium.org === Subject: Re: What Lester is on about ... >> >>Since Lester _seems_ to be talking about what mathematicians >> >>naturally >> >>generalise as a vector space (or linear space), people ask >> >>reasonable, >> >>basic questions, trying to understand the sense in which he means >> >>this, >> >>and naturally getting frustrated by the absence of any clear >> >>answers. >> >I think people are equally frustrated by the mathematical answers >> >>they >> >get. >> >>Can you give actual examples of this? (You don't mean, I hope, that >>when people come along to sci.math and say I feel that if there are >>an >>infinite number of integers some of them just must be infinite, >>that >>they feel frustrated because mathematicians try to explain to them >>that >>they are wrong?) >Circles are the set of all points equidistant from any point? >>Sounds good, and unfrustrating. (I.e. testable on a candidate circle) > That's the problem with empiricism. It doesn't show what's wrong with > an idea. This definition defines a sphere and not a circle. It only > defines a circle if you assume the points are on a plane. But in set > theory alone there is no way to keep the points equidistant on a > plane. Nor is there any way to assume equidistance without assuming > Euclidean geometric concepts to begin with. It gets worse if you change how you measure distance. points equidistant can give a square (or cube) if distance is defined as the maximum of the absolute values of the changes in the 2 (or 3) coordinates. >Lines are the integrals of points? etc. etc. etc. >>Who said that? No mathematician, I would think. > Bob Kolker said that. As to mathematician I dunno. He seems to think > he's enough of a mathematician to insult me routinely as not one. >>Does he mean 2-dimensional space? 3-dimensional space? Actually, >>line >> >>embedded in a 2-dimensional space does divide it into two parts, >>but >>a >> >>line in 3d space doesn't: well any attempt to elucidate the >>intended >>meaning is bound to fail, since as Lester demonstrated a long >>while >>ago >> >> >>understanding >> >>the abstraction that mathematicians mean by (vector) space, and >> >>quite >> >>unable to answer any simple question involving the word >>'dimension'. >More pejoration. I doubt many mathematicians understand what they >mean by space or dimension either. They by and large just draft a >> >>list >> >of properties and assume that whatever has such properties is >>space >>or >> >is a number and then complain when others claim there are >>universally >regressible properties that don't subsume all their properties. >> >>Well, your doubt is ill-founded, though I suppose I can't expect you >>to >>accept that. Making your sort of post on sci.math, and reading the >>responses isn't a very good strategy for learning what >>mathematicians >>_do_ mean by space or dimension, because you are not asking an >>elementary question, you are writing as though you understand the >>standard terms, and being surprised by responses and refutations on >>wildly differing levels. Have you tried reading an elementary >>exposition on vector spaces? I recommended that little, old book A >>concrete approach to abstract algebra by W W Sawyer to you a long >>time >>ago: have you got a copy and read Chapter 7? >Are you saying mathematics' regressions for dimensionality and space >are universally true or just axiomatically true? >>Mathematicians don't in general go in for regression, and I still >>don't understand what it means. > Tautological regression? And yet math is often described as a > tautological discipline. Go figure. Generally mathematicians start with axioms, what you may be trying to regress to, and build up from those. It's a matter of working in the opposite direction. >> If two numbers (curves?) are pointed >>out on different curves, do you have any sort of formally specified >>procedure for identifying when these numbers (curves?) are the same? >No but I imagine the straight line radius of curvature in a plane >would suffice for plane curves. >>What is the straight line radius of curvature of a general curve? A >>sine wave, for example (y = sin x in the x-y plane). (I am >>understanding correctly that Zcurves are not necessarily arcs of >>circles, yes?) > I'm not familiar with Zcurves but, no, curves in general are not > necessarily arcs of circles. I have no idea what the straight line > radius of curvature means exactly but I have seen the phrase and it > strikes me that most curves would have one or they wouldn't be curved > and we couldn't define the curvature of the curves that are curved. Zcurves are what you mean by curves. Short for Zick's curves, I suppose. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Q for subscribers to Mathematics Magazine Slightly related... Does anyone know if the Problem Index http://problems.math.umr.edu/ is coming back/relocated/abandoned/... ?? === Subject: Re: Q for subscribers to Mathematics Magazine--> bad link Link goes to the department chairman's page: ( may be a server error) http://web.umr.edu/~lmhall/ > Slightly related... > Does anyone know if the Problem Index http://problems.math.umr.edu/ is > coming back/relocated/abandoned/... ?? === Subject: Re: calculating 3+ root with only a square root function > Is it possible to calculate an nth root where n>2 on a calculator that > only has a square root function (like the HP12C)? I don't see y^x or x^y listed as one of the built-in functions on HP's website for the 12C. However they do have e^x and ln(x), so here's how to do the n-th root with logs: cuberoot(x) = e^(ln(x)/n) Enter number. Take ln(x). Divide by n Take e^x. - Randy === Subject: Re: calculating 3+ root with only a square root function > Is it possible to calculate an nth root where n>2 on a calculator > that > only has a square root function (like the HP12C)? > I don't see y^x or x^y listed as one of the built-in functions > on HP's website for the 12C. You don't see it on http://www.hpmuseum.org/12c.jpg ? Really? Dirk Vdm > However they do have e^x and ln(x), so here's how to do the > n-th root with logs: > cuberoot(x) = e^(ln(x)/n) > Enter number. > Take ln(x). > Divide by n > Take e^x. > - Randy === Subject: Re: calculating 3+ root with only a square root function > Is it possible to calculate an nth root where n>2 on a calculator > that > only has a square root function (like the HP12C)? > > I don't see y^x or x^y listed as one of the built-in functions > on HP's website for the 12C. > You don't see it on > http://www.hpmuseum.org/12c.jpg > Really? [reply to Randy's reply - which is impossible without my reformatting the who message and inserting quoting characters] Indeed, they don't mention y^x, sqrt(x), Frac, Intg and a few more. < ;-) > They don't even mention the fact that they forgot to implement the = key, so I'll have a word with them. Dirk Vdm === Subject: Re: calculating 3+ root with only a square root function > Is it possible to calculate an nth root where n>2 on a calculator > that > only has a square root function (like the HP12C)? > > I don't see y^x or x^y listed as one of the built-in functions > on HP's website for the 12C. > You don't see it on > http://www.hpmuseum.org/12c.jpg No, I didn't see it here: http://www.hp.com/calculators/financial/12c/ which is HP's page for a 12C calculator you can order today. Statistical/Mathematical Features: * Cumulative statistical analysis * Std. deviation, mean, weighted mean * Linear regression * Forecasting, correlation coefficient * Total, áx, áx2, áy, áy2, áxy * +, -, x, %, .85, 1/x, ±, LN, e^x, n! But come to think of it, they don't even list square root on that list. So as I said, I don't see it LISTED on any of the product description pages at hp.com. I still don't. However, I did find this just now: How do I... Calculate the 4th and 5th roor of a number? HP 12c To calculate the 4th root of 81: 1. Press 81, then ENTER 2. Press 4, [1/x], then [y^x] 3. Answer = 3 So obviously there is a y^x key. They just don't mention it in their great mix of mathmatical and scientific functions. > cuberoot(x) = e^(ln(x)/n) Note obvious typo. I started to describe cube root, changed it to a description of n-th root, and forgot to edit the left hand side. - Randy === Subject: Re: calculating 3+ root with only a square root function days. My association with the Department is that of an alumnus. >> Is it possible to calculate an nth root where n>2 on a calculator >that >> only has a square root function (like the HP12C)? >I don't see y^x or x^y listed as one of the built-in functions >on HP's website for the 12C. >However they do have e^x and ln(x), so here's how to do the >n-th root with logs: >cuberoot(x) = e^(ln(x)/n) ^^^^^^^^ Should be n-th root >Enter number. >Take ln(x). >Divide by n >Take e^x. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: calculating 3+ root with only a square root function >Is it possible to calculate an nth root where n>2 on a calculator that >only has a square root function (like the HP12C)? >I'm surprised to find out that the HP12C, the ogosho of financial >calculators, doesn't have this function, since to calculate a growth >rate (of, say, earnings per share), you divide the first and last >values and then take the nth root of the result, n being the number of >periods (say, years) in between. >Ex: >2002 2001 2000 1999 >$1.41 $1.27 $1.05 $0.87 >1.41/0.87 = 1.62 >Third root (because 2002-1999 = 3) of 1.62 = 0.18 = 18% growth rate >This is pretty fundamental, so why can't the HP12C do it? Is there a >workaround? >TIA, >Marc why can't the HP12C do it? because it is intended for people on a different planet, where genuine mathematics is unknown. Johan E. Mebius P.S.: Finance children and directors: please do not be hurt. === Subject: Re: calculating 3+ root with only a square root function >Is it possible to calculate an nth root where n>2 on a calculator that >only has a square root function (like the HP12C)? >I'm surprised to find out that the HP12C, the ogosho of financial >calculators, doesn't have this function, since to calculate a growth >rate (of, say, earnings per share), you divide the first and last >values and then take the nth root of the result, n being the number of >periods (say, years) in between. >Ex: >2002 2001 2000 1999 >$1.41 $1.27 $1.05 $0.87 >1.41/0.87 = 1.62 >Third root (because 2002-1999 = 3) of 1.62 = 0.18 = 18% growth rate >This is pretty fundamental, so why can't the HP12C do it? Is there a >workaround? >TIA, >Marc > > why can't the HP12C do it? > because it is intended for people on a different planet, where genuine > mathematics is unknown. > Johan E. Mebius > P.S.: Finance children and directors: please do not be hurt. Doesn't the HP12C have logarithmic and anti-logarithmic functions? If so, the numerate (mathematically literate) can still do powers and roots. === Subject: Re: calculating 3+ root with only a square root function - pic of HP12C picture of the HP12C at http://www.hpmuseum.org/12c.jpg - there is on the A21 place a y^x function key. Unfortunately there is no y^(1/x) key, nor has the HP11C such a function key. At both models the 1/x key is directly to the right of y^x, so HP12C users - financial people - can handle exponential and power functions as easily as scientific and technical people. My prank is partly unjustified. So for the cubic root of X one enters X, presses ENTER, enters 3, presses 1/x and y^x and reads off the answer. The HP12C has logarithmic and anti-logarithmic functions as well. Ciao - Johan E. Mebius >> >Is it possible to calculate an nth root where n>2 on a calculator that >only has a square root function (like the HP12C)? >I'm surprised to find out that the HP12C, the ogosho of financial >calculators, doesn't have this function, since to calculate a growth >rate (of, say, earnings per share), you divide the first and last >values and then take the nth root of the result, n being the number of >periods (say, years) in between. >Ex: >2002 2001 2000 1999 >$1.41 $1.27 $1.05 $0.87 >1.41/0.87 = 1.62 >Third root (because 2002-1999 = 3) of 1.62 = 0.18 = 18% growth rate >This is pretty fundamental, so why can't the HP12C do it? Is there a >workaround? >TIA, >Marc > > >>why can't the HP12C do it? >>because it is intended for people on a different planet, where genuine >>mathematics is unknown. >>Johan E. Mebius >>P.S.: Finance children and directors: please do not be hurt. >> >Doesn't the HP12C have logarithmic and anti-logarithmic functions? >If so, the numerate (mathematically literate) can still do powers and >roots. === Subject: Re: calculating 3+ root with only a square root function - pic of HP12C > picture of the HP12C at http://www.hpmuseum.org/12c.jpg - there is on > the A21 place a y^x function key. > Unfortunately there is no y^(1/x) key, nor has the HP11C such a function > key. At both models the 1/x key is directly to the right of y^x, so > HP12C users - financial people - can handle exponential and power > functions as easily as scientific and technical people. My prank is > partly unjustified. > So for the cubic root of X one enters X, presses ENTER, enters 3, > presses 1/x and y^x and reads off the answer. > The HP12C has logarithmic and anti-logarithmic functions as well. > Ciao - Johan E. Mebius Don't you read the other replies before you repeat the same thing that 4 people already replied? ;-) Dirk Vdm === Subject: Re: calculating 3+ root with only a square root function - pic of HP12C Not always; I like to read and to make once in a while fast and thoughtless reactions. Johan E. Mebius >>picture of the HP12C at http://www.hpmuseum.org/12c.jpg - there is on >>the A21 place a y^x function key. >>Unfortunately there is no y^(1/x) key, nor has the HP11C such a function >>key. At both models the 1/x key is directly to the right of y^x, so >>HP12C users - financial people - can handle exponential and power >>functions as easily as scientific and technical people. My prank is >>partly unjustified. >>So for the cubic root of X one enters X, presses ENTER, enters 3, >>presses 1/x and y^x and reads off the answer. >>The HP12C has logarithmic and anti-logarithmic functions as well. >>Ciao - Johan E. Mebius >> >Don't you read the other replies before you repeat the >same thing that 4 people already replied? ;-) >Dirk Vdm === Subject: Group Homomorphism Again Given a cyclic multiplicative group Q of order q, can we embed such a group into Z_n* (where n is composite) with homomorphic property, that is, there is a mapping f such that f(GH) = f(G)f(H) for all G, H in Q? Do we need to have q|n? Is there any trivial mappings in closed form? === Subject: Re: Group Homomorphism Again days. My association with the Department is that of an alumnus. >Given a cyclic multiplicative group Q of order q, can we embed such a >group into Z_n* (where n is composite) with homomorphic property, that >is, there is a mapping f such that f(GH) = f(G)f(H) for all G, H in Q? >Do we need to have q|n? No; for example, if q=2 and n is an odd prime, then you can always embed the cyclic group of order 2 into the group (Z_n)^*, even though q will not divide n. If you factor n into primes, n = p_1^{a_1} * ... * p_n^{a_n} then by the Chinese Remainder Theorem, (Z_n)^* = (Z_{p_1^{a_1}})^* x ... x (Z_{p_n^{a_n}})^*. Each Z_{p_i^{a_i}} is a cyclic group of order (p_i^{a_i-1})(p_i-1). A cyclic group of order q will be embeddable into this group if and only if, factoring q into primes, q = q_1^{b_1} * ... * q_r^{b_r} you can find a cyclic subgroup of order q_i^{b_i} in the above, for each i. This requires that either q_i^{b_i} divide p_j^{a_j} for some j, or else that it divide p_k-1 for some k. Conversely, if this happens, you can find an embedding. > Is there any trivial mappings in closed form? You don't want to use trivial here (since 'trivial mapping' usually means the map that sends everything to the identity element). You mean easy mappings, don't you? -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Group Homomorphism Again >Given a cyclic multiplicative group Q of order q, can we embed such a >group into Z_n* (where n is composite) with homomorphic property, that >is, there is a mapping f such that f(GH) = f(G)f(H) for all G, H in Q? >Do we need to have q|n? > No; for example, if q=2 and n is an odd prime, then you can always embed > the cyclic group of order 2 into the group (Z_n)^*, even though q will > not divide n. > If you factor n into primes, > n = p_1^{a_1} * ... * p_n^{a_n} > then by the Chinese Remainder Theorem, > (Z_n)^* = (Z_{p_1^{a_1}})^* x ... x (Z_{p_n^{a_n}})^*. > Each Z_{p_i^{a_i}} is a cyclic group of order > (p_i^{a_i-1})(p_i-1). > A cyclic group of order q will be embeddable into this group > if and only if, factoring q into primes, > q = q_1^{b_1} * ... * q_r^{b_r} > you can find a cyclic subgroup of order q_i^{b_i} in the above, for > each i. Sorry that I forgot to say q is prime. > Is there any trivial mappings in closed form? > You don't want to use trivial here (since 'trivial mapping' usually > means the map that sends everything to the identity element). You mean > easy mappings, don't you? Yes, I did mean an easily computable mapping. > -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu === Subject: Re: Group Homomorphism Again >>Given a cyclic multiplicative group Q of order q, can we embed such >> a group into Z_n* (where n is composite) with homomorphic property, >> that is, there is a mapping f such that f(GH) = f(G)f(H) for all >> G, H in Q? Do we need to have q|n? >> No; for example, if q=2 and n is an odd prime, then you can always >> embed the cyclic group of order 2 into the group (Z_n)^*, even though q >> will not divide n. >Sorry that I forgot to say q is prime. Still, the example above shows you don't need it. For another example, consider the case of q = 5 and n=121. The situation here is simpler. You can embed the cyclic group of order q (a prime) if and only if one of the following happens: (i) q^2 | n; or (ii) There exists a prime p such that p|n and p = 1 (mod q). Proof: If q^2 divides n, then (Z_n)^* has a cyclic factor of order q^{a-1}, where q^a is the highest power of q that divides n; since a>2, this is a multiple of q, hence contains a cyclic factor of order q. If there is a prime p which divides n and is congruent to 1 (mod q), then (Z_n)^* has a cyclic factor of order p^{a-1}(p-1), where p^a is the highest power of p that divides n. Since q divides p-1, this contains a cyclic group of order q. Conversely, proceeding as I did in the previous post, let n= p_1^{a_1}*...*p_r^{a_r} be a prime factorization of n. Then the cyclic group of prime order q is embeddable into (Z_n)^* if and only if q divides some p_i^{a_i-1} (in which case p=q and a_i>1, so q^2 divides n), or else some p_i-1 (in which case p_1 = 1 (mod q) ). I don't know if it is easy, in general, to explicitly show the embedding. I would be tempted to find a primitive root modulo p_i^{a_i} (where p=q or else q|p-1) and use that with the Chinese Remainder Theorem; but that is in general hard and there are probably smarter ways of doing it. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: countable list of countables >> Is a countably infinite disjoint set of countably infinite disjoint >> sets countably infinite? (Assuming Ken really wants to know if the union of all these sets is countable.) >Yes, if you assume the axiom of choice. Otherwise, not necessarily. Countably infinite implies you have a bijection to N, which give you a well-ordering of the set. Given the well-orderings, you can construct a bijection from N to the union of the sets using the same triangular traversal as when counting rationals, or counting NxN. (1,1) (1,2) (2,1) (1,3) (2,2) (3,1) (1,4)... I probably used AC in here somewhere without realizing. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: countable list of countables > Is a countably infinite disjoint set of countably infinite disjoint > sets countably infinite? > (Assuming Ken really wants to know if the union of all these sets is > countable.) >>Yes, if you assume the axiom of choice. Otherwise, not necessarily. > Countably infinite implies you have a bijection to N, which give you a > well-ordering of the set. Given the well-orderings, you can construct a > bijection from N to the union of the sets using the same triangular > traversal as when counting rationals, or counting NxN. You could do that if you were given a well-ordering for each set in the collection, but we are not given that. Countably infinite means the set of bijections to N is nonempty. Choosing a particular bijection for each of a countably infinite collection of countably infinite sets amounts to finding a choice function for that collection of bijections, which is exactly what AC allows you to do. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: countable list of countables > Is a countably infinite disjoint set of countably infinite disjoint > sets countably infinite? >(Assuming Ken really wants to know if the union of all these sets is >countable.) >>Yes, if you assume the axiom of choice. Otherwise, not necessarily. >Countably infinite implies you have a bijection to N, which give you a >well-ordering of the set. Given the well-orderings, you can construct a >bijection from N to the union of the sets using the same triangular >traversal as when counting rationals, or counting NxN. >(1,1) (1,2) (2,1) (1,3) (2,2) (3,1) (1,4)... >I probably used AC in here somewhere without realizing. Yes, you did. When you say we should use a triangular traversal to count the union you're supposing that we already have each set ordered (with an order isomorphic to the standard one on N). There _is_ such an order for each set without using AC. But for each set there is more than one such order - for this argument you have to pick one. For each set you can pick one without using AC. But you need to pick one for each set, simultaneously, and that requires AC. More formally: Say S_n is countable, for each n in N. Now for each n there is an order O on S_n, isomorphic to the order on N. But it takes AC to show that there is a _sequence_ (O_n) of such orders. >--Keith Lewis klewis {at} mitre.org >The above may not (yet) represent the opinions of my employer. ************************ David C. Ullrich === Subject: Re: countable list of countables > Is a countably infinite disjoint set of countably infinite disjoint > sets countably infinite? Probably not what you wanted to ask. In fact: a countably infinite disjoint set of ... is countably infinite, regardless of how you fill in ... I think you mean to ask: Is the union of a countably infinite disjoint set of countably infinite disjoint sets countably infinite? === Subject: Re: countable list of countables <020520051547403631%anniel@nym.alias.net.invalid> > I think you mean to ask: > Is the union of a countably infinite disjoint set of countably infinite > disjoint sets countably infinite? I think so. And I think the answer is yes. No? Ken === Subject: Re: 7.2569463050... Mrs Unreliable escribi.97: >> Anyone recognize this number? Hint--the next number is six... > 2^(1/3)*(9/5)^2 + 2^(5/3) > Tracie. Nope, 2^(1/3)*(9/5)^2 + 2^(5/3) ~= 7.25694630559578800334917364612... The tenth digit from decimal point is different. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Pyramid edges What has this rambling discourse to do with the question? === Subject: Re: Pyramid edges > What has this rambling discourse What discourse? > to do with the question? What question? === Subject: Re: Pyramid edges he removed the problem! anyway, it seemed to me that he was relying on an analogy from trigons, that the altitudes coincide at one point, but this is not true of tetrahedra in the general case. if I recall correctly, the special case where they *do* coincide is either orthorectangular ones, or isoceles ones. ut, you have to read _Modern Pure Solid Geometry_ for the defs, as far as I know; it's the best book of constructive geometry, totally 3d, and totally elementary. thus: uh, that is that the NYT weather miscellany box records lots of low-temperature records. just last week, on two days, they had a map with simultaneuos high and low records for the continental USA, which may occur every day, for all that I know. here's my favorite Sunday headline from the LAtribcoTimes, in the back of the Sunday paper a couple of years ago: A Hundred New Glaciers Dyscivered in the Rockies, the Continental Divide, to be more exact. alas, one cannot normally see those areas, because of near-constant cloudcover. --Chair Man George XOR Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://larouchepub.com http://members.tripod.com/~american_almanac === Subject: Re: Pyramid edges here it is: An irregular triangular pyramid (tetrahedron) stands 189 units high, and if rolled on to another face it is then 201.6 units high. Given that the six edges are integers,what are they?. All edges are of different length. are you assuming that the altitudes are concurrent? thus: uh, that is that the NYT weather miscellany box records lots of low-temperature records. just last week, on two days, they had a map with simultaneuos high and low records for the continental USA, which may occur every day, for all that I know. here's my favorite Sunday headline from the LAtribcoTimes, in the back of the paper a couple of years ago: A Hundred New Glaciers Dyscivered in the Rockies, the Continental Divide, to be more exact. alas, one cannot normally see those areas, because of near-constant cloudcover. --Chair Man George XOR Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://larouchepub.com http://members.tripod.com/~american_almanac === Subject: Re: Help with simple integral please! > is the integral of sin(x^2)= -cos(x^2)/(2x) +c? > I got this using substitution > let u = x^2 > du = 2x dx > int(sin(u)(du/2x) > now heres the question. Can you take the 2x out because x isnt the > variable u? > 1/2x int(sin(u)du)= (1/2x)(-cos(u))+c > -cos(x^2)/(2x)+c > If this is right can someone please confirm, and if not, can someone > point out the mistake? When you ask is the integral of f(x) = F(x), you should be able to answer by differentiating F(x). In this case d(-cos(x^2)/(2x))/dx =/= sin(x^2). Bernard === Subject: Re: Computer checking of math proofs tsk, in deed! > If someone knows of a proof checker capable, please point > it out. > Note, it has to be able to handle algebraic integers. > Who designs the proof checkers? thus: uh, that is that the NYT weather miscellany box records lots of low-temperature records. just last week, on two days, they had a map with simultaneuos high and low records for the continental USA, which may occur every day, for all that I know. here's my favorite Sunday headline from the LAtribcoTimes, in the back of the Sunday paper a couple of years ago: A Hundred New Glaciers Dyscivered in the Rockies, the Continental Divide, to be more exact. alas, one cannot normally see those areas, because of near-constant cloudcover. --Chair Man George XOR Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://larouchepub.com http://members.tripod.com/~ame rican_almanac === Subject: Re: Question: Choleski decomposition with Lii = 0, how to handle? > Using a Choleski decomposition, the following matrix will fail to > due zero being divided by zero, Lij = (Aji - Sum) / Lii: > A B C > A 1 1 1 > B 1 1 1 > C 1 1 1 This is a rank one, ie singular, matrix and cannot be factored by straightforward Cholesky. Increase the diagonal to 1.01 to get a soluble system rusty === Subject: Re: Extensions of continuous linear functions > Hi all, > Consider the following theorem: if V is a vector subspace of some normed > vector space and if f is a continuous linear function from V into some > normed vector space W, then f can be extended to one and only one linear > continuous function F from the closure of V into W, which happens to be > such that ||F|| = ||f||. > Can anyone provide some interesting application of the existence part > of this theorem? To be more precise, I would like to have some example > of a concrete function f between concrete normed vector spaces such that > the existence of the extension of f to the closure of V is not obvious. > And please do not tell me that what's obvious to some persons is not > obvious at all to others; I'm fully aware of it. And, besides, the same > thing can be said about the term interesting. This is a very useful theorem indeed. When I took analysis, we called it the BLT (Bounded Linear Transformation) theorem. One interesting application we discussed is to use this to define the Riemann integral. Given an interval [a,b], consider the Banach space B([a,b]) of all bounded functions f : [a,b] -> R, equipped with the uniform norm. Within this, consider the subspace S of step functions: those which can be written as a linear combination of indicator functions of intervals. In other words, S is the functions which are piecewise constant. Now for such a function f, it is obvious how to define the definite integral I(f). And it is easy to check that the functional I : S -> R is continuous. So by this theorem, it has a unique continuous linear extension to the closure of S, which we define to be the Riemann integral. Computing the closure of S is left as an exercise :), but it certainly contains all piecewise continuous functions on [a,b]. This definition gives you properties like linearity and continuity of the integral for free. Another advantage is that you can use the same approach to define the Riemann integral of functions valued in a Banach space X; just replace R with X above. It also lends itself easily to generalizations like the Riemann-Stieljtes integral. The lecture notes for that course are available online. The professor uses the BLT theorem a lot, so you can probably find some other interesting applications there. See http://math.ucsd.edu/~driver/240A-C-03-04/Lecture Notes/anal1p-new.pdf . The Riemann integral is discussed in Chapter 10. Hope this helps! === Subject: pi*(x) question Can anyone help me out with this? I know that the function pi*(x) can be expressed as: pi(x) + 1/2pi( x^1/2 ) + 1/3pi( x^1/3 ) + 1/4pi( x^1/4) + ... (Where pi(x) is the prime counting function, of course) And I also know that it can be expressed, as Riemann found, via: Li( x ) - Sum( Li( x^(zeta zeroes) ) ) - ln 2 + a tiny integral. Those two have been pretty easy to track down. Does anyone know of any other identities that equal pi*( x )? Or can anyone even point me in the direction of where I might be able to find such identities, or where I might look? Are there any of note, at all? I've scoured my books by Ivic, Ingham, and Edwards, and I've read many published papers on prime counting from the last two decades, but I haven't been able to find reference to any other identities for pi*( x ) anywhere. === Subject: mathematics for the sciences We know already when we look at the logistic function with a continuos time that we will get an exponential growth that counter exponentially stops. It's an S-curve. Not very interesting. But now the more physical approach of time in steps. We can code that in a computer. And suddenly this relatively simple function become very fascinating. So the physical direction becomes more interesting. And in this way we can go on. Like Zeno in the dichotomy problem interpreted the finite distance like it were infinite steps. The Physicists say why do you do that. Take a constant speed and you don't have a problem then. This is also closer to physics. And so we can go on the same way more. You can't measure something without a margin. So we put margins in our numbers. Normal deductive math is for reliable tools and the will stay. But the sciences need also simulation tools for the weather and a lot more but also for teaching to students. So we are going on that. Scientists who do think by gather information for the hypotheses. They could make diary and when doing it forgetting the time. At some moment they oversee the diary and they get a guess. Something is breaking through in their brain. And some strikes them. Do we hear EUREKA? This is a wider view than only accepting axioms and proof things. But in fact many mathematicians think in the same way. Only they don't accept what physicist accept. Not in the domain. For measurements are a part of the domain. So they have a wider domain. And for the solution room physicist will accept a row of measured numbers. Maybe they can approximate it for a scope with a short mathematical formula. Many see the reals as en extension of the integers. But then you see only to the production of the reals which is surely not in a recursive way exact. But with an axiom that is possible. But then motivate please that axiom. Then the scientist makes this more rational and will put it into words and some people understand it and that is science. With sudden breakthroughs. Accidents can happen, serendipity. And that in the theory as well. That we call simulation. And the secret how we do that is very simple. We introduce margins, and then we are getting for free threshold. Not right fitting into the margins. We are getting for free accumulation of energy and material. We are getting for free breakthroughs. And what then happens is another functioning. A surprise or a bad accident. Unpredictable. But we have expectations. Is this strange? Our projects are based on papers from very famous scientist that are forgotten. But with Internet we are honest and we go on with it. http://www.cs.uidaho.edu/~casey931/conway/games.html Mathematics has a wide realm. We are working on the inductive tool of math. That will also be a part of mathematics for the 21st Century. And all we do it in open source projects. Open for everybody. This is a gather story, already much to read. http://nnw.berlios.de/docs.php/scimathsto/noflash We also tackle NP problems in another way. more in the way physicists will like it. And we need a lot of members as well. We are growing fast. And pure mathematics is for playing. And great as well. of course we have to work together. Our simulation tools have to work together with the deductive analyzing tools. A great pair. http://www.usna.navy.mil/MathDept/wdj/surreal_numbers.html please have a nice day ed === Subject: It is easy to see.... The integral (I) of function f with respect to measure u is defined by I (f du) = I (f+ du) - I (f- du) where f+ and f- are the positive and negative parts of f. Now if f = v - w, where u and v are any non-negative functions, my text (Bartle) goes on to say that it is easy to see that I (f du) = I (v du) - I (w du) How is it possible to arrive at this assuming only the definition of === Subject: Re: It is easy to see.... >The integral (I) of function f with respect to measure u is defined by >I (f du) = I (f+ du) - I (f- du) >where f+ and f- are the positive and negative parts of f. Now if >f = v - w, where u and v are any non-negative functions, my text >(Bartle) goes on to say that it is easy to see that >I (f du) = I (v du) - I (w du) >How is it possible to arrive at this assuming only the definition of >(Lebesque) integral of non-negative functions? Surely at this point in the book there is more known than the definition, for example surely by this point it's been proved that int(f+g) = int(f) + int(g) for positive f,g? (And _is_ there also a hypothesis that f+, f-, u and v all have _finite_ integrals? Must be or what you're asking about is false.) ************************ David C. Ullrich === Subject: Re: It is easy to see.... >The integral (I) of function f with respect to measure u is defined by >I (f du) = I (f+ du) - I (f- du) >where f+ and f- are the positive and negative parts of f. Now if >f = v - w, where u and v are any non-negative functions, my text >(Bartle) goes on to say that it is easy to see that >I (f du) = I (v du) - I (w du) >How is it possible to arrive at this assuming only the definition of >(Lebesque) integral of non-negative functions? By definition you have I (f du) = I (v-w)+ du - I (v-w)- du. (v-w)+ is v-w when v>=w, and 0 elsewhere. (v-w)- is w-v when v<=w, and 0 elsewhere. Let A1 be the set where v>=w, A2 where v<=w. Note that on A1 intersect A2, v=w, so v-w = 0. Then I (v-w)+du = I_{A1}(v-w) du I (v-w)-du = I_{A_2}(w-v) du. We can also write I (v du) = I_{A_1} (v du) + I_{A_2} (v du) - I_{A_1cap A_2} (v du) I (w du) = I_{A_1} (w du) + I_{A_2} (w du) - I_{A_1cap A_2} (w du) so I(v du) - I(w du) = I_{A_1}(v du) - I_{A_1} (w du) +I_{A_2}(v du) - I_{A_2}(w du) = I_{A_1}(v - w)du (since v>=w in A_1) - I_{A_2}(w-v)du (since w>=v in A_2). I am assuming, of course, that you know that if 0<= f <= g, then I (g-f)du = I gdu - I fdu, but that may not be a given. I used to know all this stuff cold, but it's been too many years... -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: It is easy to see.... > I am assuming, of course, that you know that if 0<= f <= g, then > I (g-f)du = I gdu - I fdu, but that may not be a given. I used to know > all this stuff cold, but it's been too many years... responding. === Subject: Re: It is easy to see.... days. My association with the Department is that of an alumnus. > >> I am assuming, of course, that you know that if 0<= f <= g, then >> I (g-f)du = I gdu - I fdu, but that may not be a given. I used to >know >> all this stuff cold, but it's been too many years... >Actually I don't know. Is there a simple proof of this? Since 0<= f <= g, then both g-f and f are nonnegative. So I (g du) = I( (g-f) + f)du = I(g-f)du + I fdu. Therefore, I(g-f)du = I (gdu) - I (fdu). -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Can you calculate the complete set of anti-diagonals from this sublist ? > would I do that? > The diag is 0.1674.. > therefore 0.2785.. is not on that list! > 0.2785.. is an uncountable real for this list! > But there are several ways to calculate the antidiag, including > shuffling the list > as much as you want. > 0.1457.. > 0.2645.. > 0.3878.. > 0.4444.. > 0.1457.. > 0.2645.. > 0.4444.. > 0.3878.. > 0.1457.. > 0.3878.. > 0.2645.. > 0.4444.. > 0.1457.. > 0.3878.. > 0.4444.. > 0.2645.. > 0.1457.. > 0.4444.. > 0.2645.. > 0.3878.. > 0.1457.. > 0.4444.. > 0.3878.. > 0.2645.. > 0.2645.. > 0.1457.. > 0.3878.. > 0.4444.. > 0.2645.. > 0.1457.. > 0.4444.. > 0.3878.. > 0.3878.. > 0.1457.. > 0.2645.. > 0.4444.. > 0.3878.. > 0.1457.. > 0.4444.. > 0.2645.. > 0.4444.. > 0.1457.. > 0.2645.. > 0.3878.. > 0.4444.. > 0.1457.. > 0.3878.. > 0.2645.. > 0.2645.. > 0.3878.. > 0.1457.. > 0.4444.. > 0.2645.. > 0.4444.. > 0.1457.. > 0.3878.. > 0.3878.. > 0.2645.. > 0.1457.. > 0.4444.. > 0.3878.. > 0.4444.. > 0.1457.. > 0.2645.. > 0.4444.. > 0.2645.. > 0.1457.. > 0.3878.. > 0.4444.. > 0.3878.. > 0.1457.. > 0.2645.. > 0.2645.. > 0.3878.. > 0.4444.. > 0.1457.. > 0.2645.. > 0.4444.. > 0.3878.. > 0.1457.. > 0.3878.. > 0.2645.. > 0.4444.. > 0.1457.. > 0.3878.. > 0.4444.. > 0.2645.. > 0.1457.. > 0.4444.. > 0.2645.. > 0.3878.. > 0.1457.. > 0.4444.. > 0.3878.. > 0.2645.. > 0.1457.. > that's the same set of 4 reals in 24 different orders. > these are all the possible diagonals from the 4 reals (not even up to > the anti-diags yet!) > 0.1674 > 0.1648 > 0.1844 > 0.1845 > 0.1448 > 0.1475 > 0.2474 > 0.2448 > 0.3444 > 0.3445 > 0.4448 > 0.4475 > 0.2854 > 0.2458 > 0.3654 > 0.3455 > 0.4658 > 0.4855 > 0.2845 > 0.2477 > 0.3647 > 0.3447 > 0.4677 > 0.4847 > For each diag there is a family of anti-diag real numbers that are > uncountable on this list. > DIAG = 0.1674 > Each diag has a tree of numbers that contain 8 or 9 digits in each > position, as long as its different to the corresponding digit of diag > at every position. > 0. not1 not6 not7 not4 .. > There are approx. 9 * 9 * 9 * 9 antidiags for 0.1674 > 0.2 785.. (different at every digit) > 0.3 785.. (different at every digit) > 0.4 785.. > 0.5 785.. > 0.6 785.. > 0.7 785.. > 0.8 785.. > 0.9 785.. > 0.0 785.. > 0.2 885.. > 0.3 885.. > 0.4 885.. > 0.5 885.. > 0.6 885.. > 0.7 885.. > 0.8 885.. > 0.9 885.. > 0.0 885.. > 0.2 985.. > 0.3 985.. > 0.4 985.. > 0.5 985.. > 0.6 985.. > 0.7 985.. > 0.8 985.. > 0.9 985.. > 0.0 985.. > 0.2 085.. > 0.3 085.. > 0.4 085.. > 0.5 085.. > 0.6 085.. > 0.7 085.. > 0.8 085.. > 0.9 085.. > 0.0 085.. > 0.2 185.. > 0.3 185.. > 0.4 185.. > 0.5 185.. > 0.6 185.. > 0.7 185.. > 0.8 185.. > 0.9 185.. > 0.0 185.. > 0.2 285.. > 0.3 285.. > 0.4 285.. > 0.5 285.. > 0.6 285.. > 0.7 285.. > 0.8 285.. > 0.9 285.. > 0.0 285.. > 0.2 385.. > 0.3 385.. > 0.4 385.. > 0.5 385.. > 0.6 385.. > 0.7 385.. > 0.8 385.. > 0.9 385.. > 0.0 385.. > 0.2 485.. > 0.3 485.. > 0.4 485.. > 0.5 485.. > 0.6 485.. > 0.7 485.. > 0.8 485.. > 0.9 485.. > 0.0 485.. > 0.2 585.. > 0.3 585.. > 0.4 585.. > 0.5 585.. > 0.6 585.. > 0.7 585.. > 0.8 585.. > 0.9 585.. > 0.0 585.. Well, it seems like there must have been rather a lot of real numbers missing from your original list then! You're aware that the real numbers in [0,1] are uncountable? Mike. === Subject: Re: Can you calculate the complete set of anti-diagonals from this sublist ? In sci.logic, Mike Terry <42769d01$0$575$ed2619ec@ptn-nntp-reader03.plus.net>: >> would I do that? >> The diag is 0.1674.. >> therefore 0.2785.. is not on that list! >> 0.2785.. is an uncountable real for this list! >> But there are several ways to calculate the antidiag, including >> shuffling the list >> as much as you want. [listsnip] > Well, it seems like there must have been rather a lot of > real numbers missing from your original list then! You're > aware that the real numbers in [0,1] are uncountable? If there *is* a set that will enumerate all possible decimal expansions in [0,1) in a countable form, then I for one would think that T_b = {0} union {j/b^n: n, j in N, j < b^n} is a very good candidate. Of course all this set does is enumerate all *finite* expansions, base b; for example, T_10 does not contain 1/3 (since 10^n is never a multiple of 3). T_2 is an enumeration of all possible finite coinflip sequences, another of his favorites. Another candidate is of course Q01 = {0} union {p/q: p, q in N, p < q} = Q intersect [0,1). which is denumerable: {0, 1/2, 1/3, 2/3, 1/4, 1/5, 3/4, 2/5, 1/6, 3/5, 1/7, 4/5, 2/7, 1/8, 3/7, 1/9, 5/6, 4/7, 3/8, 2/9, 1/10, ... } (where I've thrown out duplicates). In any event, given a mapping f from N to R, I can construct a (base 10) d from that mapping such that the following hold true: [1] d(i) does not equal f(i)(i), and in fact abs(d(i) - f(i)(i)) > 1 (e.g., if f(i)(i) is 4, one avoids the digits 3 and 5, in order to avoid things like .xx3999... = .xx4000...). [2] d(i) is neither 0 nor 9. [3] Therefore, d is not equal to any f(i), as I can point to the i'th digit of both. In fact, I can prove there's an uncountable number of such d. For any integer n, one gets 5^n or so possible prefixes of antidiags, at least; for any alleged list of the diagonals, I can select another diagonal digit that is in *neither* list, from the 4 remaining. And yes, 1/3 makes a reasonably good d for T_10, as all of the digits of the diagonal are in fact 0: .[0]0 0 0 0 0 ... . 1[0]0 0 0 0 ... . 2 0[0]0 0 0 ... . 3 0 0[0]0 0 ... . 4 0 0 0[0]0 ... . 5 0 0 0 0[0]... I don't know what the diagonal number for Q01 might be, though one can prove without too much trouble that neither 1/e and pi/4 are therein, if one has already proven that 1/e and pi/4 are irrational (and in fact transcendental). Or one can use irrational algebraic numbers such as sqrt(2)/2, though the set of irrational algebraic numbers is also denumerable (primary enumeration key: highest exponent; secondary key: sum of absolute exponents and coefficients). Unfortunately, antidiagonal numbers are not easily computable. Given a mapping m : N -> R, where m encodes the notion of a machine run with blank input, or perhaps simply a machine with finite states which is run (N describes the state-transition matrix), the antidiagonal of all of the m(n) will probably require infinite states. I can't tell without more research/work in this area; one can make certain adjustments to generate a number R from a machine that does weird things. It's easily proven that all rational numbers are constructable (one can construct a machine that emulates long division by a fixed number), but I don't know regarding algebraics. > Mike. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Question about manifolds.. Let M be a connected smooth manifold. A closed 0-form is a smooth real-valued function f such that df = 0. Why is it true that since M is connected, df = 0 if and only if f is constant? (Why do we need connectedness?) James === Subject: Re: Question about manifolds.. > Let M be a connected smooth manifold. A closed 0-form is a smooth > real-valued function f such that df = 0. Why is it true that since M > is connected, df = 0 if and only if f is constant? (Why do we need > connectedness?) If it's not connected, you could have different constants on different components. The function would still be smooth. === Subject: de Rham basic question... Let {M_k} be a countale collection of smooth manifolds, and let M be the disjoint union of the M_k. For each p, I want to show that the inclusion maps i_k : M_k ----> M induce an isomorphism from H^p (M) to the direct product space (I will denote direct product by PROD) PROD H^p (M_k). Let A^p(M) denote the space of differential forms on M. Well, I have proven that the pullback maps i_k* : A^p(M) ---> A^p(M_k) induce an isomorphism from A^p (M) to PROD A^p (M_k). The map is w |----> (i_1*(w), i_2*(w), ...) = (w | M_1, w | M_2, ...) where w | M_1 means w restricted to M_1. I showed that this map is injective and surjective. But I can't tell if I'm done or not. H^p(M) by definition is closed p-forms modded out by exact p-forms, denoted Z^p(M) / B^p (M). Let w be a closed p-form. Would my map H^p(M) --------> PROD H^p(M_k) be w + B^p(M) |-----> (i_1*(w) + B^p(M_1), i_2*(w) + B^p(M_2), ...) ? And then I need to test injectivity, surjectivity? It seems kind of ugly. Is there an easier way to do this once I have that A^p(M) is isomorphic to PROD A^p (M_k) ? James === Subject: Re: de Rham basic question... > Let {M_k} be a countale collection of smooth manifolds, and let M be the > disjoint union of the M_k. For each p, I want to show that the inclusion > maps i_k : M_k ----> M induce an isomorphism from H^p (M) to the direct > product space (I will denote direct product by PROD) PROD H^p (M_k). > Let A^p(M) denote the space of differential forms on M. Well, I have proven > that the pullback maps i_k* : A^p(M) ---> A^p(M_k) induce an isomorphism > from A^p (M) to PROD A^p (M_k). The map is w |----> (i_1*(w), i_2*(w), ...) > = (w | M_1, w | M_2, ...) where w | M_1 means w restricted to M_1. I > showed that this map is injective and surjective. > But I can't tell if I'm done or not. H^p(M) by definition is closed p-forms > modded out by exact p-forms, denoted Z^p(M) / B^p (M). Let w be a closed > p-form. > Would my map H^p(M) --------> PROD H^p(M_k) be > w + B^p(M) |-----> (i_1*(w) + B^p(M_1), i_2*(w) + B^p(M_2), ...) > ? And then I need to test injectivity, surjectivity? It seems kind of > ugly. Is there an easier way to do this once I have that A^p(M) is > isomorphic to PROD A^p (M_k) ? Probably the *easiest* approach is this: Letting d_X denote the deRham codifferential on the forms of the manifold X and denoting your map by J, there is a commutative diagram J^p A^p(M) ---------> PROD A^p(M_i) | | | | d_M | | PROD d_M_i | | | | V J^{p+1} V A^{p+1}(M) -------> PROD A^{p+1}(M_i) where the horizontal arrows are isomorphisms. (The fact that there is such a commutative diagram follows immediately from the fact that the pullbacks of the various inclusions are cochain maps -- that is, each of *them* commutes with the corresponding codifferential.) Once you know this, the isomorphism on the cohomology level is clear .... > James === Subject: sequence of functions This is a repost to an earlier posting. I just want to note that I am looking for a method of solving the problem below that doesn't involve anti-derivatives, if such a solution exists. Let fn:[0,1]-> R be a sequence of continuous functions such that, for each n counting number, fn is differentiable on (0,1). Suppose that fn(0) converges to some number, denoted f(0), and also suppose that the sequence fn' converges uniformly on (0,1) to some function g:(0,1)->R. Prove that the sequence of fn's converges uniformly. What limit does it converge to? Any suggestions? === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> >> >> >> >> Naturally I chose names that even a yank would have heard of. >> >> Interesting experiment: Come to Texas and see how many >> people you can call yank before we have to fire up Ol' Sparky. >> >> Interesting learning experience: Realise that some words have different >> meanings in English and American (qv 'table', 'momentarily', 'pissed', >> etc etc) > >My particular favourite is fanny. I once heard the American wife of >the vicar of Wolston in the English midlands recount how she had been >caught out by fanny. > That doesn't make sense. > AFAIK > American: Fanny=Buttocks > English: Fanny=Vagina. > To be caught out by either would be strange in the extreme! > It makes sense as follows. She was American and knew only the American > meaning: buttocks. She used the word in the company of English people > (because she was living in England) and shocked people who thought she > was referring to the vagina. I think the previous poster was referring to the phrase caught out, which might be, in the meaning you intended, more common on your side of the Atlantic than on his. Or perhaps he understood your meaning yet found it amusing to pretend not... Anyhow, to add to your remarks about how it makes sense, note that the American usage is the mildest euphemism imaginable; I mean, in what other country do you even *need* a euphemism for buttocks? So, ironically, I would guess that the woman in the story, mindful of her role as vicar's wife, was merely trying to err on the side of caution. === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> >> > > > > Naturally I chose names that even a yank would have heard >of. > > Interesting experiment: Come to Texas and see how many > people you can call yank before we have to fire up Ol' >Sparky. > > Interesting learning experience: Realise that some words have >different > meanings in English and American (qv 'table', 'momentarily', >'pissed', > etc etc) >> >>My particular favourite is fanny. I once heard the American >wife of >>the vicar of Wolston in the English midlands recount how she had >been >>caught out by fanny. >> That doesn't make sense. >> AFAIK >> American: Fanny=Buttocks >> English: Fanny=Vagina. >> >> To be caught out by either would be strange in the extreme! >> It makes sense as follows. She was American and knew only the >American >> meaning: buttocks. She used the word in the company of English >people >> (because she was living in England) and shocked people who thought >she >> was referring to the vagina. >I think the previous poster was referring to the phrase >caught out, which might be, in the meaning you intended, >more common on your side of the Atlantic than on his. Or >perhaps he understood your meaning yet found it amusing >to pretend not... I honestly did not understand the meaning of the caught out.. It seems a strange circumlocution for a fart. >Anyhow, to add to your remarks about how it makes sense, >note that the American usage is the mildest euphemism >imaginable; I mean, in what other country do you even >*need* a euphemism for buttocks? So, ironically, I >would guess that the woman in the story, mindful of her >role as vicar's wife, was merely trying to err on the >side of caution. -- Jeremy Boden === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> > I honestly did not understand the meaning of the caught out.. > It seems a strange circumlocution for a fart. Well now, that's not a meaning that occurred to me at all, even when I speculated idly that you weren't being genuine. (No offense intended, btw; in my book it's perfectly OK to pretend now and then.) Caught out here just means tripped up, I think. === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> >> I honestly did not understand the meaning of the caught out.. >> It seems a strange circumlocution for a fart. >Well now, that's not a meaning that occurred to me >at all, even when I speculated idly that you weren't >being genuine. (No offense intended, btw; in my book >it's perfectly OK to pretend now and then.) >Caught out here just means tripped up, I think. It has the same meaning in the UK; hence I still don't (honestly) understand the meaning of the phrase - either UK or USA interpretation. -- Jeremy Boden === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> >> I honestly did not understand the meaning of the caught out.. >> It seems a strange circumlocution for a fart. >Well now, that's not a meaning that occurred to me >at all, even when I speculated idly that you weren't >being genuine. (No offense intended, btw; in my book >it's perfectly OK to pretend now and then.) >Caught out here just means tripped up, I think. > It has the same meaning in the UK; hence I still don't (honestly) > understand the meaning of the phrase - either UK or USA interpretation. Ah, I see. This is after all sci.math, the land of the self-referential conundrum. Clever. Almost disposes me to forgive your making me the fanny of the joke. (It was that word honestly that had me fooled. Next time you will have to say it twice in succession before I will believe you.) === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> >> >> > I honestly did not understand the meaning of the caught out.. > It seems a strange circumlocution for a fart. >> >>Well now, that's not a meaning that occurred to me >>at all, even when I speculated idly that you weren't >>being genuine. (No offense intended, btw; in my book >>it's perfectly OK to pretend now and then.) >> >>Caught out here just means tripped up, I think. >> >> It has the same meaning in the UK; hence I still don't (honestly) >> understand the meaning of the phrase - either UK or USA >interpretation. >Ah, I see. This is after all sci.math, the land of the >self-referential conundrum. Clever. Almost disposes >me to forgive your making me the fanny of the joke. >(It was that word honestly that had me fooled. Next >time you will have to say it twice in succession before >I will believe you.) Honestly, honestly... I interpreted tripped up (and hence caught out) metaphorically, not literally, sorry. -- Jeremy Boden === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> > Honestly, honestly... Were I Diogenes himself I would believe you now! > I interpreted tripped up (and hence caught out) metaphorically, not > literally, sorry. Hey, no offense taken; in my book it's perfectly OK to try to be metaphorical now and then. That you are using the past tense I take to be an excellent sign that this too too sullied thread may now be abandoned. Do have a most pleasant day. === Subject: Re: Crossword clue > Hey, no offense taken; in my book it's perfectly OK to try > to be metaphorical now and then. I never metaphor I didn't like. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability :> : Where does the English that is output come from, then? :> : C-B :> The program produces it. Is it really beyond your :> belief that a program could produce English sentences? : Highly doubtful in this case - but someone should just check (compare : input to output.) Like you, for example? Or are you going to continue shutting your eyes and putting your fingers in your ears? There is nothing doubtful about any of this. I already told you that none of the comments appear in the output. I have looked at both the input and the output, and the output contains words and phrases that do not appear in the input. The following is a small excerpt from the output. Linear arithmetic, the lemmas CAR-CONS, CDR-CONS, SUB1-ADD1, CAR-LESSEQP, CDR-LESSP, CDR-LESSEQP, ADD1-SUB1, and CAR-LESSP, and the definitions of ORDINALP, LESSP, ORD-LESSP, EQUAL, NLISTP, BTMP, UNSOLV-SUBRP, CDR, CAR, LISTP, MEMBER, and ZEROP can be used to establish that the measure: (CONS (ADD1 N) (COUNT X)) decreases according to the well-founded relation ORD-LESSP in each recursive call. Hence, EV is accepted under the principle of definition. The names of the lemmas appear in the input, but none of the words 'Linear', 'arithmetic', 'measure', 'decreases', 'relation', 'well-founded', 'recursive' or 'principle' appear in the input. All you have to do is download the program and run it yourself, which despite your claims to the contrary, you clearly never have done. Stephen === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dfeb6F6r01lmU1@news.dfncis.de> <3dk12rF6sn1r1U1@news.dfncis.de> > :> : Where does the English that is output come from, then? > :> The program produces it. Is it really beyond your > :> belief that a program could produce English sentences? > : Highly doubtful in this case - but someone should just check (compare > : input to output.) > Like you, for example? Or are you going to continue > shutting your eyes and putting your fingers in your ears? Continue? None of the problems that I have cited with the Boyer/Moore paper has anything to do with whether or not they generate English as OUTPUT. The above dialogue was an aside to the actual problems. The problem that I cited relating to English is the fact that it is in the INPUT file, showing that they developed the supposed proof before running their programs. But even without that fact, one can see that the input (section 6 and file UNSOLV.EVENTS) is based on the proof itself (section 5) independent of the presence of English. Whether they generate English or not has nothing to do with the problems: Do you agree that (1) the input file UNSOLV.EVENTS is based on the proof? (2) It doesn't generate the proof? (3) It doesn't verify the proof? (4) It doesn't generate the lemmas? (5) The only thing that the system does, at best, is to prove each lemma that you give it? These are the problems that I have been talking about, as well as the fact that neither they nor anyone here has ever established that the proof actually proves that HP is unsolvable. > There is nothing doubtful about any of this. I already told you that > none of the comments appear in the output. I have looked at both the > input and the output, and the output contains words and phrases that do > not appear in the input. Well good. I'll even assume that there isn't a table somewhere into which they manually typed these phrases. So they can generate natural language! Now all they have to do is to generate a proof, or at least verify it, and they'll live up to the title of the paper. But even then, the word mechanical in the title, and the first sentence, We describe a proof by a computer program of the unsolvability of the halting problem. is clearly saying that their program generated the proof, which is not true. > All you have to do is download the program and run > it yourself, which despite your claims to the contrary, > you clearly never have done. Now now. Aren't you just attacking the messenger? How about responding to the message - the problems that I enumerate above? C-B > Stephen === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability :> :> : Where does the English that is output come from, then? :> :> The program produces it. Is it really beyond your :> :> belief that a program could produce English sentences? :> : Highly doubtful in this case - but someone should just check : (compare :> : input to output.) :> Like you, for example? Or are you going to continue :> shutting your eyes and putting your fingers in your ears? : Continue? None of the problems that I have cited with the : Boyer/Moore paper has anything to do with whether or not they generate : English as OUTPUT. The above dialogue was an aside to the actual : problems. Until I see some evidence that you have actually tried their system and read the papers that explain their system instead of just harping on about one proof in one paper, then yes, you are continuing to shut your eyes and stick your fingers in your ears. : The problem that I cited relating to English is the fact that it is in : the INPUT file, showing that they developed the supposed proof before : running their programs. The developed a proof sketch. That is very typical of theorem provers. Theorem provers are supposed to prove given theorems, not generate them. You may disagree, but that is the common accepted idea of what a theorem prover is. A good chunk of the input are the definitions needed to actually phrase the question. They did not design their logic to prove the halting problem as you did. They do not have a primitive called HALTS. They have a somewhat low level, and very general logic, and had to build up some larger definitions before the halting question could even be expressed. : But even without that fact, one can see that the input (section 6 and : file UNSOLV.EVENTS) is based on the proof itself (section 5) : independent of the presence of English. Whether they generate English : or not has nothing to do with the problems: : Do you agree that (1) the input file UNSOLV.EVENTS is based on the : proof? (2) It doesn't generate the proof? (3) It doesn't verify : the proof? (4) It doesn't generate the lemmas? (5) The only thing : that the system does, at best, is to prove each lemma that you give it? They provide the system with definitions and a proof sketch. The proof sketch is not necessary, but for long complicated proofs the system likely will not find the proof without some assistance. The system finds a formal proof of the proof sketch and in so doing, verifies the proof. If the system finds a proof, then a proof must exist, and the theorem must be true. Anyway, a theorem with unproven lemmas has not been proven, so just the fact that it proves the lemmas means that it is proving the theorem. Given that this was originally done back in the late 70's, and given that computer speeds and memory are far far greater than what they were back then, it is quite likely that the proof sketch could be made even sketchier and the system would still find a proof in a reasonable amount of time. On my machine it only took it a minute to generate the proof. I imagine that it took a lot longer 20+ years ago. : These are the problems that I have been talking about, as well as the : fact that neither they nor anyone here has ever established that the : proof actually proves that HP is unsolvable. No. You have been talking about fraud. If you think there is something wrong with the proof, go through the output and point out exactly which of the statements in the proof is unjustified. The documentation describing the logic behind the system can be found online. It is not all contained in the one paper you have read. :> There is nothing doubtful about any of this. I already told you that :> none of the comments appear in the output. I have looked at both the :> input and the output, and the output contains words and phrases that : do :> not appear in the input. : Well good. I'll even assume that there isn't a table somewhere : into which they manually typed these phrases. So they can generate : natural language! Now all they have to do is to generate a proof, or : at least verify it, and they'll live up to the title of the paper. The system certainly verified a proof. Given that the proof provided in the input has many unproven assertions in it, it is really not a proof until the system proves that it is. The title of the paper is A Mechanical Proof of the Unsolvability of the Halting Problem. What is your definition of Mechanical Proof, and is it the generally accepted definition? Who said that a Mechanical Proof must be generated by a program? Their proof sketch certainly was verified by a program. : But even then, the word mechanical in the title, and the first : sentence, We describe a proof by a computer program of the : unsolvability of the halting problem. is clearly saying that their : program generated the proof, which is not true. It does generate a proof. A proof sketch is not a proof, and it is certainly not a formal proof. I have tried the system on much simpler conjectures and it generates a proof when given nothing but the statement of the conjecture. :> All you have to do is download the program and run :> it yourself, which despite your claims to the contrary, :> you clearly never have done. : Now now. Aren't you just attacking the messenger? How about : responding to the message - the problems that I enumerate above? : C-B Your message seems to be I do not know what a theorem prover is. You have your own personal definition that is not the definition used by people who work with theorem provers. Perhaps you have built a theorem generator. Of course, I will not believe that until you make the code available and I can try it out myself. Stephen === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dk12rF6sn1r1U1@news.dfncis.de> > Until I see some evidence that you have actually tried > their system and read the papers that explain their system > instead of just harping on about one proof in one paper, > then yes, you are continuing to shut your eyes and stick > your fingers in your ears. An author has to pass some sort of a test? What bull. > Stephen === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dfeb6F6r01lmU1@news.dfncis.de> <3dk12rF6sn1r1U1@news.dfncis.de> Obscurity, linux) > :> > :> : Where does the English that is output come from, then? > :> > :> : C-B > :> > :> The program produces it. Is it really beyond your > :> belief that a program could produce English sentences? > : Highly doubtful in this case - but someone should just check (compare > : input to output.) > Like you, for example? Or are you going to continue > shutting your eyes and putting your fingers in your ears? > There is nothing doubtful about any of this. I already told you that > none of the comments appear in the output. I have looked at both the > input and the output, and the output contains words and phrases that do > not appear in the input. > The following is a small excerpt from the output. > Linear arithmetic, the lemmas CAR-CONS, CDR-CONS, SUB1-ADD1, > CAR-LESSEQP, CDR-LESSP, CDR-LESSEQP, ADD1-SUB1, and CAR-LESSP, > and the definitions of ORDINALP, LESSP, ORD-LESSP, EQUAL, NLISTP, > BTMP, UNSOLV-SUBRP, CDR, CAR, LISTP, MEMBER, and ZEROP can be > used to establish that the measure: > (CONS (ADD1 N) (COUNT X)) > decreases according to the well-founded relation ORD-LESSP > in each recursive call. Hence, EV is accepted under the principle > of definition. > The names of the lemmas appear in the input, but none of the > words 'Linear', 'arithmetic', 'measure', 'decreases', > 'relation', 'well-founded', 'recursive' or 'principle' > appear in the input. > All you have to do is download the program and run > it yourself, which despite your claims to the contrary, > you clearly never have done. And, of course, if you want to edit the input to try and fool the system, then have a go! Stick in a negation in the conclusion of a lemma, and see what difference it makes to the English output! Surely anyone seriously interested in trying to find out what is going on would have done this some time ago ... > Stephen -- Alan Smaill === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dfeb6F6r01lmU1@news.dfncis.de> <3dk12rF6sn1r1U1@news.dfncis.de> > And, of course, if you want to edit the input to try and fool the > system, then have a go! Stick in a negation in the conclusion of a > lemma, and see what difference it makes to the English output! > Surely anyone seriously interested in trying to find out what is going on > would have done this some time ago ... Indeed. All of of those people who claim to have a sincere interest in NLP (Natural Language Processing) have been mysteriously silent! Quite suspicious . . . C-B > Stephen > -- > Alan Smaill === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dfeb6F6r01lmU1@news.dfncis.de> <3dk12rF6sn1r1U1@news.dfncis.de> Obscurity, linux) >> And, of course, if you want to edit the input to try and fool the >> system, then have a go! Stick in a negation in the conclusion of a >> lemma, and see what difference it makes to the English output! >> Surely anyone seriously interested in trying to find out what is > going on >> would have done this some time ago ... > Indeed. All of of those people who claim to have a sincere interest in > NLP (Natural Language Processing) have been mysteriously silent! Quite > suspicious . . . Of course, my comment addressed the soundness of the system: what happens if the user claims that something is a theorem when it is not? Surely anyone seriously interested in trying to find out what is going on would have done this some time ago. Quite suspicious . . . > C-B >> Stephen >> -- >> Alan Smaill -- Alan Smaill === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability :> There is nothing doubtful about any of this. I already told you that :> none of the comments appear in the output. I have looked at both the :> input and the output, and the output contains words and phrases that do :> not appear in the input. :> The following is a small excerpt from the output. :> Linear arithmetic, the lemmas CAR-CONS, CDR-CONS, SUB1-ADD1, :> CAR-LESSEQP, CDR-LESSP, CDR-LESSEQP, ADD1-SUB1, and CAR-LESSP, :> and the definitions of ORDINALP, LESSP, ORD-LESSP, EQUAL, NLISTP, :> BTMP, UNSOLV-SUBRP, CDR, CAR, LISTP, MEMBER, and ZEROP can be :> used to establish that the measure: :> (CONS (ADD1 N) (COUNT X)) :> decreases according to the well-founded relation ORD-LESSP :> in each recursive call. Hence, EV is accepted under the principle :> of definition. :> The names of the lemmas appear in the input, but none of the :> words 'Linear', 'arithmetic', 'measure', 'decreases', :> 'relation', 'well-founded', 'recursive' or 'principle' :> appear in the input. :> All you have to do is download the program and run :> it yourself, which despite your claims to the contrary, :> you clearly never have done. : And, of course, if you want to edit the input to try and fool the : system, then have a go! Stick in a negation in the conclusion of a : lemma, and see what difference it makes to the English output! If you remove all the English comments you get the exact same output. And if you modify the final theorem, by throwing an extra negation in there as you suggest, you get 9000 lines of output that end with the following. which simplifies, applying CDR-CONS, CAR-CONS, SUB1-ADD1, COROLLARY1, and EXPAND-CIRC, and unfolding CDR, CAR, LISTP, SUBLIS, ASSOC, EQUAL, CONS, APPEND, FA, UNSOLV-SUBRP, EV, BTMP, NOT, GET, NUMBERP, PAIRLIST, UNSOLV-APPLY-SUBR, IMPLIES, and AND, to the following seven new goals: Case 2.7. (NOT (EQUAL N 0)). This again simplifies, clearly, to: F, which means the proof attempt has ************** F A I L E D ************** : Surely anyone seriously interested in trying to find out what is going on : would have done this some time ago ... You would think so, wouldn't you? Stephen === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dfeb6F6r01lmU1@news.dfncis.de> <3dk12rF6sn1r1U1@news.dfncis.de> > If you remove all the English comments you get the exact same output. > And if you modify the final theorem, by throwing an extra > negation in there as you suggest, you get 9000 lines of > output that end with the following. > which simplifies, applying CDR-CONS, CAR-CONS, SUB1-ADD1, COROLLARY1, and > EXPAND-CIRC, and unfolding CDR, CAR, LISTP, SUBLIS, ASSOC, EQUAL, CONS, > APPEND, FA, UNSOLV-SUBRP, EV, BTMP, NOT, GET, NUMBERP, PAIRLIST, > UNSOLV-APPLY-SUBR, IMPLIES, and AND, to the following seven new goals: > Case 2.7. > (NOT (EQUAL N 0)). > This again simplifies, clearly, to: > F, > which means the proof attempt has > ************** F A I L E D ************** Neat! > : Surely anyone seriously interested in trying to find out what is going on > : would have done this some time ago ... > You would think so, wouldn't you? You've got 69 insincere people (well, messages that is) here. Damn! C-B > Stephen === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dk12rF6sn1r1U1@news.dfncis.de> <3dkltnF6uar10U1@news.dfncis.de> <3dm8rfF6rg17iU1@news.dfncis.de> > Right, those lemmas were coded by hand by the authors, the > verifier did not construct those automatically. The proofs of those > lemmas are derived automatically. You are curious to see the details > of the automatically derived proofs of those lemmas? Not that much. You have to write the proof. You have to write the lemmas. The system proved the lemmas? Ho-hum. Hardly worth using the system, is it? The individual lemmas are so far removed from the original theorem, what's the point? Proving the individual lemmas doesn't show that the proof represents HP or is even valid. >>No, the paper does not give >>the proof of each of these lemmas since that would make the paper unwieldy >>(with as Stephen says 9000 lines) > You're repeating the same discredited argument. > Hmmm...you may disagree with the argument, but I haven't seen > an attempt yet at it being discredited, other than a bald > statment that it is absurd. At least twice I have pointed out that any argument can be summarized* - abstracted to a higher level - thus making that argument null and void. Actually, see here: (That's bold, not bald.) > Any logical argument > can be summarized, as I say, abstracted to a higher level. > Any is a bit strong, but I'll agree informally. Couldn't you agree that > the helper lemmas given in unsolv.events are an abstraction to a higher > level than the supposed 9000 lines of (what I expect is) symbolic rewriting. Can YOU explain how the output shows us that HP must be unsolvable? > Saying that > the proof is too long to explain doesn't make sense. > Because 9000 lines is -not- too long? > I think it speaks to the inappropriateness of inclusion > in a JACM paper. LOL Nobody's saying to include 9,000 lines in a paper. I'm saying to include a justification for the assertion that those 9,0000 lines imply that HP is unsolvable. > I think the > attitude of the paper is that lots had already been > written about their system, so there was no need for > that. Not really. Lots has not been written about how that assures us that HP must be unsolvable. (I think their attitude is obfuscate, obfuscate, obfuscate.) > So section 5 (or section 6 or unsolv.events) is not enough to be > called their argument? If it were an argument, then someone could explain how it shows us that HP must be unsolvable. > Think of it like machine code. Why read that > when you can just read the original? I'm talking about the level above that - in your analogy, the specs! > It doesn't even have a concept of the overall proof. It only > processes each lemma, one at a time. The English contained in its > output that refers to a proof was of course merely input by them. > Would you feel better if they had made their title Mechanical > -Verification- of the Halting Problem? That's a step in the right direction. (Fix your wording, thought.) Next they need to show how that proves that HP is unsolvable. > I think it a shortcoming to have to see bajillions of theorems like > p or -p, > (p or -p) and (p or -p), > (p or -p) and (p or -p) and (p or -p), ... > before one gets to a cool one like UHP. If your logic is set > up such that your axioms and definitions generate exactly the > interesting theorems and none of these boring ones, that's excellent. Not the logic, the heuristics. Yes, it's easy to just keep the theorems that use only certain relations e.g. HALT(a,b). >>So, you're right, no quantifiers, but that's not a limitation, it doesn't >>make their logic a restriction to propositional logic. > I was showing that the assertion that it is not a dinky little > propositional calculus theorem-prover (at best) is not true. > How did you show that? EVAL (what they use to formalize > turing machines) takes functional arguments, that seems > pretty non-propositional to me. Either you got the quantifiers or you ain't. Excluded Middle, remember? C-B * except this thread > -- > Mitch Harris > (remove q to reply) === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability > Can YOU explain how the output shows us that HP must be unsolvable? Yes, by repeating a lot of what Boyer and Moore have already said. > LOL Nobody's saying to include 9,000 lines in a paper. I'm saying > to include a justification for the assertion that those 9,0000 lines > imply that HP is unsolvable. The justification is in the paper. >It doesn't even have a concept of the overall proof. It only >processes each lemma, one at a time. It also combines the lemmas appropriately to support the main theorem. >>Would you feel better if they had made their title Mechanical >>-Verification- of the Halting Problem? > That's a step in the right direction. (Fix your wording, thought.) > Next they need to show how that proves that HP is unsolvable. They did that. Your explanation of your denial of this makes it look like you don't understand them. >I was showing that the assertion that it is not a dinky little >propositional calculus theorem-prover (at best) is not true. >>How did you show that? EVAL (what they use to formalize >>turing machines) takes functional arguments, that seems >>pretty non-propositional to me. > Either you got the quantifiers or you ain't. So you claim that if the language of a logic has no forall or there exists quantifiers, then it must be propositional? For example, propositional modal logics are (despite the name) not equivalent to classical propositional logic. In their dialect of LISP, their definitions are essentially lambda terms which are a way of binding variables, which act very much like quantifiers. These kinds of value judgements do not seem to be supported by any technical reason. -- Mitch Harris (remove q to reply) === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dk12rF6sn1r1U1@news.dfncis.de> <3dkltnF6uar10U1@news.dfncis.de> <3dm8rfF6rg17iU1@news.dfncis.de> <3dp1snF6sec9tU1@news.dfncis.de> > Can YOU explain how the output shows us that HP must be unsolvable? > Yes, by repeating a lot of what Boyer and Moore have already said. Ok, then do it. Here's your chance. I'll leave lots of room for you: (Mitch Harris' explanation as to how the output proves that HP must be unsolvable goes here.) C-B > -- > Mitch Harris > (remove q to reply) === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris) >> Can YOU explain how the output shows us that HP must be unsolvable? >> Yes, by repeating a lot of what Boyer and Moore have already said. >Ok, then do it. Here's your chance. I'll leave lots of room for you: I am unwilling to do that. You win. -- Mitch === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dp1snF6sec9tU1@news.dfncis.de> <3dq21uF6pifksU1@news.dfncis.de> >> Can YOU explain how the output shows us that HP must be unsolvable? >> >> Yes, by repeating a lot of what Boyer and Moore have already said. >Ok, then do it. Here's your chance. I'll leave lots of room for you: > I am unwilling to do that. You win. No, we all win. Nobody gains when BS flourishes. C-B > -- > Mitch === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability The halting problem is also the G.9adel uncertainty and also informal an ugly dirty man. That is me. And after a journey I arrive in a villige with very neat people. Do you know this story? I need a barber for sure. So I look around... And then a see a so dirty man. Btw. For others we are the same. (sets with the same elements are the same sets. That is the normal set theory) But now I know he must be the barber. He can't shave himself with out harming himself. So I found the barber. This is where the program can't find a solution. Undeciable. The barber can't shave himself. He will shave me. But I am the barber as well. ---- discusion We don't know if this barber can shave us. And we can solve it with a wider view. Oh we know it. That barber is just a lazy man. He shaves his son but he doesn't trust his son and the son is the barber of this villiage. He can shave me for sure. Deductive is the limiting notion. Scientists have to gather more information. What is your opinion. === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dk12rF6sn1r1U1@news.dfncis.de> <3dkmd1F6uar10U2@news.dfncis.de> <3dm610F6ovhs0U1@news.dfncis.de> > I agree that it is sometimes nice to see the proof. The proofs > written by humans are usually much more appropriate for > understanding by humans (who have never seen the theorem before) > than the computer generated ones. You should realize that what the typical such program produces is not a proof - it's the next to the last step. An actual proof is created by mapping each line in the formal output to a logically valid inference, which of course can be expressed in English plus mathematical expressions. A complete system would do that mapping and actually produce the English proof. > But the proof verifier nqthm had years of successful testing beforehand. This is the same issue as above. You can show formalisms and you can show that they were generated by a program, but you don't have anything unless you show that they represent something. Even if each step is valid in some sense, does the final conclusion mean that HP is unsolvable? That's really the first problem: formalizing the statement of HP in a language in which we can draw inferences. In my system, it is -HALT(I,J), meaning the relation of a program halting on an input is not recursive. And the 8 rules of inference infer statements in this language. > Their only great discovery was a formalization of the argument, > not the actual proof. (Putting aside whether I agree that they actually did that,) that also leaves out the construction of that argument. And even then, can you really formalize only one thing? I am talking about Theory of Computation theorems - those related to HP. What does it take to write a computer program to output one theorem? (Answer: 1 write command.) To be authentic, you pretty much have to have a system for formalizing an infinite number of things (e.g. proofs) and the interested reader can see that it preserves the relationships among them e.g. if P => Q then the formalism for P begats the formalism for Q. This is just another reason why I see the Boyer/Moore result as being insipid. Where are all of the variations of HP e.g. Self-applicability (Does a given program halt on itself?), Membership (Does a given program and input halt yes?), Always-halting (Does a given program halt on all inputs?), Ever-halting (Does a given program ever halt on an input?) etc? > Again (again), if you really, really, -really- > want to see their -full- version of the proof, then run nqthm. I first saw that years ago. Unfortunately, the user has to write the proof in order to produce it, and they never show that it represents the Halting Problem being unsolvable. Figure it out yourself. in time proportionate to the log of the length (log) of the number: (BLAH-BLAH) Just follow through the code and you'll see. THEY need to substantiate their claim that it implies that HP is unsolvable. > And a paper that claims to have it but doesn't give it is running the > risk that (1) they are just faking it, or (2) they made a mistake and > don't really have it. > Yes, there's always that worry. Fortunately, in this situation, one can > reproduce the experiment with -very- low cost by downloading the software, > installing it, and running the software on the file in question. 9,000 lines of output is low cost? They have formalisms, but (1) how were they produced and (2) what do they signify? Answers: (1) by hand, (2) Boyer and Moore are in no hurry to show you. It's not a question of how the system works in general. It is a question of whether these particular formalisms represent this particular result (HP.) It's up to them to justify that claim. > Nobody has answered my repeated request for the logical argument that > shows us that HP is unsolvable that they present that people claim the > system created or verified. > All I can do is point out that maybe you are referring to nobody -here-. Yes. Or elsewhere, for that matter. How many of us can give Turing's proof? [all] How many of us can give Boyer/Moore's proof? [none] > The truth of the matter is, they just haven't figured out the > problem. They really haven't. If anyone is interested in reality > (and not this stupid nonsense about All it has to do is say > 'yes', that's enough for most people.), > I wouldn't say for most people. Most people have not been exposed to > formalized proofs where there are a huge number of mind- > numblingly boring logical steps. Have you ever wondered why such a simple proof as HP (requiring about 4 English sentences) would take 9,000 lines to prove? > in an authentic system: > 1. You would set it up once with ... > It's still here: http://www.arxiv.org/html/cs.lo/0003071 You can't stand in the way of progress. C-B > OK. Neat. Progress in technology. Ok, you're checked off my list of people to convince. (That's 2 in one day.) > -- > Mitch Harris > (remove q to reply) === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability >>Again (again), if you really, really, -really- >>want to see their -full- version of the proof, then run nqthm. > I first saw that years ago. Unfortunately, the user has to write the > proof in order to produce it, The user has to write -some- of the proof... > and they never show that it represents > the Halting Problem being unsolvable. Well, you're disagreeing with editors and reviewers of JACM papers. That doesn't make you wrong. > in time proportionate to the log of the length (log) of the number: > (BLAH-BLAH) Just follow through the code and you'll see. THEY need > to substantiate their claim that it implies that HP is unsolvable. The reviewers of the paper knew enough about the background of the paper to accept the explanation of their claim. > 9,000 lines of output is low cost? They have formalisms, but (1) how > were they produced and (2) what do they signify? Answers: (1) by hand, > (2) Boyer and Moore are in no hurry to show you. Yep, they're pretty relaxed about it given that they explain the significance of their formalisms (the logic, EVAL, HALTS, CIRC, LOOP, BTM, BTMP, x, fa, va, k, etc.) -in- that paper. Hmm..actually the logic is not very well explained..er.. there is the appendix in the paper..well..maybe that's not enough...they refer to their first and second bibliography entries for further explanation if needed. >The truth of the matter is, they just haven't figured out the >problem. They really haven't. If anyone is interested in reality >(and not this stupid nonsense about All it has to do is say >'yes', that's enough for most people.), >>I wouldn't say for most people. Most people have not been exposed to >>formalized proofs where there are a huge number of mind- >>numblingly boring logical steps. > Have you ever wondered why such a simple proof as HP (requiring about 4 > English sentences) would take 9,000 lines to prove? Because formalization is not as simple a task as one would intuitively expect? -- Mitch Harris (remove q to reply) === Subject: Re: A Small Blackboard > ***************************************** > * 1 7 > * 5 > * sum of numbers on the blackboard = > ***************************************** > OK Timmy, put the sum of numbers on the blackboard on the blackboard! > ***************************************** > * 1 7 > * 5 > * sum of numbers on the blackboard = 13 > ***************************************** > Bad Timmy, you got it WRONG! > The sum of numbers on the blackboard is 1 + 5 + 7 + 13 = 26 This reminds me of Finsler's paper Formale Beweise und die Entscheidbarkeit Mathematische Zeitschrift 25, 676-682. === Subject: Re: A Small Blackboard > ***************************************** > * 1 7 > * 5 > * sum of numbers on the blackboard = > ***************************************** > OK Timmy, put the sum of numbers on the blackboard on the blackboard! > ***************************************** > * 1 7 > * 5 > * sum of numbers on the blackboard = 13 > ***************************************** ***************************************** * * 1 4 * 4 * * sum of numbers on the blackboard = 11 * ***************************************** So the sum of the numbers is: 1+4+4+1+1 = 11 I win. Alun Harford === Subject: sum ((-1)^n)/(ln n + cos n) Was: Where do I begin? >Consider the related series >sum ((-1)^n)/(ln n + cos ( 2 pi n p/q ) ) >where p/q is a fraction in lowest terms. and (in a subsequent post): >Let q be even; we will show that it DIverges. >The sum of these q terms is then >sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) +O(q^2/m log m). >Without the error term, >that sum is a nonzero fraction in x; >its absolute value is at least >1/x^{1 + q/2} = 1/(log m)^{1 + q/2}. The claim here is that the absolute value of sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) is at least equal to 1/x^{1 + q/2}. But this is wrong! For instance, if p/q = 1/12 and x=2, then sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) = 0.005128... but 1/x^{1 + q/2} = 0.007812... Perhaps this is correctable. But my feeling is that the series in the original problem: sum ((-1)^n)/(ln n + cos n) converges, not because of any properties of rational approximations to pi, but because all three partial sums sum[k=m..n] (-1)^k cos k sum[k=m..n] cos 2k sum[k=m..n] cos 2k+1 out in the long run. It would be a wonder if they didn't! But I can't prove it :-( Nevertheless, as the three partial sums sum[k=m..n] (-1)^k cos(k theta) sum[k=m..n] cos(2k theta) sum[k=m..n] cos((2k+1) theta) are also bounded when theta is not a multiple of pi, I respectfully suggest that sum ((-1)^n)/(ln n + cos (n theta)) converges unless theta is a multiple of pi. (Provided, of course, that the denominator is non-zero for all n). === Subject: Re: sum ((-1)^n)/(ln n + cos n) Was: Where do I begin? >>Consider the related series >>sum ((-1)^n)/(ln n + cos ( 2 pi n p/q ) ) >>where p/q is a fraction in lowest terms. > and (in a subsequent post): >>Let q be even; we will show that it DIverges. >>The sum of these q terms is then >>sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) +O(q^2/m log m). >>Without the error term, >>that sum is a nonzero fraction in x; >>its absolute value is at least >>1/x^{1 + q/2} = 1/(log m)^{1 + q/2}. > The claim here is that the absolute value of > sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) > is at least equal to 1/x^{1 + q/2}. But this is wrong! For instance, > if p/q = 1/12 and x=2, then > sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) = 0.005128... > but 1/x^{1 + q/2} = 0.007812... > Perhaps this is correctable. But my feeling is that the series in the > original problem: > sum ((-1)^n)/(ln n + cos n) > converges, not because of any properties of rational approximations to > pi, but because all three partial sums > sum[k=m..n] (-1)^k cos k > sum[k=m..n] cos 2k > sum[k=m..n] cos 2k+1 > out in the long run. It would be a wonder if they didn't! > But I can't prove it :-( Nevertheless, as the three partial sums > sum[k=m..n] (-1)^k cos(k theta) > sum[k=m..n] cos(2k theta) > sum[k=m..n] cos((2k+1) theta) > are also bounded when theta is not a multiple of pi, I respectfully > suggest that > sum ((-1)^n)/(ln n + cos (n theta)) > converges unless theta is a multiple of pi. (Provided, of course, that > the denominator is non-zero for all n). Why not look at the cauchy sequence of the partial sums? All one has to note is that |cos(m*t)| < ln(3) (nothing special about chosing 3, just that ln(3) > 1) |1/(ln(n) + cos(n*t)) - 1/(ln(m) + cos(m*t))| <= |1/(ln(n) + cos(n*t))| + |1/(ln(m) + cos(m*t))| <= |1/(ln(n) - 1)| + |1/(ln(m) - 1| <= |1/ln(n) - ln(3)| + |1/(ln(m) - ln(3)| <= |1/ln(3n)| + |1/ln(3m)| <= 1/ln(3n) + 1/ln(3m) <= (ln(3n) + ln(3m))/(ln(3n)*ln(3m)) <= ln(9nm)/ln(3n)/ln(3m) what happens when (n,m)->infinity? well, as you can see it ends up being 0. Hence the difference between partial sums converge to 0, hence the sum converges. The main problem one might have is that in the above there might be a case for small n,m where we have problems(such as when n,m <=3), but this is not really a problem as we are only concerned with large n,m). Hope that helps. Jon === Subject: Re: sum ((-1)^n)/(ln n + cos n) Was: Where do I begin? >[...] >> Perhaps this is correctable. But my feeling is that the series in the >> original problem: >> sum ((-1)^n)/(ln n + cos n) >> converges,[...] >Why not look at the cauchy sequence of the partial sums? Uh, good plan. >All one has to note is that |cos(m*t)| < ln(3) (nothing special about >chosing 3, just that ln(3) > 1) >|1/(ln(n) + cos(n*t)) - 1/(ln(m) + cos(m*t))| Unfortunately these are not partial sums of the series. There's no problem showing that the terms tend to zero, which is the most you're proving here. ><= |1/(ln(n) + cos(n*t))| + |1/(ln(m) + cos(m*t))| ><= |1/(ln(n) - 1)| + |1/(ln(m) - 1| ><= |1/ln(n) - ln(3)| + |1/(ln(m) - ln(3)| ><= |1/ln(3n)| + |1/ln(3m)| ><= 1/ln(3n) + 1/ln(3m) ><= (ln(3n) + ln(3m))/(ln(3n)*ln(3m)) ><= ln(9nm)/ln(3n)/ln(3m) >what happens when (n,m)->infinity? >well, as you can see it ends up being 0. >Hence the difference between partial sums converge to 0, hence the sum >converges. >The main problem one might have is that in the above there might be a case >for small n,m where we have problems(such as when n,m <=3), but this is not >really a problem as we are only concerned with large n,m). >Hope that helps. >Jon ************************ David C. Ullrich === Subject: Re: sum ((-1)^n)/(ln n + cos n) Was: Where do I begin? >>[...] > Perhaps this is correctable. But my feeling is that the series in the > original problem: > sum ((-1)^n)/(ln n + cos n) > converges,[...] >>Why not look at the cauchy sequence of the partial sums? > Uh, good plan. >>All one has to note is that |cos(m*t)| < ln(3) (nothing special about >>chosing 3, just that ln(3) > 1) >>|1/(ln(n) + cos(n*t)) - 1/(ln(m) + cos(m*t))| > Unfortunately these are not partial sums of the > series. There's no problem showing that the > terms tend to zero, which is the most you're > proving here. yeah, nots ure what I was thinking ;/ thats what happens when I don't get enough sleep ;/ >><= |1/(ln(n) + cos(n*t))| + |1/(ln(m) + cos(m*t))| >><= |1/(ln(n) - 1)| + |1/(ln(m) - 1| >><= |1/ln(n) - ln(3)| + |1/(ln(m) - ln(3)| >><= |1/ln(3n)| + |1/ln(3m)| >><= 1/ln(3n) + 1/ln(3m) >><= (ln(3n) + ln(3m))/(ln(3n)*ln(3m)) >><= ln(9nm)/ln(3n)/ln(3m) >>what happens when (n,m)->infinity? >>well, as you can see it ends up being 0. >>Hence the difference between partial sums converge to 0, hence the sum >>converges. >>The main problem one might have is that in the above there might be a case >>for small n,m where we have problems(such as when n,m <=3), but this is >>not >>really a problem as we are only concerned with large n,m). >>Hope that helps. >>Jon > ************************ > David C. Ullrich === Subject: Re: sum ((-1)^n)/(ln n + cos n) Was: Where do I begin? >Consider the related series >sum ((-1)^n)/(ln n + cos ( 2 pi n p/q ) ) >where p/q is a fraction in lowest terms. > and (in a subsequent post): >Let q be even; we will show that it DIverges. >The sum of these q terms is then >sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) +O(q^2/m > log m). >Without the error term, >that sum is a nonzero fraction in x; >its absolute value is at least >1/x^{1 + q/2} = 1/(log m)^{1 + q/2}. > The claim here is that the absolute value of > sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) > is at least equal to 1/x^{1 + q/2}. But this is > wrong! For instance, > if p/q = 1/12 and x=2, then > sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) = > = 0.005128... > but 1/x^{1 + q/2} = 0.007812... > Perhaps this is correctable. But my feeling is that > the series in the > original problem: > sum ((-1)^n)/(ln n + cos n) > converges, not because of any properties of rational > approximations to > pi, but because all three partial sums > sum[k=m..n] (-1)^k cos k > sum[k=m..n] cos 2k > sum[k=m..n] cos 2k+1 > are bounded. So the deviations from sum ((-1)^n)/ln n > out in the long run. It would be a wonder if they > didn't! > But I can't prove it :-( Nevertheless, as the three > partial sums > sum[k=m..n] (-1)^k cos(k theta) > sum[k=m..n] cos(2k theta) > sum[k=m..n] cos((2k+1) theta) > are also bounded when theta is not a multiple of pi, Irrelevant, as seen below > I respectfully > suggest that > sum ((-1)^n)/(ln n + cos (n theta)) > converges unless theta is a multiple of pi. > (Provided, of course, that > the denominator is non-zero for all n). I respectfully disagree. Construct theta such that the continued fraction expansion of theta/2pi has increasingly huge entries, in such a way that for an infinite sequence of rational numbers p_k/q_k with even denominator q_k, |theta/2pi - p_k/q_k| < 1/N_k where (N_k/q_k)*(C(q_k)/(log(2N_k))^q_k) > k. Now each of the N_k/q_k blocks of q_k values of n between N_k and 2N_k contributes a value of at least C(q_k)/(log(2N_k))^q_k to the sum before the bias starts shifting in phase, so that the sum at N_k and the sum at 2_k differ by at least k. So it doesn't Cauchy-converge. N > . === Subject: Re: sum ((-1)^n)/(ln n + cos n) Was: Where do I begin? >Consider the related series >sum ((-1)^n)/(ln n + cos ( 2 pi n p/q ) ) >where p/q is a fraction in lowest terms. > and (in a subsequent post): >Let q be even; we will show that it DIverges. >The sum of these q terms is then >sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) +O(q^2/m > log m). >Without the error term, >that sum is a nonzero fraction in x; >its absolute value is at least >1/x^{1 + q/2} = 1/(log m)^{1 + q/2}. > The claim here is that the absolute value of > sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) > is at least equal to 1/x^{1 + q/2}. But this is > wrong! For instance, > if p/q = 1/12 and x=2, then > sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) = > = 0.005128... > but 1/x^{1 + q/2} = 0.007812... My mistake, but minor. The absolute value of the sum is at least C(q)/x^{1+q/2} where C(q) depends on q but not on x. C(12)=3/8. > Perhaps this is correctable. But my feeling is that > the series in the > original problem: > sum ((-1)^n)/(ln n + cos n) > converges, not because of any properties of rational > approximations to > pi, but because all three partial sums > sum[k=m..n] (-1)^k cos k > sum[k=m..n] cos 2k > sum[k=m..n] cos 2k+1 > are bounded. So the deviations from sum ((-1)^n)/ln n > out in the long run. It would be a wonder if they > didn't! > But I can't prove it :-( Nevertheless, as the three > partial sums > sum[k=m..n] (-1)^k cos(k theta) > sum[k=m..n] cos(2k theta) > sum[k=m..n] cos((2k+1) theta) > are also bounded when theta is not a multiple of pi, > I respectfully > suggest that > sum ((-1)^n)/(ln n + cos (n theta)) > converges unless theta is a multiple of pi. > (Provided, of course, that > the denominator is non-zero for all n). Do you mean unless theta is a *rational* multiple of pi? Because it does not converge when theta=2pi/12. === Subject: Re: sum ((-1)^n)/(ln n + cos n) Was: Where do I begin? >Consider the related series >sum ((-1)^n)/(ln n + cos ( 2 pi n p/q ) ) >where p/q is a fraction in lowest terms. >>and (in a subsequent post): >Let q be even; we will show that it DIverges. >The sum of these q terms is then >sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) +O(q^2/m >>log m). >Without the error term, >that sum is a nonzero fraction in x; >its absolute value is at least >1/x^{1 + q/2} = 1/(log m)^{1 + q/2}. >>The claim here is that the absolute value of >> sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) >>is at least equal to 1/x^{1 + q/2}. But this is >>wrong! For instance, >>if p/q = 1/12 and x=2, then >>sum[k=1..q] (-1)^k /(x+cos(2 pi k p/q)) = >>= 0.005128... >>but 1/x^{1 + q/2} = 0.007812... > My mistake, but minor. The absolute value of the sum is at least > C(q)/x^{1+q/2} > where C(q) depends on q but not on x. > C(12)=3/8. [...] Your approach made me think of the average case, where cos(n) is replaced by cos(nt), and we sum from n=100 to some large N: g_N(t) := sum[n=100 ... N] ((-1)^n)/(ln(n) + cos(nt)) . What I'm wondering about is bounds on d/dt g_N(.), at values of t around 1 (or elsewhere...) and large N. David Bernier === Subject: Re: What is the greatest error ever committed in math? >Surely the greatest mistake in maths is committed every time someone >or 5^100. > Can't be more common or more egregious than the Law of Universal Linearity: > (a+b)^2=a^2+b^2, sin(a=b)=sin(a)+sin(b), etc. > (Possibly actually a Universal Distributive property: f(a+b)=f(a)+f(b). > I guess this is also equal to (a+b)f, huh?) > dave Actually, the existence of a function, other than f(x) = kx, for which f(x+y) = f(x) + f(x) apparently follows from the Axiom of Choice. (Or Zorns lemma, to be precise.) === Subject: sequence of continuous functions This is a repost to an earlier posting. I just want to note that I am looking for a method of solving the problem below that doesn't involve anti-derivatives, if such a solution exists. Let fn:[0,1]-> R be a sequence of continuous functions such that, for each n counting number, fn is differentiable on (0,1). Suppose that fn(0) converges to some number, denoted f(0), and also suppose that the sequence fn' converges uniformly on (0,1) to some function g:(0,1)->R. Prove that the sequence of fn's converges uniformly. What limit does it converge to? Any suggestions === Subject: Re: sequence of continuous functions >This is a repost to an earlier posting. Actually a repost of a repost, at least. >I just want to note that I am >looking for a method of solving the problem below that doesn't involve >anti-derivatives, if such a solution exists. Well, it's possible that one could prove that the fn converge without mentioning antiderivatives (and we might also note that the proofs you've been shown using antiderivatives have had a few little gaps in them, since you didn't say that the derivatives were continuous on [0,1]...). But it's not going to be possible to say what the fn converge to without mentioning antiderivatives or integrals, because what fn converges to _is_ f(0) + int_0^x g(t) dt. >Let fn:[0,1]-> R be a sequence of continuous functions such that, for >each n counting number, fn is differentiable on (0,1). Suppose that >fn(0) converges to some number, denoted f(0), and also suppose that the >sequence fn' converges uniformly on (0,1) to some function g:(0,1)->R. >Prove that the sequence of fn's converges uniformly. What limit does >it converge to? >Any suggestions ************************ David C. Ullrich === Subject: Re: Galois Theory Problem > Let {u,v} be algebraically independant over F_p, let K = F_p(u,v) and > let a be a root of X^{2p} - uv X^p + v in K[X] in some algebraic > closure of F_p. Show that K(a)/K is not normal. > Any ideas? > nojb. Hmm, sounds like a homework problem. Anyway, let t, t' be roots of T^2 - uv.T + v = 0. Then the roots of your polynomial over K are t^(1/p) and t'^(1/p). If K(a)/K were normal, then K(a) contains both roots, and hence their sum (uv)^(1/p) and product v^(1/p). So K(a) contains u^(1/p) and v^(1/p). If p>2, we immediately get a contradiction (why?). For p=2, we must have K(a) = K(u^(1/p), v^(1/p)). But the latter is not a simple extension (again, why?). === Subject: Re: Continuity and Differentiation >Let f(x) = sum(k=1..oo) (cos pi.x.9^k)/2^k, domain f = R. >For all x in R, the sequence denoted by f(x) converges. >If x in R, > is f continuous at x? Of course (uniform convergence). > is f not differentiable at x? Consider the partial sums S_n(x) = sum_{k=1}^n cos(pi 9^k x)/2^k. Fix positive integer m, and consider x_j = j/9^m for integers j. If k >= m, cos(pi.x_j.9^k) = (-1)^j, so f(x_j) = S_{m-1}(x_j) + (-1)^j/2^{m-1}. On the other hand, |(S_{m-1})'| <= (9/2)^(m-1) pi/(1 - 2/9) so |S_{m-1}(x_{j+1}) - S_{m-1}(x_j)| <= pi/(7.2^(m-1)) and |f(x_{j+1}) - f(x_j)|/|x_{j+1}-x_j| >= (2 - pi/7).9^m/2^{m-1} For any x that is not one of the x_j, say x_j < x < x_{j+1}, at least one of |f(x_{j+1})-f(x)|/|x_{j+1}-x| and |f(x) - f(x_j)|/|x-x_j| f is nowhere differentiable. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: how to generate random process of any abitary power density function? Hi all, I have a power density function, Sxx(w), where w is the omega, from -pi to +pi, for digital random process(sequence). The Sxx(w) is arbitrarily shaped, and might be very irregular. Can anybody tell me how to generate a realization of random process to possess this power density spectrum Sxx(w)? Esp. in Matlab? For example, if I have an arbitarily generated Sxx variable in Matlab holding a vector of sampled values for the Sxx(w) from -pi to +pi, how to generate the random process correspondingly... ? === Subject: Re: how to generate random process of any abitary power density function? > Hi all, > I have a power density function, Sxx(w), where w is the omega, from -pi > to +pi, for digital random process(sequence). The Sxx(w) is arbitrarily > shaped, and might be very irregular. > Can anybody tell me how to generate a realization of random process to > possess this power density spectrum Sxx(w)? Esp. in Matlab? > For example, if I have an arbitarily generated Sxx variable in Matlab > holding a vector of sampled values for the Sxx(w) from -pi to +pi, how to > generate the random process correspondingly... ? I wouldn't bother with filter methods for a finite block discrete time process as this is a very indirect procedure. The simple way is to create a vector of the block length and fill with random numbers S_k, k=0,..,2^M, such that |S_k| = sqrt of PSD and Arg(S_k) is i.i.d and uniformly distributed on 0,2pi. Now inverse FFT and this is the random sequence. Its PSD is exactly what you started with. rusty === Subject: Re: how to generate random process of any abitary power density function? Hi Lucy You in general want a description on how to implement a simulation of a realization of a random sequence. The theoretic background of the system is as follows... Suppose, you have a white Gaussian sequence w[n] that is being fed to a LTI filter whose impulse response is h[n] to generate the random sequence x[n]. You know the PSD of the random sequence Sxx(w). Then, Sxx(w) = |H(w)|^2 * Sww(w). Without loss of generality, assume Sww(w) = 1. (Unit power assumption for a white sequence). So |H(w)| = sqrt (Sxx(w)) and then h[n] = InvFourier(sqrt(Sxx(w))) Hence, if you filter a white Gaussian sequence with filter coefficients h[n], you should get a random seqeunce that has a PSD of Sxx(w). There is one catch in the above algorithm... It assumes that the input white Gaussian sequence has started at minus infinity and extends to plus infinity; which is obviously not the case for a feasible simulation. So, you only way out is to generate a very large sequence and take a few elements at a very late point in time. The following code is an illustration. Assume that Sxx stores discrete PSD cofficients Hw = sqrt (Sxx); %Compute Fourier transform of LTI filter hn = ifft (Hw); %Get filter coefficients wn = randn (2^16, 1); %Generate a very large input WGN sequence xprime = filter (hn, 1, wn); %Filter to get output x = xprime(65281:end); %Choose only last 256 coefficients If someone else can think of a more efficient way, please let me know since I have to do this kind of operation quite often. Aditya > Hi all, > I have a power density function, Sxx(w), where w is the omega, from -pi to > +pi, for digital random process(sequence). The Sxx(w) is arbitrarily shaped, > and might be very irregular. > Can anybody tell me how to generate a realization of random process to > possess this power density spectrum Sxx(w)? Esp. in Matlab? > For example, if I have an arbitarily generated Sxx variable in Matlab > holding a vector of sampled values for the Sxx(w) from -pi to +pi, how to > generate the random process correspondingly... ? === Subject: Re: how to generate random process of any abitary power density function? > Hi Lucy > You in general want a description on how to implement a simulation of > a realization of a random sequence. The theoretic background of the > system is as follows... > Suppose, you have a white Gaussian sequence w[n] that is being fed to a > LTI filter whose impulse response is h[n] to generate the random > sequence x[n]. You know the PSD of the random sequence Sxx(w). > Then, Sxx(w) = |H(w)|^2 * Sww(w). > Without loss of generality, assume Sww(w) = 1. (Unit power assumption > for a white sequence). > So |H(w)| = sqrt (Sxx(w)) > and then h[n] = InvFourier(sqrt(Sxx(w))) this is not correct! strictly sepaking, if you want to find the LTI filter then it is not simply sqrt(Sxx(w))... the correct way is by spectral factorization. the spectral factor of Sxx(w) is not necessarily sqrt(Sxx) (this is most important! you are giving Lucy incorrect information), unless Sxx is constant (i.e. white noise). for an arbitrary psd Sxx, computing the spectral factor is not always feasible, and can usually be done only approximately. a more sensible for Lucy to do it is to approximate Sxx by a rational function and take the spectral factor of the approximant (by extraction of poles and zeros) as the LTI system to generate the signal with the desired psd. but the LTI system obtained this way is not necessarily be anything like the true LTI system which generated Sxx (due to continuity issues with the spectral factorization operation), but both systems, eventhough they are not close as LTI systems, will generate random processes with psd which are apporoximately the same. this is enough for Lucy's purpose. > Hence, if you filter a white Gaussian sequence with filter coefficients > h[n], you should get a random seqeunce that has a PSD of Sxx(w). > There is one catch in the above algorithm... It assumes that the input > white Gaussian sequence has started at minus infinity and extends to > plus infinity; which is obviously not the case for a feasible > simulation. so, you only way out is to generate a very large sequence > and take a few elements at a very late point in time. again you give a silly remark. once the spectral factor H(w) of Sxx(w) (which is once again not equal to sqrt(Sxx(w)), it is easy to generate the associated random process in simulink and no need to go from -infinity to infinity. you just need to pass a white noise sequence through the filter H(w)! just make the filter H(w) in simulink and then pass white noise through, it's very easy. fact: all zero mean second order processes (i.e. E|X_t|^2 < oo for all t >=0) can be modeled as the output of an LTI system driven by white noise. > The following code is an illustration. Assume that Sxx stores discrete > PSD cofficients > Hw = sqrt (Sxx); %Compute Fourier transform of LTI filter > hn = ifft (Hw); %Get filter coefficients > wn = randn (2^16, 1); %Generate a very large input WGN sequence > xprime = filter (hn, 1, wn); %Filter to get output > x = xprime(65281:end); %Choose only last 256 coefficients incorrect program based on an incorrect understanding. > If someone else can think of a more efficient way, please let me know > since I have to do this kind of operation quite often. i think you need to understand more about power spectral densities and what spectral factorization is..... === Subject: Re: how to generate random process of any abitary power density function? Yes, I agree with this. |H(w)|^2 = |H(w)|.|H*(w)| ------------- (. denotes multiplication and * denotes complex conjugate) You need to factor out the magnitude squared in the above form. Then go ahead and perform IFFT and get your filter coeffs. I had read about this method in a book called 'Probability and Random signals' (or something similar) by Leon Garcia. It has several solved examples on the same. Have a look at that if possible. T. === Subject: Re: how to generate random process of any abitary power density function? > Yes, I agree with this. > |H(w)|^2 = |H(w)|.|H*(w)| ------------- (. denotes multiplication and * > denotes complex conjugate) huh? why not just |H(w)|^2=H(w)H(w)*..... === Subject: how to shift the power spectrum density(PSD) of a random process? Hi all, In DSP(Digital Signal Processing), for deterministic signal, I know that if I want to flip the spectrum, i.e., the negative frequency spectrum shifts to the positive part, the positive frequency spectrum shifts to the negative part. I just need to multiply the sequence with cos(n*pi)=(-1)^(n), the spectrum will be shifted/flipped accordingly. But what if I want to manipulate the power spectrum density of the random processes. What should I do in order to flip the PSD of a random process? What about other manipulation of the PSD. For example, modulation, etc.? Do all the spectrum manipulation operations of deterministic signal carry over to the PSD of random processes? Could anybody point me to some references/books/tutorials/lecture notes about these topics? === Subject: Re: how to shift the power spectrum density(PSD) of a random process? > Hi all, > In DSP(Digital Signal Processing), for deterministic signal, I know that if > I want to flip the spectrum, i.e., the negative frequency spectrum shifts to > the positive part, the positive frequency spectrum shifts to the negative > part. I just need to multiply the sequence with cos(n*pi)=(-1)^(n), the > spectrum will be shifted/flipped accordingly. > But what if I want to manipulate the power spectrum density of the random > processes. What should I do in order to flip the PSD of a random process? > What about other manipulation of the PSD. For example, modulation, etc.? Do > all the spectrum manipulation operations of deterministic signal carry over > to the PSD of random processes? Power Spectrum Densities (PSDs) don't contain phase information about the process, so there is little point in doing anything to PSDs where the phase needs to be known. Apart from that, the *mathematical* operations that apply to a deterministic system are equally valid for PSDs. Now, one could argue that the PSD is a quadratic quantity and that it is nonlinear. In a deterministic setting, the argument becomes PSD{x(t)} = |X(f)|^2 [1] PSD{x(t) + y(t)} = |X(f) + Y(f)|^2 [2a] = (|X(f)|^2 + X^*(f)Y(f) + X(f)Y^*(f) + |Y(f)|^2) [2b] =/= PSD{x(t)} + PSD{y(t)} [2c] where X(f) = FT(x(t)) and Y(f) = FT(y(t)). A in a statistical setting, however, one argues that as long as the system under study is linear and the signals x and y are uncorrelated, there will be no cross terms in step [2b] above. This is the standard argument when analysing signals in noise: The noise and signal are uncorrelated, so the cross terms are left out of the analysis. To actually see why this happens, enclose the right-hand sides of steps [1],[2a] and [2b] in expectation operators. The expectation operators are usually left out from the equations to simplify notation, but they are the transformers that actually take the argument from the deterministic to the statistical domain. There are also practical questions involved. In a practical measurement situation, there are all sorts of complicating factors. There is noise, there signal incoherency, there are interfering sources etc, so one ought to be very careful when working with the measured data. In the case of very noisy and/or distorted data, it needs not make much sense to use a classical Fourier representation of the data. A PSD can be averaged to reduce noise, one can generate confidence intervals for the power levels etc. This statistical representation could be a lot more useful to the analyst than the deterministic representation. The theory of deterministic systems is used when one designs filters and systems. Statistical signal analysis is used when one analyzes measured data. They are two different sets of tools that, although similar, are intended for two different uses. > Could anybody point me to some references/books/tutorials/lecture notes > about these topics? Bendat & Piersol: Random Data, 3rd ed., Wiley, 2000 Kay: Modern Spectral Estimation with Applications, Prentice-Hall, 1988 Therrien: Discrete Random Signals and Statistical Signal Processing, Prentice-Hall, 1992 (This one is hard to find but a reprint was available through amazon.com two yeras ago). Rune === Subject: Re: how to shift the power spectrum density(PSD) of a random process? > Hi all, > In DSP(Digital Signal Processing), for deterministic signal, I know that > if I want to flip the spectrum, i.e., the negative frequency spectrum > shifts to the positive part, the positive frequency spectrum shifts to the > negative part. I just need to multiply the sequence with > cos(n*pi)=(-1)^(n), the spectrum will be shifted/flipped accordingly. This is not quite correct. Multiplying by (-1)^n actually gives a circular spectrum shift by half a block, but it doesn't flip it. To flip the spectrum, time reverse the waveform which effectively gives omega a negative value. If the waveform is real valued the spectrum has conjugate symmetry and its phases will be reversed. rusty === Subject: Re: how to shift the power spectrum density(PSD) of a random process? http://zone.ni.com/devzone/conceptd.nsf/2d17d611efb58b22862567a9006ffe76/c04 5a890751303a6862568650061ea98?OpenDocument is a good tutorial. Or you can find a lot more here: http://www.dsprelated.com/index.php === Subject: Re: how to shift the power spectrum density(PSD) of a random process? > Hi all, > In DSP(Digital Signal Processing), for deterministic signal, I know that if > I want to flip the spectrum, i.e., the negative frequency spectrum shifts to > the positive part, the positive frequency spectrum shifts to the negative > part. I just need to multiply the sequence with cos(n*pi)=(-1)^(n), the > spectrum will be shifted/flipped accordingly. > But what if I want to manipulate the power spectrum density of the random > processes. What should I do in order to flip the PSD of a random process? > What about other manipulation of the PSD. For example, modulation, etc.? Do > all the spectrum manipulation operations of deterministic signal carry over > to the PSD of random processes? > Could anybody point me to some references/books/tutorials/lecture notes > about these topics? Spectrum flipping and modulation are mathematical operators that don't really care if they operate on a deterministic or random sequence of time samples, so I think you can use them. They transform a deterministic sequence to another deterministic sequence, and they transform a random process to another random process. John === Subject: Re: A theorem can't be wrong [Nora Baron] > Ok, I've finished brainstorming on what to call the SFT and how > best to present it. > It's been a VERY useful few days as most importantly I've > managed to come up with a name that shows just how important > and dangerous this research is, so the people who are supposed > to pay attention to national security assuredly noticed! > But now you see posters here quibbling about whether it should > be a theorem or an observation or an identity. I thought I > made it clear that 'observation' is the right word as it is > descriptive but not overly informative. But some posters here > with low skill levels in mathematics want to call it an > identity. There is no end to the lying and cheating in naming > SFT-like things and no one seems to care. LOL! I yield -- observation it is. Well done. [... and on, and on, and on ...] > But just because you don't not believe in not naming mathematical > objects doesn't make it not so. But go figure. > You are fighting for yourselves and your own families at the > expense of others. > Think about it. > Nora Baron OK, I did. One small suggestion: go the whole way and rename it the Unprecedented Factoring Observation. First impressions count for almost everything in a field this corrupt, and UFO would help that a lot. === Subject: Bezier Values... liquify/warp effects in photoshop. I'm guessing this is roughly how to do it: If you have a 2D 3x3 quadratic bezier patch given by the formulae x = A(u,v), y = B(u,v), then how do you find out what the u & v values are given x & y? I'm not specifying any formula for A & B, as I think there are probably more than one, whichever has the cheapest most elegant solution, suits me. === Subject: =?utf-8?B?UmU6IEltcG9ydGFudCBwbGVhc2UgcmVhZCB0aGlzIFRoYW5rIHlvdSBgsLq3Li4ut7 q wYLC6ty4uLre6sGCwurcuhi4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sGCwurc u Li63urBgsLq3LoYuLre6sGCwurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4ut7qw Y LC6ty6GLi63urBgsLq3Li4ut7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCwurcuh i 4ut7qwYLC6ty4uLre6sGCwurcuLre6sGAgYLC6ty4uLre6sGCwurcuLi63urBgsLq3LoYuLre6sG C wurcuLi63urBgsLq3Li63urBgIGCwurcuLi63urBgsLq3Li4ut7qwYLC6ty6GLi63urBgsLq3Li4 u t7qwYLC6ty4ut7qwYCBgsLq3Li4ut7qwYLC6ty4uLre6sGCwurcuhi4ut7qwYLC6ty4uLre6sGCw u rcuLre6sGAgYLC6ty4uLre6sGCwurcuLi63urBgsLq3LoYuLre6sGCwurcuLi63urBgsLq3Li63u r BgIEdPT0dMRbdORVdTR1JPVVC3UE9TVLcxNTU=?= Tom > Tom sent all of his Salvation Spam to Abuse@Yahoo.com. They send back the auto-response, which means their system got it but they won't do anything about it. And Yahoo wonders why they're on my blocked ISP list. Jerry to > Tom > sent all of his Salvation Spam to Abuse@Yahoo.com. They send back > the auto-response, which means their system got it but they won't do > anything about it. And Yahoo wonders why they're on my blocked ISP > list. > Jerry Looking at full headers, it appears he is actually posting from AOL (172.143.41.97). time > to > > Tom I've > sent all of his Salvation Spam to Abuse@Yahoo.com. They send back > the auto-response, which means their system got it but they won't do > anything about it. And Yahoo wonders why they're on my blocked ISP > list. > Jerry > Looking at full headers, it appears he is actually posting from AOL > (172.143.41.97). Numi, I just pulled up full headers. His host shows 172.143.41.97. Tom's post and mine (AOL) both show 152.163.100.71. No wonder things get so confusing for us non-computer-geek-types. That notwithstanding, since he's posting under a different screen name, his old one may have been pulled by either Google or Yahoo (or maybe even AOL). Who knows. Jerry === Subject: Teaching Myself Mathematics I hope this doesn't sound exceedingly juvenile. I am a soon to graduate computer science grad student. I will be starting work soon. I hope to learn a lot of mathematics on my own before I go back to school and I might not be able to do all of it via a part-time math degree since I am not sure I will have the time to commit to degree requirements. I would really appreciate your comments and suggestions on a paced self learning course as an alternative. I have taken the following subjects: basic calculus, abstract algebra (basic group theory), combinatorics (polya's theory, a little bit of mobius inversion, incl excl, g.f's), graph theory and basic mathematical logic. I would love to learn more about rings, fields, vector spaces, topology, algebraic geometry (are these the same), combinatorial algorithms etc. Any pointers such as good texts to start with would be helpful. -vijai. === Subject: Re: Teaching Myself Mathematics > basic calculus, Courant and John Introduction to Calculus and Analysis in two volumes. === Subject: Re: Teaching Myself Mathematics > ... mathematical logic. If you're an absolute beginner, Tarski An Introduction to Logic and the Methodology of the Deductive Sciences; otherwise (or subsequently) Mendelson Introduction to Mathematical Logic > I would love to learn more about rings, fields, vector spaces, Birkhoff and Mac Lane A Survey of Modern Algebra > topology, Kelley General Topology === Subject: Re: Teaching Myself Mathematics > I am a soon to graduate computer science grad student. I will be > starting work soon. I hope to learn a lot of mathematics on my own > before I go back to school and I might not be able to do all of it via > a part-time math degree since I am not sure I will have the time to > commit to degree requirements. > I would really appreciate your comments and suggestions on a paced self > learning course as an alternative. I have taken the following subjects: > basic calculus, abstract algebra (basic group theory), combinatorics > (polya's theory, a little bit of mobius inversion, incl excl, g.f's), > graph theory and basic mathematical logic. > I would love to learn more about rings, fields, vector spaces, > topology, algebraic geometry (are these the same), No. > combinatorial algorithms etc. > Any pointers such as good texts to start with would be helpful. Well, don't waste money buying this year's textbooks. Used books are equally good. Schaum's Outlines are quite good in some cases and quite cheap. The name set theory refers to both a theory and a system of appartus and terminology. The former is a specialty but the latter is necessary for all of us. Functions, equivalence relations, order relations, etc. -- get that all down, which is not hard, and by practice get the hang of proofs and constructions by transfinite induction. For math (not computer sci) I recommend taking a good look at Galois theory, because it has been the model of many other structure theories in mathematics. If you can find Dieudonne's _Treatise on Analysis_ at a university library, take notes from the appendices on linear and multilinear algebra; it is a quick look at the algebra that is used constantly everywhere in mathematics, physics, and engineering. Analysis contains many technicalities that are necessary for rigour but not, in themselves, enlightening or useful. Avoid missing the forest because of all the trees. Books written for phyisicists can be helpful because they are light on technicalities, generally speaking. The main themes in analysis (IMO) are harmonic analysis, spectral theory, and representation theory. Lebesgue integration underlies all three. But all three also have finite analogs that involve only finite sums and no technicalities from measure theory. Such finite things can illuminate the main results of those three theories. Hope this helps, Larry H Autodidactic Extraordinaire === Subject: Re: Teaching Myself Mathematics > I hope this doesn't sound exceedingly juvenile. > I am a soon to graduate computer science grad student. I will be > starting work soon. I hope to learn a lot of mathematics on my own > before I go back to school and I might not be able to do all of it via > a part-time math degree since I am not sure I will have the time to > commit to degree requirements. > I would really appreciate your comments and suggestions on a paced self > learning course as an alternative. I have taken the following subjects: > basic calculus, abstract algebra (basic group theory), combinatorics > (polya's theory, a little bit of mobius inversion, incl excl, g.f's), > graph theory and basic mathematical logic. > I would love to learn more about rings, fields, vector spaces, > topology, algebraic geometry (are these the same), combinatorial > algorithms etc. > Any pointers such as good texts to start with would be helpful. > -vijai. Following a similar plan that a respected university follows might be a difficulty will be that, as a human being, you might tend to neglect subjects that you don't like yet that are beneficial in long run. You will need much determination. I've almost completed a B.Sc. in physics. The math courses were taught in about the following order, although some subjects were optional: Algebra and geometry, linear algebra, calculus of a single real variable, ordinary differential equations, calculus of multivariable functions, partial differential equations, calculus of a complex variable. Along the way I studied set theory on my own since it unifies all the different math subjects. Probability, statistics, and general combinatorics were taught in physics courses. Also, tensor calculus, non-euclidean geometry were taught in a course on the theory of general relativity. I'm currently learning the calculus of variations, which has shown me how vector spaces, differential equations, etc, are all interconnected. If I could, I would have learned mathematics in the following order: Logic, proofs, set theory, group theory, linear algebra, combinatorics, probability, complex variables, single variable real calculus, ordinary differential equations, multivariable calculus, partial differential equations, statistics, calculus of complex variable, tensor calculus, calculus of variations. As you can see, most of the mathematics revolves around calculus, which is due to the fact that I'm learning physics. I hope the outline of what I learned aids you. I wish you the best in your goal, Adam. === Subject: Re: Teaching Myself Mathematics Hello Adam, shall I say, the underlying unification? Or as you put it, set theory unifies the different parts. I heard someone describe that mathematics itself is built with sets at the starting point. -v. === Subject: Re: Teaching Myself Mathematics > ... that mathematics > itself is built with sets at the starting point. May be rather than is. In my other post I mentioned a book by Kelley that has an excellent appendix on set theory, the book by Mendelson has a substantial amount of set theory. === Subject: Re: Teaching Myself Mathematics > Hello Adam, > shall I say, the underlying unification? Or as you put it, set theory > unifies the different parts. I heard someone describe that mathematics > itself is built with sets at the starting point. > -v. You are welcome. Understanding how the different areas of mathematics are related makes understanding the subject easier, at least for myself. It was only when I began to learn the connections between the various areas that a true understanding and appreciation was stirred within me. That is principally why I decided to teach myself set theory. I learned basic logic, set theory and proofs over the course of last summer. Many members of this newsgroup aided me in my learning by correcting proofs and answering questions. Even after learning about vector spaces in linear algebra, I never actually understood their application and why they were taught until a short time ago when I took up the study of the calculus of variations. I bought a text at a used bookstore that I happened across. It seemed interesting, but had not been taught to me at university. The author begins by defining vector spaces, the important properties, and then defines limits, continuous functions, etc, all based on vector spaces. He does that and so develops a calculus based on function and not real numbers as is done in the calculus that is commonly taught. If you know calculus and vector spaces, reading a text on the calculus of variation may provide insight into how the subjects are related and complement each other. At least it was a benefit for me, even though I've only begun to study it. In fact, I read the text before going to bed each night. :) Basic set theory is probably a must to know if you want to be able to prove and understand the foundations of the various math subjects. Logic and proofs are also of fundamental importance. That can not be stressed enough. If you search for posts by myself, I'm sure you will find that I was clueless, but eager to learn, various areas of mathematics. Perhaps you will see how I began by not knowing ANY set theory to being able to write sound proofs. The newsgroup was instrumental in my study of the theory since I was not in university at the time and so did not have a person to ask in person. Best of luck, Adam. === Subject: Re: Points in a sphere > The integer lengths of PA,PB,PC,PD and r are required. Examples are > 1,3,11,33,21 and 7,17,21,51,33. Looking for more Quintuplets, > excluding multiples and identical values within the same > quintuplet > set. > I can guess that r is the sphere's radius, but I'm not clear what > relationship points(?) P,A,B,C,D should bear to the sphere. That's because you're coming in in mid-discussion, and people are snipping context. If I remember right, those 5 points are all supposed to be on the sphere, and the angle between Px and Py is supposed to be arccos(-1/3) (or something like that) for all choices of {x, y} from {A, B, C, D}. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Points in a sphere >> The integer lengths of PA,PB,PC,PD and r are required. Examples are >> 1,3,11,33,21 and 7,17,21,51,33. Looking for more Quintuplets, >> excluding multiples and identical values within the same >> quintuplet >> set. >> I can guess that r is the sphere's radius, but I'm not clear what >> relationship points(?) P,A,B,C,D should bear to the sphere. >That's because you're coming in in mid-discussion, >and people are snipping context. >If I remember right, those 5 points are all supposed to be on >the sphere, and the angle between Px and Py is supposed to be >arccos(-1/3) (or something like that) for all choices of {x, y} >from {A, B, C, D}. Except P isn't on the sphere. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Question about a periodic representation of 14^(1/4) >> Using a randomized variant of the Jacobi-Perron algorithm I was able to >> find the following periodic representation of 14^(1/4). >> Let >> A=[26895,13904,7188,3716;52024,26895,13904,7188;100632,52024,26895,13904;194 6 5 >> 6,100632,52024,26895]. >> Let B_n=A^n. Then lim B_n[i+1,j]/B_n[1,j]=14^(i/4) (i=1..3; j=1..4). Interesting. >> My question is whether or not there is another matrix C<>B_k, all k, >> having these same nice properties? If so how many such C's are there? >Won't something like this happen for any matrix that has 14^(1/4) as >an eigenvalue? e.g., the companion matrix for x^4 - 14? No. Note that this is comparing different entries for the same power of A. If A has a simple eigenvalue lambda that is largest in absolute value, you want a corresponding eigenvector to be [ 1 ] [ r ] [ r^2 ] [ r^3 ] where r = 14^(1/4). However, the eigenvalue doesn't have to be r, and in Rich's case it isn't. If the entries of A are a_{ij} you want sum_{j=1}^4 a_{ij} r^{j-1} = lambda r^{i-1} for each i. Eliminating lambda, you have three equations a_{21} + (a_{22}-a_{11}) r + (a_{23}-a_{12}) r^2 + (a_{24} - a_{13}) r^3 - a_{14} r^4 = 0 a_{31} + a_{32} r + (a_{33} - a_{11} r^2 + (a_{34}-a_{12}) r^3 - a_{13} r^4 - a_{14} r^5 = 0 a_{41} + a_{42} r + a_{43} r^2 + (a_{44}-a_{11}) r^3 - a_{12} r^4 - a_{13} r^5 - a_{14} r^6 If the a_{ij} are integers, since x^4 - 14 is irreducible the left sides all have to be divisible (as polynomials in r) by r^4 - 14. So from the first equation we must have a_{21} = 14 a_{14} a_{11} = a_{22} a_{12} = a_{23} a_{31} = 14 a_{13} a_{32} = 14 a_{14} a_{33} = a_{11} a_{34} = a_{12}. And from the third, a_{41} = 14 a_{12} a_{42} = 14 a_{13} a_{43} = 14 a_{14} a_{44} = a_{11} All these imply that A is a Toeplitz matrix of the form [ a b c d ] [ 14 d a b c ] [ 14 c 14 d a b ] [ 14 b 14 c 14 d a ] and any matrix of this form will do. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Question about a periodic representation of 14^(1/4) >> Using a randomized variant of the Jacobi-Perron algorithm I was able to >> find the following periodic representation of 14^(1/4). >> Let >> A=[26895,13904,7188,3716;52024,26895,13904,7188;100632,52024,26895,13904;194 65 >> 6,100632,52024,26895]. >> Let B_n=A^n. Then lim B_n[i+1,j]/B_n[1,j]=14^(i/4) (i=1..3; j=1..4). > Interesting. >> My question is whether or not there is another matrix C<>B_k, all k, >> having these same nice properties? If so how many such C's are there? >Won't something like this happen for any matrix that has 14^(1/4) as >an eigenvalue? e.g., the companion matrix for x^4 - 14? > No. Note that this is comparing different entries for the same power of A. > If A has a simple eigenvalue lambda that is largest in absolute value, you want > a corresponding eigenvector to be > [ 1 ] > [ r ] > [ r^2 ] > [ r^3 ] > where r = 14^(1/4). However, the eigenvalue doesn't have to be r, and in Rich's > case it isn't. > If the entries of A are a_{ij} you want sum_{j=1}^4 a_{ij} r^{j-1} = lambda r^{i-1} > for each i. > Eliminating lambda, you have three equations > a_{21} + (a_{22}-a_{11}) r + (a_{23}-a_{12}) r^2 + (a_{24} - a_{13}) r^3 - a_{14} r^4 = 0 > a_{31} + a_{32} r + (a_{33} - a_{11} r^2 + (a_{34}-a_{12}) r^3 - a_{13} r^4 - a_{14} r^5 = 0 > a_{41} + a_{42} r + a_{43} r^2 + (a_{44}-a_{11}) r^3 - a_{12} r^4 - a_{13} r^5 - a_{14} r^6 > If the a_{ij} are integers, since x^4 - 14 is irreducible the left sides all have to be > divisible (as polynomials in r) by r^4 - 14. So from the first equation we must have > a_{21} = 14 a_{14} > a_{11} = a_{22} > a_{12} = a_{23} > a_{31} = 14 a_{13} > a_{32} = 14 a_{14} > a_{33} = a_{11} > a_{34} = a_{12}. And from the third, > a_{41} = 14 a_{12} > a_{42} = 14 a_{13} > a_{43} = 14 a_{14} > a_{44} = a_{11} > All these imply that A is a Toeplitz matrix of the form > [ a b c d ] > [ 14 d a b c ] > [ 14 c 14 d a b ] > [ 14 b 14 c 14 d a ] > and any matrix of this form will do. original A I neglected to mention is that det(A)=1. So, is there a C such that (i) det(C)=1; (ii) C<>A^k, all k>0; and, (iii) C is of the form you describe? There is some evidence to expect the answer is yes. For example, looking at this problem for 2^(1/4), the matrix A=[7,6,5,4;8,7,6,5;10,8,7,6;12,10,8,7] and C=[471,396,333,280;560,471,396,333;666,560,471,396;792,666,560,471] seem to meet the requirements (although C is probably a power of some smaller C'). Are there any others? Knowing how many such C's there are for 2^(1/4) and/or 14^(1/4) would be quite helpful in testing the method I am working on. Rich === Subject: Re: Question about a periodic representation of 14^(1/4) >> All these imply that A is a Toeplitz matrix of the form >> [ a b c d ] >> [ 14 d a b c ] >> [ 14 c 14 d a b ] >> [ 14 b 14 c 14 d a ] >> and any matrix of this form will do. >original A I neglected to mention is that det(A)=1. So, is there a C >such that (i) det(C)=1; (ii) C<>A^k, all k>0; and, (iii) C is of the >form you describe? The determinant of the Toeplitz matrix above is product_w (a + b w + c w^2 + d w^3), the product being over the complex fourth roots of 14. I think we are looking for units in the ring Z[14^(1/4)]. Maybe it's time to call in the algebraic number theorists. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Complex Analysis How would i go about showing that if each w and z are complex numbers then E(w)E(z)=E(w+z) I know, i define a function f(z)=E(w)E(z)E(wz)=1 and take the 1st derivative. I did this and discovered that f'(z)= E(w)E(z)E(wz)=1= f(z) This implies that the entire function f is an exponential function. How do i show that f(0)=1 without using trig. identities or demoivres law and just by definition. This is by the way how an exponential functionwas defined in my notes.i.e. An exponential function E is that entire function f such that f'=f and f(0)=1. Also, how can i show E(-ipi)=-1 and E(ipi)=-1 without using trig. identities or demoivres law and just by definition. === Subject: Re: Complex Analysis > How would i go about showing that if each w and z are complex numbers > then E(w)E(z)=E(w+z) For any given w, define an analytic function by f(z) = E(w) * E(z) - E(w+z). Then a simple calculation shows f'(z) = f(z). Furthermore f(0) = 0. You should be able to show easily that f(z) = 0 for all z (for example, compute the Taylor series of f in 0 --- since f is analytic, this series does actually converge and is equal to f). Lasse --- === Subject: Re: Complex Analysis >How would i go about showing that if each w and z are complex numbers >then E(w)E(z)=E(w+z) >I know, i define a function f(z)=E(w)E(z)E(wz)=1 and take the 1st >derivative. >I did this and discovered that f'(z)= E(w)E(z)E(wz)=1= f(z) >This implies that the entire function f is an exponential function. I define E as the unique solution to the differential equation f' - f = 0 with initial condition f(0) = 1. Note that, since E is nonzero, any solution to to f' - f = 0 must be a scalar multiple of E. In particular, fixing w, d/dz E(w+z) = E(w+z), so E(w+z) = c E(z) for some constant c. Plugging in z = 0 yields c = E(w). >How do i show that f(0)=1 without using trig. identities or demoivres >law and just by definition. >This is by the way how an exponential functionwas defined in my >notes.i.e. An exponential function E is that entire function f such >that f'=f and f(0)=1. So E(0) = 1 by definitition; there is nothing to show. >Also, how can i show E(-ipi)=-1 and E(ipi)=-1 without using trig. >identities or demoivres law and just by definition. If f(x + i y) = u(x,y) + i v(x,y), then f'(x+i y) = du/dx + i dv/dx = dv/dy - i du/dy (partial derivatives). If f'(z) = f(z) and f(0) = 1, derive differentiail equations for u and v. If this is homework, cite sources of assistance in submitted work. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Complex Analysis <427734FD.3060900@netscape.net> Also, how can i show E is the limit on the number-plane of the power series sum(1/p!)I^P about 0. >How would i go about showing that if each w and z are complex numbers >then E(w)E(z)=E(w+z) >I know, i define a function f(z)=E(w)E(z)E(wz)=1 and take the 1st >derivative. >I did this and discovered that f'(z)= E(w)E(z)E(wz)=1= f(z) >This implies that the entire function f is an exponential function. > I define E as the unique solution to the differential equation f' - f > = 0 with initial condition f(0) = 1. Note that, since E is > nonzero, any solution to to f' - f = 0 must be a scalar multiple of E. > In particular, fixing w, d/dz E(w+z) = E(w+z), so E(w+z) = c E(z) for > some constant c. Plugging in z = 0 yields c = E(w). >How do i show that f(0)=1 without using trig. identities or demoivres >law and just by definition. >This is by the way how an exponential functionwas defined in my >notes.i.e. An exponential function E is that entire function f such >that f'=f and f(0)=1. > So E(0) = 1 by definitition; there is nothing to show. >Also, how can i show E(-ipi)=-1 and E(ipi)=-1 without using trig. >identities or demoivres law and just by definition. > If f(x + i y) = u(x,y) + i v(x,y), then > f'(x+i y) = du/dx + i dv/dx = dv/dy - i du/dy (partial derivatives). > If f'(z) = f(z) and f(0) = 1, derive differentiail equations for u > and v. > If this is homework, cite sources of assistance in submitted work. > -- > Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Complex Analysis > How would i go about showing that if each w and z are complex numbers > then E(w)E(z)=E(w+z) > I know, i define a function f(z)=E(w)E(z)E(wz)=1 and take the 1st > derivative. > I did this and discovered that f'(z)= E(w)E(z)E(wz)=1= f(z) > This implies that the entire function f is an exponential function. > How do i show that f(0)=1 without using trig. identities or demoivres > law and just by definition. > This is by the way how an exponential functionwas defined in my > notes.i.e. An exponential function E is that entire function f such > that f'=f and f(0)=1. > Also, how can i show E(-ipi)=-1 and E(ipi)=-1 without using trig. > identities or demoivres law and just by definition. Could you use the series definition of the exponential function and then show that it equates to -1 when the variable is -ipi? Adam. === Subject: Re: Complex Analysis >> How would i go about showing that if each w and z are complex numbers >> then E(w)E(z)=E(w+z) >> I know, i define a function f(z)=E(w)E(z)E(wz)=1 and take the 1st >> derivative. >> I did this and discovered that f'(z)= E(w)E(z)E(wz)=1= f(z) >> This implies that the entire function f is an exponential function. >> How do i show that f(0)=1 without using trig. identities or demoivres >> law and just by definition. >> This is by the way how an exponential functionwas defined in my >> notes.i.e. An exponential function E is that entire function f such >> that f'=f and f(0)=1. >> Also, how can i show E(-ipi)=-1 and E(ipi)=-1 without using trig. >> identities or demoivres law and just by definition. > Could you use the series definition of the exponential function and > then show that it equates to -1 when the variable is -ipi? > Adam. I attempted to prove exp(w)exp(z) = exp(w+z) using the series definition of the exp function. Although I haven't finished the proof since I'm not that great at series formulas, I did manage to almost prove it. I had to use the binomial theorem to expand the exp(w+z) series term. If exp is just thought of as a number raised to a power, I don't see why the proof can't be trivially done as it obeys the general rule for all numbers raised to a power. Is it because of the complex power? Adam. === Subject: Re: Complex Analysis : I attempted to prove exp(w)exp(z) = exp(w+z) using the series definition : of the exp function. Although I haven't finished the proof since I'm not : that great at series formulas, I did manage to almost prove it. I had to : use the binomial theorem to expand the exp(w+z) series term. : If exp is just thought of as a number raised to a power, I don't see why : the proof can't be trivially done as it obeys the general rule for all : numbers raised to a power. Is it because of the complex power? If you define exp(z) as e (defined in some way) raised to the power z, (where you have somehow defined raising to power z for all complex z), then the problem is to prove the general rule for all numbers raised to a power. It's obvious for integer w and z, but less obvious for rational numbers, much less for irrational or complex powers. In fact, it's not clear to me how you would define raising to a complex power without first defining exp() in some other way (you can get arbitrary real exponents with no problem, but it seems that at some point you need to use, or to define, exp(it) = cos(t) + i sin(t), to get all complex numbers.) If you use the series definition exp(z) = 1 + z + z^2/2! + ..., then you can't use the general rule for powers, because that isn't your definition. But the series expansion proof you mention does work. Ted === Subject: Re: Complex Analysis > : I attempted to prove exp(w)exp(z) = exp(w+z) using the series definition > : of the exp function. Although I haven't finished the proof since I'm not > : that great at series formulas, I did manage to almost prove it. I had to > : use the binomial theorem to expand the exp(w+z) series term. > : If exp is just thought of as a number raised to a power, I don't see why > : the proof can't be trivially done as it obeys the general rule for all > : numbers raised to a power. Is it because of the complex power? > If you define exp(z) as e (defined in some way) raised to the power z, > (where you have somehow defined raising to power z for all complex z), > then the problem is to prove the general rule for all numbers raised to a > power. It's obvious for integer w and z, but less obvious for rational > numbers, much less for irrational or complex powers. In fact, it's not > clear to me how you would define raising to a complex power without first > defining exp() in some other way (you can get arbitrary real exponents > with no problem, but it seems that at some point you need to use, or to > define, exp(it) = cos(t) + i sin(t), to get all complex numbers.) > If you use the series definition exp(z) = 1 + z + z^2/2! + ..., then you > can't use the general rule for powers, because that isn't your definition. > But the series expansion proof you mention does work. > Ted The proof I attempted used the following series definition for the exponential function: exp(x) = Sum{n = 0 to inf} x^n/n!. The logic then went as follows: exp(w)exp(z) = exp(w+z) => (Sum_{n} w^n/n!)(Sum_{m} z^m/m!) = Sum_{k} (w+z)^k/k! => Sum_{n} Sum_{m} (w^n*z^m)/(n!*m!) = Sum_{k} (w+z)^k/k! => Sum_{n} Sum_{m} (w^n*z^m)/(n!*m!) = Sum_{k} Sum_{j=0, j=k} k!*w^{k-j}*z^j/((k-j)!*j!*k!) The terms on the left appear similar to the terms on the right if (k-j) is defined as another variable, n, but I'm at a loss as to how to proceed. The last step used the binomial theorem on the (w+z)^k term. Perhaps the following? => Sum_{n} Sum_{m} (w^n*z^m)/(n!*m!) = Sum_{k} Sum_{j=0, j=k} w^n*z^j/(n!*j!) where n is defined to be k-j. The right side is now almost the same as the left side, but for the second sum where j = 0 to j = k. Am I to assume that if k goes to infinity, that, since j goes to k, then j goes to infinity? Is the following continuation mathematically correct? => Sum_{n} Sum_{m} (w^n*z^m)/(n!*m!) = Sum_{n} Sum_{j} w^n*z^j/(n!*j!) thus exp(w)exp(z) = exp(w+z)? If n = k - j, and k = 0 to inf, and j = 0 to k, then j = 0 to inf, and so n = 0 when k = j to inf when k = inf and j = 0? Again, I'm not used to infinite sums. I'd appreciate an explanation that will reduce my ignorance. === Subject: Re: Complex Analysis > The proof I attempted used the following series definition for the > exponential function: exp(x) = Sum{n = 0 to inf} x^n/n!. Is this power series really a *definition* of the exponential function? Surely *the* definition (as Stephen Herschkorn) stated is that it is the function which is its own derivative and maps zero to unity. The power series follows immediately from this definition by Taylor's Theorem but is not the definition. Mark Atherton === Subject: Re: Complex Analysis > Surely *the* definition (as Stephen Herschkorn) stated is that it is the > function which is its own derivative and maps zero to unity. That is surely not the standard definition. And anyway, why must such a function exist? === Subject: Re: Complex Analysis >>Surely *the* definition (as Stephen Herschkorn) stated is that it is the >>function which is its own derivative and maps zero to unity. >> >That is surely not the standard definition. >And anyway, why must such a function exist? It follows from basic existence and uniqueness theorems of differential equations. I admit I have only seen this developed for real functions, but I suspect one can carry through the analysis for the complex functions as well. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Complex Analysis > >>Surely *the* definition (as Stephen Herschkorn) stated is that it is the >>function which is its own derivative and maps zero to unity. >> >> >That is surely not the standard definition. >And anyway, why must such a function exist? > > It follows from basic existence and uniqueness theorems of differential > equations. It does on R, but that requires machinery (metric spaces, contraction principle, completeness, sup norm, uniform convergence for continuous functions) unavailable to most calculus students, or even most students in complex analysis. I don't think it's a good way to define e^x; it's not very well motivated. > I admit I have only seen this developed for real functions, > but I suspect one can carry through the analysis for the complex > functions as well. I guess you could look at the map f -> 1 + int_[0,z] f(w) dw, defined for entire functions f, but it looks like Cauchy's theorem would be needed. === Subject: Re: Complex Analysis >> > > >>Surely *the* definition (as Stephen Herschkorn) stated is that it is the >>function which is its own derivative and maps zero to unity. >> >> >> >> >That is surely not the standard definition. >And anyway, why must such a function exist? > > >>It follows from basic existence and uniqueness theorems of differential >>equations. >> >It does on R, but that requires machinery (metric spaces, >contraction principle, completeness, sup norm, uniform >convergence for continuous functions) unavailable to most >calculus students, or even most students in complex analysis. I >don't think it's a good way to define e^x; it's not very well >motivated. >>I admit I have only seen this developed for real functions, >>but I suspect one can carry through the analysis for the complex >>functions as well. >> >I guess you could look at the map f -> 1 + int_[0,z] f(w) dw, >defined for entire functions f, but it looks like Cauchy's >theorem would be needed. Take a look at the elementary book by Churchill, Brown, and Verhey. Though, strictlly speaking, these authors do not *define* the exponential by the differential equation, they do justify their definition that exp(z) = e^(Re z) [cos (Im z) + i sin (Im z)] by noting that this is the only entire function which satisfies f' = f and agrees with the real exponential function. I disagree that it is unmotivated. Defining the exponential function this way has always been my favorite. The exponential function is a particular eigenfunction of the differential operator. You can derive most of its salient prpoerties from this fact alone. See, for example, Finney and Ostberg's undergradute text on differential equations, where this definition of the exponential function appears in an exercise. Cf. deriving the salient properties of sine and cosine from their definition via f'' + f = 0, discussed in detail by Finney & Ostberg. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Complex Analysis > The proof I attempted used the following series definition for the > exponential function: exp(x) = Sum{n = 0 to inf} x^n/n!. > Is this power series really a *definition* of the exponential function? > Surely *the* definition (as Stephen Herschkorn) stated is that it is the > function which is its own derivative and maps zero to unity. > The power series follows immediately from this definition by Taylor's > Theorem but is not the definition. > Mark Atherton Different texts use different definitions. === Subject: Re: Complex Analysis >> The proof I attempted used the following series definition for the >> exponential function: exp(x) = Sum{n = 0 to inf} x^n/n!. > Is this power series really a *definition* of the exponential function? > Surely *the* definition (as Stephen Herschkorn) stated is that it is the > function which is its own derivative and maps zero to unity. > The power series follows immediately from this definition by Taylor's > Theorem but is not the definition. What do you mean by that? As a matter of fact, it *is* the standard definition of exponential function givin in most Complex Analysis textbooks. It's right there on the first (!) page of Rudin's Real and Complex Analysis. It's also the definition used in Lang's Complex Analysis as well as in Remmert's Theory of Complex Functions, among many other textbooks. Jose Carlos Santos === Subject: Re: Complex Analysis > How would i go about showing that if each w and z are complex numbers > then E(w)E(z)=E(w+z) > I know, i define a function f(z)=E(w)E(z)E(wz)=1 and take the 1st > derivative. > I did this and discovered that f'(z)= E(w)E(z)E(wz)=1= f(z) > This implies that the entire function f is an exponential function. > How do i show that f(0)=1 without using trig. identities or demoivres > law and just by definition. > This is by the way how an exponential functionwas defined in my > notes.i.e. An exponential function E is that entire function f such > that f'=f and f(0)=1. > Also, how can i show E(-ipi)=-1 and E(ipi)=-1 without using trig. > identities or demoivres law and just by definition. >> Could you use the series definition of the exponential function and >> then show that it equates to -1 when the variable is -ipi? >> Adam. > I attempted to prove exp(w)exp(z) = exp(w+z) using the series definition > of the exp function. Although I haven't finished the proof since I'm not > that great at series formulas, I did manage to almost prove it. I had to > use the binomial theorem to expand the exp(w+z) series term. > If exp is just thought of as a number raised to a power, I don't see why > the proof can't be trivially done as it obeys the general rule for all > numbers raised to a power. Is it because of the complex power? > Adam. Well, if you are going to go through all that might as well just use a different definition then exp(x) = lim (1+x/n)^n hence exp(a)*exp(b) = lim ((1+a/n)*(1+b/n))^n = lim (1 + a/n + b/n + ab/n^2)^n = lim (1 + (a+b)/n + ab/n^2)^n = notice that the ab/n^2 term has no effect(use whatever method you want to arrive at this) therefor we get lim (1 + (a+b)/n)^n which is just exp(a+b) its easy to show that exp(0) = 1 using this definition(as it is all definitions) since lim (1+0/n)^n = 1 === Subject: Re: Complex Analysis >>How would i go about showing that if each w and z are complex numbers >>then E(w)E(z)=E(w+z) >> >>I know, i define a function f(z)=E(w)E(z)E(wz)=1 and take the 1st >>derivative. >>I did this and discovered that f'(z)= E(w)E(z)E(wz)=1= f(z) >> >>This implies that the entire function f is an exponential function. >> >>How do i show that f(0)=1 without using trig. identities or demoivres >>law and just by definition. >> >>This is by the way how an exponential functionwas defined in my >>notes.i.e. An exponential function E is that entire function f such >>that f'=f and f(0)=1. >> >>Also, how can i show E(-ipi)=-1 and E(ipi)=-1 without using trig. >>identities or demoivres law and just by definition. >> > Could you use the series definition of the exponential function and >then show that it equates to -1 when the variable is -ipi? > Adam. >>I attempted to prove exp(w)exp(z) = exp(w+z) using the series definition >>of the exp function. Although I haven't finished the proof since I'm not >>that great at series formulas, I did manage to almost prove it. I had to >>use the binomial theorem to expand the exp(w+z) series term. >>If exp is just thought of as a number raised to a power, I don't see why >>the proof can't be trivially done as it obeys the general rule for all >>numbers raised to a power. Is it because of the complex power? >>Adam. > Well, if you are going to go through all that might as well just use a > different definition then > exp(x) = lim (1+x/n)^n > hence exp(a)*exp(b) = > lim ((1+a/n)*(1+b/n))^n = > lim (1 + a/n + b/n + ab/n^2)^n = > lim (1 + (a+b)/n + ab/n^2)^n = > notice that the ab/n^2 term has no effect(use whatever method you want to > arrive at this) > therefor we get lim (1 + (a+b)/n)^n which is just exp(a+b) > its easy to show that exp(0) = 1 using this definition(as it is all > definitions) since lim (1+0/n)^n = 1 When I did the series expansion, I used exp(x) = Sum_{n = 0, inf} x^n/n! and then the binomial expansion for when x = w + z. Do you know how to complete the proof using that definition and the binomial expansion? Adam. === Subject: Re: Complex Analysis > When I did the series expansion, I used exp(x) = Sum_{n = 0, inf} x^n/n! > and then the binomial expansion for when x = w + z. Do you know how to > complete the proof using that definition and the binomial expansion? > Adam. Lets try ;) exp(x) = sum(x^n/n!) hence exp(x)*exp(y) = sum(sum(x^n/n!*y^m/m!)) now, we must think a little, our nth term is x^(n-1)/(n-1)!*(1 + y + 1/2y^2 + 1/6y^3 + ... + 1/m!y^m + ....) x^n/n!*(1 + y + 1/2y^2 + 1/6y^3 + ... + 1/m!y^m + ....) x^(n+1)/(n+1)!*(1 + y + 1/2y^2 + 1/6y^3 + ... + 1/m!y^m + ....) Now what we need to so is find the general term that has degree N. we see that when we multiply out the subseries when n+m = N we have x^n/n!*y^m/m! for our term of degree n but for each subseries we get get 1 term, and for coefficent of the Nth degreed term we must add them all up that is a_N*z = sum(x^n/n!*y^(N-n)/(N-n),n=0..N) where degree(z) = N and some form of x^a*y^b which gives us (x+y)^N/N! But, we must sum up all degree's, so we must have sum((x+y)^N/N!,N=0..infinity) = exp(x+y) So, the trick was to look at the expansion of the series and figure out what the term that has degree N is in general. Now, we can probably work backwards and clean up the prove, but I'll leave that to you(just expand exp(x+y) out in full and then factor). Jon === Subject: Re: Standard symbol for the group of multiplicative characters? > Is there a common symbol for the group of multiplicative characters of a > certain group? (In particular, I'm considering the group of > multiplicative characters of Z_p.) The (multiplicative) group of units of a ring R is notated R^* or R^x tho the former preferable. (Z_p)^* = (Z_p)0 when p prime. (Z_4)^* = { 1,3 } === Subject: Re: Standard symbol for the group of multiplicative characters? > Is there a common symbol for the group of multiplicative characters of a > certain group? (In particular, I'm considering the group of > multiplicative characters of Z_p.) > The (multiplicative) group of units of a ring R is notated R^* > or R^x tho the former preferable. (Z_p)^* = (Z_p)0 when p prime. > (Z_4)^* = { 1,3 } Yes, but OP didn't ask about the units of a ring, but about the characters of the ring - these are the homomorphisms into the complex numbers. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Standard symbol for the group of multiplicative characters? >Is there a common symbol for the group of multiplicative characters of a >certain group? (In particular, I'm considering the group of >multiplicative characters of Z_p.) >>The (multiplicative) group of units of a ring R is notated R^* >>or R^x tho the former preferable. (Z_p)^* = (Z_p)0 when p prime. >>(Z_4)^* = { 1,3 } > Yes, but OP didn't ask about the units of a ring, but about the > characters of the ring - these are the homomorphisms into the > complex numbers. Short answer: IIUYC You can use '(+)' and '(.)' respectively to represent the additive and multiplicative groups/axiom sets, that comprise, (along with the data elements, of course) a ring. [per the Penguin dictionary of Mathematics] In books I've seen plus sign enclosed in hollow circle for '(+)' and X enclosed in a hollow circle--such as '(x) or (X)?'--for '(.)' I don't know how one would render these typographically in a code set right? *IIUYC If I Understand You Correctly Ignore the blather that follows... Boy, it sure is difficult to answer the simple questions, isn't it? I know you know all this already, sorry, I'm organizing my thoughts here. Relation := maps an n-tuple (ordered set) of elements to a value The value can, of course, be another n-tuple. # could read these here as # = -- is a # + -- has a # '{' -- collection of Ring = { Elements = set { number_system | chars } + A = group = relation + set { axioms } + M = group = relation + set { axioms } } The two groups that comprise a ring (among the instances of rings being those we know so fondly as C, R, Z), that is, the two collections A plus sign enclosed in a hollow circle is frequently used (Gullberg, Dubisch, etc.) to denote that generic/abstract operation (the group, that is) that would correspond in the specific instance of C, R, Z, whatever, to the additive properties of that ring. This lets you roll your own operation (group) or to refer in conversation to the set of properties without specifically talking about C's mapping this to an operation with an arbitrary sequence of characters, and X enclosed in a hollow circle to denote the operation (a group) that corresponds to mapping the multiplicative properties to arbitrary sequences of characters, numbers, what-have-you. Just as you say, homomorphisms of C's or R's additive and multiplicative, respectively, groups. Is that what you're referring to? I've always thought that with abstract algebra we are heirs to an undue emphasis on the data, at the expense of the flexibility we'd have if we de-coupled operations from what they act on. type-less and type-ful seeming to be preferences that are more or less useful depending on the context. but I'm probably on crack, and overlooking majestic elegancies of axiomatic structure. SL === Subject: Re: Standard symbol for the group of multiplicative characters? > Is there a common symbol for the group of multiplicative characters of a > certain group? (In particular, I'm considering the group of > multiplicative characters of Z_p.) > By multiplicative character you mean a map h : Z_p -> C, > the set of complex numbers, with h(x+y) = h(x)+h(y) and > h(xy) = h(x)h(y) ? > Or, if not that, then what? I think multiplicative character just means h(xy) = h(x) h(y). You could call it the dual group of G and denote it G-dagger. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: decreasing subsets counterexample One of the exercises I am looking at for practice asks to give a counter example to the claim: Suppose (K_i : i in N) is a sequence of decreasing closed sets in R^n, and not bounded. Show the [intersection of K_i for all i in N does not equal zero] does not hold by constructing a counter-example. In an earlier exercise, this was proved is the sets were *bounded* using the Bolzano Weierstrass theorem. === Subject: Re: decreasing subsets counterexample > One of the exercises I am looking at for practice asks to give a counter > example to the claim: > Suppose (K_i : i in N) is a sequence of decreasing closed sets in R^n, and > not bounded. Show the [intersection of K_i for all i in N does not equal > zero] does not hold by constructing a counter-example. Hint: move the closed right half of the plane to the right to infinity. === Subject: Re: Taylor Poly > Suppose f(x) = cos(x) - 1 + (x^2)/2 > I'm asked to find the corresponding Taylor poly P(sub 4). Does this > mean they want me to find out f(x) when n = 4, or do they want me to > find it when a = 4? You speak naught when you speak your text book lingo. Expand cos x by Taylor series about x = 0 and add -1 + x^2 / 2. === Subject: curious properties of functions (1) if f is continuous and differentiable is f' differentiable? (2) what sequence of differentiable functions converges pointwise to a function but its sequence of derivatives does not converge pointwise to any point of the interval on which it is defined? === Subject: Re: curious properties of functions > (1) if f is continuous and differentiable is f' differentiable? > (2) what sequence of differentiable functions converges pointwise to a > function but its sequence of derivatives does not converge pointwise to > any point of the interval on which it is defined? (2) sin(n x)/n, with x in the interval [1,2]. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: curious properties of functions > (1) if f is continuous and differentiable is f' differentiable? No. Take f:R ---> R defined by f(x) = x^2 if x >= 0 and f(x) = -x^2 otherwise. Jose Carlos Santos === Subject: Re: curious properties of functions >> (1) if f is continuous and differentiable is f' differentiable? >No. Take f:R ---> R defined by f(x) = x^2 if x >= 0 and f(x) = -x^2 >otherwise. OK, but the answer is yes if the domain of f is the complex plane. That nontrivial fact is the whole reason Complex Analysis is such a different subject from Real Analysis. BTW continuous and differentiable is redundant -- _that_ much you (i.e. the OP) should be able to prove. dave === Subject: Re: curious properties of functions >(1) if f is continuous and differentiable is f' differentiable? >>No. Take f:R ---> R defined by f(x) = x^2 if x >= 0 and f(x) = -x^2 >>otherwise. > OK, but the answer is yes if the domain of f is the complex plane. > That nontrivial fact is the whole reason Complex Analysis is such a > different subject from Real Analysis. What makes you think that the OP was interested in Complex Analysis? The word complex does not occur in his post. Besides, take a look at his posts curious properties of sets, sequence of functions (there are two of those) and sequence of continuous functions. They all make reference to the real numbers amd none of them says anything about complex numbers. Jose Carlos Santos === Subject: Re: curious properties of functions <3dordkF6vlq8iU1@individual.net> <3ds222F6tdi93U1@individual.net> >> (1) if f is continuous and differentiable is f' >> differentiable? >> >> (2) what sequence of differentiable functions >> converges pointwise to a function but its sequence >> of derivatives does not converge pointwise to any >> point of the interval on which it is defined? > No. Take f:R ---> R defined by f(x) = x^2 if x >= 0 > and f(x) = -x^2 otherwise. >> OK, but the answer is yes if the domain of f >> is the complex plane. That nontrivial fact is the >> whole reason Complex Analysis is such a different >> subject from Real Analysis. > What makes you think that the OP was interested in > Complex Analysis? The word complex does not occur > in his post. Besides, take a look at his posts > curious properties of sets, sequence of functions > (there are two of those) and sequence of continuous > functions. They all make reference to the real > numbers amd none of them says anything about > complex numbers. My interpretation is that Dave Rusin didn't think this, but he was instead simply pointing out an interesting tangent that others (and perhaps also the OP) would find interesting, at least those who didn't already know it. Along the same tangent that Dave Rusin raised, note that it's easy to define a nowhere differentiable continuous function f:C --> C. For example, f(z) = |z|. But, as most anyone who's taught a college math class (in the U.S., at least) and then read their student evaluation comments knows, tangents -- regardless of how well intended and relevant to the subject at hand -- have to be dealt with extremely carefully. However, this has not been the case in sci.math for the past 20 years. The vast majority of the interesting posts in sci.math over the years have not been those that directly answer a poster's question and nothing else. Moreover, much of the mathematical information that Dave Rusin has archived at his Mathematical Atlas, , comes from sci.math posts that *weren't* direct answers to a poster's question and nothing else. I'm sure there are some discussion groups (probably not any in Usenet, however) that only deal with answering specific questions and little else, but this has never been the format of sci.math. Dave L. Renfro === Subject: Re: curious properties of functions >>(1) if f is continuous and differentiable is f' differentiable? >No. Take f:R ---> R defined by f(x) = x^2 if x >= 0 and f(x) = -x^2 >otherwise. >> OK, but the answer is yes if the domain of f is the complex plane. >> That nontrivial fact is the whole reason Complex Analysis is such a >> different subject from Real Analysis. >What makes you think that the OP was interested in Complex Analysis? Oh, I know he's probably not -- yet. I mean, I certainly wasn't claiming your answer was wrong, or even inadequate. But I try to sneak in a By the way... into the answer to simple questions, to pique the interest of students. Hm, maybe I should have suggested he further consider what happens when applying the definition of the derivative to functions defined on the quaternions? dave === Subject: Applied mathematics, mapping from variables to ideas In science, one typically will use mathematics in various ways to understand and resolve problems. However, what is done is mapping between a physical idea and mathematical idea, equation, theory, etc. How does one know that such a mapping is true? Does it come from application and testing? That is, it seems obvious to use the calculus of differentiation to find is represented as a time-dependent function. We do it all the time in physics and it works. That is, the mapping between the concept of a physical velocity and mathematical differentiation works. The reason that I ask is because I am now learning more abstract mathematics. Specifically the calculus of variations. In its development I've learned that it is based on abstract vector spaces with abstract concepts of length called norms. Being that the ideas are mathematical abstractions and can be completely arbitrary, it made me question just which definitions of a norm, etc, will lead to a physically applicable theory. Surely not all definitions will result in an applicable theory. It seems that when we use mathematics, we create a mapping between a mathematical variable/idea and a logical/physical idea. At least for the sciences. I'm not speaking of pure mathematics where one does consider a mapping to real work concepts. How is one to know that a mathematical concept is applicable to physical concept and how to decide what mapping to use? The mappings of calculus to real world concepts seem quite obvious, which is probably because it was motivated by real world concepts. However, what about the more abstract mathematical concepts? I hope I've expressed myself clearly enough, for I truly wish to know how other people view these issues of mappings between mathematical concepts and real world concepts. === Subject: Re: Applied mathematics, mapping from variables to ideas > In science, one typically will use mathematics in various ways to > understand and resolve problems. However, what is done is mapping > between a physical idea and mathematical idea, equation, theory, etc. > How does one know that such a mapping is true? A quote from Einstein: As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality. He also made some remark about mathematical physics having to be redone because it was based on the assumption of linearity and there was no linearity in physical world. === Subject: Re: Applied mathematics, mapping from variables to ideas Try equity valuation. They map applied math to ideas for evaluating stocks and bonds and making cold, hard cash money: === Subject: Re: Applied mathematics, mapping from variables to ideas >In science, one typically will use mathematics in various ways to >understand and resolve problems. However, what is done is mapping >between a physical idea and mathematical idea, equation, theory, etc. >How does one know that such a mapping is true? Does it come from >application and testing? >That is, it seems obvious to use the calculus of differentiation to find >is represented as a time-dependent function. We do it all the time in >physics and it works. That is, the mapping between the concept of a >physical velocity and mathematical differentiation works. Whether it works or not is not entirely clear. The mathematicians ask the physicists, Is it true that there is one moment in time for every real number? Is it true that there is one position on the track for every real number? Does the moving object 'have' a position at each time (i.e. does the motion describe a function x = f(t) )? When you speak of 'displacement in space', do you mean x2 - x1? Likewise is the notion of 'time elapsed' captured by t2 - t1 ? Does 'average velocity' mean the ratio of displacement to time elapsed? Does 'instantaneous velocity' mean the limit of the average velocities as the time elapsed tends to zero? (pause for gulp of air) Ah! THEN instantaneous velocity is measured by the derivative f'(t). (Note that along the way the physicist might want to check what we mean by real number or limit before saying yes or no to one of the questions.) So the extent to which the derivative measures the instantaneous velocity is limited by a few things: we have to ask the physicists what they _mean_ by things like velocity; and we have to accept their _assumptions_ about some more fundamental aspects of the system being modeled. So for example, derivatives will cease to do a good job of capturing the notion of velocity when we move to quantum mechanics, where the answer to the third question becomes 'no'. We have to work harder to make derivatives give the right answer when passing to Relativity, where the second question becomes 'no' (in the sense that space is no longer simple Euclidean space). And finally there is some speculation that time itself need not be properly modeled by the real line; if the answer to the very first question is 'no' then I don't see how derivatives will necessarily provide an understanding of velocity. Some say that mathematicians make up the rules of the game and then deduce the consequences, while physicists have to observe the consequences of the universe's rules and have to guess what the rules are. That means that the conversation with the physicists must be continuously renewed as they (P.) continue to perform new experiments which might reveal that they were making assumptions which, at least in some situations, are not really valid. Typically (not always) we (M.) have other mathematical systems ready to go which satisfy different sets of axioms, and whose consequences we know; so there is a give-and-take going on with the M. and P. working together to determine which mathematical constructs best model the newly emergent understanding of the universe. dave === Subject: Re: Applied mathematics, mapping from variables to ideas > So the extent to which the derivative measures the instantaneous > velocity is limited by a few things: we have to ask the physicists > what they _mean_ by things like velocity; and we have to accept > their _assumptions_ about some more fundamental aspects of the > system being modeled. > So for example, derivatives will cease to do a good job of capturing > the notion of velocity when we move to quantum mechanics, where the > answer to the third question becomes 'no'. We have to work harder > to make derivatives give the right answer when passing to Relativity, > where the second question becomes 'no' (in the sense that space is > no longer simple Euclidean space). And finally there is some speculation > that time itself need not be properly modeled by the real line; > if the answer to the very first question is 'no' then I don't see > how derivatives will necessarily provide an understanding of velocity. I feel a need to better explain myself because I think the velocity example did give the proper idea. When mathematical formula are written down, a meaning will be given to the symbols used and a general idea to the mathematical operations used. We then perform typographical manipulation to arrive at, typically, one symbol equal to a combination of many others. We then map that symbol and the combination from the mathematical domain to the real world domain. For instance, F = Integral f(x,y,z)dV We can manipulate the formula using mathematics and given values, but that is not dependent on what F, f, dV actually represent when being used to calculate something. The mapping that I'm asking about is when we say that f is mass density, dV a volume element, and F the total mass in the volume. OR, f can be charge density, and F the total charge. Those are obvious mappings, but how does one know that if we let f represent length, and dV atmospheric density, that F will be total mass? Obviously we know that such a mapping of ideas to the symbols in the last example does not work, what I ask is how one can know that such a mapping does work and so get useful information from the mathematics. === Subject: Re: Applied mathematics, mapping from variables to ideas Most abstraction in mathematics are a fallout of several studies of real life phenomena. It is in an attempt to get the essence in these real life examples that abstraction comes about. In my case, I cannot appreciate a mathematical abstraction without getting adequate motivation. I like to ask questions like why was this choice of definition or concept used? Could I improve or get a different slant to it? And more often than not I cannot improve such concepts which tends to dramatically increase my appreciation for such. Others may have a different view but the true test of a mathematical abstraction, in my own opinion, is how much insight it gives when applied to concrete problems. > In science, one typically will use mathematics in various ways to > understand and resolve problems. However, what is done is mapping > between a physical idea and mathematical idea, equation, theory, etc. > How does one know that such a mapping is true? Does it come from > application and testing? > That is, it seems obvious to use the calculus of differentiation to find > is represented as a time-dependent function. We do it all the time in > physics and it works. That is, the mapping between the concept of a > physical velocity and mathematical differentiation works. > The reason that I ask is because I am now learning more abstract > mathematics. Specifically the calculus of variations. In its development > I've learned that it is based on abstract vector spaces with abstract > concepts of length called norms. Being that the ideas are mathematical > abstractions and can be completely arbitrary, it made me question just > which definitions of a norm, etc, will lead to a physically applicable > theory. Surely not all definitions will result in an applicable theory. > It seems that when we use mathematics, we create a mapping between a > mathematical variable/idea and a logical/physical idea. At least for the > sciences. I'm not speaking of pure mathematics where one does consider a > mapping to real work concepts. > How is one to know that a mathematical concept is applicable to physical > concept and how to decide what mapping to use? The mappings of calculus > to real world concepts seem quite obvious, which is probably because it > was motivated by real world concepts. However, what about the more > abstract mathematical concepts? > I hope I've expressed myself clearly enough, for I truly wish to know > how other people view these issues of mappings between mathematical > concepts and real world concepts. === Subject: Re: Applied mathematics, mapping from variables to ideas 05/03/2005 >Most abstraction in mathematics are a fallout of several studies of >real life phenomena. What gives you that idea? -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Applied mathematics, mapping from variables to ideas days. My association with the Department is that of an alumnus. >05/03/2005 >>Most abstraction in mathematics are a fallout of several studies of >>real life phenomena. >What gives you that idea? Most abstract mathematics are done by people in real life, hence they are the consequence of real life phenomena... (-: (My dad says that once they asked his Abstract Automata Theory professor if there were any practical applications to what he was teaching. After thinking about it for a few seconds, he said As far as I know, there are 10 people in the world, plus me, who make a living by studying it. I cannot think of anything more practical than providing a living for someone.) -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Applied mathematics, mapping from variables to ideas >>05/03/2005 >Most abstraction in mathematics are a fallout of several studies of >real life phenomena. >>What gives you that idea? > Most abstract mathematics are done by people in real life, hence they > are the consequence of real life phenomena... (-: > (My dad says that once they asked his Abstract Automata Theory > professor if there were any practical applications to what he was > teaching. After thinking about it for a few seconds, he said As far > as I know, there are 10 people in the world, plus me, who make a > living by studying it. I cannot think of anything more practical than > providing a living for someone.) Also, is it just coicidental that the axioms of set theory, upon which modern mathematics is based, happen to be logical in our reality? Adam. === Subject: Weierstrass M test Can the sequence of functions gn(x)=x/(1+nx^2) be made to pass the Weierstrass M test for some sequence,Mn, involving n's such that the sum of the sequence Mn converges? If yes, then what is your Mn? === Subject: Re: Weierstrass M test > Can the sequence of functions > gn(x)=x/(1+nx^2) > be made to pass the Weierstrass M test for some sequence,Mn, involving > n's such that the sum of the sequence Mn converges? If yes, then what > is your Mn? === Subject: tiny sample Bayesian inference My equity valuation group is looking for an introductory text or econometric techniques to use for tiny samples (e.g., less than 10 data points.) Mostly we pick stocks, but we are also interested in the standard methods used in all applications. Double Dribble === Subject: Re: When I was studying about Mobius transformation.... I would like hearing of you, Do you agree with the answer I gave you . === Subject: Just started a blog on Fermat's Last Theorem Hi Math experts, I just started a blog on Fermat's Last Theorem: http://fermatslasttheorem.blogspot.com My goal is to trace the history of the problem and present proofs for each of the major developments. This will include blogs on major results, theorems, and individuals who make up the story. I would greatly appreciate any comments that anyone has. As an amateur, it is my hope to really get the details correct. Here is roughly the major milestones that I intend to cover with the blog. 1. Solution for Pythagorean Triples 2. Proof for n=4 (Fermat's method of infinite descent) 3. Euler's proof for n=3 4. Gauss's proof for n=3 5. Sophie Germain's proof for n=5 6. Lame's proof for n=7 7. Kummer's proof for cyclotomic integers 8. Wiles' first proof with error 9. Wiles' refined proof in 1995. I would be very interested to hear what people think of this project. -Larry === Subject: Re: Just started a blog on Fermat's Last Theorem > Hi Math experts, > I just started a blog on Fermat's Last Theorem: > http://fermatslasttheorem.blogspot.com > My goal is to trace the history of the problem and present proofs for > each of the major developments. This will include blogs on major > results, theorems, and individuals who make up the story. > I would greatly appreciate any comments that anyone has. As an > amateur, it is my hope to really get the details correct. What would surpass other websites on the matter is to include much on failed proofs. There is a lot of information on particular proofs, and those key failed proofs, such as that by Kummer. But it would be nice to read about all the dead-end avenues. Practically all the work is disaster-prone on this matter but, because of that, it doesn't get published (readily?). There is a lot on FLT on the web already. To be frank, you probably wouldn't add much if you stuck to the standard format. I think you might make it much more interesting to include where the standard amateur proofs fail. Of course, Fermat was himself an 'amateur'. Good luck Richard Miller === Subject: Re: Just started a blog on Fermat's Last Theorem > I just started a blog on Fermat's Last Theorem: > http://fermatslasttheorem.blogspot.com > > My goal is to trace the history of the problem and present proofs for > each of the major developments. This will include blogs on major > results, theorems, and individuals who make up the story. > What would surpass other websites on the matter is to include much on failed > proofs. I have made an attempt of this. But even showing that some proof is a failure goes beyond the comprehension of the amateur. E.g., why was Euler's proof for n=3 wrong? It is very subtle and was not found until much later. See my musings on it on: (the link does not yet work. I have more to do...) Much of the history has already been outlined clear enough by Paul Ribenboim in his book: Fermat's Last Theorem for Amateurs. Although the showing of the wrongness of Lame's proof is way beyond the comprehension of the amateurs (as is his proof). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Just started a blog on Fermat's Last Theorem > Hi Math experts, > I just started a blog on Fermat's Last Theorem: > http://fermatslasttheorem.blogspot.com > My goal is to trace the history of the problem and > present proofs for each of the major developments. > This will include blogs on major results, theorems, > and individuals who make up the story. > I would greatly appreciate any comments that anyone > has. As an amateur, it is my hope to really get the > details correct. > Here is roughly the major milestones that I intend to > cover with the blog. > 1. Solution for Pythagorean Triples > 2. Proof for n=4 (Fermat's method of infinite descent) > 3. Euler's proof for n=3 > 4. Gauss's proof for n=3 > 5. Sophie Germain's proof for n=5 > 6. Lame's proof for n=7 > 7. Kummer's proof for cyclotomic integers > 8. Wiles' first proof with error > 9. Wiles' refined proof in 1995. > I would be very interested to hear what people think of > this project. You forgot ... 10. Heckman's proof for n=6. 8-) Seriously, though, I'm relieved to hear that someone is interested in the problem as a whole, and I was glad when I realized you weren't another FLT crank, like Mark Provo. There may be a book out with all these proofs in it, though. (Later:) There should be a bunch in section QA244 of your local university library. (About a dozen here at Arizona State, for instance.) --- Christopher Heckman === Subject: Re: Interesting curve - is it ellipse? > Well for a start, the number of opening braces doesn't agree with the number > of closing braces. But even > if I put a missing brace in the most likely place (just before /) the > results is unfortunatelly not right. Well, sorry for the typo, but you got it right: the missing closing brace should be located just before the / sign. > In case you are going to review your calculation, please let me know about > the right result. The right result is the inverse of the square root of the previous one. In order to avoid ambiguities, the length of the semimajor axe is the square root of 2(A B sin(C))^2/(A^2 + B^2 - sqrt((A^2 - B^2)^2 + 4(A B cos(C))^2)) and the length of the semiminor axe is the square root of 2(A B sin(C))^2/(A^2 + B^2 + sqrt((A^2 - B^2)^2 + 4(A B cos(C))^2)) Jose Carlos Santos === Subject: Re: Interesting curve - is it ellipse? These times the expressions are correct! And for those who might be interested, here is also the angle of ellipse rotation (I got it by finding minimum of radius of ellipse written in spherical coordinates): c= arctan((2 * A / B * Cos(C)) / ((A / B) ^ 2 - 1)) / 2 Marko Pinteric >> Well for a start, the number of opening braces doesn't agree with the >> number of closing braces. But even >> if I put a missing brace in the most likely place (just before /) the >> results is unfortunatelly not right. > Well, sorry for the typo, but you got it right: the missing closing > brace should be located just before the / sign. >> In case you are going to review your calculation, please let me know >> about the right result. > The right result is the inverse of the square root of the previous one. In > order to avoid ambiguities, the length of the semimajor axe is the > square root of > 2(A B sin(C))^2/(A^2 + B^2 - sqrt((A^2 - B^2)^2 + 4(A B cos(C))^2)) > and the length of the semiminor axe is the square root of > 2(A B sin(C))^2/(A^2 + B^2 + sqrt((A^2 - B^2)^2 + 4(A B cos(C))^2)) > Jose Carlos Santos === Subject: Re: Interesting curve - is it ellipse? Assuming that A <> 0 and B <> 0 one has cos(t) = x/A, cos(t+C) == cos(t)cos(C) - sin(t)sin(C) = y/B, sin(t) = some expression linear in x and y; 1 = cos^2(t) + sin^2(t) = some quadratic polynomial P(x, y) in x and y; Finally prove that the ax^2 + bxy + cy^2 part of P is positive-definite. Happy studies: Johan E. Mebius >Hello mathematicians! >I have the following parametric definition of a curve: >x = A cos(t), y = B cos(t+C), >where A, B and C are constants. >I am almost certain that this curve is an ellipse, rotated >by a certain angle from its normal position. I have succeeded >to write this curve in polar coordinates and to find the angles >of maximum and minimum radii from the center of the coordinate >system and its appropriate radii values. So if I draw a >rotated ellipse with these parameters and they graphically seem >to fit perfectly. >I would like to know if there is a way to PROVE that this is >really an ellipse? Is it possible to find three expressions >that give the angle of rotation c, and ellipse's semimajor axe >a and semiminor axe b, that is >a=f1(A,B,C), b=f2(A,B,C), c=f3(A,B,C)? >Marko Pinteric (physicist) === Subject: Re: Epistemology 202: Advanced Topics > Curiouser and curiouser. There seems to be no difference between > a 'space filling curve' and plane. Yes, it's one of the interesting things that come from defining a curve the calculus way. Geometrical curves can't do that. -- Giuseppe Oblomov Bilotta I weep for our generation -- Charlie Brown === Subject: Re: Epistemology 202: Advanced Topics >>Curiouser and curiouser. There seems to be no difference between >>a 'space filling curve' and plane. > Yes, it's one of the interesting things that come from defining a > curve the calculus way. Geometrical curves can't do that. Does Hilbert's axioms allow this? Or does 'interesting' trump axioms? -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics >Curiouser and curiouser. There seems to be no difference between >a 'space filling curve' and plane. >> Yes, it's one of the interesting things that come from defining a >> curve the calculus way. Geometrical curves can't do that. > Does Hilbert's axioms allow this? Hilber axioms don't define curves. And I don't have the time to check if the calculus definition of curve contradicts Hilbert axioms. > Or does 'interesting' trump > axioms? Interesting is in the eye of the beholder. -- Giuseppe Oblomov Bilotta Da grande lotter.98 per la pace A me me la compra il mio babbo (Altan) (When I grow up, I will fight for peace I'll have my daddy buy it for me) === Subject: Re: Epistemology 202: Advanced Topics >>Curiouser and curiouser. There seems to be no difference between >>a 'space filling curve' and plane. >Yes, it's one of the interesting things that come from defining a >curve the calculus way. Geometrical curves can't do that. >>Does Hilbert's axioms allow this? > Hilber axioms don't define curves. And I don't have the time to check > if the calculus definition of curve contradicts Hilbert axioms. So defining a 5 diameter circle does not rely on Hilbert's axioms? There must be a different set of axioms explaining point usage, other than Hilbert's that you are using. I wonder why you all sent me on this wild goose chase in Hilbert's axioms. Where might I find the axioms that you are *really* using? >>Or does 'interesting' trump >>axioms? > Interesting is in the eye of the beholder. Yes, indeed. Assumed truth and defined truth hold no interest for me, in that even madmen can and do assume and define truth, and then weave fantastic fictions from such assumptions and definitions. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics >> >> >Curiouser and curiouser. There seems to be no difference between >a 'space filling curve' and plane. >> >>Yes, it's one of the interesting things that come from defining a >>curve the calculus way. Geometrical curves can't do that. >> >Does Hilbert's axioms allow this? >> Hilber axioms don't define curves. And I don't have the time to check >> if the calculus definition of curve contradicts Hilbert axioms. > So defining a 5 diameter circle does not rely on Hilbert's > axioms? There must be a different set of axioms explaining point > usage, other than Hilbert's that you are using. I wonder why you > all sent me on this wild goose chase in Hilbert's axioms. We didn't send you anywhere. We just showed you that 5 points were not in contrast with H's axioms. IOW, we showed that the axiom: F.I Points are 5 is consistent with H's axioms; just as much as the axiom you usually use F.I' Points are dimensionless is. > Where > might I find the axioms that you are *really* using? When you say really, you mean in real life or to show you what does it mean to deal with an axiomatic system? >Or does 'interesting' trump >axioms? >> Interesting is in the eye of the beholder. > Yes, indeed. Assumed truth and defined truth hold no interest > for me, in that even madmen can and do assume and define truth, > and then weave fantastic fictions from such assumptions and > definitions. Indeed. -- Giuseppe Oblomov Bilotta Hic manebimus optime === Subject: Re: Epistemology 202: Advanced Topics >> >> > > > >>Curiouser and curiouser. There seems to be no difference between >>a 'space filling curve' and plane. > >Yes, it's one of the interesting things that come from defining a >curve the calculus way. Geometrical curves can't do that. > >> >>Does Hilbert's axioms allow this? >Hilber axioms don't define curves. And I don't have the time to check >if the calculus definition of curve contradicts Hilbert axioms. >>So defining a 5 diameter circle does not rely on Hilbert's >>axioms? There must be a different set of axioms explaining point >>usage, other than Hilbert's that you are using. I wonder why you >>all sent me on this wild goose chase in Hilbert's axioms. > We didn't send you anywhere. We just showed you that 5 points were > not in contrast with H's axioms. You showed no such thing. You merely continued to assert it ignoring the inconsistencies in Hilbert's axioms that dimensioned points cause. IOW, we showed that the axiom: > F.I Points are 5 > is consistent with H's axioms; just as much as the axiom you usually > use > F.I' Points are dimensionless > is. >> Where >>might I find the axioms that you are *really* using? > When you say really, you mean in real life or to show you what does > it mean to deal with an axiomatic system? The axioms you use in combination with set theory that allow you to make the statement that a circle or a line segment can be defined as being composed of dimensionless points, as in the set of all points... >>Or does 'interesting' trump >>axioms? >Interesting is in the eye of the beholder. >>Yes, indeed. Assumed truth and defined truth hold no interest >>for me, in that even madmen can and do assume and define truth, >>and then weave fantastic fictions from such assumptions and >>definitions. > Indeed. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics >> >> > > >> >> >> >Curiouser and curiouser. There seems to be no difference between >a 'space filling curve' and plane. >> >>Yes, it's one of the interesting things that come from defining a >>curve the calculus way. Geometrical curves can't do that. >> > >Does Hilbert's axioms allow this? >> >>Hilber axioms don't define curves. And I don't have the time to check >>if the calculus definition of curve contradicts Hilbert axioms. >So defining a 5 diameter circle does not rely on Hilbert's >axioms? There must be a different set of axioms explaining point >usage, other than Hilbert's that you are using. I wonder why you >all sent me on this wild goose chase in Hilbert's axioms. >> We didn't send you anywhere. We just showed you that 5 points were >> not in contrast with H's axioms. > You showed no such thing. You merely continued to assert it > ignoring the inconsistencies in Hilbert's axioms that dimensioned > points cause. You keep repeating there is an inconsistency, but each time I ask you to point it out, show how it's inconsistent, you just call me a contentious asshole. How about actually showing the inconsistency? Using logic, not just asserting they're inconsistent. >> IOW, we showed that the axiom: >> F.I Points are 5 >> is consistent with H's axioms; just as much as the axiom you usually >> use >> F.I' Points are dimensionless >> is. > Where >might I find the axioms that you are *really* using? >> When you say really, you mean in real life or to show you what does >> it mean to deal with an axiomatic system? > The axioms you use in combination with set theory that allow you > to make the statement that a circle or a line segment can be > defined as being composed of dimensionless points, as in the set > of all points... This isn't relevant to the discussion here since I didn't use those axioms nor those definitions here. -- Giuseppe Oblomov Bilotta I'm never quite so stupid as when I'm being smart --Linus van Pelt === Subject: Re: Epistemology 202: Advanced Topics > You keep repeating there is an inconsistency, but each time I ask you > to point it out, show how it's inconsistent, you just call me a > contentious asshole. How about actually showing the inconsistency? > Using logic, not just asserting they're inconsistent. I've already wasted my time doing this for Guenther Von Krackpot. Go find it. >>Where >>might I find the axioms that you are *really* using? >When you say really, you mean in real life or to show you what does >it mean to deal with an axiomatic system? >>The axioms you use in combination with set theory that allow you >>to make the statement that a circle or a line segment can be >>defined as being composed of dimensionless points, as in the set >>of all points... > This isn't relevant to the discussion here since I didn't use those > axioms nor those definitions here. It became relevant to the discussion when I asked the question. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics > But when _I_ talked about > space, I meant empty space. And it seems odd to me to claim that a > region > of space that has a line drawn on it is still (empty) space. >> Oho, now we have this new empty qualifier jumping in ... >> Would you mind defining empty space now? > You know what you see between two objects? That's it. Nothing? (Plus, I thought we were talking abstract concepts) > And if you start > talking about air molecules, you'll know that you've just gotten into an > argumentative mode. Eh eh eh. -- Giuseppe Oblomov Bilotta Hic manebimus optime === Subject: Re: Epistemology 202: Advanced Topics > Universal truth is > vastly superior to mere axioms because it ensures terminological > regression is not just to assumptions and circular definitions. Agreed on this. > The > Peano axioms, for example, just assume everything they're supposed > to prove. The Pean axioms don't prove anything. They *define* things. > Why not just assume the real numbers and be done with it? Because the less you assume the cleaner everything is. >>Without considering that math usage of terms is defined *when it's >>introduced*, not when other people start asking what you mean because >>what you're saying doesn't make sense. > What I'm saying doesn't make sense to you because your sense is > defined by current mathematical thinking defined most recently over > the course of the last hundred years. Uh, no. They don't make sense because your thinking only barely resembles logic, and it's full of hidden assumptions. And yes, I've read your initial posts and they don't make sense either. > That's why your terms are > defined at the beginning because everyone knows what the critical > cognitive path is. I had no idea supposedly critical thinkers could > have such problems with contradiction and lines. Even so I spelled out > my definitions for true, false, and not in terms of universal > tautological regressions to self contradictory alternatives right in > the very root post to the E201 thread in the section Ur Regressions > for True, False, and Not. Actually you haven't. Quoting: > Ur Regressions for True, False, and Not > -------------- >True, False, and Not are defined in reciprocal terms in the following >way. For any empirical observation [subject] the proposition >p:[subject][not subject] is always true. And the proposition >p:[subject not subject] is always false. And the empirical observation >P:[not] is always true because the proposition >P:[not not] is always false. Is this the right part? Where you claim to have spelled out your definitions for true, false and not in terms of universal tautological regressions to self contradictory alternatives? Let's see. Those can be taken are definitions for true, false and not all right. Are they in terms of universal tautological regressions to self contradictory alternatives? Hm. I don't see how. Or are you making such a claim on the basis that (quoting you from the subsequent paragraph on that same post): > These seem to be the only reducible definitions for true, false, and > not. So because you cannot find any other definitions for true, false and not? -- Giuseppe Oblomov Bilotta [W]hat country can preserve its liberties, if its rulers are not warned from time to time that [the] people preserve the spirit of resistance? Let them take arms...The tree of liberty must be refreshed from time to time, with the blood of patriots and tyrants. -- Thomas Jefferson, letter to Col. William S. Smith, 1787 === Subject: Re: Epistemology 202: Advanced Topics >> Universal truth is >> vastly superior to mere axioms because it ensures terminological >> regression is not just to assumptions and circular definitions. >Agreed on this. >> The >> Peano axioms, for example, just assume everything they're supposed >> to prove. >The Pean axioms don't prove anything. They *define* things. But what they define is cardinal integer counting and not numbers. There is even a more comprehensive set of axiomatic assumptions: Suc(none)=first, suc(first)=second, etc. from which Peano counting can be deduced as a subset by assuming differences between ordinal intervals are equal. >> Why not just assume the real numbers and be done with it? >Because the less you assume the cleaner everything is. And cleanliness is next to godliness? >Without considering that math usage of terms is defined *when it's >introduced*, not when other people start asking what you mean because >what you're saying doesn't make sense. >> What I'm saying doesn't make sense to you because your sense is >> defined by current mathematical thinking defined most recently over >> the course of the last hundred years. >Uh, no. They don't make sense because your thinking only barely >resembles logic, and it's full of hidden assumptions. And yes, I've >read your initial posts and they don't make sense either. What can I say? >> That's why your terms are >> defined at the beginning because everyone knows what the critical >> cognitive path is. I had no idea supposedly critical thinkers could >> have such problems with contradiction and lines. Even so I spelled out >> my definitions for true, false, and not in terms of universal >> tautological regressions to self contradictory alternatives right in >> the very root post to the E201 thread in the section Ur Regressions >> for True, False, and Not. >Actually you haven't. Quoting: >> Ur Regressions for True, False, and Not >> -------------- >>True, False, and Not are defined in reciprocal terms in the following >>way. For any empirical observation [subject] the proposition >>p:[subject][not subject] is always true. And the proposition >>p:[subject not subject] is always false. And the empirical observation >>P:[not] is always true because the proposition >>P:[not not] is always false. >Is this the right part? Where you claim to have spelled out your >definitions for true, false and not in terms of universal tautological >regressions to self contradictory alternatives? Yes. >Let's see. Those can be taken are definitions for true, false and >not all right. Are they in terms of universal tautological >regressions to self contradictory alternatives? Hm. I don't see how. The tautological propostion p:[subject][not subject] is true because it exhausts all possibilities for truth. The tautological proposition p:[subject not subject] is false because it is self contradictory. And the tautological regression for p:[not] is P:[not not] and is finite because it's self contradictory. I don't know what better definitions for true, false, and not you could want. >Or are you making such a claim on the basis that (quoting you from the >subsequent paragraph on that same post): >> These seem to be the only reducible definitions for true, false, and >> not. >So because you cannot find any other definitions for true, false and >not? Oh I imagine there could be a multitude of other definitions. Just not a multitude of other universally true definitions finitely regressible to self contradictory alternatives by means of tautologies. QED. === Subject: Re: Epistemology 202: Advanced Topics > Universal truth is > vastly superior to mere axioms because it ensures terminological > regression is not just to assumptions and circular definitions. >>Agreed on this. > The > Peano axioms, for example, just assume everything they're supposed > to prove. >>The Pean axioms don't prove anything. They *define* things. > But what they define is cardinal integer counting and not numbers. > There is even a more comprehensive set of axiomatic assumptions: > Suc(none)=first, suc(first)=second, etc. from which Peano counting can > be deduced as a subset by assuming differences between ordinal > intervals are equal. See also my reply to Wagner. Peano axioms are necessary and sufficient to define the natural numbers; they characterize it. In the sense that *any* set with *any* function that satisfy Peano axioms can be mapped into the naturals and on it you can define two operations that map to the natural addition and multiplication. i.e. you can say: ok I define natural numbers by any other way, whichever way you want. Yet, if you do, you'll find that the set you define will obey Peano axioms. > Why not just assume the real numbers and be done with it? >>Because the less you assume the cleaner everything is. > And cleanliness is next to godliness? No, it's just good housekeeping. Helps spotting problems. >>Without considering that math usage of terms is defined *when it's >>introduced*, not when other people start asking what you mean because >>what you're saying doesn't make sense. > > What I'm saying doesn't make sense to you because your sense is > defined by current mathematical thinking defined most recently over > the course of the last hundred years. >>Uh, no. They don't make sense because your thinking only barely >>resembles logic, and it's full of hidden assumptions. And yes, I've >>read your initial posts and they don't make sense either. > What can I say? You could try being clearer. And no, repeating the same things over and over again with the same words and the same sentences expressed in the same way does not help being clearer. > That's why your terms are > defined at the beginning because everyone knows what the critical > cognitive path is. I had no idea supposedly critical thinkers could > have such problems with contradiction and lines. Even so I spelled out > my definitions for true, false, and not in terms of universal > tautological regressions to self contradictory alternatives right in > the very root post to the E201 thread in the section Ur Regressions > for True, False, and Not. >>Actually you haven't. Quoting: > Ur Regressions for True, False, and Not > -------------- >True, False, and Not are defined in reciprocal terms in the following >way. For any empirical observation [subject] the proposition >p:[subject][not subject] is always true. And the proposition >p:[subject not subject] is always false. And the empirical observation >P:[not] is always true because the proposition >P:[not not] is always false. >>Is this the right part? Where you claim to have spelled out your >>definitions for true, false and not in terms of universal tautological >>regressions to self contradictory alternatives? > Yes. Good, at least something we can agree on :) >>Let's see. Those can be taken are definitions for true, false and >>not all right. Are they in terms of universal tautological >>regressions to self contradictory alternatives? Hm. I don't see how. > The tautological propostion p:[subject][not subject] is true > because it exhausts all possibilities for truth. The tautological > proposition p:[subject not subject] is false because it is self > contradictory. I assume the first has an OR between the brackets and the second and AND before the NOT. Your non-use of conjunctions except for NOT makes your statemes much less clear. I don't agree with your use of tautological. Of the two proposition, the first is a tautology and the second is a (self) contradiction. The second is not tautological in any way. Regardless, I do agree that we can accept (as an axiom) that any statement in the form (P) or (not P) is true and that (as an axiom again) any statement in the form (P) and (not P) is false. > And the tautological regression for p:[not] is > P:[not not] and is finite because it's self contradictory. Where is the regression? > I don't know what better definitions for true, false, and not you > could want. The problem here is that you're giving a meaning to not as a proposition alone. *This* doesn't make sense to me. >>Or are you making such a claim on the basis that (quoting you from the >>subsequent paragraph on that same post): > These seem to be the only reducible definitions for true, false, and > not. >>So because you cannot find any other definitions for true, false and >>not? > Oh I imagine there could be a multitude of other definitions. Just not > a multitude of other universally true definitions finitely regressible > to self contradictory alternatives by means of tautologies. QED. The only thing you're proven is that you're taking your definitions to be universal because you can't think of others. Arrogant to say the least. -- Giuseppe Oblomov Bilotta E la storia dell'umanit.88, babbo? Ma niente: prima si fanno delle cazzate, poi si studia che cazzate si sono fatte (Altan) (And what about the history of the human race, dad? Oh, nothing special: first they make some foolish things, then you study what foolish things have been made) === Subject: Re: Epistemology 202: Advanced Topics >> Universal truth is >> vastly superior to mere axioms because it ensures terminological >> regression is not just to assumptions and circular definitions. >Agreed on this. >> The >> Peano axioms, for example, just assume everything they're supposed >> to prove. >The Pean axioms don't prove anything. They *define* things. >> But what they define is cardinal integer counting and not numbers. >> There is even a more comprehensive set of axiomatic assumptions: >> Suc(none)=first, suc(first)=second, etc. from which Peano counting can >> be deduced as a subset by assuming differences between ordinal >> intervals are equal. >See also my reply to Wagner. >Peano axioms are necessary and sufficient to define the natural >numbers; they characterize it. They characterize operations on the natural numbers such as counting and not the natural numbers themselves. >In the sense that *any* set with *any* function that satisfy Peano >axioms can be mapped into the naturals and on it you can define two >operations that map to the natural addition and multiplication. Yeah, but you can't define subtraction or division. >i.e. you can say: ok I define natural numbers by any other way, >whichever way you want. Yet, if you do, you'll find that the set you >define will obey Peano axioms. Who cares. I never said a correct defintion for natural numbers would result in something that couldn't be counted. I just said that Peano axioms define counting of natural numbers and not the natural numbers. >> Why not just assume the real numbers and be done with it? >Because the less you assume the cleaner everything is. >> And cleanliness is next to godliness? >No, it's just good housekeeping. Helps spotting problems. >Without considering that math usage of terms is defined *when it's >introduced*, not when other people start asking what you mean because >what you're saying doesn't make sense. >> >> What I'm saying doesn't make sense to you because your sense is >> defined by current mathematical thinking defined most recently over >> the course of the last hundred years. >Uh, no. They don't make sense because your thinking only barely >resembles logic, and it's full of hidden assumptions. And yes, I've >read your initial posts and they don't make sense either. >> What can I say? >You could try being clearer. And no, repeating the same things over >and over again with the same words and the same sentences expressed in >the same way does not help being clearer. So the fact that you don't understand means they're not understandable? >> That's why your terms are >> defined at the beginning because everyone knows what the critical >> cognitive path is. I had no idea supposedly critical thinkers could >> have such problems with contradiction and lines. Even so I spelled out >> my definitions for true, false, and not in terms of universal >> tautological regressions to self contradictory alternatives right in >> the very root post to the E201 thread in the section Ur Regressions >> for True, False, and Not. >Actually you haven't. Quoting: >> Ur Regressions for True, False, and Not >> -------------- >> >>True, False, and Not are defined in reciprocal terms in the following >>way. For any empirical observation [subject] the proposition >> >>p:[subject][not subject] is always true. And the proposition >> >>p:[subject not subject] is always false. And the empirical observation >> >>P:[not] is always true because the proposition >> >>P:[not not] is always false. >Is this the right part? Where you claim to have spelled out your >definitions for true, false and not in terms of universal tautological >regressions to self contradictory alternatives? >> Yes. >Good, at least something we can agree on :) >Let's see. Those can be taken are definitions for true, false and >not all right. Are they in terms of universal tautological >regressions to self contradictory alternatives? Hm. I don't see how. >> The tautological propostion p:[subject][not subject] is true >> because it exhausts all possibilities for truth. The tautological >> proposition p:[subject not subject] is false because it is self >> contradictory. >I assume the first has an OR between the brackets and the second and >AND before the NOT. Your non-use of conjunctions except for NOT makes >your statemes much less clear. Yeah well I also mentioned this I think to Randy. Is an expression like red car any the less precise and exact than red and car? Is an expression like red; car any the less precise and exact than red or car? If so I think we'd need to revise the language entirely. >I don't agree with your use of tautological. What do you mean you don't agree with my use of tautological. Show where it's any different from your use. > Of the two proposition, >the first is a tautology and the second is a (self) contradiction. The >second is not tautological in any way. Regardless, I do agree that we >can accept (as an axiom) that any statement in the form >(P) or (not P) >is true >and that (as an axiom again) any statement in the form >(P) and (not P) >is false. >> And the tautological regression for p:[not] is >> P:[not not] and is finite because it's self contradictory. >Where is the regression? Between the empirical observation [not] and its tautological contradiction [not not]. >> I don't know what better definitions for true, false, and not you >> could want. >The problem here is that you're giving a meaning to not as a >proposition alone. *This* doesn't make sense to me. Well if you expect ever to regress propositions to universal truth instead of axioms you'd better get used to it. >Or are you making such a claim on the basis that (quoting you from the >subsequent paragraph on that same post): >> These seem to be the only reducible definitions for true, false, and >> not. >So because you cannot find any other definitions for true, false and >not? >> Oh I imagine there could be a multitude of other definitions. Just not >> a multitude of other universally true definitions finitely regressible >> to self contradictory alternatives by means of tautologies. QED. >The only thing you're proven is that you're taking your definitions to >be universal because you can't think of others. Arrogant to say the >least. The only arrogance involved is your beliefs concerning what I've taken my definitions to. === Subject: Re: Epistemology 202: Advanced Topics > >> > Universal truth is > vastly superior to mere axioms because it ensures terminological > regression is not just to assumptions and circular definitions. >> >>Agreed on this. >> > The > Peano axioms, for example, just assume everything they're supposed > to prove. >> >>The Pean axioms don't prove anything. They *define* things. > > But what they define is cardinal integer counting and not numbers. > There is even a more comprehensive set of axiomatic assumptions: > Suc(none)=first, suc(first)=second, etc. from which Peano counting can > be deduced as a subset by assuming differences between ordinal > intervals are equal. >>See also my reply to Wagner. >>Peano axioms are necessary and sufficient to define the natural >>numbers; they characterize it. > They characterize operations on the natural numbers such as counting > and not the natural numbers themselves. Wrong. See below. >>In the sense that *any* set with *any* function that satisfy Peano >>axioms can be mapped into the naturals and on it you can define two >>operations that map to the natural addition and multiplication. > Yeah, but you can't define subtraction or division. Eager to show your ignorance, uh? Subtraction and division are not proper operations (in the algebraic sense); they are just the inverse of addition and multiplication. Once you define the latter, the former are automatically defined. >>i.e. you can say: ok I define natural numbers by any other way, >>whichever way you want. Yet, if you do, you'll find that the set you >>define will obey Peano axioms. > Who cares. I never said a correct defintion for natural numbers would > result in something that couldn't be counted. I just said that Peano > axioms define counting of natural numbers and not the natural numbers. No, the Peano axioms state that anything that can be counted is isomorphic to the natural numbers, since the natural numbers are *defined* by this counting thing. >>Without considering that math usage of terms is defined *when it's >>introduced*, not when other people start asking what you mean because >>what you're saying doesn't make sense. > > What I'm saying doesn't make sense to you because your sense is > defined by current mathematical thinking defined most recently over > the course of the last hundred years. >> >>Uh, no. They don't make sense because your thinking only barely >>resembles logic, and it's full of hidden assumptions. And yes, I've >>read your initial posts and they don't make sense either. > > What can I say? >>You could try being clearer. And no, repeating the same things over >>and over again with the same words and the same sentences expressed in >>the same way does not help being clearer. > So the fact that you don't understand means they're not > understandable? It means it's not understandable to me. > That's why your terms are > defined at the beginning because everyone knows what the critical > cognitive path is. I had no idea supposedly critical thinkers could > have such problems with contradiction and lines. Even so I spelled out > my definitions for true, false, and not in terms of universal > tautological regressions to self contradictory alternatives right in > the very root post to the E201 thread in the section Ur Regressions > for True, False, and Not. >> >>Actually you haven't. Quoting: >> > Ur Regressions for True, False, and Not > -------------- > >True, False, and Not are defined in reciprocal terms in the following >way. For any empirical observation [subject] the proposition > >p:[subject][not subject] is always true. And the proposition > >p:[subject not subject] is always false. And the empirical observation > >P:[not] is always true because the proposition > >P:[not not] is always false. >> >>Is this the right part? Where you claim to have spelled out your >>definitions for true, false and not in terms of universal tautological >>regressions to self contradictory alternatives? > > Yes. >>Good, at least something we can agree on :) >>Let's see. Those can be taken are definitions for true, false and >>not all right. Are they in terms of universal tautological >>regressions to self contradictory alternatives? Hm. I don't see how. > > The tautological propostion p:[subject][not subject] is true > because it exhausts all possibilities for truth. The tautological > proposition p:[subject not subject] is false because it is self > contradictory. >>I assume the first has an OR between the brackets and the second and >>AND before the NOT. Your non-use of conjunctions except for NOT makes >>your statemes much less clear. > Yeah well I also mentioned this I think to Randy. Is an expression > like red car any the less precise and exact than red and car? Is > an expression like red; car any the less precise and exact than > red or car? If so I think we'd need to revise the language entirely. Try the same trick with words that can be both adjectives and nouns. >>I don't agree with your use of tautological. > What do you mean you don't agree with my use of tautological. Show > where it's any different from your use. Tautological is an adjective that refers to propositions which are true regardless of the truth value of its components. (Which implies that a tautological proposition cannot be *false*, as it is for you above.) >> Of the two proposition, >>the first is a tautology and the second is a (self) contradiction. The >>second is not tautological in any way. Regardless, I do agree that we >>can accept (as an axiom) that any statement in the form >>(P) or (not P) >>is true >>and that (as an axiom again) any statement in the form >>(P) and (not P) >>is false. > And the tautological regression for p:[not] is > P:[not not] and is finite because it's self contradictory. >>Where is the regression? > Between the empirical observation [not] and its tautological > contradiction [not not]. [not] is an empirical observation? > I don't know what better definitions for true, false, and not you > could want. >>The problem here is that you're giving a meaning to not as a >>proposition alone. *This* doesn't make sense to me. > Well if you expect ever to regress propositions to universal truth > instead of axioms you'd better get used to it. Why? >>Or are you making such a claim on the basis that (quoting you from the >>subsequent paragraph on that same post): >> > These seem to be the only reducible definitions for true, false, and > not. >> >>So because you cannot find any other definitions for true, false and >>not? > > Oh I imagine there could be a multitude of other definitions. Just not > a multitude of other universally true definitions finitely regressible > to self contradictory alternatives by means of tautologies. QED. >>The only thing you're proven is that you're taking your definitions to >>be universal because you can't think of others. Arrogant to say the >>least. > The only arrogance involved is your beliefs concerning what I've taken > my definitions to. You *claim* your definitions take to universal truth. That's what *you*, not *me*. -- Giuseppe Oblomov Bilotta They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety. Benjamin Franklin === Subject: Re: Epistemology 202: Advanced Topics >> > >> Universal truth is >> vastly superior to mere axioms because it ensures terminological >> regression is not just to assumptions and circular definitions. > >Agreed on this. > >> The >> Peano axioms, for example, just assume everything they're supposed >> to prove. > >The Pean axioms don't prove anything. They *define* things. >> >> But what they define is cardinal integer counting and not numbers. >> There is even a more comprehensive set of axiomatic assumptions: >> Suc(none)=first, suc(first)=second, etc. from which Peano counting can >> be deduced as a subset by assuming differences between ordinal >> intervals are equal. >See also my reply to Wagner. >Peano axioms are necessary and sufficient to define the natural >numbers; they characterize it. >> They characterize operations on the natural numbers such as counting >> and not the natural numbers themselves. >Wrong. See below. >In the sense that *any* set with *any* function that satisfy Peano >axioms can be mapped into the naturals and on it you can define two >operations that map to the natural addition and multiplication. >> Yeah, but you can't define subtraction or division. >Eager to show your ignorance, uh? More eager to show your ignorance. >Subtraction and division are not proper operations (in the algebraic >sense); How so? > they are just the inverse of addition and multiplication. Do tell. So where are inverse operations defined in the Peano axioms? > Once >you define the latter, the former are automatically defined. So where is this automata defined in the Peano axioms? Or is this just another puerile assumption on the part of Peano axiomatists? >i.e. you can say: ok I define natural numbers by any other way, >whichever way you want. Yet, if you do, you'll find that the set you >define will obey Peano axioms. >> Who cares. I never said a correct defintion for natural numbers would >> result in something that couldn't be counted. I just said that Peano >> axioms define counting of natural numbers and not the natural numbers. >No, the Peano axioms state that anything that can be counted is >isomorphic to the natural numbers, since the natural numbers are >*defined* by this counting thing. And isomorphisms do not have the same cause so all the Peano axioms define is counting and not the natural numbers counted with since the natural numbers are only isomorphic to and not caused by this counting thing. >Without considering that math usage of terms is defined *when it's >introduced*, not when other people start asking what you mean because >what you're saying doesn't make sense. >> >> What I'm saying doesn't make sense to you because your sense is >> defined by current mathematical thinking defined most recently over >> the course of the last hundred years. > >Uh, no. They don't make sense because your thinking only barely >resembles logic, and it's full of hidden assumptions. And yes, I've >read your initial posts and they don't make sense either. >> >> What can I say? >You could try being clearer. And no, repeating the same things over >and over again with the same words and the same sentences expressed in >the same way does not help being clearer. >> So the fact that you don't understand means they're not >> understandable? >It means it's not understandable to me. >> That's why your terms are >> defined at the beginning because everyone knows what the critical >> cognitive path is. I had no idea supposedly critical thinkers could >> have such problems with contradiction and lines. Even so I spelled out >> my definitions for true, false, and not in terms of universal >> tautological regressions to self contradictory alternatives right in >> the very root post to the E201 thread in the section Ur Regressions >> for True, False, and Not. > >Actually you haven't. Quoting: > >> Ur Regressions for True, False, and Not >> -------------- >> >>True, False, and Not are defined in reciprocal terms in the following >>way. For any empirical observation [subject] the proposition >> >>p:[subject][not subject] is always true. And the proposition >> >>p:[subject not subject] is always false. And the empirical observation >> >>P:[not] is always true because the proposition >> >>P:[not not] is always false. > >Is this the right part? Where you claim to have spelled out your >definitions for true, false and not in terms of universal tautological >regressions to self contradictory alternatives? >> >> Yes. >Good, at least something we can agree on :) >Let's see. Those can be taken are definitions for true, false and >not all right. Are they in terms of universal tautological >regressions to self contradictory alternatives? Hm. I don't see how. >> >> The tautological propostion p:[subject][not subject] is true >> because it exhausts all possibilities for truth. The tautological >> proposition p:[subject not subject] is false because it is self >> contradictory. >I assume the first has an OR between the brackets and the second and >AND before the NOT. Your non-use of conjunctions except for NOT makes >your statemes much less clear. >> Yeah well I also mentioned this I think to Randy. Is an expression >> like red car any the less precise and exact than red and car? Is >> an expression like red; car any the less precise and exact than >> red or car? If so I think we'd need to revise the language entirely. >Try the same trick with words that can be both adjectives and nouns. Red can't be a noun, Tolstoij? >I don't agree with your use of tautological. >> What do you mean you don't agree with my use of tautological. Show >> where it's any different from your use. >Tautological is an adjective that refers to propositions which are >true regardless of the truth value of its components. (Which implies >that a tautological proposition cannot be *false*, as it is for you >above.) But a tautological proposition can be self contradictory. So can a self contradictory proposition be false. Probably not in your dialect. > Of the two proposition, >the first is a tautology and the second is a (self) contradiction. The >second is not tautological in any way. Regardless, I do agree that we >can accept (as an axiom) that any statement in the form >(P) or (not P) >is true >and that (as an axiom again) any statement in the form >(P) and (not P) >is false. >> And the tautological regression for p:[not] is >> P:[not not] and is finite because it's self contradictory. >Where is the regression? >> Between the empirical observation [not] and its tautological >> contradiction [not not]. >[not] is an empirical observation? Why not? It's input to a tautology. >> I don't know what better definitions for true, false, and not you >> could want. >The problem here is that you're giving a meaning to not as a >proposition alone. *This* doesn't make sense to me. >> Well if you expect ever to regress propositions to universal truth >> instead of axioms you'd better get used to it. >Why? Why what? >Or are you making such a claim on the basis that (quoting you from the >subsequent paragraph on that same post): > >> These seem to be the only reducible definitions for true, false, and >> not. > >So because you cannot find any other definitions for true, false and >not? >> >> Oh I imagine there could be a multitude of other definitions. Just not >> a multitude of other universally true definitions finitely regressible >> to self contradictory alternatives by means of tautologies. QED. >The only thing you're proven is that you're taking your definitions to >be universal because you can't think of others. Arrogant to say the >least. >> The only arrogance involved is your beliefs concerning what I've taken >> my definitions to. >You *claim* your definitions take to universal truth. That's what >*you*, not *me*. Cute if not smart. === Subject: Re: Epistemology 202: Advanced Topics <1htanmg0a58sl.1a1mrqlqtp2e7.dlg@40tude.net> <426e8266.45555685@netnews.att.net> <426fa325.62833321@netnews.att.net> <1q8a4jdycgivi.11z4lpzyo0kqk.dlg@40tude.net> <426fec49.70591893@netnews.att.net> <1AVbe.174$Uz4.145@okepread04> <427236e1.88642499@netnews.att.net> > Don't flatter yourself, Tolstoij. Tolstoy did not write Oblomov. I believe it was Goncharov. Shall we talk about different railway gauges now, or does that topic not come into play until post # 5782? MP === Subject: Re: Epistemology 202: Advanced Topics On 2 May 2005 14:45:45 -0700, Mitch Perkins >> Don't flatter yourself, Tolstoij. > Tolstoy did not write Oblomov. I believe it was Goncharov. Shall we >talk about different railway gauges now, or does that topic not come >into play until post # 5782? Comic relief is like motions to adjourn, always in order. === Subject: Re: Epistemology 202: Advanced Topics <1htanmg0a58sl.1a1mrqlqtp2e7.dlg@40tude.net> <426e8266.45555685@netnews.att.net> <426fa325.62833321@netnews.att.net> <1q8a4jdycgivi.11z4lpzyo0kqk.dlg@40tude.net> <426fec49.70591893@netnews.att.net> <1AVbe.174$Uz4.145@okepread04> <427236e1.88642499@netnews.att.net> <42779147.61026524@netnews.att.net> > On 2 May 2005 14:45:45 -0700, Mitch Perkins >> >> Don't flatter yourself, Tolstoij. > Tolstoy did not write Oblomov. I believe it was Goncharov. Shall we >talk about different railway gauges now, or does that topic not come >into play until post # 5782? > Comic relief is like motions to adjourn, always in order. ~:?) Mp === Subject: Re: Epistemology 202: Advanced Topics On 3 May 2005 15:27:17 -0700, Mitch Perkins >> On 2 May 2005 14:45:45 -0700, Mitch Perkins >> > > Don't flatter yourself, Tolstoij. >> >> Tolstoy did not write Oblomov. I believe it was Goncharov. Shall >>talk about different railway gauges now, or does that topic not come >>into play until post # 5782? >> Comic relief is like motions to adjourn, always in order. > ~:?) Exactly. === Subject: Re: Epistemology 202: Advanced Topics >> >> > > > >> > >[...] > > > >I'm familiar with universal truth and my terminology is more than >adequate to address things universally true whether or not that >includes your understanding of mathematics being more difficult to >say. >> >> >> >>If you want to replace modern math, wouldn't it help if your work >>addresses the same concepts? >> > >Will, Lester believes in universal truth, and he also believes he can >find it just by thinking about it. It keeps him happy. >> >>That much I've figured out... I'm just trying to figure out how lost a >>cause he is. Right now it appears that he is unwilling to engage in >>discussions at a technical level, even though he proposes revamping >>technical subject matter. >You don't consider contradiction, finite tautological regression, >angular momentum, Planck's constant, hermit functions and SR >technical? My mistake. >>When I ask for definitions of your use of contradiction and finite >>tautological regression I get something equally vague or insults. > Of course I never get insults or vague definitions. But then I deserve > it whereas those who criticize me without understanding material are > the soul of virtue and wisdom. I gave you and others definitions of > contradiction yesterday which if not comprehensive show what > contradiction means in relation to self contradiction. Finite > tautological regression you ought to be able to figure out for > yourself. Do you know what a tautological regression is? Well then > tautological regression is finite if it produces self contradictory > alternatives. Do some thinking for yourself. I'll be honest: I don't think I heard the term tautological regression before meeting you. I believe I'm figuring out what you mean by the term, though. >>I asked for a statement of a theorem and its proof, I got told it's in >>this stuff you dismissed. > It still is. I'll look back, but if it's not clearly labeled, I may not be able to tell what it is. >>In a mass of text that makes little sense, I stop reading. > In a mass of what text? It's 107 lines. I didn't say massive. >>the relevent text, I can start asking specific questions and perhaps >>achieve clarity. Right now, though, I am getting very little from you >>that I can ascribe meaning to. > How fast do you read exactly? You want me to point to the right text > segment so you can argue whether I've pointed to the right text > segment? If you can't read 107 lines of material for yourself, what is > it you expect? I figured the statement and proof would be about 10 lines. I'm sorry if it was difficult to copy and paste. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Epistemology 202: Advanced Topics >> >> >> > > >> >> >> > >> >>[...] >> >> >> >>I'm familiar with universal truth and my terminology is more than >>adequate to address things universally true whether or not that >>includes your understanding of mathematics being more difficult to >>say. > > > >If you want to replace modern math, wouldn't it help if your work >addresses the same concepts? > >> >>Will, Lester believes in universal truth, and he also believes he can >>find it just by thinking about it. It keeps him happy. > >That much I've figured out... I'm just trying to figure out how lost a >cause he is. Right now it appears that he is unwilling to engage in >discussions at a technical level, even though he proposes revamping >technical subject matter. >> >> >>You don't consider contradiction, finite tautological regression, >>angular momentum, Planck's constant, hermit functions and SR >>technical? My mistake. >When I ask for definitions of your use of contradiction and finite >tautological regression I get something equally vague or insults. >> Of course I never get insults or vague definitions. But then I deserve >> it whereas those who criticize me without understanding material are >> the soul of virtue and wisdom. I gave you and others definitions of >> contradiction yesterday which if not comprehensive show what >> contradiction means in relation to self contradiction. Finite >> tautological regression you ought to be able to figure out for >> yourself. Do you know what a tautological regression is? Well then >> tautological regression is finite if it produces self contradictory >> alternatives. Do some thinking for yourself. >I'll be honest: I don't think I heard the term tautological regression >before meeting you. I believe I'm figuring out what you mean by the >term, though. You know this is an interesting point because mathematics is often described as a tautological discipline. It's also interesting that tautologies are considered true but useless. To be honest I never even realized my theory of different from differences was tautological until someone suggested it. Of course the person was just trying to disparage the idea but wound up providing a massive final clue. >I asked for a statement of a theorem and its proof, I got told it's in >this stuff you dismissed. >> It still is. >I'll look back, but if it's not clearly labeled, I may not be able to >tell what it is. Try the section labeled Ur Regressions for True, False, and Not. >In a mass of text that makes little sense, I stop reading. >> In a mass of what text? It's 107 lines. >I didn't say massive. Yeah, as always I'm torn between accessibility and readibility. You simply ideas to the extent no one understands the ideas. >the relevent text, I can start asking specific questions and perhaps >achieve clarity. Right now, though, I am getting very little from you >that I can ascribe meaning to. >> How fast do you read exactly? You want me to point to the right text >> segment so you can argue whether I've pointed to the right text >> segment? If you can't read 107 lines of material for yourself, what is >> it you expect? >I figured the statement and proof would be about 10 lines. I'm sorry if >it was difficult to copy and paste. No the statement and proof are only about ten lines: Tautologies are labeled p and P for propositions. > Ur Regressions for True, False, and Not > -------------- >True, False, and Not are defined in reciprocal terms in the following >way. For any empirical observation [subject] the proposition >p:[subject][not subject] is always true. And the proposition >p:[subject not subject] is always false. And the empirical observation >P:[not] is always true because the proposition >P:[not not] is always false. This is a theorem and proof for the meanings of true, false, and not by tautological regression to self contradictory alternatives. === Subject: Re: Epistemology 202: Advanced Topics >> > >> > > >I'm familiar with universal truth and my terminology is more than >adequate to address things universally true whether or not that >includes your understanding of mathematics being more difficult to >say. >> >> >> >>If you want to replace modern math, wouldn't it help if your work >>addresses the same concepts? >> > >Will, Lester believes in universal truth, and he also believes he can >find it just by thinking about it. It keeps him happy. >> >>That much I've figured out... I'm just trying to figure out how lost a >>cause he is. Right now it appears that he is unwilling to engage in >>discussions at a technical level, even though he proposes revamping >>technical subject matter. > > >You don't consider contradiction, finite tautological regression, >angular momentum, Planck's constant, hermit functions and SR >technical? My mistake. >> >>When I ask for definitions of your use of contradiction and finite >>tautological regression I get something equally vague or insults. >Of course I never get insults or vague definitions. But then I deserve >it whereas those who criticize me without understanding material are >the soul of virtue and wisdom. I gave you and others definitions of >contradiction yesterday which if not comprehensive show what >contradiction means in relation to self contradiction. Finite >tautological regression you ought to be able to figure out for >yourself. Do you know what a tautological regression is? Well then >tautological regression is finite if it produces self contradictory >alternatives. Do some thinking for yourself. >>I'll be honest: I don't think I heard the term tautological regression >>before meeting you. I believe I'm figuring out what you mean by the >>term, though. > You know this is an interesting point because mathematics is often > described as a tautological discipline. I don't think I've ever heard it described that way before. It strikes me as a description that is correct from a certain perspective, but completely misleading. > It's also interesting that > tautologies are considered true but useless. To be honest I never even > realized my theory of different from differences was tautological > until someone suggested it. Of course the person was just trying to > disparage the idea but wound up providing a massive final clue. It may not have been that useful a clue. >>the relevent text, I can start asking specific questions and perhaps >>achieve clarity. Right now, though, I am getting very little from you >>that I can ascribe meaning to. >How fast do you read exactly? You want me to point to the right text >segment so you can argue whether I've pointed to the right text >segment? If you can't read 107 lines of material for yourself, what is >it you expect? >>I figured the statement and proof would be about 10 lines. I'm sorry if >>it was difficult to copy and paste. > No the statement and proof are only about ten lines: > Tautologies are labeled p and P for propositions. >> Ur Regressions for True, False, and Not >> -------------- >>True, False, and Not are defined in reciprocal terms in the following >>way. For any empirical observation [subject] the proposition >>p:[subject][not subject] is always true. And the proposition I would say subject OR not subject is true. This is a definition of the [] symbols. >>p:[subject not subject] is always false. And the empirical observation I would say subject AND not subject is false. This is also a definition. >>P:[not] is always true because the proposition This is not even close to clear. I could understand [ ][not ] being true from the above, however. Or this might be a definition again. >>P:[not not] is always false. Umm... pehaps that is interpretted as [ not not ]? Or if the above is a definition, then [not not] = not true = false? > This is a theorem and proof for the meanings of true, false, and not > by tautological regression to self contradictory alternatives. The above is not a theorem or proof in any conventional sense. They appear to be definitions of notation. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Epistemology 202: Advanced Topics >> > >> > >> >> >>I'm familiar with universal truth and my terminology is more than >>adequate to address things universally true whether or not that >>includes your understanding of mathematics being more difficult to >>say. > > > >If you want to replace modern math, wouldn't it help if your work >addresses the same concepts? > >> >>Will, Lester believes in universal truth, and he also believes he can >>find it just by thinking about it. It keeps him happy. > >That much I've figured out... I'm just trying to figure out how lost a >cause he is. Right now it appears that he is unwilling to engage in >discussions at a technical level, even though he proposes revamping >technical subject matter. >> >> >>You don't consider contradiction, finite tautological regression, >>angular momentum, Planck's constant, hermit functions and SR >>technical? My mistake. > >When I ask for definitions of your use of contradiction and finite >tautological regression I get something equally vague or insults. >> >> >>Of course I never get insults or vague definitions. But then I deserve >>it whereas those who criticize me without understanding material are >>the soul of virtue and wisdom. I gave you and others definitions of >>contradiction yesterday which if not comprehensive show what >>contradiction means in relation to self contradiction. Finite >>tautological regression you ought to be able to figure out for >>yourself. Do you know what a tautological regression is? Well then >>tautological regression is finite if it produces self contradictory >>alternatives. Do some thinking for yourself. >I'll be honest: I don't think I heard the term tautological regression >before meeting you. I believe I'm figuring out what you mean by the >term, though. >> You know this is an interesting point because mathematics is often >> described as a tautological discipline. >I don't think I've ever heard it described that way before. It strikes >me as a description that is correct from a certain perspective, but >completely misleading. Not sure. I'm pretty sure I've heard Neil Rickert, who claims to be a mathematician, describe it exactly that way. But I don't know what he meant. > It's also interesting that >> tautologies are considered true but useless. To be honest I never even >> realized my theory of different from differences was tautological >> until someone suggested it. Of course the person was just trying to >> disparage the idea but wound up providing a massive final clue. >It may not have been that useful a clue. Well it turns out to have been immensely useful in framing the general idea in terms which I should have thought to familiar to critical thinkers of all kinds. >the relevent text, I can start asking specific questions and perhaps >achieve clarity. Right now, though, I am getting very little from you >that I can ascribe meaning to. >> >>How fast do you read exactly? You want me to point to the right text >>segment so you can argue whether I've pointed to the right text >>segment? If you can't read 107 lines of material for yourself, what is >>it you expect? >I figured the statement and proof would be about 10 lines. I'm sorry if >it was difficult to copy and paste. >> No the statement and proof are only about ten lines: >> Tautologies are labeled p and P for propositions. > Ur Regressions for True, False, and Not > -------------- >True, False, and Not are defined in reciprocal terms in the following >way. For any empirical observation [subject] the proposition >p:[subject][not subject] is always true. And the proposition >I would say subject OR not subject is true. This is a definition of >the [] symbols. The problem is that if you want to go the conjunction route you can't just get away with one conjunction. You really need to phrase such things as red car as and red and car or the phrase red; car as or red or car. Of course then the interesting problem arises as to what all the conjunctions actually conjoin besides each other. >p:[subject not subject] is always false. And the empirical observation >I would say subject AND not subject is false. This is also a definition. True enough. >P:[not] is always true because the proposition >This is not even close to clear. I could understand [ ][not ] >being true from the above, however. Or this might be a definition again. >P:[not not] is always false. >Umm... pehaps that is interpretted as [ not not ]? Or if the >above is a definition, then [not not] = not true = false? No. P:[not not] is the proof of P:[not] given previous definitions. >> This is a theorem and proof for the meanings of true, false, and not >> by tautological regression to self contradictory alternatives. >The above is not a theorem or proof in any conventional sense. They >appear to be definitions of notation. P:[not] as universally true is certainly a theorem proven by the tautological regression to P:[not not] which is self contradictory. === Subject: Re: Epistemology 202: Advanced Topics >> Do you know what a tautological regression is? Well then >> tautological regression is finite if it produces self contradictory >> alternatives. Do some thinking for yourself. >I'll be honest: I don't think I heard the term tautological regression >before meeting you. How could you have failed to learn such an omnipresent and important term? Why, Google's review of zillions of web pages has found: 62 pages containing the term. Every one of them mentions Zick. Sounds like LZ is mounting a revival of Project Cogno-Intellectual . dave === Subject: Re: Epistemology 202: Advanced Topics > Do you know what a tautological regression is? Well then > tautological regression is finite if it produces self contradictory > alternatives. Do some thinking for yourself. >>I'll be honest: I don't think I heard the term tautological regression >>before meeting you. >How could you have failed to learn such an omnipresent and important >term? Why, Google's review of zillions of web pages has found: >62 pages containing the term. Every one of them mentions Zick. >Sounds like LZ is mounting a revival of Project Cogno-Intellectual . Way beyond that, Dave. Strange that mathematicians practice what is often characterized as a tautological discipline without ever having practiced tautologies. Go figure. === Subject: Re: Epistemology 202: Advanced Topics >>Do you know what a tautological regression is? Well then >>tautological regression is finite if it produces self contradictory >>alternatives. Do some thinking for yourself. >I'll be honest: I don't think I heard the term tautological regression >before meeting you. >>How could you have failed to learn such an omnipresent and important >>term? Why, Google's review of zillions of web pages has found: >>62 pages containing the term. Every one of them mentions Zick. >>Sounds like LZ is mounting a revival of Project Cogno-Intellectual . > Way beyond that, Dave. Strange that mathematicians practice what is > often characterized as a tautological discipline without ever having > practiced tautologies. Go figure. Mathematicians don't use the word with the same meaning you do. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Epistemology 202: Advanced Topics >Do you know what a tautological regression is? Well then >tautological regression is finite if it produces self contradictory >alternatives. Do some thinking for yourself. >> >>I'll be honest: I don't think I heard the term tautological regression >>before meeting you. >How could you have failed to learn such an omnipresent and important >term? Why, Google's review of zillions of web pages has found: >62 pages containing the term. Every one of them mentions Zick. >Sounds like LZ is mounting a revival of Project Cogno-Intellectual . >> Way beyond that, Dave. Strange that mathematicians practice what is >> often characterized as a tautological discipline without ever having >> practiced tautologies. Go figure. >Mathematicians don't use the word with the same meaning you do. But the meaning of tautology is very well defined. I use the term in a generic universal sense. So if mathematicians use it in some other way I would have to suspect they don't use it generically. === Subject: Re: Epistemology 202: Advanced Topics >Do you know what a tautological regression is? Well then >tautological regression is finite if it produces self contradictory >alternatives. Do some thinking for yourself. >>I'll be honest: I don't think I heard the term tautological regression >>before meeting you. > How could you have failed to learn such an omnipresent and important > term? Why, Google's review of zillions of web pages has found: > 62 pages containing the term. Every one of them mentions Zick. > Sounds like LZ is mounting a revival of Project Cogno-Intellectual . I'm disappointed in you, Dave. I thought you were here to learn and there you go flame-baiting Lester. It doesn't take long to get into the swing of things, does it? -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions === Subject: Re: Epistemology 202: Advanced Topics > > I suppose I was curious what you thought a 'concept' was and if and > how you would distinguish between a 'concept' and a 'system'. >> >> >> >> To be honest, I view both sets and set theory as concepts. Granted, >> set theory is also a system when formalized, but their are various >> formalizations which each reflect a different view on the basic >> concept of set theory. YMMV. >> > Well, let's go from there: What do you think is 'the basic concept > of set theory'? >> Best for doing what? I'm partial to ZFC, but that's a matter of taste >> and familiarity. > I said 'basic', not 'best'. I would say the basic concept of set theory is the concept of a set. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Epistemology 202: Advanced Topics > I would say the basic concept of set theory is the concept of a set. Can you explain that concept to me, keeping in mind that I am a programmer and not a mathematician? The only set concept that I am aware of is that of a container, meaning to me a closed boundary isolating the things inside from the things outside, and containing no duplicates. -- ...how an individual invents a new way of giving order to data now all assembled must here remain inscrutable and may be permanently so... Almost always the men who achieve these fundamental inventions of new paradigm have either been very young or very new to the field whose paradigm they change... (they) are particularly likely to see that those rules no longer define a playable game and to conceive another set that can replace them. Thomas Kuhn The Structure of Scientific Revolutions k[k l [kp