mm-8757 Subject: Re: limitation to the halting proof > Working on a similar project, constructing a computer model with a simple UTM. > Some functional computer models are tiny, just a couple lines, and TMs are > trivial, but the smallest UTM is about 20kb. Can¹t do any models with that > need a 50 byte UTM Þrst, a really primitive notion of computing that a small > mechanical system can emulate. Got close about 5 years ago, had a 2 cell > machine scanning a list for a match for a symbol, almost solved the second Claudio Baiocchi and Y. Rogozhin each have constructed UTMs with a two-symbol alphabet and 22 states. David Bernier see ref.: Baiocchi, C.: Three Small Universal Turing Machines in Lecture Notes in Computer Science, vol. 2055, Springer, 2001. === Subject: Re: limitation to the halting proof In sci.logic, David Bernier >> Working on a similar project, constructing a computer model with a simple UTM. >> Some functional computer models are tiny, just a couple lines, and TMs are >> trivial, but the smallest UTM is about 20kb. Can¹t do any models with that >> need a 50 byte UTM Þrst, a really primitive notion of computing that a small >> mechanical system can emulate. Got close about 5 years ago, had a 2 cell >> machine scanning a list for a match for a symbol, almost solved the second > > Claudio Baiocchi and Y. Rogozhin each have constructed UTMs with > a two-symbol alphabet and 22 states. That does sound rather pretty. :-) Of course it doesn¹t do much for |-|erc¹s halting problem. > David Bernier > see ref.: > Baiocchi, C.: Three Small Universal Turing Machines in > Lecture Notes in Computer Science, vol. 2055, > Springer, 2001. -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: limitation to the halting proof >> Working on a similar project, constructing a computer model with a simple UTM. >> Some functional computer models are tiny, just a couple lines, and TMs are >> trivial, but the smallest UTM is about 20kb. Can¹t do any models with that >> need a 50 byte UTM Þrst, a really primitive notion of computing that a small >> mechanical system can emulate. Got close about 5 years ago, had a 2 cell >> machine scanning a list for a match for a symbol, almost solved the second > > Claudio Baiocchi and Y. Rogozhin each have constructed UTMs with > a two-symbol alphabet and 22 states. > That does sound rather pretty. :-) Of course it doesn¹t do much > for |-|erc¹s halting problem. that¹s your halting problem remember not mine, you¹re the one diagonalising at every opportunity claiming sequences are imaginary and invisible numbers are real. Herc === Subject: Re: limitation to the halting proof In sci.logic, |-|erc : >> > Working on a similar project, constructing a computer model with a simple UTM. > Some functional computer models are tiny, just a couple lines, and TMs are > trivial, but the smallest UTM is about 20kb. Can¹t do any models with that > need a 50 byte UTM Þrst, a really primitive notion of computing that a small > mechanical system can emulate. Got close about 5 years ago, had a 2 cell > machine scanning a list for a match for a symbol, almost solved the second >> >> >> Claudio Baiocchi and Y. Rogozhin each have constructed UTMs with >> a two-symbol alphabet and 22 states. >> That does sound rather pretty. :-) Of course it doesn¹t do much >> for |-|erc¹s halting problem. > that¹s your halting problem remember not mine, you¹re the one > diagonalising at every opportunity claiming sequences are > imaginary and invisible numbers are real. Numbers are invisible. Or have you ever seen a 2? All you¹ve seen are lit pixels in the shape of a 2, darkened spots on paper in the shape of a 2, strips or threads of fabric woven in the shape of a 2, ... Of course one sees pairs of things all the time -- but that¹s not the same thing. As for the sequence of functions provably halting: that¹s constructible and can be proven to be a valid sequence. Trouble is, it¹s not complete. Unless you can prove the Collatz Conjecture...? > Herc -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: limitation to the halting proof The Ghost In The Machine > Working on a similar project, constructing a computer model with a simple UTM. > Some functional computer models are tiny, just a couple lines, and TMs are > trivial, but the smallest UTM is about 20kb. Can¹t do any models with that > need a 50 byte UTM Þrst, a really primitive notion of computing that a small > mechanical system can emulate. Got close about 5 years ago, had a 2 cell > machine scanning a list for a match for a symbol, almost solved the second >> >> >> Claudio Baiocchi and Y. Rogozhin each have constructed UTMs with >> a two-symbol alphabet and 22 states. >> >> That does sound rather pretty. :-) Of course it doesn¹t do much >> for |-|erc¹s halting problem. >> > that¹s your halting problem remember not mine, you¹re the one > diagonalising at every opportunity claiming sequences are > imaginary and invisible numbers are real. > Numbers are invisible. Or have you ever seen a 2? > All you¹ve seen are lit pixels in the shape of a 2, > darkened spots on paper in the shape of a 2, strips or > threads of fabric woven in the shape of a 2, ... > Of course one sees pairs of things all the time -- but > that¹s not the same thing. > As for the sequence of functions provably halting: that¹s > constructible and can be proven to be a valid sequence. > Trouble is, it¹s not complete. > Unless you can prove the Collatz Conjecture...? no you¹re þying off on tangents at every turn like always. that¹s a simulation nothing to do with haltability, just rephrase the problem as does it halt in x cycles? you claim the uncountable irrationals are not able to be put on paper as opposed to 2, that they go Œinbetween¹ the numbers we can *write* hence *see*. you use a faulty premise of open diagonalisation and when its shown to be nonsense you shift the burden. you have no mathematical opposition to countable functions and countable reals is just a step further. UTM(n), from n e N IS the countable reals, it doesn¹t miss a beat, not all n halt thats a petty copout requirement of YOUR argument, now even that is invalid. Herc === Subject: Re: [OT] Re: Bondy & Murty¹s book available on-line (was: Re: Eulerianpath in inÞnite graph) X-URL: http://mygate.mailgate.org/mynews/talk/talk.bizarre/ c518a20f1e79483b10d8d1577 4b6ef76.48257%40mygate.mailgate.org > You¹ve got to admit, No, I don¹t. You are, besides making a fool of yourself by knee-jerk reactions not mediated by sufÞcient prior research, also incorrect. > he¹s got a point there. The point of Julian Waldby when he is in the throes of voluntarily suffered bipolar disorder cycles, is strictly in the eye of the beholder. The more pointiform Julian¹s text seem to you, the blinder your eye appears to others. > 1) Your grammar is just _terrible_. Compared to that of someone, say, who answers an ŒNet, and uses pronouns he and you without giving clue one of the antecedents of those pronouns? Perhaps you have trouble differentiating bad grammar from text _you_ cannot understand, due to poor reading skills and no clue what good grammar is. > 2) You insult someone for calling you on it. Julian and I have been having this little conversation for somewhat over a decade. Before commenting further about what you think is being said, go gain sufÞcent understanding and context to follow the gist without making yourself a laughing-stock to clueful readers. > 3) You rant for several dozen lines about > absolutely nothing. Again, your lack of English comprehension skills does not equate to my lack of writing skills. > Clearly, Shotgun Squad¹s assessment was not too > far off the mark. The manic cycle sufferer has it right is your Þnal assessment? Snort. > I would only like to add that it it clear to me > that your are an incredible narcissist. You haven¹t seen what I have to face in my mirror. I have a splendid ego, more than sufÞcient to survive maggot-tossed slings and arrows from grubs of your ilk, but like most long term severe depressives, my self-image is one with the benthic slime. > Symptoms: Oh, splendid, and I thought David Longley was more than sufÞcient tele-psyhchologists for any one Usenet. Seems we must suffer another dunce who pretends to clairvoyance to do diagnoses. > Everything Shotgun Squad described, plus your > transparent attempt at showing us how great a > person you are indicates a severe need for > acceptance and adoration by those around you. There are dozens if not hundreds of ŒNet-virtual former-person-shaped piles of þame-cinders who, if they were able, would jump, leap, even spring to disagree with you. > What, are you trying to make him feel guilty for > not thinking you¹re a god among men? _Julian_, think _someone_else_ a god???? Bwahahahahaha! My, I hope the shuttle back to your home planet remembers to pick you up at your stop. > Œcid Œooh One more poster qualiÞes for a spot on the list of Usenet¹s mental munchkins. Too bad that list is so extensive your addition to it won¹t even be noticed. xanthian, as always, vastly amused by the unintentionally amusing. For those following the background story, Jesus V.-S. is back from the hospital after 8 days on interveinous antibiotics, plus two transfusions, just to clear up a fairly simple infection. A body mostly unused doesn¹t have much in the way of immune system self-defenses. -- === Subject: Re: [OT] Re: Bondy & Murty¹s book available on-line (was: Re: Eulerianpath in inÞnite graph) > You¹ve got to admit, > No, I don¹t. You are, besides making a fool of > yourself by knee-jerk reactions not mediated by > sufÞcient prior research, also incorrect. > he¹s got a point there. > The point of Julian Waldby when he is in the > throes of voluntarily suffered bipolar disorder > cycles, is strictly in the eye of the beholder. The > more pointiform Julian¹s text seem to you, the > blinder your eye appears to others. > 1) Your grammar is just _terrible_. > Compared to that of someone, say, who answers an > ŒNet, and uses pronouns he and you without > giving clue one of the antecedents of those > pronouns? Perhaps you have trouble differentiating > bad grammar from text _you_ cannot understand, > due to poor reading skills and no clue what good > grammar is. my previous message, it is clear to whom I was referring. You certainly understood it. I¹m sure most people who bothered to read my post did. If you don¹t have a clue, you need to get one. > 2) You insult someone for calling you on it. > Julian and I have been having this little > conversation for somewhat over a decade. Before > commenting further about what you think is being > said, go gain sufÞcent understanding and context to > follow the gist without making yourself a > laughing-stock to clueful readers. And why, pray-tell, should I care what people on talk.bizarre or misc.misc care? > 3) You rant for several dozen lines about > absolutely nothing. > Again, your lack of English comprehension skills > does not equate to my lack of writing skills. I wasn¹t refering to your grammar, but your rant¹s content--or lack thereof. Spewing vitriol in defense that the following sentence is grammatical is ridiculous: >> You honestly are that thick, that you think it >> was and still is appropriate for you, repeatedly, >> to claim that anyone who had the temerity to be >> posting in this comp.theory thread was _obliged_, >> _on your say-so_, to be able to recognize Paul >> Erdos in a framed image hung on a wall behind the >> main subject of a home page author¹s online >> photograph, must be lying to claim not to have >> recognized the image, and to continue that >> insistence time after time, despite that poster¹s >> repeated insistances, that no, he¹d never >> happened to see a picture of Paul Erdos that >> identiÞed itself as such? > Clearly, Shotgun Squad¹s assessment was not too > far off the mark. > The manic cycle sufferer has it right is your > Þnal assessment? Snort. > I would only like to add that it it clear to me > that your are an incredible narcissist. > You haven¹t seen what I have to face in my mirror. > I have a splendid ego, more than sufÞcient to > survive maggot-tossed slings and arrows from grubs > of your ilk, but like most long term severe > depressives, my self-image is one with the benthic > slime. > Symptoms: > Oh, splendid, and I thought David Longley was > more than sufÞcient tele-psyhchologists for any one > Usenet. Seems we must suffer another dunce who > pretends to clairvoyance to do diagnoses. Your age is showing. Who is David Longley, what does he have to do with the issue at hand, and why should I care? > Everything Shotgun Squad described, plus your > transparent attempt at showing us how great a > person you are indicates a severe need for > acceptance and adoration by those around you. > There are dozens if not hundreds of ŒNet-virtual > former-person-shaped piles of þame-cinders who, if > they were able, would jump, leap, even spring to > disagree with you. > What, are you trying to make him feel guilty for > not thinking you¹re a god among men? > _Julian_, think _someone_else_ a god???? > Bwahahahahaha! > My, I hope the shuttle back to your home planet > remembers to pick you up at your stop. Your response isn¹t one. In particular, a course of action¹s lack of effectiveness does not entail suspension of such a practice. > One more poster qualiÞes for a spot on the list of > Usenet¹s mental munchkins. Too bad that list is so > extensive your addition to it won¹t even be noticed. I take it you¹re on talk.bizarre? That makes some sense. Well, I can assure you, I¹m not posting from misc.misc. I suggest that if you¹re going to use a single sample of a person¹s behavior with which to judge them, to make sure that the sample actually supports your conclusion. Or you might want to try to follow your own advice to me. (Atleast one of us ought to--it¹s not bad advice) Œcid Œooh > xanthian, as always, vastly amused by the > unintentionally amusing. > For those following the background story, Jesus > V.-S. is back from the hospital after 8 days on > interveinous antibiotics, plus two transfusions, > just to clear up a fairly simple infection. A body > mostly unused doesn¹t have much in the way of immune > system self-defenses. === Subject: Re: [OT] Re: Bondy & Murty¹s book available on-line (was: Re: Eulerianpath in inÞnite graph) X-URL: http://mygate.mailgate.org/mynews/talk/talk.bizarre/ c2ce0db25e4758fcd18761be9 e85bf7d.48257%40mygate.mailgate.org > I wasn¹t refering to your grammar, but your rant¹s content--or lack > thereof. And yet, grammar was your subject, and to the sane person¹s eye, considering your continuing sentence, just below, still _is_ your subject. Do you have these problems with truthfulness frequently? > Spewing vitriol in defense that the following sentence is grammatical > is ridiculous: > > You honestly are that thick, that you think it was and still is > appropriate for you, repeatedly, to claim that anyone who had the > temerity to be posting in this comp.theory thread was _obliged_, _on > your say-so_, to be able to recognize Paul Erdos in a framed image > hung on a wall behind the main subject of a home page author¹s > online photograph, must be lying to claim not to have recognized the > image, and to continue that insistence time after time, despite that > poster¹s repeated insistances, that no, he¹d never happened to see a > picture of Paul Erdos that identiÞed itself as such? > Perhaps, since you still fail to distinguish richly textured text which you simply fail to understand, from ungrammatical text, you would be willing to point out the grammatical errors no doubt rife in that sentence? I have no trouble at all diagramming it, and cannot Þnd a single such error. Maybe you can point out, as a lesson in writing good English, and strictly for the beneÞt of the onlookers, of course, where your education fails to let you do the same? Note that neither complexity nor pleonasm count as errors, merely as foibles, so you must do much better than that to be thought anything but a constantly carping incompetent critic. I saw the rest of your post, I¹m deliberately ignoring it since it seems still to be based upon your ignorance is evaluating the above text. That you repeated, rather than defending, the rest of your errors merely increases your entertainment value as the butt of your own unintentional jokes. xanthian, wondering just which age that is that I was showing, mental, physical, emotional, or aspirational? I just wanted to call everyones attention to this particularly agonizing and downright Kentish work of grammatical convolution. (I have no brain...) -- hamlet@chopin.udel.edu (Chris Adams) -- === Subject: Re: Another mass question. > > Mass is volume multiplied by density. > It¹s the integral of density over a volume. Density is > not the factor of a product, but the kernel of a 3-cochain. > Mass is volume multiplied by density. It¹s the integral of density over a volume. Mass is not the product of factors, but the value of a 3-cochain on a 3-chain. === Subject: Re: Stationary action in physics > I recently looked at some intros and overviews of string theory, > and they all start with the principle of stationary action, as they should. > I recall studying this in classical theory, and the universal applicability > of this principle and the Lagrangian is amazing. The theory actually simplÞes > in the relativistic formulation. It also gives the > eqns. for þuids, MHD, and other continua, also EM theory and GR. The Þrst time I realised how beautiful the concept of a stationary action is was when I looked at minimal surfaces imbedded in 3-dim manifolds (with positive-deÞnite metric since I¹m a chicken). Proving that the Þrst variation leads to a stationary point is usually a no-brainer, but it¹s when you try to prove that the second variation of the action is strictly positive that all the magic starts to show itself. There are some *really* beautiful ways to apply this to the positive mass theorem and the location of apparent horizons. See Schoen & Yau¹s papers on the positive mass theorem and black hole formation from the early 80s for more info. davidoff === Subject: Re: I Need Help With This Equation > Let A={real numbers x such that |x|<1} > Let S={real numbers Z such that 1 < Z < inÞnity} > DeÞne ^ on A by the rule x^y =(x+y)/(1+xy). > It follows that (A, ^) is an Abelian group: > x^y=y^x > x^0=x > x^(-x)=0 > x^(y^z)=(x^y)^z > I¹m looking for a function from AxS->S (also written ^) such that, > for any x, y, in A and any Z in S: > x^(y^Z) = (x^y)^Z > 0^Z=Z > x^Z > Z if x>0 > x^Z < Z if x<0 I have a particular application in mind. I¹d like there to be two functions of a single variable (F and G) such that y^Z = F[(y+Z)G(Z)/sqrt(1-y*y)] Does that help? Eugene Shubert http://www.everythingimportant.org === Subject: Re: 640,000,000,000 TO 1 Paul Nutteing asks: > Do you have a reference ( URL ) to this info accessible in the > public domain ? The original URL was http://www.smh.com.au/breaking/2002/02/28/FFXJCGNF7YC.html but the webpage is long gone now. michael === Subject: Re: 640,000,000,000 TO 1 > Paul Nutteing asks: > Do you have a reference ( URL ) to this info accessible in the > public domain ? > The original URL was > http://www.smh.com.au/breaking/2002/02/28/FFXJCGNF7YC.html > but the webpage is long gone now. > michael It started me thinking if there is a 7 loci/14 datapoint match in 5,500 proÞles - how many 10 loci / 20 point matches are likely in 2 million proÞles. Especially when Prof Chaseling refused to answer my emails concerning her Þnding. What they aren¹t telling you about DNA proÞles and what Special Branch don¹t want you to know. http://www.nutteing2.freeservers.com/dnapr.htm or nutteingd in a search engine email nonarevers@yahoo.co.....uk (remove 4 of 5 dots) === Subject: Re: 640,000,000,000 TO 1 Paul Nutteing reveals: 2/text/review8.html> Yeah, I was the unnamed ŒJustice Action¹ spokesman Linda Doherty several others that she declined to mention - including Raymond Easton, which she refused to believe as it hadn¹t appeared in any ofÞcial reports at that time). > It started me thinking if there is a 7 loci/14 datapoint match > in 5,500 proÞles - how many 10 loci / 20 point matches > are likely in 2 million proÞles. Don¹t forget it was 14 out of 18 - which is far more likely than 14 out of 14. Pretty sure Chaseling was referring to the early stages of data collection for what was to be published as Analysis of Australian ProÞler Plus Data (see email below). She was also wrong when she said that the Red Cross donors were Œinformed¹. They knew nothing about what was to be done with their blood. (see http://tinyurl.com/3bmno & http://tinyurl.com/2wrz4 ) Alastair¹s insistence that Œa new sample¹ will be taken before an investigative cold hit is used as court evidence is pretty meaningless of course. The errors almost always occur during the handling or analysis of crime scene samples - not those taken from suspects - and even if there is sufÞcient material for multiple tests, contaminated results will often be eminently reproducible (as was the case in the VFSC and ESR contaminations). His assurances regarding the accreditation procedure is best judged by reference to the Jaidyn Leskie debacle at the fully accredited VFSC labs (the ESR lab was also accredited when its contamination scandal occured). And the NT forensic lab that tested the samples in the Falconio case is not accredited at all. michael ------------------------------------------------------------- ------------- === Subject: Analysis of Australian ProÞler Plus Data > Hi All, > Recently received a pre-print Sept 2002 copy of > Analysis of Australian ProÞler Plus Data by > Bruce Weir, Aleksander Bagdonavicius, Barry > Blair, Carmen Eckhoff, Christopher Pearman, > Peta Stringer, Julie Sutton, Laszlo Szabo and > John West. > As seems to be traditional in this kind of paper > it makes consistent claims of statistical conservatism > throughout while actually adopting several presumptions > which serve to totally over-ride and reverse those > measures in practice. > 14,763 9 loci ProÞler Plus proÞles from across all > Australian jurisdictions are checked against each other > for a total of 108,965,703 pairwise comparisons. > Some interesting things which emerge include are 13 > pairs matching at 7 loci of which 8 partially matched at > one other locus and 5 partially match at the two other loci > (for a total of 16 out of 18 points - or a Œhit¹ according > to default Crimtrac matching criteria, which can be > overridden by participating jurisdictions of course). > The paper notes that 28 fully matching 9 loci pairs were > initially discovered but that one of these was between > identical twins and the other 27 attributed to the same > person giving samples in two jurisdictions. It is not clear > what method was used to determine that 27 were duplicates > but it is interesting to note that no duplicates were noted > *within* jurisdictions - which seems to indicate that some > method is used by each participating database to eliminate > them. > Two pairs of proÞles were found to match at all except one > allele (i.e. 8 full loci + 1 partial). One of these seems to > have resulted from differing allele designations for D21 between > participating labs and the other one was written off as either > the same contributor + lab error (i.e. Œvery similar names¹ and > a subsequent retest showed the same proÞle) or a Œrare event¹. > The possibility of an error detected in this way is then used to > claim an overall lab error rate for the samples of one in 29,526 - > were retested (no point in searching for what you don¹t want to Þnd > I guess). > It is also claimed that Œcold hits¹ would always result in such a retest, > although this presumes that there is enough crime stain left for such > a test and that the error wasn¹t systematic and repeatable (e.g. dropout > due to binding site mutation or polymer contamination of crime stain). > It also presumes that all databases adopt such a policy (Crimtrac, for > instance, would not normally have access to state based samples for > such a retest and would have to leave it up to participating jurisdictions > to work it out themselves). > Incredibly, the authors suggest that based on the data observed, a > nine loci Œmatch by chance¹ should only be expected as the dataset > approaches 600,000 proÞles. This would give a discrimination for > ProÞler Plus of about 180 billion to one - higher than is usually given > using the product rule for proÞles from unrelated individuals. > They also say that 100,000 samples should be expected before an > 8 loci match is observed - even though their data suggests that they > have probably found at least one with less than 15,000 samples. > If anyone has any idea as to what kind of maths they used to reach > these conclusions I would appreciate enlightenment as they do not > the need to consider siblings or other close or inbred rellies. > that there is little difference between allele distribution found in > Australian and US samples and that subpopulation variations > in frequencies and departures from H-W and LE can be adequately > compensated for by using relatively modest theta values (<=0.03) > plugged into NRC-II equation 4.10. (But don¹t hold your breath > waiting for the contributors to abandon raw product rule calculations > in the labs they work in). > This is achieved by using very gross binning for subpopulations > (i.e. ŒAsians¹ [from Istanbul to Irian Jaya], ŒAborigines¹ and ŒCaucasians¹ > from each state) to smooth out the actual variations between the smaller > groups likely to constitute a Œsuspect pool¹ for a given Œcold hit¹ > investigation. > cheers, > michael === Subject: Re: 640,000,000,000 TO 1 > Paul Nutteing reveals: 2/text/review8.html> > Yeah, I was the unnamed ŒJustice Action¹ spokesman Linda Doherty > several others that she declined to mention - including Raymond Easton, > which she refused to believe as it hadn¹t appeared in any ofÞcial > reports at that time). > It started me thinking if there is a 7 loci/14 datapoint match > in 5,500 proÞles - how many 10 loci / 20 point matches > are likely in 2 million proÞles. > Don¹t forget it was 14 out of 18 - which is far more likely than > 14 out of 14. > Pretty sure Chaseling was referring to the early stages of data > collection for what was to be published as Analysis of Australian > ProÞler Plus Data (see email below). She was also wrong when > she said that the Red Cross donors were Œinformed¹. They knew > nothing about what was to be done with their blood. (see > http://tinyurl.com/3bmno & http://tinyurl.com/2wrz4 ) > Alastair¹s insistence that Œa new sample¹ will be taken before an > investigative cold hit is used as court evidence is pretty meaningless > of course. The errors almost always occur during the handling or > analysis of crime scene samples - not those taken from suspects - > and even if there is sufÞcient material for multiple tests, contaminated > results will often be eminently reproducible (as was the case in > the VFSC and ESR contaminations). > His assurances regarding the accreditation procedure is best > judged by reference to the Jaidyn Leskie debacle at the fully > accredited VFSC labs (the ESR lab was also accredited when > its contamination scandal occured). > And the NT forensic lab that tested the samples in the Falconio > case is not accredited at all. > michael > ------------------------------------------------------------- ------------- === > Subject: Analysis of Australian ProÞler Plus Data > Hi All, > Recently received a pre-print Sept 2002 copy of > Analysis of Australian ProÞler Plus Data by > Bruce Weir, Aleksander Bagdonavicius, Barry > Blair, Carmen Eckhoff, Christopher Pearman, > Peta Stringer, Julie Sutton, Laszlo Szabo and > John West. > As seems to be traditional in this kind of paper > it makes consistent claims of statistical conservatism > throughout while actually adopting several presumptions > which serve to totally over-ride and reverse those > measures in practice. > 14,763 9 loci ProÞler Plus proÞles from across all > Australian jurisdictions are checked against each other > for a total of 108,965,703 pairwise comparisons. > Some interesting things which emerge include are 13 > pairs matching at 7 loci of which 8 partially matched at > one other locus and 5 partially match at the two other loci > (for a total of 16 out of 18 points - or a Œhit¹ according > to default Crimtrac matching criteria, which can be > overridden by participating jurisdictions of course). > The paper notes that 28 fully matching 9 loci pairs were > initially discovered but that one of these was between > identical twins and the other 27 attributed to the same > person giving samples in two jurisdictions. It is not clear > what method was used to determine that 27 were duplicates > but it is interesting to note that no duplicates were noted > *within* jurisdictions - which seems to indicate that some > method is used by each participating database to eliminate > them. > Two pairs of proÞles were found to match at all except one > allele (i.e. 8 full loci + 1 partial). One of these seems to > have resulted from differing allele designations for D21 between > participating labs and the other one was written off as either > the same contributor + lab error (i.e. Œvery similar names¹ and > a subsequent retest showed the same proÞle) or a Œrare event¹. > The possibility of an error detected in this way is then used to > claim an overall lab error rate for the samples of one in 29,526 - > were retested (no point in searching for what you don¹t want to Þnd > I guess). > It is also claimed that Œcold hits¹ would always result in such a retest, > although this presumes that there is enough crime stain left for such > a test and that the error wasn¹t systematic and repeatable (e.g. dropout > due to binding site mutation or polymer contamination of crime stain). > It also presumes that all databases adopt such a policy (Crimtrac, for > instance, would not normally have access to state based samples for > such a retest and would have to leave it up to participating jurisdictions > to work it out themselves). > Incredibly, the authors suggest that based on the data observed, a > nine loci Œmatch by chance¹ should only be expected as the dataset > approaches 600,000 proÞles. This would give a discrimination for > ProÞler Plus of about 180 billion to one - higher than is usually given > using the product rule for proÞles from unrelated individuals. > They also say that 100,000 samples should be expected before an > 8 loci match is observed - even though their data suggests that they > have probably found at least one with less than 15,000 samples. > If anyone has any idea as to what kind of maths they used to reach > these conclusions I would appreciate enlightenment as they do not feel > the need to consider siblings or other close or inbred rellies. > that there is little difference between allele distribution found in > Australian and US samples and that subpopulation variations > in frequencies and departures from H-W and LE can be adequately > compensated for by using relatively modest theta values (<=0.03) > plugged into NRC-II equation 4.10. (But don¹t hold your breath > waiting for the contributors to abandon raw product rule calculations > in the labs they work in). > This is achieved by using very gross binning for subpopulations > (i.e. ŒAsians¹ [from Istanbul to Irian Jaya], ŒAborigines¹ and > ŒCaucasians¹ > from each state) to smooth out the actual variations between the smaller > groups likely to constitute a Œsuspect pool¹ for a given Œcold hit¹ > investigation. > cheers, > michael I would like to Þnd somewhere what , if any , correlation there is between theta values and the spread of allele frequencies for different population diversity. I¹ve looked at my results for 10 loci simulations and scaled them for the Oz 9 loci situation. For absolutely no relatedness I would expect on 9 loci / 18 datapoint one match in 940,000. Factoring in >= 8 per cent allele frequency for relatedness then for 9 loci 1 in 60,000. If you allow any 8 matching pairs, and that is any 8 pairs ie no speciÞc order, and one pair of alleles matching in the remaining 2 pairs of alleles ( 17 datapoint ) then you can reduce these (60,000 to 940,000 Þgures ) by a factor of 30 to 50. No one , (apparently ) investigates these unrelated matches in DNA databases. It is very easy to cross-correlate mug-shots or even better friction ridge Þngerprint data to rule out genuine repeats of one individual. The FBI deletes any matches from any data they pass on to academe because it is OBVIOUSLY erroneous. I go back to my MP next week to try and get the number of 6 and 10 loci matches in the arrestee side of the UK NDNAD written into Hansard. Pigs might þy What they aren¹t telling you about DNA proÞles and what Special Branch don¹t want you to know. http://www.nutteing2.freeservers.com/dnapr.htm or nutteingd in a search engine email nonarevers@yahoo.co.....uk (remove 4 of 5 dots) === Subject: Opinions about Synergetics What is the current mathematical opinion of Buckminster Fuller¹s two books Synergetics 1 and 2 and Synergetics coordinates? Here is an old 1998 opinion from Princeton. Opinions might have changed since then. On Wed, 4 Feb. 1998 10:01:12 -0500 (EST) in Geometry-Research, But Bucky Fuller, though a wonderful architect, was close to crazy, as his books clearly show. He regularly used phrases (such as the fundamental structure of the plane is hexagonal) that sound wonderful but have no meaning. In fact it¹s a mere matter of convenience whether we use orthogonal or hexagonal coordinates - neither is intrinsically better than the other; it¹s just that some coordinate-systems are better suited to some problems than others. I¹ve used dozens of different coordinate-systems in my life, as have most other professional mathematicians, and I prefer not to waste time by muttering meaningless mumbo-jumbo to show how one is somehow more moral than the others. --- End of quote. Webster¹s deÞnition of mathematics: Math`e*matics, n. [F. math[Œe]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See Mathematic, and -ics.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations. Note: Mathematics embraces three departments, namely: 1. Arithmetic. 2.Geometry, including Trigonometry and Conic Sections. 3. Analysis, in which letters are used, including Algebra, Analytical Geometry, and Calculus. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations. Source: Webster¹s Revised Unabridged Dictionary, © 1996, 1998 MICRA, In --- If you use Webster¹s deÞnition Note about mathematics, the Cartesian coordinate system should be considered part of pure or abstract mathematics, and the Synergetics coordinate system part of mixed or applied mathematics, even though the Synergetics coordinate system was developed in pure principle, because If you make things with string pulled through equal length soda straws, the triangle holds its shape and the square does not. The tetrahedron with four triangular faces holds its shape and the cube with six square faces does not, so the Synergetics coordinate system is interwoven with physical considerations, and the Cartesian coordinate system is not, really. A brief description of Synergetics coordinates is at: http://mathworld.wolfram.com/SynergeticsCoordinates.html The Mathematica notebook SynergeticsApplication7 at: http://library.wolfram.com/infocenter/MathSource/600/ or the SynergeticsApplication7 notebook as html at: http://users.adelphia.net/~cnelson9/ R. Buckminster Fuller¹s Synergetics 1 and 2 at: http://www.rwgrayprojects.com/synergetics/synergetics.html Cliff Nelson === Subject: Re: Opinions about Synergetics > What is the current mathematical opinion of Buckminster Fuller¹s two > books Synergetics 1 and 2 and Synergetics coordinates? What is the current mathematical opinion of James Joyce¹s two books Ulysses and Finnegans Wake and the stream of consciousness? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Opinions about Synergetics > What is the current mathematical opinion of Buckminster Fuller¹s two > books Synergetics 1 and 2 and Synergetics coordinates? > What is the current mathematical opinion of James Joyce¹s two > books Ulysses and Finnegans Wake and the stream of consciousness? I know what you mean, like Arthur L. Loeb¹s preface, but, the books were commissioned about 1960 and published in 1975 and 1979 and are about the discovery Fuller made before 1900 and got a patent for about 1960 and then the discovery was called for by name by NASA in 1978. More time went into the books than most books, and the preface below is based on a little time thinking about the books. --- Preface of Synergetics by Arthur L. Loeb ...Fuller expresses himself metaphorically: his poems sometimes convey his meaning more lucidly than his prose. Gertrude Stein¹s language really becomes unintelligible only when analyzed; the sentence Entropy is not random; it is always one negative tetrahedron (Synergetics, Sec. 345) is worthy of a place in American literature next to Miss Stein¹s. And if happiness be a warm puppy, why should not entropy be a negative tetrahedron? I have learned never to reject one of Fuller¹s outrageous statements without careful consideration, and even hesitated to call Sum of angels around each vertex in the rough manuscript a misprint... --- End of quote. I¹ve read the books for about twenty years and I¹ve just about decided that Bucky was literal about geometric models of physics. It isn¹t poetry. It is unintelligible when not analyzed, that¹s for sure. Cliff Nelson === Subject: Re: f:Z_9 -> z_5 cannot be homomorphism >> I just read f:Z_9 -> z_5 cannot be homomorphism >> Is it true ? >> Depends. What¹s f? Have you read and posted the complete context? >I believe that one of the properties of f: is that is missing in the >original post is that it is surjective. >If so, then f(1) = 1 (identity to identity) No. Z_9 and Z_5 are not multiplicative groups. The operations you are looking at are Addition (mod 9, mod 5 resp.) So f(0) = 0. But that doesn¹t help you very much, does it? But, supposing f is surjectiv, then there is an a<>0 such that f(a) = 1. >Therefore f(3) = 3 due to homomorphism >f(0) = f(3x3) = f(3)xf(3) = 3x3 = 4 in Z_5 Now you bring 0 back into the game - are we talking about surjective ring-homomorphisms now?? But f(0) = 4mod5 follows from f(9a) = 9 = 4 mod 5 (since f is homomorphism) and 9a=0 mod 9. So you don¹t need a ring structure. >but note that f(0) = f(0x0) = 4x4 = 1 in Z_5 >so 0 gets mapped to 2 values of Z_5 (including the identity) OK. So there is a contradiction. Thomas >-Tralfaz === Subject: Re: 111,111,111 x 111,111,111 = 12,345,678,987,654,321 > Extra credit: > 111,111,111 / 9 = 12,345,679 > Why is 8 missing? > Because 8 = 987654312/123456789 ;-) Wow!! That earns you full credit! :-) === Subject: Hartogg¹s aleph function An exercise from Chapter I of Kunen: Let A(x) = sup{a: a is an ordinal and there exists an injection from a to x}. In ZF - Foundation, show that A(x) < A(P(P(P(x)))), where P denotes the power set. Kunen marks this as a more difÞcult exercise and notes that it is easier to show that A(x) < A([P^4](x)). I am working through these exercises for my own ediÞcation (i.e., this is not homework), and I have been scratching my head about this one (even the easy part) for some time. Can anyone offer a hint? Of course, I refer to the case where x is inÞnite. My meagre thoughts on the easy part so far: - A(x) is a cardinal. In fact, A(x) is the set whereof it is the supremum. - It sufÞces to show that there is an injection from A(x) to [P^4](x) . - Might there be a way to go from x to [P^2](x) and compose this result with itself? - I have played with the equivalence classes of P(x) determined by bijections. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Hartogg¹s aleph function >An exercise from Chapter I of Kunen: >Let A(x) = sup{a: a is an ordinal and there exists an injection from >a to x}. In ZF - Foundation, show that >A(x) < A(P(P(P(x)))), where P denotes the power set. This should read A(x) <= (P(P(P(x)))) This is the original proof of Hartogs, and uses the representation of a reþexive antisymmetric transitive relation R as {a: for some x, a = {y: yRx}}. This precedes the use of ordered pairs to represent relations. Using ordered pairs, the inequality would read A(x) <= (P(P(x) x P(x))). Getting rid of the = in these relations is harder. It follows because on can get, instead of A(x), P(A(x)) with the =. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Hartogg¹s aleph function boundary=------------010209020701040804090508 ------------------------------------------------------------- -------- >>An exercise from Chapter I of Kunen: >> >>Let A(x) = sup{a: a is an ordinal and there exists an injection from >>a to x}. In ZF - Foundation, show that >>A(x) < A(P(P(P(x)))), where P denotes the power set. >> >This should read > A(x) <= (P(P(P(x)))) I assume you mean A(x) <= A(P(P(P(x)))). Or did you mean the curly inequality, i.e., an injection into [P^3](x)? give them consideration to see which inequality I can get. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Hartogg¹s aleph function > An exercise from Chapter I of Kunen: > Let A(x) = sup{a: a is an ordinal and there exists an injection > from a to x}. In ZF - Foundation, show that > A(x) < A(P(P(P(x)))), where P denotes the power set. > Kunen marks this as a more difÞcult exercise and notes that it is > easier to show that A(x) < A([P^4](x)). > I am working through these exercises for my own ediÞcation (i.e., > this is not homework), and I have been scratching my head about this > one (even the easy part) for some time. Can anyone offer a hint? > Of course, I refer to the case where x is inÞnite. Hartogg¹s is an unusual misspelling. The usual misspelling is Hartog¹s. The correct spelling is Hartogs¹ or maybe Hartogs¹s, I¹m not sure; the guy¹s name was Hartogs. All right, here¹s a hint. Forget about [P^3](x) and [P^4](x). Just focus on injecting A(x) into *something* built up from x. After you¹ve done that it will be time to count iterations of the power set operation. If it¹s more than you want, try to do what you did more economically. Another hint: forget about Von Neumann ordinals. Ordinal number are the isomorphism types of well-ordered sets; like any other isomorphism types, they are equivalence classes. (Give me that old time set theory, it¹s good enough for me.) === Subject: Re: Hartogg¹s aleph function >Hartogg¹s is an unusual misspelling. The usual misspelling is >Hartog¹s. The correct spelling is Hartogs¹ or maybe Hartogs¹s, I¹m not >sure; the guy¹s name was Hartogs. hint; I will consider them in my efforts. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Þnite open cover for vector bundle >> It is a corollary of the (at Þrst equally surprising) fact >> that if M is a connected differentiable n-manifold, compact >> or not (with empty boundary; though that can be worked >> around), then M has a cover by n+1 open sets each diffeomorphic >> to the open n-ball. This is in turn a corollary of the fact that >> M has a handle decomposition (because M supports a Morse function), >> coupled with the fact that (at least when n > 2; you can make >> special arguments for smaller n) there are plenty of disjoint >> arcs in M. >> > Elements of Noncommutative Geometry to extend Swan¹s > lemma: every vector bundle over M is direct summand of a > trivial bundle, to paracompact M. Can¹t this be proved in a more elementary way, by using the embedding theorem? Suppose M is connected. Let E be a vector bundle over M. Then E is also paracompact and connected and can be embedded in R^N where N only depends on the dimension of E. Consider M as the zero-section in E. Now the vertical tangents to M in E form a subbundle of the trivial bundle M x R^n, and this subbundle is isomorphic to E. By a vertical tangent at P in M I mean a tangent at P to pi^{-1}(P) where pi : E -> M is the projection and E is regarded as a subspace of R^N. Then E splits off as a summand of the trivial bundle of dimension N. As a paracompact M may have lots of connected components, we apply this method to each of them, but as N can be chosen independently of the component, the result extends to this general case. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Probability / Combinatorical Problem... Actually No Way¹s hint only worked for this small problem...I¹m still stuck with a bigger version though... This time it¹s two counters A and B that start at 777 and max out at 999. We have an experiment X, and this time we deÞne two events eA and eB that aren¹t mutually exclusive and increase the respective counter by 1 if it hasn¹t reached 999 yet. Then there¹s two events eA1 and eB1 that trigger win mechanisms(eA1 and eA are mutually exclusive, same for B). You get three different type of wins depending on where the counter is located, one from 777 to 887, another one from 888 to 998 and a last one at 999. After you score a win, the respective counter is reset to 777 and the other one is set to max {937, current_value}. Same question, what are the expected values for eA1 and eB1 if you rerun X inÞnitely. Well, even if you don¹t consider the states that are only passed once, you get a huge state matrix, if I¹m not mistaken there are (999-777+1)*(999-937+1) + (999-937+1)*(936-777+1) = 24129 possible states which is entirely too many for a quick solution. So maybe there is a mistake here and I don¹t need to consider all those states or there is a way to break this up into smaller subproblems. Any help would be appreciated... >Hello... >I have 3 counters A, B and C that start at 1. For an experiment X, we >deÞne 3 mutually exclusive events eA, eB and eC that occur with >probabilities p(eA), p(eB) and p(eC). >Each time an event eA occurs, you gain wA1 if A <=4 and wA2 if A = 5, >for eB it¹s wB1 if B <= 4 and wB2 if B=5, and for eC it¹s wC1 if C <= >4 and wC2 if C=5. >Then the respective counter is increased by 1 if it¹s smaller than 5. >If it already had reached 5, it is reset to 1 and the other two >counters are set to 5. >If you choose to rerun X inÞnitely, what are the expected gains for >eA, eB and eC? >Well if I only had one counter it would be easy, but how do I deal >with those several interacting counters? My Þrst idea was to consider >a combination of the 3 counters as one state and then solve the >problem with Markov chains, but that would lead to a >(5^3)*(5^3)-matrix > It leads to a 13 by 13 matrix. Since you are going to rerun X > inÞnitely, you do not need to consider the states that are only > passed once. > and I don¹t think that¹s a clever idea considering >that I have a similar problem with much larger counters. >Does anyone know which trick is needed for this problem? > I certainly don¹t. :) === Subject: calculus question Is there an instance of non-injective function f:R->R such that for every x in R oo>|df(x)/dx|>0 ? === Subject: Re: calculus question > Is there an instance of non-injective function f:R->R such that for > every x in R oo>|df(x)/dx|>0 ? No, call a and b two points in R such that f(a)=f(b). If f were differentiable on R (and therefore continuous on [a,b] differentiable on ]a,b[) then Rolle¹s theorem could be applied and there would be a point c in ]a,b[, df(c)/dx = 0. -- Julien Santini === Subject: Re: new laws on what Unions can exist in companies Re: vonNeumann Gametheory > So, what I am proposing is that any company in the USA cannot have Union > workers unless the Union of workers own a minimum amount of the shares of > the company itself. How many union workers does SBC have? And how many > shares of SBC in aggregate are owned by those union workers? > SBC has 168,000 employees and its market cap is $78B. So if each > employee would by $500,000 worth of stock, they would own the place. > What I am saying is that Congress should make a law about Corporations that > the company cannot have Union workers unless those workers own a percentage > of the company shares. Say perhaps 1% in aggregate. Otherwise all the > workers at that company would be nonUnion workers. > SBC¹s insiders own 0.77% of the stock. Why require more for the union? > -- > Ron One of the reasons companies outsource overseas is to escape yearly Union demands. So I wonder how much of the outsourcing is attributable to Union demands. The USA needs a fresh look at Unions because the century in which they were created and evolved in a business environment seem to no longer exist. It makes more sense that a company should have its Shareholders and Workers moving in synch with the betterment of the company. But in the case of SBC with its Unions and with its shareholders they are diametrically opposite such as the recent strike. So, the 1% aggregate rule if a company wants to apply the rule (some companies can opt for Union labor force regardless of whether they have 1% of the shares) is a rule that promotes a harmony between Shareholders and with Union Workers. If they owned 1% or more at SBC, they could still go on strike, but those Unions would tend to see how their actions inþuence the share prices. And it is not proper for a group of people to be able to inþuence a company to such an extent that Unions can and will do, yet have no ownership stake in the company itself. And how Unions can place so much demands on a company whereas the Shareholders themselves, by the laws in place, cannot place noteworthy demands on the company. There is a huge imbalance of the power of Unions compared to the power of the actual owners of the company itself. So the 1% rule would try to put the Unions on the same track as the Shareholders and not so diametrically opposite one another. So that if a group of workers of a company want to form a Union within that company they must own 1% of the shares of that company in aggregate. Archimedes Plutonium www.archimedesplutonium.com www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: new laws on what Unions can exist in companies Re: vonNeumann Gametheory >> SBC has 168,000 employees and its market cap is $78B. So if each >> employee would by $500,000 worth of stock, they would own the place. >> SBC¹s insiders own 0.77% of the stock. Why require more for the union? > One of the reasons companies outsource overseas is to escape yearly Union > demands. So I wonder how much of the outsourcing is attributable to Union > demands. Very little is due to union demands. Help desk people aren¹t unionized and professionals aren¹t either. > The USA needs a fresh look at Unions because the century in which they were > created and evolved in a business environment seem to no longer exist. Without the threat of Communism, the right has no need to allow unions. > It makes more sense that a company should have its Shareholders and Workers > moving in synch with the betterment of the company. But in the case of > SBC with its Unions and with its shareholders they are diametrically > opposite such as the recent strike. The main issue of bargaining is working conditions, not wages. Workers will vote for candidates that support the industry they work for over their own interests. > So, the 1% aggregate rule if a company wants to apply the rule (some > companies can opt for Union labor force regardless of whether they > have 1% of the shares) is a rule that promotes a harmony between > Shareholders and with Union Workers. If they owned 1% or more at SBC, > they could still go on strike, but those Unions would tend to see how > their actions inþuence the share prices. $5,000 per worker is too small to make a difference, besides management owns less than 1% of the company. SBC needs to empower their workforce instead of trying to make mid-level management into pocket dictators. -- Ron === Subject: Diffeomorphisms of subsets of R^3 I¹ve got a stupid question: Are the sets { (x,y,z) in R^3 | 1 < |(x,y,z)| < 2 and ( x^2 + y^2 > 0 for z > 0 ) } and { (x,y,z) in R^3 | |(x,y,z)| < 2 and ( x^2+z^2 > 1 for y=0 ) ) and and ( x^2 + y^2 > 0 for z > 0 ) } both diffeomorphic to the open unit ball of R^3? How to prove that? Your sincerely, Tobias === Subject: Diffeomorphisms of subsets of R^3 (continued) Let n be a positive integer. Is { (x,y,z) | |(x,y,z)| < n and ( |(x,y)| > 0 if z > 0 ) and and ( |(x,y,z-k)| > 0.25 for k=0,1,...,n-1 ) } diffeomorphic to the unit ball? How do you prove your answear? TN === Subject: Re: Diffeomorphisms of subsets of R^3 (continued) > Let n be a positive integer. > Is > { (x,y,z) | |(x,y,z)| < n and ( |(x,y)| > 0 if z > 0 ) and > and ( |(x,y,z-k)| > 0.25 for k=0,1,...,n-1 ) } > diffeomorphic to the unit ball? Do you mean |(x,y,-k)| > 0.25 for k=0,1,...,n-1 ? === Subject: Re: Diffeomorphisms of subsets of R^3 > I¹ve got a stupid question: > Are the sets > { (x,y,z) in R^3 | 1 < |(x,y,z)| < 2 and ( x^2 + y^2 > 0 for z > 0 ) } > and > { (x,y,z) in R^3 | |(x,y,z)| < 2 and ( x^2+z^2 > 1 for y=0 ) ) and > and ( x^2 + y^2 > 0 for z > 0 ) } > both diffeomorphic to the open unit ball of R^3? Yes... I¹m sure. > How to prove that? the Þrst is easy: that¹s diffeomorphic to an interval times the sphere with one point removed. The sphere with one point removed is diffeomorphic to the plane, e.g., stereographic projection. The second is a bit Þddly. It can be deformed radially into the Þrst, and such a deformation can be smoothed out but I don¹t really want to think about the horrible little details of how one does that :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Best Þt with exponentials >I doubt that this can be done in closed form, and I have an iterative >approximation, but it is just too slow, so here it is: >y = a * exp(bx) + c * exp(-dx) >I need to Þnd a, b, c, d, (all positive) that minimizes the sum of >the squares of the errors against a set of (at least 4) points (x,y). >Here is what I have done so far. If S is the sum of the squares of >the errors, the partial derivative of S with respect to a and to b is >linear in a and b. Therefore, for any c and d, I can easily Þnd the >a and b for which those partial derivatives are zero. So what I do is >perform a two-dimensional trial and error search over c and d. When >the total error stop going down, I stop trying. I need to implement >this on a Palm PDA, and the þoating point calculations are taking >about 10 times longer than what is tolerable for my application. Does >anyone have any idea on how to make the process converge faster, or >(gasp!) to solve for a,b,c,d in closed form? > Closed form, no. I suggest you use some form of Newton¹s > method or gradient methods. It can be quite ill conditioned. > Also, the minimum might be at end points. This can be > taken into account. It was solved in closed form on 1795, as long as one looks for exact Prony¹s Method Prony¹s method is a technique for extracting the sinusoid or exponential signals by solving a set of linear equations for the coefÞcients of the recurrence equation that the signals satisfy. It is closely related to Pisarenko¹s method, which Þnds the smallest eigenvalue of an estimated covariance matrix. The classical method of Count de Prony models a sequence of 2p observations made at equally spaced times by a linear combination of p exponential functions. Prony¹s ingenious method converted the problem to a system of linear equations. See the MacTutor biography of Baron Gaspard Riche de Prony. The original reference is de Prony, Baron Gaspard Riche (1795). Essai .8experimental et analytique: sur les lois de la dilatabilit.8e de þuides .8elastique et sur celles de la force expansive de la vapeur de l¹alkool, .88 diff.8erentes temp.8eratures. Journal de l¹.83cole Polytechnique, volume 1, cahier 22, 24-76. A contemporary treatment of modern Prony methods can be found in Marple, S. L. (1987). Digital Spectral Analysis with Applications. Prentice-Hall. === Subject: Cuthill Mc Kee Do you know a good book, where the algorithm of Cuthill Mc Kee is described. Yours sincerely, TN === Subject: Re: DFT and FFT >Hi again >Everyone seems to think I want to analyse the maths here, I don¹t think >that¹s what my tutor meant. Here is exactly what he said (about making a >website about FFTs): >The Web-site should explain the background theory and allow visitors to >perform FFT¹s on-line, as well as cover recent research in this area. You >might also attempt, for example, the uploading of sample data (such as >music) and performing some kind of FFT analysis on that data... Windowing >is also an important concept in this area... Elsewhere you say you don¹t know what calculus or polynomials are. I really don¹t see how you can possibly do what it says in the above paragraph - to explain the background theory you¹ll need to _know_ the background theory, and the background theory is several years ahead of you, mathematically. Same for reading through quality academic texts, etc. Is the person who suggested this topic _aware_ that you don¹t know what calculus or polynomials are? Not trying to be nasty, honest - just seems like it would a shame to waste time on something that you¹re really not equipped to do. Nothing wrong with not knowing what calculus and polynomials are, but if you don¹t there¹s no way you¹re going to be able to Þgure out what windowing is, for example... >[and] >Background research work would involve reading through quality academic >texts on the subject of FFT/DFT, looking for recent research papers, >reading up on PHP/ASP/JSP, Web-servers, creating graphs using the >aforementioned scripting languages on a Web-page... >[and] >Ideally, after doing all the research, you will think of some novel >application for this subject area, which could potentially allow you to >attain the highest marks... >So, like someone said, it¹s more about the application than the theory? >Sorry if I gave anyone the wrong idea. >Clare >~ Hello >~ I¹m a Computer Science student and my tutor gave me the idea of doing FFT >~ for my Þnal year project. I¹ve read some stuff about it but it¹s all in >~ very mathy language. >~ I gather that it¹s an algorithm for turning... something... into a >~ frequency? But what use is that, practically? >~ So I was wondering if someone could tell me what FFT does and what it¹s >~ used for, in very simple and normal language, so that I have a base for >~ understanding what I¹m reading about. >~ Clare ************************ David C. Ullrich === Subject: Re: DFT and FFT >[erm, snip] >~ I suspect that Clare is at an American university. >No, sorry, I¹m English. >~ As I teach at one myself, I >~ suspect that what happens at her university is similar to my experience. >~ Computer science programs usually don¹t require a lot of math, indeed some of >~ them only require cook-book calculus as in differentiating and integrating >~ polynomials. I am thinking that many of the courses in the computer science >~ course are in things like object oriented concepts, designing applications, as >~ well as stuff like sorting, databases, computer graphics, etc. (I don¹t >~ actually know - I am a math teacher, but I suspect that I am close to being >~ correct.) >I don¹t know what calculus or polynominals are but the rest of what you¹ve >said is correct. I didn¹t do Maths A Level. In the Þrst year here we >did Basic Maths for Computer Science which was about different bases and >the OR and NOT gates and stuff. Also in the Þrst year I did Discrete >Maths which was about big equations, which was a bit difÞcult but I >ended up doing quite well. This semester (my second year) I¹m doing >networks and intranet system admin. You should go back to your advisor or whoever and explain that you need to do your paper on another topic. If it¹s supposed to be something with no content at all Þne, but if it¹s supposed to be something that actually has something to do with the FFT then you have several years of math to study before you can possibly attain even a vague understanding of the topic. (The content-free paper that you¹d be able to write right now would be very short, just a few sentences listing Þelds where the FFT is used.) Hope you¹re not offended by that - it¹s simply true, from your present point of view the FFT is a topic in _very_ advanced mathematics, if you don¹t know what calculus or polynomials are the topic is simply not appropriate. >Clare ************************ David C. Ullrich === Subject: Re: DFT and FFT > So I was wondering if someone could tell me what FFT does > and what it¹s used for, The explanation that has been easiest for me to follow is in Sara Baase, Computer Algorithms - Introduction to Design and Analysis, Second Edition, Addison-Wesley, 1988, pp. 268ff. For an application to problems of insurance (with a corrected algorithm), see my paper at: http://www.casact.org/pubs/proceed/proceed92/92057.pdf John Robertson === Subject: Re: DFT and FFT ~ ~ > So I was wondering if someone could tell me what FFT does ~ > and what it¹s used for, ~ ~ The explanation that has been easiest for me to follow is in ~ ~ Sara Baase, Computer Algorithms - Introduction to Design and ~ Analysis, Second Edition, Addison-Wesley, 1988, pp. 268ff. ~ ~ For an application to problems of insurance (with a corrected algorithm), see ~ my paper at: ~ ~ http://www.casact.org/pubs/proceed/proceed92/92057.pdf Clare === Subject: Re: Consecutive ingegers: Coprime > It¹s harder than that, since it¹s false. > Consider the numbers 2184, 2185, ..., 2200. > No one of those 17 consecutive integers > is coprime to all of the other 16. > More generally, for any integer n greater than or equal to 17. there > is a set of n consecutive integers, none of which is coprime to all > the others. I wonder if this is in the Wells¹s Dictionary of Curious Numbers, or whatever that book is that lists curious properties of integers. The number 17 has another interesting property, almost certainly not related to the above: It is the length of the longest sequence of fractions {a_i} in the interval [0, 1) such that for each j = 1, 2, .. the j fractions in the leftmost subsequence {a_1, a_2, .. a_j} can be found one in each interval [(k-1)/j, k/j) for k = 1, 2, j. In other words, it is the length of the longest sequence of fractions which can be built up by throwing successive fractions into the pot, so to speak, while at each stage keeping all the fractions as equally spread as possible. John R Ramsden === Subject: Re: Consecutive ingegers: Coprime > consecutive integers: > I¹ve been trying to show that in any sequence of n consecutive > integers, there existe at least one of them which is comprime > with all the others (in pairs). > Do you have any hint? This is Bertrands Postulate. Between n and 2n there is a prime. Do a Google . You will be able to Þnd Erdos elementary proof of this proposition. Bob Kolker === Subject: Re: Consecutive ingegers: Coprime ... > This is Bertrands Postulate. Between n and 2n there is a > prime. Do a Google . You will be able > to Þnd Erdos elementary proof of this proposition. Should this not be: between n and 2n - 2, n > 3? -- Mail sent to this email address is automatically deleted (unread) on the server. Send replies to the newsgroup. === Subject: Re: Consecutive ingegers: Coprime > consecutive integers: > > I¹ve been trying to show that in any sequence of n consecutive > integers, there existe at least one of them which is comprime > with all the others (in pairs). > > This is Bertrands Postulate. Between n and 2n there is a prime. True, but irrelevant - I think you¹ve misunderstood the question. Have you seen the other replies in the thread? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Consecutive ingegers: Coprime > > > consecutive integers: > > > > I¹ve been trying to show that in any sequence of n consecutive > > integers, there existe at least one of them which is comprime > > with all the others (in pairs). > > > This is Bertrands Postulate. Between n and 2n there is a prime. > True, but irrelevant - I think you¹ve misunderstood the question. > Have you seen the other replies in the thread? Thaks to everyone!! You¹ve been very helpfull. === Subject: Proof of twin-prime conjecture? Arenstorf has uploaded an MS purporting to prove the twin prime conjecture to the arXiv: http://uk.arxiv.org/abs/math.NT/0405509 . This does not look completely stupid, in fact (whether or not it is all correct) it appears to be a substantial piece of analytic number theory. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: 3^n - 2^n and relatives / an attempt for the proof of the inequality Am 27.05.04 17:47 schrieb Gottfried Helms: >> Does this attempt sufÞce for a disproof of the given inequality? >> Since the following is a very short form, how I tried to reformulate >> the problem, I¹ll just post it here. >> Could someone please point out an error, if there is one (maybe >> even trivial...;-) my own job... >> (3^N-1) div Z + (3^N-1) mod Z >= Z - 1 >> which is simply >> (3^N-1) / Z >= Z - 1 is not, since I mixed the sum of the digits with the decomposition of a number w.r. to a base by base* (x div base) + x mod base So the following considerations do not hold. However, they show, that the digit-sum of a power of 3 in the z-system is smaller than z-1 (with the same probability as the approximations, which were posted here.) Gottfried Helms === Subject: Re: Cosets > > > > Can anyone suggest a good text book to study about cosets....I am > > totally confused about them. > > > > Fraleigh¹s book is pretty good. > ?? > An entire book all about cosets? Are we thinking of > the same notions of cosets? Subsets of a group of the > form x.H where H is some subgroup? > We are thinking of the same notion. I didn¹t say > Fraleigh¹s book was entirely about cosets. I didn¹t > imagine that OP wanted a text that was entirely > about cosets - I took it for granted that OP wanted > a text that contained material about cosets, and > wouldn¹t be upset if the book also contained information > that wasn¹t about cosets. > Either I misunderstood OP, or you misunderstood both of us. Actually, i only asked because i was afraid i had missed some exciting new emerging Þeld. Cosets - that sounds like something very exciting, sort of in the same league as coalgebras, cohomology, cooperative games... ;-) Herman Jurjus === Subject: Re: Peano¹s space-Þlling curve message >> >> >message >> >>> > Can you even _state_ the axiom of choice precisely? No. However, I can quote various published forms, some of which I can relate to, others that seem too obscure for me to follow all the consequences One that I can make deductions from, using my own - possibly idiosyncratic - form of logic, is from Kac and Ulam, who I believe are, or were, somewhat highly thought of among their fellow mathies. It reads as follows: Given a collection C of disjoint sets one can form a set Z by choosing one element out of each set in the collection. I quote verbatim from their book Mathematics and Logic in which they attribute this form of the axiom to Zermelo. > But it¹s clear that (at least when you¹re talking about AC) > you have _no_ acquaintance with the relevant ideas! > You have acquaintance with the _words_. So this deÞnition by Kac and Ulam is wrong? I Þnd it difÞcult to believe that I read such a simple string of words and do not understand the idea contained in them. I quote from one of your posts that gave me a map from two reals to one real viz. > x = 0.a1a2a3... > y = 0.b1b2b3..., > then (x,y) corresponds to the number > t = 0.a1b1a2b2... . I proposed that you created this function by performing what K and U say is AC, because it seemed to me that you made the (Þnite or inÞnite) string, t, as follows: Take two real numbers x and y; create two sets x¹ and y¹ whose members (subscripted according to decimal expansion place value and whether they originate in x or y) are in one-to-one correspondence with the decimal expansions of x and y respectively; take one element from x¹ and one from y¹ to create a set of two elements; repeat the selection as needed or ad inÞnitum; order the results to make set t¹; equate string t¹ with the real number t whose decimal expansion has digits in one-to-one correspondence with t¹. Apologies for not phrasing this in mathspeak. I hope the plain English makes sense. Now you can tell me what you actually did to make the real number t. You suggest that I view this creation in the same way I view f: x---->2x but I have a certain blind spot when I try this. I referred to it as a choice function but I now recognise that you have a reserved meaning in mathematics for this word so I¹ll switch and call it a choosing function, even though, to me, it seems to do what the Zermelo deÞnition says for AC. > You have no idea what the phrase choice function > means. By what seems to be your interpretation of > the axiom of choice, if I deÞne f(x) = 2x I must be > using the axiom of choice, because I¹m choosing > to map x to 2x. It seems for the function f:x --->2x we don¹t actually need to take a peek inside the domain a priori, in order to see what the x we¹re going to use looks like. I can start with a random number and invite a computer to compute a value in the co-domain. To use a choosing function of the kind you propose, at the very least I must tell the computer whether I have reals, complex numbers or hypercomplex numbers in the domain. In another way f:x --->2x seems to me part of a generalised function f:a,b ---> ab, with domain and range unlimited, where a and b can be pairs of reals, complexes, quaternions, etc. etc, and I need tell the computer no more than to multiply them. (Of course, I may need to deÞne what kind of multiplication is desired) > AC says this: Suppose that A is a set and each element > of A is a nonempty set. Let S be the union of the elements > of A. Then there exists a function f : A -> S such that > f(x) is an element of x for every x in A. I have tried to square this deÞnition with the deÞnition by Zermelo (quoted by K and U) and the latter seems to be more general. I suppose this can be proved by displaying objects that Þt Zermelo but do not Þt the deÞnition above? John === Subject: Re: Peano¹s space-Þlling curve Originator: hack@watson.ibm.com (hack) >One that I can make deductions from, using my own - possibly >idiosyncratic - form of logic, is from Kac and Ulam, who I >believe are, or were, somewhat highly thought of among their >fellow mathies. It reads as follows: > Given a collection C of disjoint sets one can form a >set Z by choosing one element out of each set in the >collection. >I quote verbatim from their book Mathematics and Logic in >which they attribute this form of the axiom to Zermelo. If that¹s verbatim it¹s indeed wrong. It should say disjoint NON-EMPTY sets. These little details matter in mathspeak. Others (e.g. Dave Seaman) have already explained the real point of AC, namely that it asserts the existence of something even in cases where one cannot describe in detail what this thing is. In the case at hand, there *is* a way to describe the result of those choices, so AC is not needed. (AC asserts the existence of something you¹re already holding in your hand -- wowee.) Michel. === Subject: Re: Peano¹s space-Þlling curve >[...] I Þnd it >difÞcult to believe that I read such a simple string of >words and do not understand the idea contained in them. Well, since the technicalities have already been answered by two other people, I¹ll just say that this is precisely why you don¹t appear to be making much progress in understanding this stuff: you can¹t believe you don¹t understand things that in fact you don¹t understand. You¹re not _going_ to understand them _until_ you realize that you _don¹t_. >I quote from one of your posts that gave me a map from two >reals to one real viz. >> x = 0.a1a2a3... >> y = 0.b1b2b3..., >> then (x,y) corresponds to the number >> t = 0.a1b1a2b2... . >I proposed that you created this function by performing what >K and U say is AC, because it seemed to me that you made the >(Þnite or inÞnite) string, t, as follows: > Take two real numbers x and y; create two sets x¹ and >y¹ whose members (subscripted according to decimal expansion >place value and whether they originate in x or y) are in >one-to-one correspondence with the decimal expansions of x >and y respectively; take one element from x¹ and one from y¹ >to create a set of two elements; repeat the selection as >needed or ad inÞnitum; order the results to make set t¹; >equate string t¹ with the real number t whose decimal >expansion has digits in one-to-one correspondence with t¹. > Apologies for not phrasing this in mathspeak. I hope the >plain English makes sense. Now you can tell me what you >actually did to make the real number t. >You suggest that I view this creation in the same way I >view f: x---->2x but I have a certain blind spot when I try >this. I referred to it as a choice function but I now >recognise that you have a reserved meaning in mathematics >for this word so I¹ll switch and call it a choosing >function, even though, to me, it seems to do what the >Zermelo deÞnition says for AC. >> You have no idea what the phrase choice function >> means. By what seems to be your interpretation of >> the axiom of choice, if I deÞne f(x) = 2x I must be >> using the axiom of choice, because I¹m choosing >> to map x to 2x. >It seems for the function f:x --->2x we don¹t actually need >to take a peek inside the domain a priori, in order to see >what the x we¹re going to use looks like. I can start with a >random number and invite a computer to compute a value in >the co-domain. This is nonsense. The only difference is that one procedure looks more complicated than the other. In _fact_ programming a computer to calculate 2x is if anything _more_ complicated than programming a computer to do that digit-interleaving operation above! (Not all that much more complicated.) You haven¹t thought about this, you¹re just assuming that something you¹re familiar with must be simple. Suppose you actually want to calculate 2x. The number x is going to be given as a (Þnite) decimal (or rather in binary, but that doesn¹t matter.) In order to calculate the Þrst ten digits of 2x your computer has to peek inside x to see what the Þrst ten digits of x are, in _exactly_ the same way that it has to peek inside to look at a1, a2, b1, b2 before it can output .a1b1a2b2. >To use a choosing function of the kind you propose, at the >very least I must tell the computer whether I have reals, >complex numbers or hypercomplex numbers in the domain. In >another way f:x --->2x seems to me part of a generalised >function f:a,b ---> ab, with domain and range unlimited, >where a and b can be pairs of reals, complexes, quaternions, >etc. etc, and I need tell the computer no more than to >multiply them. (Of course, I may need to deÞne what kind of >multiplication is desired) >> AC says this: Suppose that A is a set and each element >> of A is a nonempty set. Let S be the union of the elements >> of A. Then there exists a function f : A -> S such that >> f(x) is an element of x for every x in A. >I have tried to square this deÞnition with the deÞnition >by Zermelo (quoted by K and U) and the latter seems to be >more general. I suppose this can be proved by displaying >objects that Þt Zermelo but do not Þt the deÞnition >above? Of course this cannot be proved that way. It can¹t be proved at all - what I said is precisely equivalent to (a precise version of) what you quoted! You _really_ have to get over this idea that if something is not immediately clear to you, in a Þeld you have no competence in at all, it must be false. It would be a much better idea to say that one looks more general to you, and _ask_ whether that¹s actually so. Saying one looks more general to you and then making a conjecture on how it can be proved is telling me that I don¹t know what AC says. I do, and for _you_ to conjecture I don¹t is a little irritating. Um, note: It¹s not irritating enough that I¹d bother to mention it, except for a recent exchange we had, where I suggested that you should make fewer assertions and instead ask more questions. You claimed you weren¹t asserting things... >John ************************ David C. Ullrich === Subject: Re: Peano¹s space-Þlling curve >> Can you even _state_ the axiom of choice precisely? > No. However, I can quote various published forms, some of > which I can relate to, others that seem too obscure for me > to follow all the consequences > One that I can make deductions from, using my own - possibly > idiosyncratic - form of logic, is from Kac and Ulam, who I > believe are, or were, somewhat highly thought of among their > fellow mathies. It reads as follows: > Given a collection C of disjoint sets one can form a > set Z by choosing one element out of each set in the > collection. > I quote verbatim from their book Mathematics and Logic in > which they attribute this form of the axiom to Zermelo. >> But it¹s clear that (at least when you¹re talking about > AC) >> you have _no_ acquaintance with the relevant ideas! >> You have acquaintance with the _words_. > So this deÞnition by Kac and Ulam is wrong? I Þnd it > difÞcult to believe that I read such a simple string of > words and do not understand the idea contained in them. Question: Does it ever happen in mathematics that a statement may be proved in more than one way? In particular, is it possible that for some speciÞc collection C of disjoint sets, we may be able to prove the existence of a choice function in more than one way? One way would be to actually specify the value of f(x) for each x in C according to some rule, while the other way would be to invoke the axiom of choice. If this happens, do we conclude that the construction of f *requires* the axiom of choice? Obviously not. It is not sufÞcient to show that AC *could* be used; you must show that no alternative exists. > I quote from one of your posts that gave me a map from two > reals to one real viz. >> x = 0.a1a2a3... >> y = 0.b1b2b3..., >> then (x,y) corresponds to the number >> t = 0.a1b1a2b2... . > I proposed that you created this function by performing what > K and U say is AC, because it seemed to me that you made the > (Þnite or inÞnite) string, t, as follows: I propose that he created this function by specifying exactly what the n¹th digit should be for each n. There is an explicit rule, depending on whether n is even or odd, that Þnds a speciÞc digit of x or y and makes that digit the n¹th digit of t. No AC is required, because a speciÞc rule is presented. > Take two real numbers x and y; create two sets x¹ and > y¹ whose members (subscripted according to decimal expansion > place value and whether they originate in x or y) are in > one-to-one correspondence with the decimal expansions of x > and y respectively; take one element from x¹ and one from y¹ > to create a set of two elements; repeat the selection as > needed or ad inÞnitum; order the results to make set t¹; > equate string t¹ with the real number t whose decimal > expansion has digits in one-to-one correspondence with t¹. Instead of take one element I would say take the next element, or take the n¹th element. This makes it clear that the selection is not arbitrary. > Apologies for not phrasing this in mathspeak. I hope the > plain English makes sense. Now you can tell me what you > actually did to make the real number t. For each n, take the n¹th digit of x and the n¹th digit of y, and make those the (2n-1)th and (2n)th digits of t. I didn¹t leave you any room to make a choice. If you carry out this procedure 1000 times you will get precisely the same number t each time. A characteristic property of AC is that there is no other way to describe the set whose existence is guaranteed by the axiom. >> You have no idea what the phrase choice function >> means. By what seems to be your interpretation of >> the axiom of choice, if I deÞne f(x) = 2x I must be >> using the axiom of choice, because I¹m choosing >> to map x to 2x. > It seems for the function f:x --->2x we don¹t actually need > to take a peek inside the domain a priori, in order to see > what the x we¹re going to use looks like. I can start with a > random number and invite a computer to compute a value in > the co-domain. In the same way, in deÞning t from x and y according to the odd/even scheme above, we don¹t need to look inside x and y to see what the actual digits are. We are able to represent them as a_i and b_j, precisely because it doesn¹t matter to the argument. That¹s why AC is not needed. -- Dave Seaman Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Peano¹s space-Þlling curve >> Can you even _state_ the axiom of choice precisely? > No. However, I can quote various published forms, some of > which I can relate to, others that seem too obscure for me > to follow all the consequences > One that I can make deductions from, using my own - possibly > idiosyncratic - form of logic, is from Kac and Ulam, who I > believe are, or were, somewhat highly thought of among their > fellow mathies. It reads as follows: > Given a collection C of disjoint sets one can form a > set Z by choosing one element out of each set in the > collection. The phrase choosing one element is rather weaselly: better to assert the existence of Z with Z intersecting each element of C in a singleton. The loose formulation you cite is due to Kac/Ulam writing a populist potboiler, rather than an academic text. > So this deÞnition by Kac and Ulam is wrong? I Þnd it > difÞcult to believe that I read such a simple string of > words and do not understand the idea contained in them. You are not alone in failing to understand simple strings of words. > I quote from one of your posts that gave me a map from two > reals to one real viz. >> x = 0.a1a2a3... >> y = 0.b1b2b3..., >> then (x,y) corresponds to the number >> t = 0.a1b1a2b2... . > I proposed that you created this function by performing what > K and U say is AC, What you propose is beside the point. The fact is that the proof that this is a function does not require the axiom of choice. > Take two real numbers x and y; create two sets x¹ and > y¹ whose members (subscripted according to decimal expansion > place value and whether they originate in x or y) are in > one-to-one correspondence with the decimal expansions of x > and y respectively; take one element from x¹ and one from y¹ > to create a set of two elements; repeat the selection as > needed or ad inÞnitum; order the results to make set t¹; > equate string t¹ with the real number t whose decimal > expansion has digits in one-to-one correspondence with t¹. > Apologies for not phrasing this in mathspeak. I hope the > plain English makes sense. Mathematics (not mathspeak) is written in plain English (at least by English-speaking mathematicians). But why do you use this loaded phrase to refer to your own text? --- that smacks of self-congratulation. This text makes little sense. You say create two sets x¹ and y¹ etc.. This is a fabrication of your own, not appearing in David¹s text. > You suggest that I view this creation in the same way I > view f: x---->2x but I have a certain blind spot when I try > this. You said it. > I referred to it as a choice function but I now > recognise that you have a reserved meaning in mathematics > for this word so I¹ll switch and call it a choosing > function, even though, to me, it seems to do what the to me... me, me, me, me! > To use a choosing function of the kind you propose, No, David did not speak of choosing function; that¹s your own private language term. > at the > very least I must tell the computer whether I have reals, > complex numbers or hypercomplex numbers in the domain. The computer? Ach! a worshipper of The Machine :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Peano¹s space-Þlling curve message > I note some points exist that will never be traced by any >number of countable recursions. I also Þnd points that are >included more than once and topology tells me that I no >longer have a structure that is homeomorphic with a line >(0,1). Any wonder then that I am confused by your assertions >that a piece of abstract algebra can be used to show that >neither of these concrete discoveries are correct. > Nobody has denied either of those facts. On the contrary, I have been told several times on this newsgroup that the Peano construct (a) passes through every point in the square and (b) it is an image of the [0,1] continuum. > Exactly where did I assert what you say I¹ve > asserted? You asserted that everything I had been told was proved. >perhaps you should try to understand it nevertheless. > Understand _what_? The ordinary language that I use when I try to express my mathematical thoughts. >Despite this, the reals can be used as adequate models for a >huge range of natural objects. > That seems to be true, at least if we don¹t look at things at > a Þne enough scale. When we start talking about a very > tiny scale it¹s not at all clear to physicists what space and > time look like - they _could_ be something like the > reals, they could instead be discrete/quantized. Certainly correct. I think I said something vaguely related to this latter thought of yours in another part of the thread, though I was thinking of the cosmological end of the scale and the geometry thereof.. John. === Subject: Re: Peano¹s space-Þlling curve >> >message > >> I note some points exist that will never be traced by >any >>number of countable recursions. I also Þnd points that >are >>included more than once and topology tells me that I no >>longer have a structure that is homeomorphic with a line >>(0,1). Any wonder then that I am confused by your >assertions >>that a piece of abstract algebra can be used to show that >>neither of these concrete discoveries are correct. >> Nobody has denied either of those facts. >On the contrary, I have been told several times on this >newsgroup that the Peano construct (a) passes through every >point in the square and (b) it is an image of the [0,1] >continuum. That¹s not on the contrary. My best guess is that the reason for the confusion here is that you don¹t know what the word image means. A function has an image. By deÞnition, if f is a function with domain A then the image of f is just the set of all values f takes, ie {f(x) x is in A}. The Peano curve is indeed a function f with domain equal to [0,1] and image equal to the unit square - that does not say that it passes through each point exactly once, does not say that it looks like the unit interval, it just says that it passes through each point of the square at least once. >> Exactly where did I assert what you say I¹ve >> asserted? > You asserted that everything I had been told was proved. > >>perhaps you should try to understand it nevertheless. >> Understand _what_? >The ordinary language that I use when I try to express my >mathematical thoughts. > >>Despite this, the reals can be used as adequate models >for a >>huge range of natural objects. >> That seems to be true, at least if we don¹t look at things >> a Þne enough scale. When we start talking about a very >> tiny scale it¹s not at all clear to physicists what space >and >> time look like - they _could_ be something like the >> reals, they could instead be discrete/quantized. >Certainly correct. I think I said something vaguely related >to this latter thought of yours in another part of the >thread, though I was thinking of the cosmological end of the >scale and the geometry thereof.. >John. ************************ David C. Ullrich === Subject: Re: Peano¹s space-Þlling curve >> I note some points exist that will never be traced by > any >>number of countable recursions. I also Þnd points that > are >>included more than once and topology tells me that I no >>longer have a structure that is homeomorphic with a line >>(0,1). Any wonder then that I am confused by your > assertions >>that a piece of abstract algebra can be used to show that >>neither of these concrete discoveries are correct. >> Nobody has denied either of those facts. > On the contrary, I have been told several times on this > newsgroup that the Peano construct (a) passes through every > point in the square and (b) it is an image of the [0,1] > continuum. Both of those statements are correct, and neither contradicts anything you said in that previous paragraph. The Peano curve is the uniform limit of a particular sequence of functions. Each function in the sequence fails to be a surjection, but the Peano curve itself is, nevertheless, a surjection. An image is not the same thing as a homeomorphic image. The unit square is not homeomorphic to the unit interval, but the Peano curve is a surjection from the unit interval onto the unit square. It takes more than a surjection to make a homeomorphism; a homeomorphism is a bicontinuous bijection. -- Dave Seaman Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Peano¹s space-Þlling curve >> > message > >> I note some points exist that will never be traced by > any >>number of countable recursions. I also Þnd points that > are >>included more than once and topology tells me that I no >>longer have a structure that is homeomorphic with a line >>(0,1). Any wonder then that I am confused by your > assertions >>that a piece of abstract algebra can be used to show that >>neither of these concrete discoveries are correct. >> Nobody has denied either of those facts. > On the contrary, I have been told several times on this > newsgroup that the Peano construct (a) passes through every > point in the square and (b) it is an image of the [0,1] > continuum. There is a continuous function f: [0,1] -> R with image [0,1] x [0,1]. What¹s your problem with that? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: is this group an ideal in Z? >Hello! >Let J be subgroup of Z generated by 2 primes p1 and p2. >(a,b) will mean (a*p1+b*p2), where a,b are natural numbers. >If c is natural, c*(a,b) is in J. >(a,b)+(c,d) is also from J. >So, is J an ideal? > This is confusing. > Is J the SUBGROUP of Z generated by 2 primes, or is it equal to the > set of all elements of the form a*p1 + b*p2 with a and b natural numbers? > Because the two things are not the same. > The smallest subgroup of Z that contains the primes p1 and p2 will be > either: > (a) all multiples of p1, if p1=p2; or > (b) all integers, if p1 <> p2. > So if J is meant to be a SUBGROUP, then you should note that it is NOT > equal to the set of all elements of the form a*p1 + b*p2 with a and b > natural numbers; rather, it is the set of all elements of the form > a*p1 + b*p2 with a and b INTEGERS (positive, negative, or zero). >is such group generated by 2 and 3 equal Z-{1} > No: the smallest subgroup generated by 2 and 3 is all of Z: 1 = 3-2 > must be in such a group. > If you mean, the collection of all elements of the form 2a+3b, with a > and b natural numbers, then this is equal either to the set of all > positive integers greater than 1 (if you accept 0 as a natural > number), or else is equal to the set {5} U {n : n an integer, n>6} if > you do not take 0 to be a natural number. It is also not even a subgroup. > An ideal must satisfy more than just being closed under sums and > absorbing multiplications: it must be a GROUP under addition, so it > must contain 0, and must contain x-y for any x and y in the set. One > way to check is to check all three of the following for a subset S of > a ring R: > (i) S is nonempty; > (ii) If x in S and r in R, then rx and xr are in S; > (iii) If x and y are in S, then x-y is in S. >an ideal of Z? > The set Z-{1} is certainly not an ideal: it is not even a subgroup: it > contains 3 and contains -2, but does not contain their sum. > The set of all elements of the form 2a+3b with a and b nonnegative is > also not an ideal (it is not a subgroup), and neither is it a subgroup > if we require a and b to be positive. If you allow them to be any > integers, then this set is all of Z, and so is the total ideal. > -- > It¹s not denial. I¹m just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu A couple of additional comments: As it happens, every subgroup of Z is an ideal. One way of looking at that is that multiplication by a positive element is just repeated addition and by a negative element, the negative of that. Every subgroup/ideal of Z is generated by a single element (its smallest positive element, unless the subgroup is {0}). And the subgroup generated by m and n is generated by the gcd of m and n and this is true for any number of elements. === Subject: NLS equation with Matlab Hi all, I have troubles with Þnding the solution of the NLS equation using Maltab. The equation is: -i * du/dt + d^2u/dx^2 + (|u|^2-1)*u=0 how can i recover Œu¹ ? I tried to use rk4 and central differences in space (x), but the solution doesn¹t seem to be correct. maybe I made a mistake in the combination of these but i can¹t Þnd it. === Subject: Re: NLS equation with Matlab > Hi all, > I have troubles with Þnding the solution of the NLS equation using Maltab. > The equation is: > -i * du/dt + d^2u/dx^2 + (|u|^2-1)*u=0 > how can i recover Œu¹ ? > I tried to use rk4 and central differences in space (x), but the solution > doesn¹t seem to be correct. > maybe I made a mistake in the combination of these but i can¹t Þnd it. Are you looking for a numerical solution? You need to consider the boundary conditions (x->inÞnity); presumably you want |u| -> 0 for x->inÞnity, for all times Œt¹. What are you initial conditions (i.e. t=0)? rk4 (I presume it is fourth order Runge Kutta) is Þne, but there are exact methods as well for the NLS. -Michael. === Subject: Re: NLS equation with Matlab Yes, I need a numerical solution. I consider periodic boundary conditions u(x+L,t)=u(x,t) and a soliton as an initial condition u(x,0) = 4em exp(-2i er x)sech(2em x) can you suggest exact methods for the NLS? is there a matlab code for it? > Hi all, > I have troubles with Þnding the solution of the NLS equation using > Maltab. > The equation is: > -i * du/dt + d^2u/dx^2 + (|u|^2-1)*u=0 > how can i recover Œu¹ ? > I tried to use rk4 and central differences in space (x), but the solution > doesn¹t seem to be correct. > maybe I made a mistake in the combination of these but i can¹t Þnd it. > Are you looking for a numerical solution? You need to consider the boundary > conditions (x->inÞnity); presumably you want |u| -> 0 for x->inÞnity, for > all times Œt¹. What are you initial conditions (i.e. t=0)? > rk4 (I presume it is fourth order Runge Kutta) is Þne, but there are exact > methods as well for the NLS. > -Michael. === Subject: Four-Color-Theorem I recently heard about the four-color-theorem and I wonder, why no one found a not-computational proof yet. Can anyone tell me some good literature about this topic- maybe some attempts for a proof and why they didn¹t work out. The four color theorem is valid for every 2-dimensional map/surface. In a map colored with 4 colors only, color is surrounded by max. 3 other colors. Seen from the color max. 3 colors are necessary to Þll any region of neighboring Þelds. Neighbors are deÞned as Þelds with a common border, a common junction is not enough. Neighbors cannot have the same color. If the number of neigbors is even, only 2 colors are needed for neighbors. Fields can be set/removed freely at the map. A Þeld can have at least 3 different neighboring colors. Therefore four colors are needed. If this special Þeld does no longer exist, it turns into a junction point of three Þelds. Two neighboring Þelds can only have two colors as common neighbors. And three neighboring Þelds can only have only special color as common neighbor. If they have a common neighbor (on the junction or on the outside) this means that the this neighbor is limited by the others. Either it has no other neighbor (if its on the junction) or it is surrounding the other three countries completely. This is a property of the 2-dimensionality. A conjunction is deÞned by at least 3 neighbors, a border by two. If that would be different, the four-color-theorem wouldn¹t be true. It doesn¹t matter how big the map is or how Þelds form a junction. If you have a map, the end of the paper is surrounding the map completely, that means, that the outer ring of Þeld of map should be colored by three colors only. The fourth color is the and of the sheet and therefor it is valid for every 2-dimensional space (e.g. sphere) with or without borders. Looking forward to your answers! Alexandra === Subject: Re: Four-Color-Theorem > I recently heard about the four-color-theorem and > I wonder, why no one > found a not-computational proof yet. Can anyone > tell me some good literature > about this topic- maybe some attempts for a proof > and why they didn¹t work out. > The four color theorem is valid for every > 2-dimensional map/surface. > In a map colored with 4 colors only, color is > surrounded by max. 3 other colors. > Seen from the color max. 3 colors are necessary to > Þll any region of neighboring > Þelds. Neighbors are deÞned as Þelds with a > common border, a common junction > is not enough. Neighbors cannot have the same > color. If the number of neigbors > is even, only 2 colors are needed for neighbors. > Fields can be set/removed freely > at the map. A Þeld can have at least 3 different > neighboring colors. Therefore four > colors are needed. > If this special Þeld does no longer exist, it > turns into a junction point of three Þelds. > Two neighboring Þelds can only have two colors as > common neighbors. And three > neighboring Þelds can only have only special > color as common neighbor. If > they have a common neighbor (on the junction or on > the outside) this means that the > this neighbor is limited by the others. Either it > has no other neighbor (if its on the > junction) or it is surrounding the other three > countries completely. > This is a property of the 2-dimensionality. A > conjunction is deÞned by at least 3 > neighbors, a border by two. If that would be > different, the four-color-theorem > wouldn¹t be true. > It doesn¹t matter how big the map is or how Þelds > form a junction. > If you have a map, the end of the paper is > surrounding the map completely, > that means, that the outer ring of Þeld of map > should be colored by three colors only. > The fourth color is the and of the sheet and > therefor it is valid for every 2-dimensional > space (e.g. sphere) with or without borders. Did you intend the collection of statements above to be a proof? === Subject: Re: Two traits common to all masses snip > For matter in bulk the same comment applies. Mass equals inertia, not > amount of Œstuff¹. > Inertia, i.e. resistance to acceleration is a measure of the amount of > stuff i.e. the material substance, or mass that makes up an object; > body, or mass of matter. It¹s what the customers want. The more for > the money, the merrier(:-) An object¹s inertia is a measure of an object¹s inertia and that¹s all! Concepts like Œamount of stuff¹, or Œquantity of matter¹ are illusions of the senses; useful in everyday life but less useful when contemplating the nature of matter at the fundamental level. i.e a bunch of point/string-like TG === Subject: Re: Two traits common to all masses > Hey Thomas! It¹s Œquantity of matter¹ that _I¹m_ talking about: > Merchants and their customers are concerned with quantities of matter, > including length, area and volume; not whether or not it¹s called > mass, inertia or weight. Mass (fundamental) http://scienceworld.wolfram.com/physics/Mass.html Weight (derived) http://scienceworld.wolfram.com/physics/Weight.html === Subject: Re: Two traits common to all masses >> These two traits are common to all masses of matter; by which we >> measure their quantity: >> #1) All masses have three dimensional extent so that we can use their >> linear length; their area, or their bulk, as measures of their >> quantity. > Hey Dumb Donny Head, what is the mass of a Menger-Sierpinski > sponge? It has an arbitrarily large physical extent in all dimensions > but asymptotically zero mass overall. >> #2) All matter has inertia: > Hey Dumb Donny Head, tell us the differences among inertial mass, > gravitational mass, active mass, and passive mass, you ing > imbecile. > [snip idiocy] Uncle Al, take a hike!!! if you want to swear at someone, swear at them using email, this is a physics newsgroup. Do you mind? -- Dodik === Subject: Mirrored S curve Thinking of the logistic function that plots an S curve -- y=1/(1+exp(-x)) -- as a nice way to model new technology adoption, is there a function that turns the curve down again, corresponding to the decline phase of a technology¹s life? A mirror image would be perfect. I¹ve considering a sine curve and a normal distribution but they lack the particular shape of the logistic function that makes it a good match for this type of modelling. SpeciÞcally, a sine curve does not vary enough and normal distribution peaks too sharply. Hope this makes sense and TIA. === Subject: Re: Mirrored S curve >Thinking of the logistic function that plots an S curve -- >y=1/(1+exp(-x)) -- as a nice way to model new technology adoption, is >there a function that turns the curve down again, corresponding to the >decline phase of a technology¹s life? A mirror image would be >perfect. I¹ve considering a sine curve and a normal distribution but >they lack the particular shape of the logistic function that makes it >a good match for this type of modelling. SpeciÞcally, a sine curve >does not vary enough and normal distribution peaks too sharply. Hope >this makes sense and TIA. y = 1 / (1+x^2) ? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Mirrored S curve >Thinking of the logistic function that plots an S curve -- >y=1/(1+exp(-x)) -- as a nice way to model new technology adoption, is >there a function that turns the curve down again, corresponding to the >decline phase of a technology¹s life? A mirror image would be >perfect. I¹ve considering a sine curve and a normal distribution but >they lack the particular shape of the logistic function that makes it >a good match for this type of modelling. SpeciÞcally, a sine curve >does not vary enough and normal distribution peaks too sharply. Hope >this makes sense and TIA. You might try the difference of two such curves, one translated from the other: y = 10/(1+exp(-(x-5)))-10/(1+exp(-(x-25))) where you can put a, b, c in for the 5, 25, and 10 and also a decay factor k in front of the negative (x - a) and (x - b) terms. I have just chosen some values that look nice. --Lynn === Subject: AI Theory [Was: ...] XPident: uj797 [...] > I wonder: any of his other theories close to the mark? > Were his theories consistent? Or did he spew enough > different theories that it¹s not too surprising if > one of them was correct in some detail? http://mentifex.virtualentity.com/theory5.html is the theory of mind. Arthur T. Murray -- http://doi.acm.org/10.1145/307824.307853 -- ACM SIGPLAN/Mind.Forth http://mentifex.virtualentity.com/mentifex_faq.html -- by mentifex http://osgaming.net/Downloads/Documentation/ OSReferenceBook.pdf AI http://www.amazon.com/exec/obidos/ASIN/0595654371/ -- AI4U AI Text === Subject: Re: AI Theory 2kpKShAQELm5uc578GgAAAJ5SURBVHichdNBb9sgFABgvMblarSwXDMcKdfUps o1td1xDWqfc020 mV5JoHt/f0Dc1pU2jUskvrz3eA9D6n8s8h+ QnVSzv0G3zUDPlOzqut3N2g9YcwDQHgm1zJj9O8gc IshWIu+ NMe4NEICFGKdoySHArzd4AI0ZQLNyZR7hNIJ8gL3YcLiIRuQQcg3te8S+ CfVdpa4Rx8UI Z9DUcuiEXKcax8OkOAunEuUyZjJHN0IR24D9jlCVjjuMxbcm9ZGYh31zGOHR7C PkwILFiPYKsjdD PsCCg14AfDPmZRzJXWyp8GueQwH8bMzXK8jbmNbsFQ8hALSPo4qwTfv9M4YT6F 7j7RvETGbgM5vp DPge1XmEHymi8NZqVrCiaB5vrpBKDNhYZEwznTWv8yvEg5sXL5QPY8kY8w+ YoEuZ5nVlfRYiWOEs SQ1uUqaLKK0jnDG28K+nBPepCe9q1RGGXFN/Zgl+h/ 8PmsxqidnCEkZbe0gQDhUGW9C6tCQstF7p +JWECRrdE+pljZYUAXCXPh8ZMjGwrkKPWQCF+ JpgF8fB81mzDVUpUmkxTxBGOMADLLxqS9Uhlkrd pouKt2QQTtWqEbWizarD/pQ6fzRD/ yXvDyKsuq065ccbVP3QPy9hmJetRSSsVebpOsTfpj8pgIEU GSuyU931Hxc1KGS6VJY03wPQ8UXF8iqcVoiyEuvj5Knl5qTEmjax/ PZ5Alt29GJFLmF/tZxPH+fd S8hjGyGVWz5N4f5gK7HzXWfd5ucnoOoidliV2H6qUZ9R3ohNqI7N1k3Bo3Chuh ey2vgpzGh4lSv0 ohQKp7BwAUrESohOuQngtbf4s9qNIX8At0MbK5iXDI4AAAAASUVORK5CYII= Discussion, linux) > [...] >> I wonder: any of his other theories close to the mark? >> Were his theories consistent? Or did he spew enough >> different theories that it¹s not too surprising if >> one of them was correct in some detail? > http://mentifex.virtualentity.com/theory5.html is the theory of mind. I was interested in your theories about the Unabomber prior to the arrest of Kaczynski. -- [T]here¹s no point in telling any of you what mathematicians I¹m in email contact with, just like there¹s no point in going into detail about my contacts in a major news organization. [...] [P]olite disinterest is what I¹ve found. --JSH on his important contacts. === by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SDtUG14028; >I am looking for the distribution or at least the mean of the largest >eigenvalue of covariance matrix A, where A follows a Wishart >Mike There is Wigner¹s Semicircle Law: http://mathworld.wolfram.com/WignersSemicircleLaw.html Is that helpful? Don Coppersmith === >>I am looking for the distribution or at least the mean of the largest >>eigenvalue of covariance matrix A, where A follows a Wishart >>Mike >There is Wigner¹s Semicircle Law: >http://mathworld.wolfram.com/WignersSemicircleLaw.html I believe the joint distribution of all of the characteristic values of the Wishart matrix with the covariance matrix being the identity can be found in any good mathematical multivariate book. Wigner¹s Semicircle Law is not for Wishart matrices. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: I Need Help With This Equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SDtTt13994; Stick on inverses. We may write 1/(x^y) = 1/x +1/y ,except x^0 = x, 0^y=y . Thence x^y = y^x ,x^(-x)=0 ;we also obtain: 1/(x^(y^z))= 1/x + 1/(y^z) = 1/x + 1/y +1/z =1/((x^y)^z);and x^z > z if x>0 , x^z < z if x<0. Your last formula implies by symmetry g(z)=c/sqrt(1-z^2) ? Once more:does that help? ALAIN. === Subject: Sequence for experts, this was never solved before by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SDtVU14070; Find the next number of this sequence, and explain why ;) 2 4 4 6 5 4 4 4 4 === Subject: Re: Sequence for experts, this was never solved before > Find the next number of this sequence, and explain why ;) > 2 4 4 6 5 4 4 4 4 A057697 ;-) Dirk Vdm === Subject: Re: Proof of twin-prime conjecture? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SDtUd14051; >Arenstorf has uploaded an MS purporting to prove the twin >prime conjecture to the arXiv: >http:// uk.arxiv.org/abs/math .NT/0405509 . >This does not look completely stupid, in fact >(whether or not it is all correct) it appears to be >a substantial piece of analytic number theory. I have started reading it. The general approach seems as if it should work. Indeed it might even extend to lambda(n-1)lambda(n+1)lambda(n+5) etc. i.e. to 3 tuples or even k-tuples by suitably modifying T(s). I am not fully competent to judge all the details (although I do understand a fair bit of it; I am reading it now). The people to ask are A. Granville, A, Odlyzko, H. Montgomery, etc. Even if some details are wrong, they might be correctable. It is seemingly a substantial piece of work. === Subject: Re: On the partial sums of reciprocals of primes by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SDtWJ14116; >>In Prime Numbers: A computational Perspective by Crandall and >> Pomerance >>you will Þnd this fact on page 32. Their Theorem 1.4.2 (Mertens) >> states >>that: >> >>As x -> inf, prod_{p le x}(1 - 1/p) ~ e^{-gamma}/ln(x) >>where gamma is the Euler constant. Taking the logarithm of this >> relation, we >>have, >>sum_{p le x}{1/p} = ln(ln(x)) + B + o(1), >>for the Mertens constant B deÞned as >>B = gamma + sum_{p}{(ln(1 - 1/p) + 1/p) >> A lot of people say that the proof of >>sum_{p le x}{1/p} = ln(ln(x)) + B + o(1), >> can be found in Hardy and Wright. I know that. Please note that >> I said >> S(x)< lnlnx + B + 1/(logx)^2. >> The point is the explicit formula for the upper bound. I think this >> inequality is very stronger than the asymptotic formula. >If S(x) = log log x + B + o(1) then for some C it¹s also >true that S(x) < log log x + C. That sufÞces to make it >true that S(x) < log log x + constant + 1/(log x)^2; the >last term is irrelevant, isn¹t it? >If the result you were interested in said, say, that >log log x + B < S(x) < log log x + B + 1/(log x)^2 >(same B on both sides) then that would indeed be much >stronger than the result proved in H&W. >-- >Gareth McCaughan >sig under construc >log log x + B < S(x) < log log x + B + 1/(log x)^2 I have thought of it once. But this can not be possible, since 1/(logx)^2 goes to zero much more rapidly than O(1/logx) erdos fan === Subject: Re: Sigma-ring proof > > OK. Can you please indicate some subsets of R^k that are left out > > of this smallest sigma-ring, which would be included in, say, the > > power set? Clearly all open and all closed subsets are included in B, > > so that would leave those which are neither open nor closed. > > > > Well, any non-measurable subset of R^k would do ... let¹s take k = 1, the > classical example is that of the set of the classes of the equivalence > relation: xRy <=> (x-y) is rational. You¹ll Þnd a proof in any book. > > Number of Borel sets in R is c. Since the number of open subsets of R is uncountable, what exactly is c? > Number of all subsets of R is 2^c. > I guess a few sets are non-Borel, then. > In particular, AC is not required to prove the existence of non-Borel > sets. OK, but it would be helpful to know what some of those non-Borel === Subject: Re: Sigma-ring proof > > > OK. Can you please indicate some subsets of R^k that are left out > > of this smallest sigma-ring, which would be included in, say, the > > power set? Clearly all open and all closed subsets are included in B, > > so that would leave those which are neither open nor closed. > > > > > > Well, any non-measurable subset of R^k would do ... let¹s take k = 1, the > > classical example is that of the set of the classes of the equivalence > > relation: xRy <=> (x-y) is rational. You¹ll Þnd a proof in any book. > > > > Number of Borel sets in R is c. > Since the number of open subsets of R is uncountable, what exactly is > c? c = 2^(aleph_0) = cardinal of the continuum > Number of all subsets of R is 2^c. > I guess a few sets are non-Borel, then. > In particular, AC is not required to prove the existence of non-Borel > sets. > OK, but it would be helpful to know what some of those non-Borel === Subject: Re: Sigma-ring proof > OK. Can you please indicate some subsets of R^k that are left out > of this smallest sigma-ring, which would be included in, say, the > power set? Clearly all open and all closed subsets are included in B, > so that would leave those which are neither open nor closed. > Well, any non-measurable subset of R^k would do ... let¹s take k = 1, the > classical example is that of the set of the classes of the equivalence > relation: xRy <=> (x-y) is rational. You¹ll Þnd a proof in any book. === Subject: Mass is a quantity of matter; inertia is its measure A quantity of matter is not a fundamental measure; the measures of the mass in a quantity of matter are the net force [f] exerted on, and/or by it, divided by the acceleration [a] that is caused; which is equal to the ratio of its weight [w], divided by the acceleration of free fall [g] at the location of the scale on which it is weighed: For any quantity or mass of matter, anywhere, anytime: These _ratios_ are Constants; called inertia [f/a = w/g]; which may be substituted for any quantity or mass of matter, anywhere, anytime. The equations: f = ma, or w = mg are strawmen: f = [w/g]a, and w = [f/a]g! Wake up and get a good breath of truth. === Subject: Re: Mass is a quantity of matter; inertia is its measure Shead, off and die. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Mass is a quantity of matter; inertia is its measure > Shead, off and die. And Gene Nygaard needs to escort him. === Subject: Re: Mass is a quantity of matter; inertia is its measure Inertia http://scienceworld.wolfram.com/physics/Inertia.html Weight http://scienceworld.wolfram.com/physics/Weight.html Mass http://scienceworld.wolfram.com/physics/Mass.html The time has come, the Walrus said, To talk of many things: Of shoes, and ships, and sealing wax, of cabbages and kings. And why the sea is boiling hot and whether pigs have wings. Makes about as much sense as Shead eternal struggle with inertia, weight and mass. === Subject: Re: Mass is a quantity of matter; inertia is its measure >Inertia > http://scienceworld.wolfram.com/physics/Inertia.html >Weight > http://scienceworld.wolfram.com/physics/Weight.html >Mass > http://scienceworld.wolfram.com/physics/Mass.html Note that the pound is not a unit of mass, but rather one of weight. http://scienceworld.wolfram.com/physics/ AvoirdupoisSystemofUnits.html The system of masses used in the United States for most common commercial purposes based on the pound-mass and ounce and summarized in the following table. Eric Weisstein, the author of the pages you cite, struggles with simple facts nearly as much as Shead does. Gene Nygaard http://ourworld.compuserve.com/homepages/Gene_Nygaard/ === Subject: Re: Panu Raatikainen¹s review of two of Chaitin¹s books. > > > Contrary to Chaitin¹s assertion : > > The complexity of a Þnite sequence of numbers can be mesured by the > > length of the minimum program that reproduce it. (Randomness and > > Mathematical Proof) > > That is utterly false. > > Actually, this is not Chaitin¹s assertion, but the basic deÞnition > of Kolmogorov complexity. You may or may not Þnd it satisfactory, but > that¹s what complexity means in the theory. Chaitin¹s assertion about > a dependence between the complexity of a theory and its power to prove > theorems is a different matter altogether. > thoughtful response. Eray and Torkel forgets that the deÞnitions are subject to laws. To the Aristotelian dictum: Proximal genus and speciÞc difference If the periodic sequence produced by the iteration of: x = t + sin(5*y) y = t - cos(2*x) Have the same complexity that the almost chaotic sequence produced by: x = t + sin(5*x) y = t - cos(2*y) Then the same deÞntion is applied to two differents specimens. L. Rodriguez === Subject: Re: Panu Raatikainen¹s review of two of Chaitin¹s books. > No, no, no. That the value of Omega is such and such can certainly be > a mathematical fact. I don¹t dispute that. It is that this fact is > true for no reason that is dubious. At the least it requires > explanation and argument. (Note: I¹m not saying that all mathematical > facts are true for some reason either. I haven¹t a clue what either > claim would mean.) You seem to misunderstand his argument. The value of Omega alone is not enough to assert what Chaitin does. He also takes into account that it comes up in basic number theory. By for no reason, he seems to be referring to the fact that Omega is a random real. Otherwise, there is an uncountably inÞnite number of random reals. The argument is not only that it is a random real, there are many such numbers. -- Eray === Subject: Re: Panu Raatikainen¹s review of two of Chaitin¹s books. <87vÞmo2t2.fsf@phiwumbda.org> <87wu2zmbzi.fsf@phiwumbda.org> <87zn7ul8x8.fsf@phiwumbda.org> 2kpKShAQELm5uc578GgAAAJ5SURBVHichdNBb9sgFABgvMblarSwXDMcKdfUps o1td1xDWqfc020 mV5JoHt/f0Dc1pU2jUskvrz3eA9D6n8s8h+ QnVSzv0G3zUDPlOzqut3N2g9YcwDQHgm1zJj9O8gc IshWIu+ NMe4NEICFGKdoySHArzd4AI0ZQLNyZR7hNIJ8gL3YcLiIRuQQcg3te8S+ CfVdpa4Rx8UI Z9DUcuiEXKcax8OkOAunEuUyZjJHN0IR24D9jlCVjjuMxbcm9ZGYh31zGOHR7C PkwILFiPYKsjdD PsCCg14AfDPmZRzJXWyp8GueQwH8bMzXK8jbmNbsFQ8hALSPo4qwTfv9M4YT6F 7j7RvETGbgM5vp DPge1XmEHymi8NZqVrCiaB5vrpBKDNhYZEwznTWv8yvEg5sXL5QPY8kY8w+ YoEuZ5nVlfRYiWOEs SQ1uUqaLKK0jnDG28K+nBPepCe9q1RGGXFN/Zgl+h/ 8PmsxqidnCEkZbe0gQDhUGW9C6tCQstF7p +JWECRrdE+pljZYUAXCXPh8ZMjGwrkKPWQCF+ JpgF8fB81mzDVUpUmkxTxBGOMADLLxqS9Uhlkrd pouKt2QQTtWqEbWizarD/pQ6fzRD/ yXvDyKsuq065ccbVP3QPy9hmJetRSSsVebpOsTfpj8pgIEU GSuyU931Hxc1KGS6VJY03wPQ8UXF8iqcVoiyEuvj5Knl5qTEmjax/ PZ5Alt29GJFLmF/tZxPH+fd S8hjGyGVWz5N4f5gK7HzXWfd5ucnoOoidliV2H6qUZ9R3ohNqI7N1k3Bo3Chuh ey2vgpzGh4lSv0 ohQKp7BwAUrESohOuQngtbf4s9qNIX8At0MbK5iXDI4AAAAASUVORK5CYII= Discussion, linux) >> No, no, no. That the value of Omega is such and such can certainly be >> a mathematical fact. I don¹t dispute that. It is that this fact is >> true for no reason that is dubious. At the least it requires >> explanation and argument. (Note: I¹m not saying that all mathematical >> facts are true for some reason either. I haven¹t a clue what either >> claim would mean.) > You seem to misunderstand his argument. > The value of Omega alone is not enough to assert what Chaitin does. He > also takes into account that it comes up in basic number theory. > By for no reason, he seems to be referring to the fact that Omega is > a random real. You seem to misunderstand my point. Remarkably. The fact that Omega is random has no apparent connection to the claim that it is true for no apparent reason. Or at least you have to explain what the connection is. I¹m not arguing that there are no random reals. I¹m asking why the fact that there are random real numbers (which are solutions to Diophantine equations or whatever) imply that some mathematical facts are true for no reason. > Otherwise, there is an uncountably inÞnite number of random reals. > The argument is not only that it is a random real, there are many such > numbers. I¹m not disputing that. -- Quincy, would you rather do epistemology or conceptual analysis? You know what? I¹d rather Þght on an aircraft carrier.... And Mama and Baba (Papa) would Þght on an aircraft carrier, too. -- Quincy P. Hughes, age 3 1/2 === Subject: Re: Panu Raatikainen¹s review of two of Chaitin¹s books. | As emphasized by Raatikainen, to claim that statements about the |bits of Omega are true by accident or that there is no reason for |their truth is to claim that statements of the form the Turing |machine T eventually halts are true by accident or for no reason. |It is entirely obscure what this claim is supposed to mean or on what |it is based. There have been various attempts to analyze the notion of explanation, or cause in mathematics and science, which I don¹t think can be said to be entirely successful. It¹s true, however, that typically the reason why something is true is supposed to be prior, more basic, or simpler somehow than what it¹s supposed to exaplain. I think the reasoning here may be roughly that the Þrst N bits of Omega have no explanation because there¹s no (signiÞcantly) simpler set of facts from which they can be deduced. An explanation ordinarily requires having some kind of pattern which implies or at least makes likely the fact being explained, but here we have a collection of facts in which computable patterns are lacking. I¹m not sure I would consider it a good thing to discourage the kind of speculative thinking Chaitin engages in, but in a sense I wish it could be done in a different sort of medium, one which was less prone to spotlight a small number of people whose speculations had achieved this kind of fame. Keith Ramsay === Subject: Re: Panu Raatikainen¹s review of two of Chaitin¹s books. > I¹m not sure I would consider it a good thing to discourage the kind > of speculative thinking Chaitin engages in, [...] The problem with Chaitin¹s remarks about Omega and accidental truth is not that they are speculative, but that they have no apparent content. You suggest that > the reasoning here may be roughly that the Þrst N bits of Omega have no > explanation because there¹s no (signiÞcantly) simpler set of facts > from which they can be deduced. An explanation ordinarily requires > having some kind of pattern which implies or at least makes likely > the fact being explained, but here we have a collection of facts in > which computable patterns are lacking. In this sense, then, any Þnite Kolmogorov-random string has no explanation. To add that statements of the form the i:th bit of s is 0, for some mathematically deÞned random s, are true by accident or for no reason is at best a highly misleading way of saying that there is no pattern in the bits of the string. We can contrast Chaitin¹s obfuscatory twaddle about true by accident with other uses of the concepts of Kolmogorov complexity to elucidate matters pertaining to randomness and explanation in qualitative terms. Alexander Shen, who takes the view that complexity considerations should not be considered as something really practical, they can only give some hints for real applications and motivate our decisions gives the following example (I¹m quoting his paper Discussion on Kolmogorov Complexity and Statistical Analysis): There is a general rule saying that a statistical hypothesis P could be rejected if P assigns a small probability to a simple set T containing the data string x. We can support this by proving an inequality saying that the randomness deÞciency in x with respect to P is at least log (1/P(T))-K(T), so if P(T) is small and T is simple, the randomness deÞciency is large and the hypothesis P does not explain the data string x. === Subject: Re: Panu Raatikainen¹s review of two of Chaitin¹s books. >> I¹m not sure I would consider it a good thing to discourage the kind >> of speculative thinking Chaitin engages in, [...] > The problem with Chaitin¹s remarks about Omega and accidental truth > is not that they are speculative, but that they have no apparent > content. You suggest that > > the reasoning here may be roughly that the Þrst N bits of Omega > have no > explanation because there¹s no (signiÞcantly) simpler > set of facts > from which they can be deduced. Mostly, I disagree strongly with this interpretation. The problem is not one of explanation, but of value. The fact that the 7th decimal of pi is a 6 may be said to have no simple explanation in that sense (it is certainly simpler that a proof that pi is between 3.1415926 and 3.145927). But the fact is that the 100th (even the 5th one, i believe) decimal of Omega is *for ever* inattainable : ie. I can take as an axiom that it is a 3, *and there is a proof* that I will never be contradicted !! An explanation > ordinarily requires > having some kind of pattern which implies or > at least makes likely > the fact being explained, but here we have > a collection of facts in > which computable patterns are lacking. > In this sense, then, any Þnite Kolmogorov-random string has no > explanation. To add that statements of the form the i:th bit of s is > 0, for some mathematically deÞned random s, are true by accident > or for no reason is at best a highly misleading way of saying that > there is no pattern in the bits of the string. > We can contrast Chaitin¹s obfuscatory twaddle about true by > accident with other uses of the concepts of Kolmogorov complexity to > elucidate matters pertaining to randomness and explanation in > qualitative terms. Alexander Shen, who takes the view that complexity > considerations should not be considered as something really practical, > they can only give some hints for real applications and motivate our > decisions gives the following example (I¹m quoting his paper > Discussion on Kolmogorov Complexity and Statistical Analysis): There > is a general rule saying that a statistical hypothesis P could be > rejected if P assigns a small probability to a simple set T containing > the data string x. We can support this by proving an inequality saying > that the randomness deÞciency in x with respect to P is at least log > (1/P(T))-K(T), so if P(T) is small and T is simple, the randomness > deÞciency is large and the hypothesis P does not explain the data > string x. === Subject: Re: Panu Raatikainen¹s review of two of Chaitin¹s books. > The fact that the 7th decimal of pi is a 6 may > be said to have no simple explanation in that sense (it is certainly simpler > that a proof that pi is between 3.1415926 and 3.145927). But the fact is > that the 100th (even the 5th one, i believe) decimal of Omega is *for ever* > inattainable : ie. I can take as an axiom that it is a 3, *and there is a > proof* that I will never be contradicted !! No there isn¹t. What is provable is that in any consistent effectively axiomatizable theory T, only Þnitely many decimals of an Omega number can be determined - speciÞcally, Solovay showed that for a suitable Omega number, no decimal at all can be determined in ZF. This general phenomenon has been known since Post: for any set A which is simple, in the recursion-theoretic sense, and any consistent effectively axiomatizable theory T, only Þnitely many of the inÞnitely many true statements of the form k is not in A are provable in T. As always when we are talking about incompleteness phenomena, it is an open question whether a particular problem can be settled in an extension of T that we can recognize as correct. How would you say that being true by accident or being true for no reason is connected with this incompleteness phenomenon? === Subject: Re: Polynomial system apparently for champs... > I have now been working for a couple of days on the following system without Þnding any real clue... > Could someone give me a help? > If there wouldn¹t be any mathematical way to solve (we never know), could someone detail me how to solve it programmatically? > Didier > -x + y + z + 2A sin(y-z) = C > -x + y - z + 2A sin(x-z) = C > x + y - z + 2A sin(x-y) = C > where A, C are constant and A, x, y, z are [B][U]not equal [/U] [/B] to 0 Are you sure you have the right sign in the second equation? It looks like it should be Œ-C¹ instead of ŒC¹. Anyway, subtracting the Þrst equation from the last eliminates C and gives: A * [sin(y-z) - sin(x-y)] = x-z This looks very much like a transcendental equation without a closed form solution. I suggest you simplify the equations by introducing a = x-y and b=x-z. Then y-z = b-a, and you only have two unknowns (a and b) instead of three. === Subject: Re: Old Mac-based graphing calculator from Harvard, circa 1998? > Does anyone have a copy of a graphing calculator application from > Harvard University, circa 1989? It was standard issue for Math 21a, > Multivariable Calculus. > It may be the one named simply Graphing Calculator, > which has since gone commercial, > in which case see > http://www.paciÞct.com/ Unfortunately, it¹s not this one. I emailed the author(s), and their app derives from a Stanford Physics app, circa 1987. Did anyone else here take Math 21a at Harvard? === Subject: Comparing m x n matrices I want to compare two m x n matrices whose elements are rational numbers. What I want to determine is whether one matrix can be converted to the other by means of elementary row and/or column operations. I¹m thinking of reducing both to normalized row-echelon form and checking for identity, but this only takes into account the row operations. Should I also Þrst reduce to normalized column-echelon form, or is there an easier way to do the comparison? I will need to know the sequence of elementary operations which convert one matrix to the other, if such exists. For example, (-1 -1 1 0) and (-1 1 0 -1) can be inter-converted by permuting the columns. ( 2 0 0 1) ( 0 0 1 2) (1 1 -1 0) can be obtained from the Þrst matrix above by elementary row operations. (0 1 -1 1/2) (-2 -1 1 0) cannot be obtained from either matrix by any elementary operations ( 2 0 0 1) === Subject: a+b+c= Pi and cos(a)+cos(b)+cos(c) < = 3/2 Hi all. I know that it is a very basic problem, but i have to explain it to a 14 years old friend. I can¹t use calculus. But I have somme problems doing that just with trigonometry. May be you can help me! I am trying to show that if a,b and c are the angles of a triangle, then cos(a)+cos(b)+cos(c) < = 3/2 just using trigonometry (with calculus is easy). Do you have a hint? === Subject: Re: a+b+c= Pi and cos(a)+cos(b)+cos(c) < = 3/2 schrieb Jesus Rogelio Perez Buendia : > I know that it is a very basic problem, but i have to explain it > to a 14 years old friend. I can¹t use calculus. But I have somme > problems doing that just with trigonometry. May be you can help > me! > I am trying to show that if a,b and c are the angles of a > triangle, then cos(a)+cos(b)+cos(c) < = 3/2 just using > trigonometry (with calculus is easy). Use cos(a)+cos(b) = 2 * s * cos[(a-b)/2] <= 2 * |s| and cos(c) = 1 - 2 * s^2 with the abbreviation s := sin(c/2) All you have to do then is to maximize a quadratic function in |s|. BTW: I think, I¹ve senn a elementary-geometric proof for this inequality, unfortunately however I cannot recall it... === Subject: Quick refresh please I¹m reverse engineering a program and its been so long since I¹ve taken any math, I¹m having trouble remembering how to resolve a function given the points on a curved line. Here are the points. A quick reminder would be appreciated! 1 7619 2 6196 3 4773 4 3350 5 1928 6 1682 7 1186 8 590 9 420 10 299 11 253 12 207 13 162 14 93 15 25 16 -43 17 -112 18 -181 19 -249 20 -386 === Subject: Re: Quick refresh please >I¹m reverse engineering a program and its been so long since I¹ve >taken any math, I¹m having trouble remembering how to resolve a >function given the points on a curved line. Here are the points. A >quick reminder would be appreciated! [20 data points omitted] You need to decide what kind of curved line you want. You could Þt the twenty points exactly to a 19th-degree polynomial by plugging in and solving the linear system for the unknown coefÞcents. If you have another parametric form in mind, use regression (i.e., least squares). -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Quick refresh please > I¹m reverse engineering a program and its been so long since I¹ve > taken any math, I¹m having trouble remembering how to resolve a > function given the points on a curved line. By resolve a function I assume you mean you want to know a function which will produce these points; i e, determine y(x) for x=1,2,... and y=7619,6196,... in your example. The Þrst step is to Þnd the form of the function, which is most easily done by graphing the points you have. If the graph is a straight line, for example, y = ax + b is appropriate. If the graph is a curve, you try to Þgure out from its appearance whether it¹s a circle, parabola, etc. Once you have assumed a form, you can Þt for the parameters (e g, a and b in y = ax + b) using least squares. However, since you are dealing with a program and apparently there are only 20 points, you could just use a lookup table to produce exactly the desired output. === Subject: Re: Quick refresh please > I¹m reverse engineering a program and its been so long since I¹ve > taken any math, I¹m having trouble remembering how to resolve a > function given the points on a curved line. By resolve a function I assume you mean you want to know a function which will produce these points; i e, determine y(x) for x=1,2,... and y=7619,6196,... in your example. The Þrst step is to Þnd the form of the function, which is most easily done by graphing the points you have. If the graph is a straight line, for example, y = ax + b is appropriate. If the graph is a curve, you try to Þgure out from its appearance whether it¹s a circle, parabola, etc. Once you have assumed a form, you can Þt for the parameters (e g, a and b in y = ax + b) using least squares. However, since you are dealing with a program and apparently there are only 20 points, you could just use a lookup table to produce exactly the desired output. === Subject: Re: Sequence for experts, this was never solved before by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SGcer30274; >Find the next number of this sequence, and explain why ;) >2 4 4 6 5 4 4 4 4 That¹s easy. Those are Hyun Soon Park¹s golÞng round number and then golÞng scores, by hole, during the 2nd round of the 2002 CJ Nine Bridges Classic tournament. Round 2: 4 4 6 5 4 4 4 4 5 5 4 4 3 4 5 5 3 6 So the next number is a 5. === Subject: Re: Sum of two approximation polynomials question. >If one approximates two curves, each by a cubic (Bezier) polynomial >then does adding the two cubic Bezier approximations give a polynomial >that approximates the sum of the two true curves. To be a bit more >speciÞc, if I directly derive a cubic Bezier from the sum of the two >true curves and compare that to the sum of two cubic Bezier each >derived from one of the two true curves, will these two approximations >be the same? A cubic Bezier curve is represented parametrically as P(t) = A (1-t)^3 + 3 B t (1-t)^2 + 3 C t^2 (1-t) + D t^3, 0 <= t <= 1. This could be used to approximate a curve X(t) where X(0) = A, X(1) = D, X¹(0) = 3(B-A) and X¹(1) = 3(D-C). I assume that¹s how you derive the Bezier curve from the true curve. So... if X_1 is approximated in this sense by P_1, and X_2 by P_2, what do you suppose would be the A,B,C,D for X_1 + X_2? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Design in n-dimensional space Richard Hamming, in The Art of Doing Science and Engineering observed that even though we build three-dimensional objects their _design_ is in a high-dimensional space. One dimension for each parameter that is considered during the design. This seems correct to me. Thus it seems reasonable one should have a good understanding for higher dimensional space. Then he talks a little about the volume of a sphere as a measue of a piece of restriced space. He shows the volume of the unit sphere tends to zero as the number of dimensions increase. He shows most of the volume of a sphere is close to the surface, the more so the higher dimension we got. He talks about the four circle paradox. But what does this really mean? What concrete example can be given when it is valuable to have this understanding in the design process? What would be a typical trap for someone without the understanding of the properties of higher dimensional spaces? I believe I understand the philosophical aspect of this, but I don¹t see how to put it into use. Maybe some smart mathematicians can help me? === Subject: Re: Great-circle radius of ellipsoid >> Okay, but what is the signiÞcance of this object and how does it >> solve the problem the OP originally described? > What¹s not clear? >> Primarily what the problem is, and why your solution solves this >> problem and why mine doesn¹t (assuming it doesn¹t). >> The object you describe is most likely valid (given your >> derivation), but you have not stated the signiÞcance and utility >> of this object relative to the original problem described, or why >> it solves the problem. Time out--let¹s take a step back and take a breath. The OP is looking for a great circle radius equivalent for a spheroid/ellipsoid. My approach has been to average all of the spheroid¹s great ellipses together and extract the mean arcradius from it, the root-mean-average of just the meridian and equator being: [.5 * (a^2 + .5*)]^.5 = [.25 * (3a^2 + b^2)]^.5 The consensus response has been no, [a*b]^.5 is a better choice. I have since attempted to justify and legitimize my interpretation by focusing on the radii: While I truly believe this model is related to the mean GE arcradius--likely beacause of the root-mean- average of the meridion and equator--perhaps my other, much more complex isopathic method is the one and only mean great ellipse equation for Þnding a spheroid¹s great circle radius of best Þt. HOWEVER, [.25 * (3a^2 + b^2)]^.5 is also valid as a great circle radius of best Þt, not based on the mean great ellipse, but based on the mean graticular coordinates, using the non-elliptic, graticular/spherical/geographical transverse components: Oz = acos{b/a} (e = sin{Oz}, f = ver{Oz} = 1 - cos{Oz}); AP = Arc Path = Transverse Longitude; TvL = Transverse Latitude (equatorially concentric rings); Lat = Graticular/Geographical/Spherical Latitude, = asin{cos{AP}*sin{TvL}}; Long = Graticular/Geographical/Spherical Longitude, = atan{sin{AP}*tan{TvL}}; All we need to be concerned with is Lat: sin{Lat} = cos{AP} * sin{TvL}; The elliptic integrand of the second kind is: E2¹{Lat} = [1 - (sin{Lat} * sin{Oz})^2]^.5, = [1 - (cos{AP} * sin{TvL} * sin{Oz})^2]^.5, = ME¹{AP:TvL}; Now let¹s consider the arcradius for the transverse graticular midpoint: Gr = a * ME¹{45¡:45¡} = a * E2¹{30¡}, = a * [1- (cos{45¡} * sin{45¡} * sin{Oz})^2]^.5, = [a^2 - .5^2 * (a^2 - b^2)]^.5, = [a^2 - (.5*a)^2 + (.5*b)^2]^.5, = [a^2 - .25*a^2 + .25*b^2]^.5, = [.75*a^2 + .25*b^2]^.5, = [.25 * (3a^2 + b^2)]^.5 While the apparent solution would be to Þnd the divided difference (mean value theorem) of ME¹{AP:TvL}, Gr = a * ME¹{0 (They argue that they might have trouble giving stability > to our retirement accounts if they¹re shut out of an > entire industry. Good point, I can use the bigger retirement > account to pay for my chemo.) Stop smoking, its cheaper. Those who smoke beneÞt the rest of us through the taxes they pay to use the Evil Weed. Some of them die before collecting Social Security. That is good too. Bob Kolker === Subject: Re: OT: TIAA/CREF Proxies and Tobacco >> (They argue that they might have trouble giving stability >> to our retirement accounts if they¹re shut out of an >> entire industry. Good point, I can use the bigger retirement >> account to pay for my chemo.) >Stop smoking, its cheaper. Those who smoke beneÞt the rest of us >through the taxes they pay to use the Evil Weed. Some of them die before >collecting Social Security. That is good too. I started smoking tobacco about four years ago for health reasons. Honest. I have this nasty condition which my doctor told me tobacco is actually good for. He didn¹t advise me to start smoking. I checked - the standard line is that people with this condition shouldn¹t smoke just for that reason because the other problems caused by smoking are worse. But there exist doctors who do advise their patients to smoke. I thought about it - people usually start smoking when they¹re teenagers and get cancer much later in life, I Þgure if I start smoking when I¹m already this old I¹ll get lung cancer after I¹m already dead. And just a few days ago we got the results from a recent procedure - the doctor was simply amazed how good things looked, even with a microscopic examination of a biopsy there was no sign of the condition, which doesn¹t usually happen, usually it comes and goes but when it¹s gone you can still detect it in the lab. Not to go on about me, I just wanted to point out that if I die before collecting any Social Security I won¹t agree that was a good thing. The condition isn¹t fatal, so even if the tobacco takes care of it forever it stlil could be that the tobacco shortens my life some, but it¹s deÞnitely making it a lot more pleasant, for reasons a lot more [can¹t come up with the right word] than just because I enjoy the weed. (Could be it lengthens my life, because the condition in question _can_ lead to other conditions that _can_ be fatal, regardless, although this condition doesn¹t kill you it really can make you very miserable in a lot of ways; if it gets really bad they cure you, and regarding the cure let me just say I hope never to be cured.) Ok, I guess I really do have to say, or nobody will be able to verify my unbelievable claims here. It¹s ulcerative colitis, egch. >Bob Kolker ************************ David C. Ullrich === Subject: Re: OT: TIAA/CREF Proxies and Tobacco >I started smoking tobacco about four years ago for health >reasons. ... If you decide at some point that smoking tobacco is just too gross, there might be alternatives: http://www.theiowachannel.com/health/3330596/detail.html Lee Rudolph === Subject: Re: OT: TIAA/CREF Proxies and Tobacco > I started smoking tobacco about four years ago for health > reasons. Have you enjoyed it? -- G.C. === Subject: Re: OT: TIAA/CREF Proxies and Tobacco > Ok, I guess I really do have to say, or nobody will > be able to verify my unbelievable claims here. It¹s > ulcerative colitis, egch. A fairly uncommon disease. Most people are much better off tobacco free. Obligatory What-If: If Martin Luether had tobacco at hand, would he have remained Catholic. Bob Kolker === Subject: Re: OT: TIAA/CREF Proxies and Tobacco >> (They argue that they might have trouble giving stability >> to our retirement accounts if they¹re shut out of an >> entire industry. Good point, I can use the bigger retirement >> account to pay for my chemo.) > Stop smoking, its cheaper. Those who smoke beneÞt the rest of us > through the taxes they pay to use the Evil Weed. Some of them die before > collecting Social Security. That is good too. At the risk of perpetuating a pointless thread, I point out that the taxes generated by cigarettes is a molehill compared to the additional medicare/social security cigarettes cause. Drug dealers are predators and I remain Þrmly against them. This isn¹t the jungle, it is civilization (at least for the time being.) Bart === Subject: mass-density or weight-density > Hello. You are wrong. Who the heck are you? The Puke? What am I wrong about? Surely; by now you know that a mass is an aggregation or quantity of matter, and inertia is its accurate measure. >>When Newton said at the beginning of his Principia that mass is the >> bulk and density of a body jointly: What did he mean? >>The product of bulk [volume] and density is either the volume x the >> mass-density [mass/unit volume]; or volume x weight-density >> [weight/unit volume]: >> Either way, the volume Œdrops out¹; leaving: Mass equals mass; or Mass >> equals weight. >> Will somebody who _knows_, please explain that to me? === Subject: Re: mass-density or weight-density > What am I wrong about? Surely; by now you know that a mass is an > aggregation or quantity of matter, and inertia is its accurate > measure. Shead, did you know that an electron has mass? It¹s not an aggregate of anything so your deÞnition is obviously incomplete and naive. Inertia isn¹t a measure of anything as has been pointed out to you for years now. Inertia http://scienceworld.wolfram.com/physics/Inertia.html Weight http://scienceworld.wolfram.com/physics/Weight.html Mass http://scienceworld.wolfram.com/physics/Mass.html === Subject: Re: Inverse image misunderstanding days. My association with the Department is that of an alumnus. [.snip.] >> The function f* is a function from P(B) to P(A). Given a subset of Q >> of B, f*(Q) is the subset of A consisting of all elements of A that >> map into Q under f. > If I view f* as a map from P(B) to P(A) then I can call f* a function, Yes. Mostly because it ->IS<- a function: a function which takes subsets of B as arguments, and answers with subsets of A. >but if I don¹t, then it is a set? Any function between sets may be described, within set theory, as a set (explicitly, as a set of ordered pairs, where the pair (x,y) means that the image of x under the function is y). In set theory, everything is a set. But in this case, we want to emphasize that f* is a function between the power sets, going in the opposite direction from the original function (i.e., if f mapped A to B, then f* maps P(B) to P(A)). So it¹s a function. To emphasize that it is a function whose very deÞnition depends on another function f, we sometimes say that f* is an induced function, or a function induced by f. -- It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Inverse image misunderstanding > Yes. Mostly because it ->IS<- a function: a function which takes > subsets of B as arguments, and answers with subsets of A. >but if I don¹t, then it is a set? > Any function between sets may be described, within set theory, as a > set (explicitly, as a set of ordered pairs, where the pair (x,y) means > that the image of x under the function is y). In set theory, > everything is a set. But in this case, we want to emphasize that f* is > a function between the power sets, going in the opposite direction > from the original function (i.e., if f mapped A to B, then f* maps > P(B) to P(A)). So it¹s a function. To emphasize that it is a function > whose very deÞnition depends on another function f, we sometimes say > that f* is an induced function, or a function induced by f. I see now. Since I am now learning the formal and more abstract side of mathematics, my old ways of thinking about functions, etc, are causing problems. However, your explanations were very helpful and much appreciated. === Subject: Re: Inverse image misunderstanding > As has been said, you¹re looking at it wrong: f* > is not a function, and f* does not map elements > of B to anything. > Is f* a function if I view it as a mapping from P(B) to P(A)? > [Note: when I said f* is not a mapping I lied. > It _is_ a mapping, but it¹s a mapping from > sets to sets.] > Yes, I now understand my misunderstanding. I was seeing it as mapping > from elements to elements like a regular function. Mistake corrected. If you don¹t mind my adding a few words to this thread, here is another description of the situation. Suppose f: A --> B is a function. Then f^*: P(B) --> P(A) is deÞned as above and f_*: P(A) --> P(B) is the direct image function. These are related as follows. If S is a subset of A and T is a subset of B, then f_*(S) is included in T iff A is included in S is included in f^*(T). Less well known is another function f_!: P(A) --> P(B) with the reverse property that f^*(T) is included in S iff T is included in f_!(S). f_!(S) consists of all points of B whose inverse image is included in S. It can also be deÞned by the formula f_!(S) = B - f_*(A - S). It can be especially useful in topology since f_* takes closed sets to closed sets iff f_! takes open sets to open sets. === Subject: Re: Inverse image misunderstanding >> As has been said, you¹re looking at it wrong: f* >> is not a function, and f* does not map elements >> of B to anything. > Is f* a function if I view it as a mapping from P(B) to P(A)? Yes. >> [Note: when I said f* is not a mapping I lied. >> It _is_ a mapping, but it¹s a mapping from >> sets to sets.] > Yes, I now understand my misunderstanding. I was seeing it as mapping >from elements to elements like a regular function. Mistake corrected. ************************ David C. Ullrich === Subject: Re: Multiplication of negative binary numbers. This post not CC¹d by email >As I said: for each negative factor, subtract the other factor from >the high word of the result. Then you get: > 01111110 >- 1110 >- 1001 >---------- > 00001110 >which is 14. >1001*0011 gives > 00011011 >-0011 >--------- > 11101011 >which is -21. >-- >David Kastrup, Kriemhildstr. 15, 44793 Bochum G¹day G¹day David, OK, I am most impressed. The examples have helped me understand more clearly how the process is performed. It quite happily gave the correct answers to some examples I tried. Still, I am left with that sense of awe that children have when a magician performs a magic trick. In this instance I don¹t so much as an inkling as to where to start looking for a proof. Best wishes, -- Quentin Grady ^ ^ / New Zealand, >#,#< [ / / ... and the blind dog was leading. http://homepages.paradise.net.nz/quentin === Subject: Re: Multiplication of negative binary numbers. > OK, I am most impressed. The examples have helped me understand more > clearly how the process is performed. It quite happily gave the > correct answers to some examples I tried. Still, I am left with that > sense of awe that children have when a magician performs a magic > trick. In this instance I don¹t so much as an inkling as to where to > start looking for a proof. Just remember that extending a negative twos-complement number to the left consists of appending 1¹s as needed, and a string of 1¹s looks like a -1 in twos-complement. Or another way of putting it: -2 in four bits looks like a 14 because -2 = 14 - 2^4. Likewise, the -7 looks like a 9 because -7 = 9 - 2^4. 9*14 - 2^4 * 9 - 2^4 * 14 = 9*14 - 2^4 * 9 - 2^4 * 14 + 2^8 - 2^8 = (9 - 2^4) * (14 - 2^4) - 2^8 = (-7)*(-2) - 2^8 In an eight-bit representation, the 2^8 is as invisible as the 2^4 was in the four-bit representation. Another approach to a proof starts with: (2^8 - 2) * (2^8 - 7) = (14 + 2^8 - 2^4) * (9 + 2^8 - 2^4) again remembering that multiples of 2^8 are dropped at the end. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: Four people crossing a bridge mumbles... >Four people, A, B, C, and D, are on one side of a bridge, and they all want >to cross the bridge. However, it¹s late at night, so you can¹t cross without >a þashlight. They only have one þashlight. Also, the bridge is only strong >enough to support the weight of two people at once. The four people all walk >at different speeds: A takes 1 minute to cross the bridge, B takes 10 >minutes, C takes 11 minutes, and D takes 12 minutes. When two people cross >together, sharing the þashlight, they walk at the slower person¹s rate. How >quickly can the four cross the bridge? Depends on three further parametres: 1) what¹s their name 2) what¹s their quest 3) what¹s their favourite colour -- *MM Replace Œdot¹ with a dot to reply === Subject: Great Circle Distance Formula rework... I am working on a proximity searching algorithm where I know a given point in latitude/longitude in decimal degrees and a search radius which I¹ll call Distance below... I reworked the Great Circle Distance Formula (using decimal degrees) from ( bad attempt at ascii representation for conversion ): Distance = 3963.0 * arccos ( sin ( lat1 / 57.2958 ) * sin ( lat2 / 57.2958 ) + cos( lat1 / 57.2958 ) * cos( lat2 / 57.2958 ) * cos( long2 / 57.2958 - long1 / 57.2958 ) ) to: / ( cos ( Distance / 3963.0) - ( sin( lat1 / 57.2958) * sin( lat2 / 57.2958) ) ) long2 = arccos ------------------------------------------------------------- ------------- -------- | - (long1 / 57.2958 ) cos ( lat1 / 57.2958 ) * cos ( lat2 / 57.2958 ) / Can anyone verify that this is in fact a correct derivation of the 1st equation? I will already know lat1, long1, and lat2 (because change in latitutde degrees per mile is constant), and I will need to solve for long2 so that I can determine change in degrees latitude and change in degrees longitude. Then, given a starting latitude/ longitude, I can query a database to give all other locations that are within the bounding latitude and longitude. Andy === Subject: Re: contravariant tensor Þelds, differential operators and quantisation >> You could also consider the symmetrized combination >> X v Y = X (x) Y + Y (x) X >> Perhaps this is the not so subtle way that commutativity of derivatives >> is recovered. It seemed to me that you were looking for a natural >> action of higher rank tensors on functions. Well, it¹s possible that >> there is more than one such action. Do you have any other speciÞc >> criteria that you are looking for except natural? > Probably it was not really clear in my OP, but I thought also about this > already. Can you prove that a symmetric tensor of rank r always gets mapped > to a differential operator of degree r? (It is intended that I do not > really specify what a diff op of degree r is, but it should be so that > acting on a function in local coordinates, the result depends on the Þrst > r derivatives). If such a diffeomorphism-equivariant map from symmetric r-tensors to differential operators of degree r exists, it should have a connection to the short exact sequence of 0 ---> symmetric r-tensors ---> r-jets ---> (r-1)-jets ---> 0 > The other thing is that the symmetric algebra is commutative, so the > connection to quantisation will probably get lost. Mmm, Rafe Mazzeo was recently mentioning a connection between (Pseudo-)Differential Operators and the different formalisms of quantizations in his lectures, but I don¹t recall the details and I do not have my notes at hand right now... Maybe googling for Mazzeo +quantization or even Mazzeo +Weyl quantization might give you a clue? === Subject: Re: contravariant tensor Þelds, differential operators and quantisation > In the meantime, I worked out the transformation rule for the components of > a second order differential operator in coordinates. By this, I mean > something of the form > a^ij d^2/(dx^i dx^j) > where the a^ij are the component functions and d is the curly d partial > differential operator. The transformation will get pretty complicated for > higher orders, and it does not seem to have to do anything with tensors - > what kind of structure is it then? I don¹t know enough about these things, so I maybe talking my head off, but I think that objects which transform under coordinate changes like your second-order differential operator above are called *2-jets*. I can¹t explain about jet bundles in detail, but they play an important role in the formal theory of PDE. Here are some links which might (or might not) be useful to you: http://www.mrao.cam.ac.uk/agacse2001/Abstracts/Baker.htm http://www.mrao.cam.ac.uk/~rcdb100/research/jetbundles.ps http://arxiv.org/abs/math.AP/0106090 http://www.mat.univie.ac.at/~michor/dgbook.pdf === Subject: Re: Turing Machine with Tape > Oh, oh. Should I call in a quantum mechanic? I looked in the phone > book for a quantum mechanic but all I could Þnd were quantum mechanics. > Are they unionized, quarktized or something that requires more than one > for a single simple job? You¹re out of luck anyhow -- the phone at the quantum mechanics¹ ofÞce doesn¹t work if you already know the phone number. === Subject: Re: Turing Machine with Tape >> >> >The tape on my Turing machine broke, >where can I get a new replacement tape? >> >>Why bother replacing it? Just shift it an inÞnite number of places >>to the left. >> > Takes too long. > Not if you shift 1 in 1/2 sec, 1 in 1/4 sec, 1 in 1/8 sec, ... > After 1 second, voila! But by then it will be traveling at inÞnite speed, and will take another second to stop. === Subject: Re: Turing Machine with Tape > >> > >The tape on my Turing machine broke, > >where can I get a new replacement tape? > >> > >>Why bother replacing it? Just shift it an inÞnite number of places > >>to the left. > > > > Takes too long. > Not if you shift 1 in 1/2 sec, 1 in 1/4 sec, 1 in 1/8 sec, ... > After 1 second, voila! > But by then it will be traveling at inÞnite speed, and will take another > second to stop. Just a nanosec or two. If tape wasn¹t Þctionless, it would burn up. If tape was frictionless, brakes would be ineffective. Futhermore, unlike tape players, no Turing machine has been produced with fast forward. === Subject: Re: Turing Machine with Tape ... > But by then it will be traveling at inÞnite speed, and will take another > second to stop. > > Just a nanosec or two. > If tape wasn¹t Þctionless, it would burn up. > If tape was frictionless, brakes would be ineffective. > Futhermore, unlike tape players, no Turing machine > has been produced with fast forward. Doesn¹t ooM produce suitable tape? -- 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: Turing Machine with Tape 2kpKShAQELm5uc578GgAAAJ5SURBVHichdNBb9sgFABgvMblarSwXDMcKdfUps o1td1xDWqfc020 mV5JoHt/f0Dc1pU2jUskvrz3eA9D6n8s8h+ QnVSzv0G3zUDPlOzqut3N2g9YcwDQHgm1zJj9O8gc IshWIu+ NMe4NEICFGKdoySHArzd4AI0ZQLNyZR7hNIJ8gL3YcLiIRuQQcg3te8S+ CfVdpa4Rx8UI Z9DUcuiEXKcax8OkOAunEuUyZjJHN0IR24D9jlCVjjuMxbcm9ZGYh31zGOHR7C PkwILFiPYKsjdD PsCCg14AfDPmZRzJXWyp8GueQwH8bMzXK8jbmNbsFQ8hALSPo4qwTfv9M4YT6F 7j7RvETGbgM5vp DPge1XmEHymi8NZqVrCiaB5vrpBKDNhYZEwznTWv8yvEg5sXL5QPY8kY8w+ YoEuZ5nVlfRYiWOEs SQ1uUqaLKK0jnDG28K+nBPepCe9q1RGGXFN/Zgl+h/ 8PmsxqidnCEkZbe0gQDhUGW9C6tCQstF7p +JWECRrdE+pljZYUAXCXPh8ZMjGwrkKPWQCF+ JpgF8fB81mzDVUpUmkxTxBGOMADLLxqS9Uhlkrd pouKt2QQTtWqEbWizarD/pQ6fzRD/ yXvDyKsuq065ccbVP3QPy9hmJetRSSsVebpOsTfpj8pgIEU GSuyU931Hxc1KGS6VJY03wPQ8UXF8iqcVoiyEuvj5Knl5qTEmjax/ PZ5Alt29GJFLmF/tZxPH+fd S8hjGyGVWz5N4f5gK7HzXWfd5ucnoOoidliV2H6qUZ9R3ohNqI7N1k3Bo3Chuh ey2vgpzGh4lSv0 ohQKp7BwAUrESohOuQngtbf4s9qNIX8At0MbK5iXDI4AAAAASUVORK5CYII= Discussion, linux) >> >> > > >>The tape on my Turing machine broke, >>where can I get a new replacement tape? > >Why bother replacing it? Just shift it an inÞnite number of places >to the left. > >> >> Takes too long. >> Not if you shift 1 in 1/2 sec, 1 in 1/4 sec, 1 in 1/8 sec, ... >> After 1 second, voila! > But by then it will be traveling at inÞnite speed, and will take another > second to stop. Now that is a problem. By the time you stop, you¹ll have shift twice an inÞnite number of spaces and you¹ll be past the end of the tape. What? Why¹s everyone looking at me like that? -- At the Microsoft-sponsored cocktail reception in the Galaxy Ballroom that evening, Robert Dees urges us Œto network on behalf of the people of Iraq,¹ -- Naomi Klein reports on Microsoft¹s efforts to further democracy. === Subject: Re: Turing Machine with Tape >> >The tape on my Turing machine broke, >where can I get a new replacement tape? >> >>Just use the portion that is inÞnitely long in one direction, your >>Turing machine won¹t notice the difference. > Because of budgetary considerations we opted for a model with > the inÞnitely long in one direction tape and then instead of > purchasing and storing the whole tape, we opted for a just in > time inventory contract. But now that it¹s broken, the short > end has gone thru the machine and the long end has yet to get > to the machine. Yes we could toss away the short end. Still > we¹re puzzled about the long end as it¹s not now in the shop. > The problem is that you put bureaucrats in charge, What choice does anyone have these days in our hyper-commercial society when dealing with a mega-mania-merged corporate conglomerate? > and when you accepted delivery of the short end, Of course, MMM-CC¹s give everybody the short end. > they assumed the long end was mis-delivered and returned it to the > manufacturer. Fortunately, since the manufacturer only had recorded the > sale of the inÞnitely long in one direction tape, they gave you a full > refund. As a result, after the annual audit you will Þnd that you have > unallocated funds sufÞcient for the purchase of a new tape. Except for handling, shipping, deduction for portion sent, inþation, devaluation of currency and misc. charges such as locating old (3-month old) model tapes. Likely I¹d have sufÞcient funds for just half a tape, but a tape 1/2, or 1/3 the width because of cost of new case, would perform unreliably. === Subject: Re: Turing Machine with Tape >> >The tape on my Turing machine broke, >where can I get a new replacement tape? >> >>Why bother replacing it? Just shift it an inÞnite number of places >>to the left. > Takes too long. > Not if you shift 1 in 1/2 sec, 1 in 1/4 sec, 1 in 1/8 sec, ... > After 1 second, voila! Dangerous as this tape isn¹t a cryogenic super conducting frictionless tape. Perhaps in the further a frictionless nanoteck tape will become available, but even then being of smaller size, it would be a bad Þt. === Subject: counting certain relative-frequency distributions charset=Windows-1252 Suppose a 3-sided die with faces 0,1,2 is tossed n times. Then the set of possible 3-tuples of relative frequencies (r0,r1,r2) of the respective face-values is S_n = {(r0,r1,r2) in {0/n,1/n,...,n/n}^3: r0+r1+r2 = 1}. S_n is a Þnite subset of the simplex which is a triangle with vertices (1,0,0), (0,1,0), (0,0,1). The number of points in S_n is, if I¹m not mistaken, |S_n|=(n+1)(n+2)/2. Now consider the subset of relative-frequency 3-tuples for which the *average* face-value is, say, x: A(x,n) = {(r0,r1,r2) in S_n: 0*r0 + 1*r1 + 2*r2 = x}. A(x,n) is nonempty iff (x is a rational in the interval [0,2], say x=p/q in lowest terms, and n in {k*q: k in N}). So, given x=p/q, let n be restricted to {k*q: k in N}. By inspecting the geometry of A(x,n) in S_n, I believe that if x is a rational in [0,2], then limit(n->oo) |A(x,n)|/n = x/2 (Correct?) If the face-values were arbitrary integers i0,i1,i2, i.e., if A(x,n) = {(r0,r1,r2) in S_n: i0*r0 + i1*r1 + i2*r2 = x} would the corresponding limit be expressible as a function of i0,i1,i2? What if the die were m-sided with rational face-values? --r.e.s. === Subject: Newbie question - Neumann¹s expansion I have a question concerning Neumann¹s expansion (I-A) ^(-1)= I + A + A^2 + ... I don¹t understand why this is correct. Suppose A = diagonal matrix with entries (-1) Leftside gives diagonal matrix with entries (1/2) Rightside gives alternating zero matrix or unity matrix. This isn¹t the same. Where does the difference come from? Alfonso. === Subject: Some basic math questions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SKXI420517; Iâm trying to determine the mind-set I should be adopting upon trying to solve certain problems. E.g., consider the challenge of factoring the following statement (it came up as part of another problem): 2x^2 + 5x - 3 Some of the things that run through my mind upon trying to solve this are: (1) OK...hereâs a quadratic function. (2) Factoring it will identify two algebraic expressions in the form ã(term1 +/- term2) * (term3 +/- term4)ä that may be multiplied together to yield the original function. (3) Hhhmm...term 1 multiplied by term 3 yields ã2x^2ä. So, Iâll try substituting in 2x and x, respectively; thereby leaving me with (2x +/- term2) * (x +/- term4). (4) Next, term2 multiplied by term4 yields ã-3ä. So, Iâll systematically substitute positive and negative values of 1 and 3 into the empty Þelds until I get a working ãanswer,ä such as: (2x - 1) * (x + 3) (5) Along the way some thoughts about ãthe distributive law,ä ãcompleting the square,ä ãthe quadratic equationä and ãconicsä come to mind. My questions relate to the idea that I donât Þnd the above-described methodology at all that satisfying.* (And I bet some of the cognoscenti out there are thinking: ãYou damn well shouldnât!!!ä Ha!) It seems more like an exercise in persistence than an exercise in math. So, Iâm gradually wandering into what are for me more advanced levels of math, and at the same time thinking: ãWhy is it that the solution is right, but I donât feel that I learned much of anything by solving the problem, since I really just spent a lot of time ãplugging awayä at the answer?ä ãIs there another way?ä ãWill exploring more advanced topics change the way I think about these types of problems in general?ä ãIs How to perform a proof the area Iâm looking for?ä Any thoughts would be appreciated. * At some point, even discouraging. === Subject: Re: Some basic math questions > IÕm trying to determine the mind-set I should be adopting upon trying to solve certain problems. E.g., consider the challenge of factoring the following statement (it came up as part of another problem): > 2x^2 + 5x - 3 > Some of the things that run through my mind upon trying to solve this are: Think about what use you have of the zeros of the expression. === Subject: Proof: If f: A->B is a function, then f*(B) = A. below is formal and correct. Let f* denote the inverse image of the function f. Theorem. Let f: A -> B be a function. f*(B) = A. Proof. We will prove that each side of the equality is a subset of the other. First, let x in f*(B). Then f(x) in B. Since f is a mapping from A to B, x in A. Hence f*(B) <= A. Second, let x in A. Then f(x) in B, since f is a mapping from A to B. Therefore, x in f*(B). Hence A <= f*(B). Since f*(B) <= A and A <= f*(B), f*(B) = A. QED. === Subject: Re: Proof: If f: A->B is a function, then f*(B) = A. days. My association with the Department is that of an alumnus. >below is formal and correct. Let f* denote the inverse image of the function >Theorem. Let f: A -> B be a function. f*(B) = A. Stylistic comment: Mathematical formulas should make sense when you read them aloud. What you have here is let f from A to B be a function. f star of B is equal to A. As you can see, the second statement is a bit terse. There¹s a then missing. It sounds better if you rewrite it as Let f: A->B be a function. Then f*(B) = B. Even better, given the length, if you write it as a single if - then statement: If f:A->B is a function, then f*(B)=A. >Proof. We will prove that each side of the equality is a subset of the >other. The inclusion f*(B) contained in A is from the very deÞnition of f*: f* is a function from P(B) to P(A); hence, the value of f* on any subset of B (including B itself) must be an element of P(A), hence a subset of A. So you don¹t have to prove it. > First, let x in f*(B). Then f(x) in B. perhaps by deÞnition of f* might be appropriate here. > Since f is a mapping from A to >B, x in A. Hence f*(B) <= A. Correct, but unnecessary. > Second, let x in A. Then f(x) in B, since f is >a mapping from A to B. Therefore, x in f*(B). Hence A <= f*(B). Since f*(B) ><= A and A <= f*(B), f*(B) = A. QED. This is correct. -- It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Basic question on functions > Bare with me I have not had this in 12 years. My textbook is vague. > f(x) = 2x^2 + 3 > I need to Þnd the domain and range. ... How can the domain be found? It can be any set of things for which the arithmetical operations make sense. -- G.C. === Subject: Re: Basic question on functions >Bare with me What, on a Þrst date?! === Subject: Re: On the partial sums of reciprocals of primes by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SLAXH24110; >If S(x) = log log x + B + o(1) then for some C it¹s also >true that S(x) < log log x + C. That sufÞces to make it >true that S(x) < log log x + constant + 1/(log x)^2; the >last term is irrelevant, isn¹t it? I gave another amount of time to the argument above, and yes I would agree with it; after all, since O(1/logx) approaches 0, there should be a constant C > B, such that B + O(1/logx) < C or B + O(1/logx) < C < C + 1/log(x)^2. But everyone, could the argument above really work? If yes, then it seems that I can complete my paper. erdos fan === Subject: SpeciÞc Gravity or Relative Density Copied from : ŒThe density of water is often taken as a standard and the density of other materials compared relative to it. The relative density of water is the density of water over the density of water, 62.4 pounds per cubic foot over 62.4 pounds per cubic foot, so that it is 1.0. The relative density of a rock that is 2 times as dense as water is the density of the rock (124.8 pounds per cubic foot) divided by the density of water (62.4 pounds per cubic foot) to give 2.0. The relative density of steel is 7.7 and that of mercury is 13.6.¹ SpeciÞc gravity is the same in either the customary, or the SI system of weights and measures. === Subject: Re: SpeciÞc Gravity or Relative Density > Copied from > ŒThe density of water is often taken as a standard and the density of > other materials compared relative to it. The relative density of water > is the density of water over the density of water, 62.4 pounds per > cubic foot over 62.4 pounds per cubic foot, so that it is 1.0. The > relative density of a rock that is 2 times as dense as water is the > density of the rock (124.8 pounds per cubic foot) divided by the > density of water (62.4 pounds per cubic foot) to give 2.0. The > relative density of steel is 7.7 and that of mercury is 13.6.¹ > SpeciÞc gravity is the same in either the customary, or the SI system > of weights and measures. Donald, SG is deÞnately NOT a fundamental quantity these days. Useful sometimes, but not fundamental. What point are you trying to make? Owen P.S. keep to SI units and you avoid confusion when mixing standards. for example AFAIK the Œpound¹ is NOT a unit of mass, although the Œounce¹ may be such. Since density is always mass/volume, we shouldn¹t mix Œpounds¹ and Œcubit feet¹, coz that¹s just confusing to normal folks. Anyway, back to the important QQQ What point are you trying to make? === Subject: Re: SpeciÞc Gravity or Relative Density > The density of water is often taken as a standard and the density of > other materials compared relative to it. Not any more Shead-- The SI is founded on seven SI base units for seven base quantities assumed to be mutually independent. http://physics.nist.gov./cuu/Units/units.html Other quantities, called derived quantities, are deÞned in terms of the seven base quantities via a system of quantity equations. The SI derived units for these derived quantities are obtained from these equations and the seven SI base units. http://physics.nist.gov./cuu/Units/units.html Mass Density is in kg/m^3 http://scienceworld.wolfram.com/physics/Density.html === Subject: Re: SpeciÞc Gravity or Relative Density > [...] the customary, or the SI system > of weights and measures. SI _is_ customary... unless you¹re an idiot. -- G.C. === Subject: Re: SpeciÞc Gravity or Relative Density Your posts are off-topics in sci.math. Although you have been asked to stop posting here, and have consistently refused, please consider honoring the purpose of this forum. > Copied from > ŒThe density of water is often taken as a standard and the density of > other materials compared relative to it. The relative density of water > is the density of water over the density of water, 62.4 pounds per > cubic foot over 62.4 pounds per cubic foot, so that it is 1.0. The > relative density of a rock that is 2 times as dense as water is the > density of the rock (124.8 pounds per cubic foot) divided by the > density of water (62.4 pounds per cubic foot) to give 2.0. The > relative density of steel is 7.7 and that of mercury is 13.6.¹ > SpeciÞc gravity is the same in either the customary, or the SI system > of weights and measures. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Trigonometry: Problem for you If x, y and z are positiv Reals such that xyz=x+y+z then we have that 1/(sqrt(1+x^2) + 1/(sqrt(1+y^2)+ 1/(sqrt(1+z^2)<= 3/2. === Subject: Re: Trigonometry: Problem for you > If x, y and z are positiv Reals such that > xyz=x+y+z then we have that > 1/(sqrt(1+x^2) + 1/(sqrt(1+y^2)+ 1/(sqrt(1+z^2)<= 3/2. One suggestion: make x=cot(A), y=cot(B), z=cot(C), (where A, B, C go from 0 to pi), and the condition x*y*z=x+y+z translates into A+B+C= some constant. And 1/sqrt(1+x^2) etc. also simpliÞes. Now you are on your own. === Subject: Re: Proof of twin-prime conjecture? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4SLlXT26985; >Arenstorf has uploaded an MS purporting to prove the twin >prime conjecture to the arXiv: >http:// uk.arxiv.org/abs/math .NT/0405509 . >This does not look completely stupid, in fact >(whether or not it is all correct) it appears to be >a substantial piece of analytic number theory. >-- >Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html >Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 >Francis Wheen, _How Mumbo-Jumbo Conquered the World_ Finally, a discussion about some real and signiÞcant developments in math on sci.math. Anthony J. Natoli === Subject: Re: Journal editors and reviewers, speak up > > > > snip > > > >>Oh, right! The Harris-basher¹s favorite weapon of all: James is > >>CrAzY!!! Isn¹t this growing tiresome? Can¹t you guys come up with > >>something new? > > > > How about this for originality ? ... > > > > James Harris is Nuts(TM) > > > > :) > > > > P.S. Well, he is! > Professor Harris landmark paper (Advanced Polynomial Factorization, > http://rattler.cameron.edu/swjpam/vol2-03.html) is totally ground > breaking. Professor Harris is neither crazy nor nuts. > Perhaps you ought to show some respect for your intellectual superiors. > Lance Lamboy > May I, as a complete outsider, suggest No. === Subject: Re: Journal editors and reviewers, speak up >> My point was to observe that the concatenation of Dr. and Harris >> seemed to provoke the gag reþex in Doug Norris, > Ah, so pointing out deliberate falsehoods is a gag reþex? > Listen - I spent the Þrst twenty-eight years of my life working towards > a doctorate in mathematics, and if Harris wants one, he can go to grad > school just like anyone else. He doesn¹t need you conferring an > honorary degree upon him. > Doug OK. I get it. You¹re a burned out academic. You wasted the best years of your life on a worthless degree. Now you¹re bitter because Professor Harris¹s achievements have exceeded your own. Get over it. Professor Harris¹s seminal work on factorization has many applications in such diverse Þelds as the War Against Terror (TM), energy policy, and family planning. It is deÞnitely Noble-worthy. To Professor Harris, my hats off to you for an outstanding paper. Keep up the good work. We expect many more papers from you. -- Lance Lamboy === Subject: Re: Journal editors and reviewers, speak up > OK. I get it. You¹re a burned out academic. Actually, I¹m an actuary. You must be some detective. Doug === Subject: Re: Please help, I¹m getting frustrated with this math puzzle. [Top-quoted since quoted portion is long.] In my dynamic programming formulation, I changed the speciÞcation of the problem slightly. My n is the OP¹s n + x + y. This does not change the solution method. >> The Setting: >> 1) A person has four pails (A, B, C, D). >> 2) In a pile off to one side is a stack of Œn¹ marbles. >> 3) In pail A there are Œx¹ additional marbles >> 4) In pail B there are Œy¹ additional marbles >> >> The Rules >> 5) The existing marbles in pails A and B must remain and represent the >> minimum number of marbles that the respective pails will contain >> 6) The pile of marbles may be allocated to any of the four pails as >> you see Þt. >> >> The Problem >> 7) What is the optimal allocation of marbles that will have the >> greatest >> 20%A, 30%B, 25%C, and 25%D component (by this I mean that 20% of the >> marbles end up in pail A, 30% in pail B, etc....) > You could use a dynamic programming approach. Here is the general problem. > minimize sum(i = 1..m, d(x_i, a_i)) > subject to > sum(i = 1..m, x_i) = n > x_i >= c_i > x_i integer. > Here, d is your componentwise norm, e.g., the absolute or squared > difference. > For k = 1...m, deÞne > V_k(y) = the minimal total error for components k through m given > sum(i = k...m, x_i) = y. (I.e., there are y marbles remaining to be > allocated to buckets k through m). > Then > V_(m+1) (y) = 0 if y = 0, inÞnity otherwise > V_k(y) = min {d(x,a_k) + V_(k+1) (y-k): c_k <= x <= y} > We then wish to compute V1(n). Keep track of the argmins in your > recursion. > Instead of the terminal condition V_(m+1), we could use > V_m(y) = d(y,a_m) if c_m <= y, inÞnity otherwise. > For V_k, use the convention that the minimum of an empty set is > inÞnity, and/or modify the range of x to > c_k <= x <= y - sum(j = k+1...m, c_j). Actually, by using this > restricted range, we never have to refer to inÞnity as long as sum(i > = 1..m, c_i) <= n. > In your particular problem m = 4, c = (x, y, 0, 0), and a = n (.2, > .3, .25, .25). -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Please help, I¹m getting frustrated with this math puzzle. > I would appreciate any guidance, because I¹ve managed solutions for > these problems through basic trial and error but a method to solve the > problem type eludes me. > Let me Þrst mention, I¹m not a student (twenty years past that stage of > life!). I¹ve been bantering back and forth with a friend as to how to > best tackle this problem and we¹d very much appreciate a resolution to > how to solve these kinds of problems. > The Setting: > 1) A person has four pails (A, B, C, D). 2) In a pile off to one side is > a stack of Œn¹ marbles. 3) In pail A there are Œx¹ additional marbles > 4) In pail B there are Œy¹ additional marbles > The Rules > 5) The existing marbles in pails A and B must remain and represent the > minimum number of marbles that the respective pails will contain 6) The > pile of marbles may be allocated to any of the four pails as you see > Þt. > The Problem > 7) What is the optimal allocation of marbles that will have the greatest > 20%A, 30%B, 25%C, and 25%D component (by this I mean that 20% of the > marbles end up in pail A, 30% in pail B, etc....) > As a matter of course, I¹ve looked through some old math books (modeling > data, boundary value problems, etc...), but I¹m not sure where I¹ll Þnd > my answer. Even a point to the relevant category of mathematics would be > gratefully accepted as I¹m interested to see how such puzzles are > solved. > Karen SlowyGoingMad Hasler ; ) I recommend that you read Professor Harris¹s landmark paper on factorization. It is very applicable to your problem. -- Lance Lamboy === Subject: Re: mass-density or weight-density > > > > What I don¹t get is why do you keep posting these obvious questions > about > > mass? > > > > Donny¹s demonstrating the limits of his education. > > > > I¹m only an every day layman and I understand these concepts as > > simple common sense. Is it that you need some complex calculus > formula > > before you can understand it? > > > > Donny don¹t do calculus. > I thought he didn¹t even believe in calculus? > He uses x = (1/2)*a*t*t a lot, but I don¹t think he knows how to derive it > in calculus. I think that you¹ll Þnd that just off the top of your head: 1/2at^2 = (vt-vi)/(2t) x t^2 = (vt-vi)t/2 = at: Anyways it sure doesn¹t require the calculus. === Subject: Re: mass-density or weight-density > I think that you¹ll Þnd that just off the top of your head: 1/2at^2 > = (vt-vi)/(2t) x t^2 = (vt-vi)t/2 = at: Anyways it sure doesn¹t > require the calculus. The formulas are derived with calculus, Shead. Assuming the force is constant and the mass remains constant, we have constant acceleration: F/m = dv/dt = a = constant direct integration (with respect to t) gives formulas such as: v - v_o = at v = at + v_o A second integration results in: x - xo = 1/2 at^2 + v_ot x = 1/2 at^2 + v_ot + x_o eliminating t, we get 2a(x-x_o) = v^2 - v_o^2 === Subject: Re: mass-density or weight-density > > > When Newton said at the beginning of his Principia that mass is the > > > bulk and density of a body jointly: What did he mean? > > > > He meant that the mass of the body *equals* the product of the bodies > > density and the bodies volume. He did not mean, or it is not clear > that > he > > meant, that mass is *deÞned* as such. > > > > Pmb > > > > body¹s <> bodies > http://www.vscht.cz/mat/Pavel.Pokorny/physics/newsgroups.html > Please don¹t waste bandwidth by correcting or criticizing peoples > spelling > or grammar unless it obscures what was said. Pointless spelling > corrections > almost always draw responses which correct spelling errors in the > correction! > I Þnd the difference between bodies and body¹s in the post above to be > confusing. Try using common sense; instead of making a big deal(;^) === Subject: Re: mass-density or weight-density >> Will somebody who _knows_, please explain that to me? > Mass (fundamental) > http://scienceworld.wolfram.com/physics/Mass.html > Weight (derived) > http://scienceworld.wolfram.com/physics/Weight.html The cited references do not substantiate your stated claims, Sam. Furthermore, if you look at connected pages on the same site, you Þnd things such as http://scienceworld.wolfram.com/physics/Pound.html Unfortunately, there is a great deal of confusion about the deÞnition of the pound, with many authors using the unit to denote the mass-equivalent of 4.448 newtons, namely 453.592 grams. This is from the same wolfram.com, Eric Weisstein¹s statement equivalent of Dense Donny¹s often-repeated claims that force is fundamental, and mass is derived. At least to true believers in magical qualities of base units, as Sam Wormley has shown himself to be. Gene Nygaard http://ourworld.compuserve.com/homepages/Gene_Nygaard/ === Subject: Re: mass-density or weight-density >> Will somebody who _knows_, please explain that to me? > Mass (fundamental) > http://scienceworld.wolfram.com/physics/Mass.html That doesn¹t claim any fundamentalness for mass. Like I told you before, Sam, you don¹t help things one bit by attributing magical quantities to the base units of SI (one of several metric systems) which are the same as the magical qualities which Shead attributes to the base units of one of the several English systems, the gravitational foot-pound-second system. But do you pay any attention whatsoever to what the sources you cite actually say? Eric Weisstein, on the cited page, makes the þat-out statement Note that the pound is not a unit of mass, but rather one of weight. That is patently false. Its falseness doesn¹t even depend on the ambiguities in the meaning of the word weight in the second clause, since he actually claims that the pound is not a unit of mass in his Þrst clause. Note, of course, that it is in fact those ambiguities in the word weight as used in the second clause which are the PRIMARY STUMBLING BLOCK in most of Dense Donny Shead¹s postings on this subject. Dishonest Don loves to complain about the use of kilograms for weight, and idiots like you are far too obliging in playing along with him, misapplying a deÞnition of weight which almost never applies to any weight measured in kilograms. Eric Weisstein himself provides prove that this statement is false. Learn to do a better job of evaluating the credibility of your authorities. Internal inconsistency is one indicator of shoddy work. claims under the entry for pound. http://scienceworld.wolfram.com/physics/Pound.html But while he did make some corrections to that entry, he did not correct the same misstatement in the entry for mass. The confusion between pound-mass and pound-force alone is a good reason to use metric units wherever possible (in the MKS system, the base unit of mass is the kilogram, while the unit of force is the newton). This does, of course, directly contradict what he says in your cited entry under mass. But Weisstein still doesn¹t understand. His entry under pound still contains many errors. For example, he claims Unfortunately, there is a great deal of confusion about the deÞnition of the pound, with many authors using the unit to denote the mass-equivalent of 4.448 newtons, namely 453.592 grams. Just what in the world does he imagine the standard for a pound to be, do you suppose? Pounds are ofÞcially deÞned as 0.45359237 newtons. Pounds force are a spinoff from them--something that Eric Weisstein has totally ass-backwards. The pound force is the force-equivalent (at some midrange value of the acceleration of free fall on Earth) of 453.59237 grams. Not the other way around, as Weisstein has it. The pound force is a recent spinoff, something that was never a well-deÞned unit before the 20th century. -- Gene Nygaard http://ourworld.compuserve.com/homepages/Gene_Nygaard/ It¹s not the things you don¹t know what gets you into trouble. It¹s the things you do know that just ain¹t so. Will Rogers === Subject: Re: mass-density or weight-density >> >> Will somebody who _knows_, please explain that to me? > Mass (fundamental) > http://scienceworld.wolfram.com/physics/Mass.html > That doesn¹t claim any fundamentalness for mass. > Like I told you before Who cares what you have told people. Science will progress despite you, Gene. Go away, crackpot. === Subject: Re: mass-density or weight-density >Who cares what you have told people. Science will progress despite you, >Gene. >Go away, crackpot. Stange comments from a former crackpot, no longer even worthy of that title because he hasn¹t even had anything substantive to say about science for 31 months! Gene Nygaard Gene Nygaard http://ourworld.compuserve.com/homepages/Gene_Nygaard/ === Subject: Re: mass-density or weight-density > When Newton said at the beginning of his Principia that mass is the > bulk and density of a body jointly: What did he mean? > The product of bulk [volume] and density is either the volume x the > mass-density [mass/unit volume]; or volume x weight-density > [weight/unit volume]: > Either way, the volume Œdrops out¹; leaving: Mass equals mass; or Mass > equals weight. > Will somebody who _knows_, please explain that to me? > Why bother? You¹re not going to listen to anything that contradicts your own > opinion? You are right about that; because my opinion is that mass and inertia are _not_ fundamental measures: Mass is the ratio of net force [f] divided by the acceleration [a] that it causes; where the ratio of a body¹s net weight [w] divided by the deceleration [g] at which it is restrained at the location of the weight-scale on which it is weighed; is a special; particular case. Try explaining the question with something more compatible to my opinion. That I might listen to. === Subject: Re: mass-density or weight-density > Try explaining the question with something more compatible to my > opinion. That I might listen to. That pretty much sums up Donny¹s usenet life. === Subject: Re: mass-density or weight-density Hello. You are wrong. > When Newton said at the beginning of his Principia that mass is the > bulk and density of a body jointly: What did he mean? > The product of bulk [volume] and density is either the volume x the > mass-density [mass/unit volume]; or volume x weight-density > [weight/unit volume]: > Either way, the volume Œdrops out¹; leaving: Mass equals mass; or Mass > equals weight. > Will somebody who _knows_, please explain that to me? === Subject: Re: Unabomber see the Fall 2000 ŒWAND¹ Review, for much of the administration¹s current policy -- talk about Orwellian! I call it, The Invisible John, because it¹s actually a hype for the emmissions trading scheme of the Kyoto Protocol, dysguised as a love-story taht got the Oscar for not being the worst such effort. fortunately, the movie does not depict the Public Bathroom sequence, which could have been a set-up; that¹s just speculation, but it is what seems to have pushed him over the edge, and out of the perview of MacNamara, Kissinger et al as vomitorium. of course, they used his math in their war-gaming, thereafter. thus quoth: >In actual history didn¹t that person actually do work for the CIA and the >NSA ? The movie¹s depiction of John Nash¹s delusions was complete Þction, with no resemblance to the actual case. Nash did work on game theory as a consultant for the RAND Corporation. > Finally, is there a missing fundamental of a McCarty era political network > combining with aggressive personnel management systems such that individuals > not favored or not trusted would Þnd themselves in unexpected conþict ? > I mean this if someone were independent and aloof while working for defense > industries or agencies... --Give Earth a Trickier Dick Cheeny -- out of ofÞce, after gigayears! http://www.benfranklinbooks.com/ http://members.tripod.com/~american_almanac http://www.wlym.com/pdf/iclc/HowTheNation.PDF http://www.rand.org/publications/randreview/issues/rr.12.00/ http://www.rwgrayprojects.com/synergetics/plates/Þgs/plate02. html === Subject: Re: Unabomber >Well, BM was on network TV a few nights ago and in that production the >mathematician had a delusional side life where he pretended to work for the >U.S. government on a code breaking mission and notes of the code breaking >were the towers of insanity. >In actual history didn¹t that person actually do work for the CIA and the >NSA ? The movie¹s depiction of John Nash¹s delusions was complete Þction, with no resemblance to the actual case. Nash did work on game theory as a consultant for the RAND Corporation. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Unabomber > It might be difÞcult to say why T. Kaczynski¹s mathematical background > was not given much media attention, although I heard about it quite some > time ago (don¹t remember how) and discovered this link to some of his > works. > http://www.rpi.edu/~bulloj/tjk/tjk.html > Who says it wasn¹t given much media attention? _____________________________________________________ I think the original poster did. I¹m not sure I agree, but then that¹s a matter of opinion. _____________________________________________________ It received rather a > lot of attention at the time I thought. > Of course, the OP was pointing to posts prior to the identiÞcation of > Kaczynski as the Unabomber. _____________________________________________________ I hadn¹t considered that possibility -- I was thinking of only of after the Unabomber¹s capture. _____________________________________________________ > -- > Jesse F. Hughes > That¹s the base tautological space where by tautological space I mean > a region of truth. -- James S. Harris does philosophy of mathematics. > JSH is a renaissance man. === Subject: Re: Unabomber > I think the original poster did. I¹m not sure I agree, but then that¹s > a matter of opinion. I didn¹t say it wasn¹t given much media attention. I thought it got more than enough attention (in fact I thought they concentrated on the fact he was a mathematician) > I hadn¹t considered that possibility -- I was thinking of only of after the > Unabomber¹s capture. I originally said I wonder was he the Þrst to suspect the unabomber was a mathematician. This of course means before they -knew- he was a mathematician. -- InterBang === Subject: Re: Stupid question Expires: 28 days >Cut< >> I wonder if you claim that if that wall is the wall of a centrifuge, that >> you are not exerting a centrifugal force (= away from t he center or axis) on >> the wall. >> Harald >Indeed, I do not claim that the walls of centrifuges and spin-dry >washing machines are not being subjected to considerable centrifugal >force: Nor do I deny that þywheels burst because of the centrifugal >force that tends to make them þy away from their axis; in proportion >to their angular momentum. Not so. Their molecules simply want to continue in a straight line but are forced to accelerate inwards by a mechanical force. >A bucket¹s bottom restrains the centrifugal force exerted by the water >in it when it¹s whirled over our head; otherwise the water, if it >didn¹t exert centrifugal force, would fall to the ground; when the >bucket is bottomside up(;^) No wrong again. Yo can accelerate a bucket of water straight upwards and get the same effect. The water merely resists acceleration. Henri Wilson. www.users.bigpond.com/hewn/index.htm See how three orbiting bodies interact: www.users.bigpond.com/hewn/threebody.exe See proof that light speed is source dependent. www.users.bigpond.com/hewn/variablestars.exe === Subject: Re: Stupid question Expires: 28 days >>I wonder if you claim that if that wall is the wall of a centrifuge, that >>you are not exerting a centrifugal force (= away from the center or axis) >>the wall. >> >>Harald >> >> Centrifugal force is an imaginary force that exists in rotating frames. >> For instance, if you twirl a mass around on a string, you feel the >centripetal >> force pulling on your Þnger. >I disagree! It is correct to state that your Þnger exerts a centripetal >force on the mass, as it pulls towards the centre. But inversely, the force >that pulls on your Þnger pulls it *away* from the centre. That¹s a trap. You Þnger also rotates (ever so slightly) around the centre of gravity of the system. So the frce on your Þnger is also TOWARDS the barycentre. >> The mass feels the centripetal force pulling it >> towards the centre. The tension in the string equals the centripetal >force. >Exactly. >> In a frame rotating with the mass, there is no straightforward reason for >the >> existence of any tension in the string, since nothing is moving wrt the >frame. >> The centrifugal force was devised to explain this (like the Coriolis >force) . >> PS: I don¹t like rotating frames, at all. They should never have been >invented. >I don¹t think they were invented, they were discovered! (Just try a >merry-go-round). >If this madness continues, one day the word centrifuge will be banned, and >at home we¹ll only have centripetes for drying our wash... I agree that it does sound like madness but it actualy isn¹t. There is always a force balancing a centripetal one but it isn¹t Œcentrifugal¹. It is a tension in a string or a Œforce like gravity¹. >Harald Henri Wilson. www.users.bigpond.com/hewn/index.htm See how three orbiting bodies interact: www.users.bigpond.com/hewn/threebody.exe See proof that light speed is source dependent. www.users.bigpond.com/hewn/variablestars.exe === Subject: Re: neurons and artiÞcial intelligence 2 > Perhaps calling Minsky an idiot is going too far, but the truth is that > he has, time and again, come up with forecasts about AI that proved to be > wildly off the mark. He clearly got carried away by the initial successes > of the 60s, when a number of problems (which with the beneÞt of > hindsight we know to be very simple) were spectacularly solved by the then > nascent AI discipline. He should have known better than to be so > exuberant in his predictions; circumspection is a virtue in general, much > more so when doing science. > In no small measure, Minsky¹s (and others¹) unbridled prognosticating > enthusiasm is one of the reasons why the reputation of the AI world > remains tarnished, when viewed from other scientiÞc endeavors. That¹s a nice theory, but do you have some examples of such Œtime and again¹ forecasts written by me, or are you repeating some rumors you have heard, or careless secondhand attributions? Good theories need good evidence. === Subject: Re: neurons and artiÞcial intelligence 2 > now the 2 parrots of behaviorpism are calling the father of AI an idiot. > you guys are too true for words. > Perhaps calling Minsky an idiot is going too far, but the truth is that > he has, time and again, come up with forecasts about AI that proved to be > wildly off the mark. It is absolutely going too far. I¹ve heard him direct some criticism towards some of the research, but failing forecasts are not uncommon among AI researchers. How many AI forecasts do you think truly predicted decades ahead? Can you count speech recognition software in 2001: A Space Odyssey movie a good prediction? Who made the forecasts that hit the mark? Be speciÞc. Tell us which forecasts of Minsky are wildly off the mark, otherwise you appear to be in support of our famous net.kooks. > He clearly got carried away by the initial successes > of the 60s, when a number of problems (which with the beneÞt of > hindsight we know to be very simple) were spectacularly solved by the then > nascent AI discipline. He should have known better than to be so > exuberant in his predictions; circumspection is a virtue in general, much > more so when doing science. Which predictions? Try to be speciÞc. > In no small measure, Minsky¹s (and others¹) unbridled prognosticating > enthusiasm is one of the reasons why the reputation of the AI world > remains tarnished, when viewed from other scientiÞc endeavors. Who are these others? And in what sense do you think reputation of the AI world remains tarnished? I¹d personally attribute damage in reputation to the excessive optimism of people who saw genetic algorithms and neural networks as something magical. Now, it has turned out they are not. What we need is more fundamental research instead of toy applications. Why is that surprising? -- Eray Ozkural === Subject: Re: neurons and artiÞcial intelligence 2 <2Lj29fS7vnsAFwNQ@longley.demon.co.uk> >> now the 2 parrots of behaviorpism are calling the father of AI an idiot. >> you guys are too true for words. >> Perhaps calling Minsky an idiot is going too far, but the truth is that >> he has, time and again, come up with forecasts about AI that proved to be >> wildly off the mark. >It is absolutely going too far. >I¹ve heard him direct some criticism towards some of the research, but >failing forecasts are not uncommon among AI researchers. How many AI >forecasts do you think truly predicted decades ahead? Can you count >speech recognition software in 2001: A Space Odyssey movie a good >prediction? Who made the forecasts that hit the mark? >Be speciÞc. Tell us which forecasts of Minsky are wildly off the >mark, otherwise you appear to be in support of our famous net.kooks. >> He clearly got carried away by the initial successes >> of the 60s, when a number of problems (which with the beneÞt of >> hindsight we know to be very simple) were spectacularly solved by the then >> nascent AI discipline. He should have known better than to be so >> exuberant in his predictions; circumspection is a virtue in general, much >> more so when doing science. >Which predictions? Try to be speciÞc. >> In no small measure, Minsky¹s (and others¹) unbridled prognosticating >> enthusiasm is one of the reasons why the reputation of the AI world >> remains tarnished, when viewed from other scientiÞc endeavors. >Who are these others? And in what sense do you think reputation of the >AI world remains tarnished? I¹d personally attribute damage in >reputation to the excessive optimism of people who saw genetic >algorithms and neural networks as something magical. Now, it has >turned out they are not. What we need is more fundamental research >instead of toy applications. Why is that surprising? >Eray Ozkural Because it is quite clear to people older and more experienced than you that Minsky and others (such as Miller, Chomsky, Bruner and other cognitivists) did not understand what Skinner, Quine and many of those who worked with them were saying and doing. This led them to repeat what little they *did* (tacitly) glean albeit intensionally, and get a lot else very *wrong* besides. This happened, probably, for no other reason that 1) what Skinner and Quine etc had to say was far more subtle than they appreciated and 2) young people who choose these disciplines are far too easily led by rhetoric. You, at present, are too young, too inexperienced and too naive to grasp any of this. You should evaluate what has been done by the respective groups by what has actually been demonstrably achieved, not by what they claim they have achieved. Failure to base your evaluation on that is just to be seduced by intensional heuristics and nefarious rhetoric. In the Þnal analysis, you have to make a choice. -- David Longley === Subject: Re: neurons and artiÞcial intelligence 2 > >>GS:Nice try Rickert. Even mentalists will recognize Thorndike¹s and >>Skinner¹s law of effect (even if idiots like Minsky do not). But tell >>me something, Mr. Originality, what IS the role of the setting in which >>an animal stimulates the environment and gets it to give it food? >> >> >> >>You really are pitiful. >> >> >> >> >> DL: To state the obvious (to others), Rickert¹s behaviour (and Minsky¹s >> perhaps) are clear examples of how people (*authoritatively* in the >> latter¹s case) propagate and reinforce myths. > > > now the 2 parrots of behaviorpism are calling the father of AI an idiot. > you guys are too true for words. > Perhaps calling Minsky an idiot is going too far, but the truth is that > he has, time and again, come up with forecasts about AI that proved to be > wildly off the mark. He clearly got carried away by the initial successes > of the 60s, when a number of problems (which with the beneÞt of > hindsight we know to be very simple) were spectacularly solved by the then > nascent AI discipline. He should have known better than to be so > exuberant in his predictions; circumspection is a virtue in general, much > more so when doing science. > In no small measure, Minsky¹s (and others¹) unbridled prognosticating > enthusiasm is one of the reasons why the reputation of the AI world > remains tarnished, when viewed from other scientiÞc endeavors. easier to forgive exuberance than rude parrots who preach only neagativity. === Subject: Re: neurons and artiÞcial intelligence 2 <2Lj29fS7vnsAFwNQ@longley.demon.co.uk> >> >GS:Nice try Rickert. Even mentalists will recognize Thorndike¹s and >Skinner¹s law of effect (even if idiots like Minsky do not). But tell >me something, Mr. Originality, what IS the role of the setting in which >an animal stimulates the environment and gets it to give it food? > > > >You really are pitiful. > > > > > DL: To state the obvious (to others), Rickert¹s behaviour (and Minsky¹s > perhaps) are clear examples of how people (*authoritatively* in the > latter¹s case) propagate and reinforce myths. >> >> >> now the 2 parrots of behaviorpism are calling the father of AI an idiot. >> you guys are too true for words. >> Perhaps calling Minsky an idiot is going too far, but the truth is that >> he has, time and again, come up with forecasts about AI that proved to be >> wildly off the mark. He clearly got carried away by the initial successes >> of the 60s, when a number of problems (which with the beneÞt of >> hindsight we know to be very simple) were spectacularly solved by the then >> nascent AI discipline. He should have known better than to be so >> exuberant in his predictions; circumspection is a virtue in general, much >> more so when doing science. >> In no small measure, Minsky¹s (and others¹) unbridled prognosticating >> enthusiasm is one of the reasons why the reputation of the AI world >> remains tarnished, when viewed from other scientiÞc endeavors. >easier to forgive exuberance than rude parrots who preach only neagativity. It wasn¹t exuberance that I was correcting, it was the propagation and reinforcement (intentionally or not) of speciÞc falsehoods. To an ignorant, undiscriminating reader like you, that appears, no doubt, to be no more than a slur. Careful criticism is fundamental to the practice of science. It¹s anathema to pseudo-scientiÞc mythologies like Cognitivism. -- David Longley