mm-157 I just gave a post which *can* clear up the main issues if you follow> it all the way through, but it's long, the issues are advanced, and I> don't think many of you care what the mathematical truth is.> The primary point that settles things is that 22 is coprime to 7 in> the ring of algebraic integers. Posters using various primitive> arguments, and mostly just *claiming* one position that they cannot> prove, are stuck with their position forcing them to need 22 if you> look one way, and 7 if you believe them, and look another.> But you see 22 is not 7.> By showing> (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) 300125 x^3 - 18375 x^2 - 360 x + 22> where f_1(0) = f_2(0) = f_3(0) = 0, I'm pushing the underlying> polynomial to the fore, and for *that* polynomial the constant term is> 22.> Make no mistake, what I do is use special tools of analysis on the> polynomial 300125 x^3 - 18375 x^2 - 360 x + 22.> It doesn't take much to see that if the f's are linear functions> making the factorization into polynomial factors that multiplying> f_1(x) and f_2(x) by 22 will give you algebraic integer functions.> But I've also used a more complicated solution for the f's where> multiplying them times 7 gives algebraic integer functions.> What gives?> That's what I'm not going to explain in detail to you, except to say> that the simplest explanation is that the de'nition for the ring of> algebraic integers is screwed up enough to let you do some wacky> stuff.> Now then, consider the *demonstrated* apparent contradiction with> (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) 300125 x^3 - 18375 x^2 - 360 x + 22> where f_1(0) = f_2(0) = f_3(0) = 0, as you look for 22 times f_1(x)> and f_2(x) in one case, while in another you look at 7 times them with> the guarantee that you get algebraic integer functions.> What contradiction? The two cases have no connection. Nor would you expect them to.You have noted that there are an in'nite number of solutions to (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22Indeed, let f_1(x) be *any* function such that f_1(v) is not equal to -1/5for any v not a root of (300125 x^3 - 18375 x^2 - 360 x + 22). Thenf_1(x) is part of a triplet that solves the equation. Two possibilitiesare the constants, f_1(x) = 3/2 and f_1(x)=5/3. In the 'rst casewhen we multiply by 2 we get an integer, in the second case when wemultiply by 3 we get an integer. The integers 2 and 3 are coprime.Do you think this is a contradiction? -William Hughes But you see 22 is not 7.> I'm learning so much from this posting, my head is about to explode ...'22 is not 7'. I need a time to digest this notion. If you have any sense at all, you might be now wondering, what does 7> have to do with anything?Lets see:It is at least the number of years that you have been spewing nonsense invarious newsgroups?Yeah, that probably it! = > If you have any sense at all, you might be now wondering, what does 7 > have to do with anything? > Lets see: > It is at least the number of years that you have been spewing nonsense in > various newsgroups? > Yeah, that probably it!No. Multiplied by 6 we get that answer to everything.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ I just gave a post which *can* clear up the main issues if you follow> it all the way through, but it's long, the issues are advanced, and I> don't think many of you care what the mathematical truth is.You don't think -- period.> The primary point that settles things is that 22 is coprime to 7 in> the ring of algebraic integers. Posters using various primitive> arguments, and mostly just *claiming* one position that they cannot> prove, are stuck with their position forcing them to need 22 if you> look one way, and 7 if you believe them, and look another. But you see 22 is not 7.May I quote you? But you see 22 is not 7. -- James Harris> By showing (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 where f_1(0) = f_2(0) = f_3(0) = 0, I'm pushing the underlying> polynomial to the fore, and for *that* polynomial the constant term is> 22. Make no mistake, what I do is use special tools of analysis on the> polynomial 300125 x^3 - 18375 x^2 - 360 x + 22.You made the mistake, unless you'd care to cite which tools of analysisyou are claiming to use here.> It doesn't take much to see that if the f's are linear functions> making the factorization into polynomial factors that multiplying> f_1(x) and f_2(x) by 22 will give you algebraic integer functions.It would help if you gave the values for f_1(x) and f_2(x).> But I've also used a more complicated solution for the f's where> multiplying them times 7 gives algebraic integer functions. What gives? That's what I'm not going to explain in detail to you, except to say> that the simplest explanation is that the de'nition for the ring of> algebraic integers is screwed up enough to let you do some wacky> stuff.Wrong. The simplest explanation is that you are an ignorant blowhard withthe intelligence of an imbecile and the cognitive faculties of a hockypuck.> Now then, consider the *demonstrated* apparent contradiction with (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 where f_1(0) = f_2(0) = f_3(0) = 0, as you look for 22 times f_1(x)> and f_2(x) in one case, while in another you look at 7 times them with> the guarantee that you get algebraic integer functions. If you have any sense at all, you might be now wondering, what does 7> have to do with anything?I'm more interested in whether you might have eaten too much lead paint asa child.--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 =22 wins, right?If not, then I guess I'll never understand football[1].Footnotes: [1] American football, of course.-- Now I'm informing all of you that the people arguing against me are EVIL,yes they are real, live EVIL people as mathematics is that important, soit's important enough for Evil itself to send minions like them. -- James Harris on Evil's interest in Algebraic Number Theory =So I've given(5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22where f_1(0) = f_2(0) = f_3(0) = 0and talked about *two* of the in'nite number of solutions for thef's, but how do you translate between those functions?The answer is *more* functions.Consider(5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = (5 f'_1(x)+ 1)(5 f'_2(x) + 1)(5 f'_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22.You end up with three additional functions for the translations.(5 f_1(x) + 1) = J_1(x)(5 f'_1(x) + 1)(5 f_2(x) + 1) = J_2(x)(5 f'_2(x) + 1)(5 f_3(x) + 1) = J_3(x)(5 f'_3(x) + 1)Now then, any of you have *any* idea what the J functions look like?James HarrisMy math discoveries, found for pro'thttp://mathforpro't.blogspot.com/ So I've given> (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) 300125 x^3 - 18375 x^2 - 360 x + 22> where f_1(0) = f_2(0) = f_3(0) = 0> > and talked about *two* of the in'nite number of solutions for the> f's, but how do you translate between those functions?> The answer is *more* functions.> Consider> (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = > (5 f'_1(x)+ 1)(5 f'_2(x) + 1)(5 f'_3(x) + 22) 300125 x^3 - 18375 x^2 - 360 x + 22.> You end up with three additional functions for the translations.> (5 f_1(x) + 1) = J_1(x)(5 f'_1(x) + 1)> (5 f_2(x) + 1) = J_2(x)(5 f'_2(x) + 1)> (5 f_3(x) + 1) = J_3(x)(5 f'_3(x) + 1)> I wonder ... maybe that last one is supposed to be (5 f_3(x) + 22) = J_3(x)(5 f'_3(x) + 22).> Now then, any of you have *any* idea what the J functions look like?> One good candidate for the J's might be the constant functions J_1(x) = J_2(x) = J_3(x) = 1. Are there others? Do you have any explicit solutions for f_1(x),f_2(x), f_3(x), other than the ones I gave in another thread? More general question: given a number of theform 5*E + 22,where E is an algebraic integer, is it alwayspossible to 'nd *nonzero* algebraic integers A, B, and Csuch that (5 A + 1)(5 B + 1)(5 C + 22) = 5 E + 22 ?Or are there counterexamples? Just curious.Andrzej > James Harris> My math discoveries, found for pro't> http://mathforpro't.blogspot.com/ Now then, any of you have *any* idea what the J functions look like?I certainly don't. Are you asking for help or do you already know theanswer?-- Richard-- FreeBSD rules! So I've given (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 where f_1(0) = f_2(0) = f_3(0) = 0 and talked about *two* of the in'nite number of solutions for the> f's, but how do you translate between those functions?You've talked a lot, but have not produced and values for the ïf's fromthe in'nitue number of solutions you claim exist. Give us *one*solution from that in'nite set, James. Just ONE.> The answer is *more* functions. Consider (5 f_1(x)+ 1)(5 f_2(x) + 1)(5 f_3(x) + 22) = (5 f'_1(x)+ 1)(5 f'_2(x) + 1)(5 f'_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22. You end up with three additional functions for the translations. (5 f_1(x) + 1) = J_1(x)(5 f'_1(x) + 1) (5 f_2(x) + 1) = J_2(x)(5 f'_2(x) + 1) (5 f_3(x) + 1) = J_3(x)(5 f'_3(x) + 1)You haven't changed the values of the ïf's at all. You have merelysubsituted strings of other symbols in place of the ïf's. This does*not* change the number or values of the solutions for the ïf's.Obviously you can invent strings of symbols and stick them in asreplacements for other quantities -- so what? Give us a solution for thevalues of the ïf's. Just ONE.> Now then, any of you have *any* idea what the J functions look like?Who cares?> James Harris My math discoveries, found for pro't> http://mathforpro't.blogspot.com/--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 =appreciated.sum from n=1 to N : sinnx ... I am assuming it wants the answer interms of sinx's.just before this question on the sheet, i had to do sum cosnx up to5.... but no the general case. the current topic is complex numbersso i used the standard methods for the speci'c cases... but I amhaving trouble spotting patterns/generalising to N. (for cos and sinin fact).I am only a 'rst year, so please use ïsimple' methods.Jackie appreciated. sum from n=1 to N : sinnx ... I am assuming it wants the answer in> terms of sinx's. just before this question on the sheet, i had to do sum cosnx up to> 5.... but no the general case. the current topic is complex numbers> so i used the standard methods for the speci'c cases... but I am> having trouble spotting patterns/generalising to N. (for cos and sin> in fact). I am only a 'rst year, so please use ïsimple' methods.We haveexp(inx) = cos(nx) + i sin(nx)sosum(n= 1 to N: exp(inx)) = sum(n = 1 to N: cos(nx)) + i sum(n = 1 to N: sin(nx))Now, the left hand side is a geometric series which can be summed togive(exp((N+1)x) - exp(ix))/(exp(ix) - 1) = (exp(-ix) - 1)(exp(i(N+1)x) - exp(ix))/ (2 - exp(ix) - exp(-ix)) = (exp(iNx) - 1 - exp(i(N+1)x) + exp(ix)) / (2 - 2 cos(x))Taking real and imaginary parts gives the series for cos(nx) and sin(nx)respectively.-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will 'nd these people and we will bring them to justice. =Jackie Tung> appreciated. sum from n=1 to N : sinnx ... I am assuming it wants the answer in> terms of sinx's. just before this question on the sheet, i had to do sum cosnx up to> 5.... but no the general case. the current topic is complex numbers> so i used the standard methods for the speci'c cases... but I am> having trouble spotting patterns/generalising to N. (for cos and sin> in fact). I am only a 'rst year, so please use ïsimple' methods.> We have> exp(inx) = cos(nx) + i sin(nx)> so> sum(n= 1 to N: exp(inx)) = sum(n = 1 to N: cos(nx))> + i sum(n = 1 to N: sin(nx))> Now, the left hand side is a geometric series which can be summed to> give> (exp((N+1)x) - exp(ix))/(exp(ix) - 1)> = (exp(-ix) - 1)(exp(i(N+1)x) - exp(ix))/ (2 - exp(ix) - exp(-ix))> = (exp(iNx) - 1 - exp(i(N+1)x) + exp(ix)) / (2 - 2 cos(x))> Taking real and imaginary parts gives the series for cos(nx) and sin(nx)> respectively. appreciated.> sum from n=1 to N : sinnx ... I am assuming it wants the answer in> terms of sinx's.sin nx = Im (e^(inx))So sum from n=1 to N : sinnx = Im (sum from n=1 to N : e^(inx))You cann see the sum of terms of a geometric serie. I let you conclude.-- =In case you have not already seen it:We are trying to set up a new newsgroup misc.metric-systemdedicated to discussions about the metric system and its introductionin areas where it is not yet widely used.Related threads have §ared up in this group in the past. So if youare interested in this topic and group, please participate in thisballot. (At least ~150 votes are necessary before a group has arealistic chance of being created.)The second Call for Votes with a description of the proposed groupand exact instructions on how to vote has recently been posted to news.announce.newgroupsIt is also available by simply replying to this message or by sending metric-cfv-request@netagw.comand it can be downloaded from the USENET Volunteer Votetakers websiteon http://www.uvv.org/cgi-bin/getmsg/2471the instructions given in the Call for Votes. Your vote has beencounted only if you receive a con'rmation reply.Markus-- Markus Kuhn, Computer Laboratory, University of Cambridgehttp://www.cl.cam.ac.uk/~mgk25/ || CB3 0FD, Great Britain How does 2x(2sinx + 3cosx) + (x^2 - 3)(2cosx - 3sinx) become (2x^2 + 6x - 6)cosx + (-3x^2 + 4x + 9)sinx sure that I understand how tomake my data more Laplacian... again, this is not for cheating purpose andit is a problem of proper cause.My question is: suppose in literature people know well that this data shouldbe in Laplacian distribution... but due to some distortion my data is notLaplacian distortion and unfornutely I don't know what the distribution itis... but in order to restore as much as possible, we need to restore theLaplacian distribution...So the question is:How can we go from unknown distributed data to Laplacian distributed data?Please give me some help! I am urgently in need of this,-Walala still not sure that I understand how to>make my data more Laplacian... again, this is not for cheating purpose and>it is a problem of proper cause.>My question is: suppose in literature people know well that this data should>be in Laplacian distribution... but due to some distortion my data is not>Laplacian distortion and unfornutely I don't know what the distribution it>is... but in order to restore as much as possible, we need to restore the>Laplacian distribution...>So the question is:>How can we go from unknown distributed data to Laplacian distributed data?>Please give me some help! I am urgently in need of this,>-WalalaDon't. It is only ignorance which would lead one to evenconsider doing this.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue =How does(3x + 1) / root xbe manipulated to(3 root x) + (1 / ( root x))>What are the steps? Let me try again but shorter than my original answer:(3x + 1) / root(x) = 3x/root(x) + 1/root(x)Multiply one of those terms by 1 to rationalize the denominators ... 'll inthe steps =I sent a response to sci.math for this question and it never appeared. Now,I'm just checking to see if this message reaches the newsgroup. I sent a response to sci.math for this question and it never> appeared. Now, I'm just checking to see if this message reaches> the newsgroup. Maybe it's of interest to you that news@cs.com =sdl@msn.com askes:>How does(3x + 1) / root xbe manipulated to(3 root x) + (1 / ( root x))>What are the steps?they were merely counting the same denomination. You do just the reverse ofthat here:(3x +1)/root(x) = 3x/root(x) + 1/root(x)Maybe you do not want a square root expression in one of your denominators, butfor some unstated reason, you still are comfortable having square rootretained in the other term. OK, 'ne,[3x/root(x)][root(x) / root(x)] + 1/root(x)that is just multiplying top and bottom of the 'rst term by root(x). Continue the multiplication and simpli'cation and you obtain3root(x) + 1/root(x)which is what you wanted.G C =It's not knowledge itself that has power as *social* forces rule, andI have a simple example to show you what I mean.A little while back I found a nifty way to *'nd* and count primenumbers that you can't 'nd in any math textbook:With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you getS(x,y) when y doesn;'t equal one by summingdS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,sqrt(j-1))],incrementing j from 2 to y in steps of 1.That little bit of math not only counts prime numbers, but 'ndsindividual prime numbers along the way--a 'rst in recorded humanhistory for a discrete closed function.That's *pure* knowledge which has run into a *social* block, which isa hatred between me and mathematicians.Mathematicians hate me for many reasons, and they have the socialpower to ignore or attack knowledge because I discovered it.I hate mathematicians because they are capable of attacking knowledgefor social reasons.However, *society* gives power, so the knowledge is held hostagebetween mathematicians and me. Many of you, pawns to society, areincapable of appreciating the pure knowledge if you're told that it'snot important, so mathematicians tell you that it's not, though it'spowerful in ways that no one has *ever* seen before.Human beings are social creatures. You need society to feel safe, andto tell you what is important.So you see, knowledge is NOT power. Society is.James HarrisMy math discoveries, found for pro'thttp://mathforpro't.blogspot.com/ Mathematicians hate me for many reasonsNo, no, no, James, I don't _hate_ you. I _like_ you. If I see a post by youor a thread started by you, I know I'm in for some 'rst class nonsense, andthere's nothing like 'rst class nonsense for entertainment value.-- Wolf Kirchmeir, Blind River ON CanadaNature does not deal in rewards or punishments, but only in consequences.(Robert Ingersoll) Mathematicians hate me for many reasons, and they have the social> power to ignore or attack knowledge because I discovered it.If mathematicians are such powerful and effective conspirators, why ain't we rich? Mathematicians hate me for many reasons,No, no, no.Mathematicians don't *hate* you at all! You're simply not signi'cantenough to cause such a strong emotional reaction in anyone.-jcr Mathematicians hate me for many reasons,> No, no, no.> Mathematicians don't *hate* you at all! You're simply not signi'cant> enough to cause such a strong emotional reaction in anyone.> -jcrAhem.He elicits lots of emotional reactions, some of them quite vehement,right here on sci.math, which is where I read him.Yeah, I guess I'm sortof addicted to JSH myself. But I can quit anytime I want to! Really I can!-- Chris HenrichMoral indignation is jealousy with a halo. -- H. G. Wells It's not knowledge itself that has power as *social* forces rule, and> I have a simple example to show you what I mean.> A little while back I found a nifty way to *'nd* and count prime> numbers that you can't 'nd in any math textbook:> With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you get> S(x,y) when y doesn;'t equal one by summing> dS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,> sqrt(j-1))],> incrementing j from 2 to y in steps of 1.> That little bit of math not only counts prime numbers, but 'nds> individual prime numbers along the way--a 'rst in recorded human> history for a discrete closed function.> That's *pure* knowledge which has run into a *social* block, which is> a hatred between me and mathematicians.> Mathematicians hate me for many reasons, and they have the social> power to ignore or attack knowledge because I discovered it.> I hate mathematicians because they are capable of attacking knowledge> for social reasons.> However, *society* gives power, so the knowledge is held hostage> between mathematicians and me. Many of you, pawns to society, are> incapable of appreciating the pure knowledge if you're told that it's> not important, so mathematicians tell you that it's not, though it's> powerful in ways that no one has *ever* seen before.> Human beings are social creatures. You need society to feel safe, and> to tell you what is important.> So you see, knowledge is NOT power. Society is.> > James Harris> My math discoveries, found for pro't> http://mathforpro't.blogspot.com/Dude, if youd gone to college all this time youve been pesteringsci.*, by now youd have your Ph.D. and people would take you seriouslybartmanbartmanbartmanbartmanbartmanbartmanbartmanbartm an It's not knowledge itself that has power as *social* forces rule, and> I have a simple example to show you what I mean.> A little while back I found a nifty way to *'nd* and count prime> numbers that you can't 'nd in any math textbook:> > With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you get> S(x,y) when y doesn;'t equal one by summing> dS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,> sqrt(j-1))],> incrementing j from 2 to y in steps of 1.> That little bit of math not only counts prime numbers, but 'nds> individual prime numbers along the way--a 'rst in recorded human> history for a discrete closed function.> That's *pure* knowledge which has run into a *social* block, which is> a hatred between me and mathematicians.> Mathematicians hate me for many reasons, and they have the social> power to ignore or attack knowledge because I discovered it.> I hate mathematicians because they are capable of attacking knowledge> for social reasons.> However, *society* gives power, so the knowledge is held hostage> between mathematicians and me. Many of you, pawns to society, are> incapable of appreciating the pure knowledge if you're told that it's> not important, so mathematicians tell you that it's not, though it's> powerful in ways that no one has *ever* seen before.> Human beings are social creatures. You need society to feel safe, and> to tell you what is important.> So you see, knowledge is NOT power. Society is.> > James Harris> My math discoveries, found for pro't> http://mathforpro't.blogspot.com/> Dude, if youd gone to college all this time youve been pestering> sci.*, by now youd have your Ph.D. and people would take you seriouslyNo they wouldn't. Look at Halton Arp, a distinguished scientist witha PH.D, and besides, why isn't *knowledge* important?Why should I kiss mathematician's butts to get a Ph.D when I have PUREKNOWLEDGE itself in the form of the discovery?You make my point. Knowledge isn't power. Society is power.To many of you, the result doesn't matter. It's what someone elsetells you about it that does.Yet mathematicians CLAIM to care about math for math's sake, and callit pure math but I present a wonderful pure math result and theyattack me and the result.Mathematicians are liars. And all their talk about beauty and purityin mathematics is just a con game.James Harris Dude, if youd gone to college all this time youve been pestering> sci.*, by now youd have your Ph.D. and people would take you seriously> No they wouldn't. Look at Halton Arp, a distinguished scientist with> a PH.D, and besides, why isn't *knowledge* important?Harlan Arp IS taken seriously. Knowlwdge is importqant, but people who do not have the knowlegde they claim to have, are not.> Why should I kiss mathematician's butts to get a Ph.D when I have PURE> KNOWLEDGE itself in the form of the discovery?You may have pure knowledge of some arcane sort, but as you are totally incapable of explaining what you say you have to anyone else, it does you no more good than if you did not have it.> You make my point. Knowledge isn't power. Society is power.The ability to communicate one's knowledge or to apply knowledge one's knowledge is what gives that knowledge value and JSH has neither ability.> To many of you, the result doesn't matter. It's what someone else> tells you about it that does.In mathematics, it is results that one can verify for ones self that matter. And JSH does not present any results that one can verify for oneself, except for those he takes from others.> Yet mathematicians CLAIM to care about math for math's sake, and call> it pure math but I present a wonderful pure math result and they> attack me and the result.All such attacks are counterattacks, only after JSH has attacked the mathematical community wholesale. Mathematical questions about mathematical derivations are not attacks, except to the paranoid.> Mathematicians are liars. And all their talk about beauty and purity> in mathematics is just a con game.Then the have certainly conned you into making an ass of yourself publicly and repeatedly.> James Harris > Dude, if youd gone to college all this time youve been pestering> sci.*, by now youd have your Ph.D. and people would take you seriously>not to metnion the fact that he would have actually learned some math...> No they wouldn't. Look at Halton Arp, a distinguished scientist with> a PH.D, and besides, why isn't *knowledge* important?>As always, the stupifying illogical mind of JSH strikes again, this timeinferring the nature of a population from a rare event.This is the mathematical equivalent of estimating the mean from an outlier.Do you ever read your own posts over and realize how stupid and illogicalthe diarrhea spouting from you really is?MB It's not knowledge itself that has power as *social* forces rule, and> I have a simple example to show you what I mean. A little while back I found a nifty way to *'nd* and count prime> numbers that you can't 'nd in any math textbook: With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you get> S(x,y) when y doesn;'t equal one by summing dS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,> sqrt(j-1))], incrementing j from 2 to y in steps of 1. That little bit of math not only counts prime numbers, but 'nds> individual prime numbers along the way--a 'rst in recorded human> history for a discrete closed function. That's *pure* knowledge which has run into a *social* block, which is> a hatred between me and mathematicians. Mathematicians hate me for many reasons, and they have the social> power to ignore or attack knowledge because I discovered it. I hate mathematicians because they are capable of attacking knowledge> for social reasons. However, *society* gives power, so the knowledge is held hostage> between mathematicians and me. Many of you, pawns to society, are> incapable of appreciating the pure knowledge if you're told that it's> not important, so mathematicians tell you that it's not, though it's> powerful in ways that no one has *ever* seen before. Human beings are social creatures. You need society to feel safe, and> to tell you what is important. So you see, knowledge is NOT power. Society is.> James Harris My math discoveries, found for pro't> http://mathforpro't.blogspot.com/ Dude, if youd gone to college all this time youve been pestering> sci.*, by now youd have your Ph.D. and people would take you seriously bartmanbartmanbartmanbartmanbartmanbartmanbartmanbartmanWe've been telling him that for so long, but he says he's not interested ingetting a Ph.D. in Math. In his case, we really shouldn't feel sorry for hisignorance because he refuses to learn.David Moran It's not knowledge itself that has power as *social* forces rule, and> I have a simple example to show you what I mean.If knowledge *did* have power, it would have crushed you long ago. You'vebeen clobbered by the 2x4-of-truth often enough to leave dents in castiron. But social forces don't seem to affect you, either. I suppose youshould be treated like petri'ed stump -- umoved by truth and as smart asa box of hammers.> A little while back I found a nifty way to *'nd* and count prime> numbers that you can't 'nd in any math textbook: With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you get> S(x,y) when y doesn;'t equal one by summing dS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,> sqrt(j-1))], incrementing j from 2 to y in steps of 1. That little bit of math not only counts prime numbers, but 'nds> individual prime numbers along the way--a 'rst in recorded human> history for a discrete closed function.No it isn't. A simple sieve can do that.> That's *pure* knowledge which has run into a *social* block, which is> a hatred between me and mathematicians.You are a hateful person. What do you expect?> Mathematicians hate me for many reasons, and they have the social> power to ignore or attack knowledge because I discovered it. I hate mathematicians because they are capable of attacking knowledge> for social reasons.The record shows that you hate them for 'nding your errors.> However, *society* gives power, so the knowledge is held hostage> between mathematicians and me. Many of you, pawns to society, are> incapable of appreciating the pure knowledge if you're told that it's> not important, so mathematicians tell you that it's not, though it's> powerful in ways that no one has *ever* seen before. Human beings are social creatures. You need society to feel safe, and> to tell you what is important.I assume you are *not* a human being, from that last paragraph. (I hadsome clues from your past behavior, but this seems like an admission.)> So you see, knowledge is NOT power. Society is.This is a math newsgroup, not a power group. If you want power, seek apolitical of'ce.--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 = It's not knowledge itself that has power as *social* forces rule, and> I have a simple example to show you what I mean. If knowledge *did* have power, it would have crushed you long ago. You've> been clobbered by the 2x4-of-truth often enough to leave dents in cast> iron. But social forces don't seem to affect you, either. I suppose you> should be treated like petri'ed stump -- umoved by truth and as smart as> a box of hammers.LOL. Somehow, your anaology reminds me of the old bumper-sticker Forget theugly stick . You was beat by an ugly FOREST! In Jim's case you'd replaceugly with stupid.MB It's not knowledge itself that has power as *social* forces rule, and>I have a simple example to show you what I mean.>A little while back I found a nifty way to *'nd* and count prime>numbers that you can't 'nd in any math textbook:>With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you get>S(x,y) when y doesn;'t equal one by summing>dS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,>sqrt(j-1))],>incrementing j from 2 to y in steps of 1.>That little bit of math not only counts prime numbers, but 'nds>individual prime numbers along the way--a 'rst in recorded human>history for a discrete closed function.Put it to the test.Demonstrate the usefulness of your formula, by using it to break RSAencryption.If you succeed in doing that, I guarantee that people will starttaking notice. It's not knowledge itself that has power as *social* forces rule, and>I have a simple example to show you what I mean.>A little while back I found a nifty way to *'nd* and count prime>numbers that you can't 'nd in any math textbook:>With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you get>S(x,y) when y doesn;'t equal one by summing>dS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,>sqrt(j-1))],>incrementing j from 2 to y in steps of 1.>That little bit of math not only counts prime numbers, but 'nds>individual prime numbers along the way--a 'rst in recorded human>history for a discrete closed function.> Put it to the test.> Demonstrate the usefulness of your formula, by using it to break RSA> encryption.> If you succeed in doing that, I guarantee that people will start> taking notice.My point is that my discovery is PURE MATH, and PURE KNOWLEDGE,representing a previously unknown formula for both 'nding andcounting primes.Mathematicians claim that beauty and mathematical knowledge areimportant in and of themselves, but here with an *actual* case of apure math discovery people keep hitting me with practical application.Mathematicians are cons and liars.I have the pure math result that shows the reality behind the lies.When they talk about caring about beauty in mathematics, they'relying.You just don't get much more beautiful or purer than a closed functionfor counting prime numbers, previously unknown to humanity.Mathematicians are LIARS!!!James Harris It's not knowledge itself that has power as *social* forces rule, and>I have a simple example to show you what I mean.A little while back I found a nifty way to *'nd* and count prime>numbers that you can't 'nd in any math textbook:With p(x, y) = §oor(x) - S(x, y) - 1, where S(x,1) = 0, and you get>S(x,y) when y doesn;'t equal one by summingdS(x,j) = [p(x/j, j-1) - p(j-1, sqrt(j-1))][ p(j, sqrt(j)) - p(j-1,>sqrt(j-1))],incrementing j from 2 to y in steps of 1.That little bit of math not only counts prime numbers, but 'nds>individual prime numbers along the way--a 'rst in recorded human>history for a discrete closed function.That's *pure* knowledge which has run into a *social* block, which is>a hatred between me and mathematicians.Mathematicians hate me for many reasons, You think so? If they hate you it's for exactly one reason.You illustrated the reason in another thread just now, whenyou said>If you ing morons think that I will let you get away with not>giving me credit for my ing math discoveries then you have another>ing thing coming.What the ??!!!Where's pure math now, huh? Where's loving math for the ing>beauty of it now you ing s??!!!LOOK AT IT!!!Here is the partial difference equation and instructions for>integrating.>dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,>sqrt(y-1))],S(x,1) = 0.And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dS>from dS(x,2) to dS(x,y).http://mathforpro't.blogspot.com/>You ing s. I will get credit for my discovery and get ing>paid, and you best believe that I will not ing let you stupid>s get away with your ing stupid bull--pure math my>ass--without me coming at you ers with some ing PURE ing>PURE AS math that you stupid s have been ting on for over>a ing YEAR!!!You goddamn S!!!What the is wrong with you s??!!! Don't you even believe in>your own stupid ? Where's pure math now?Where is it?>James HarrisYou've got that prime counting thing down pat. Time to start workingon the problem of counting reasons mathematicians hate you - thereare still a few bugs in whatever routine returned the estimate many above.David C. Ullrich =If you ing morons think that I will let you get away with notgiving me credit for my ing math discoveries then you have anothering thing coming.What the ??!!!Where's pure math now, huh? Where's loving math for the ingbeauty of it now you ing s??!!!LOOK AT IT!!!Here is the partial difference equation and instructions forintegrating.dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,sqrt(y-1))],S(x,1) = 0.And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dSfrom dS(x,2) to dS(x,y).http://mathforpro't.blogspot.com/You ing s. I will get credit for my discovery and get ingpaid, and you best believe that I will not ing let you stupids get away with your ing stupid bull--pure math myass--without me coming at you ers with some ing PURE ingPURE AS math that you stupid s have been ting on for overa ing YEAR!!!You goddamn S!!!What the is wrong with you s??!!! Don't you even believe inyour own stupid ? sgomori@host174.subnet75.battelle.org = paid,Money is one of two things mathmeticians do not get much of grammar usage. The lively discussion between another thing and> another think has taught me the actual use of the term (I always> heard it as thing, cpompletely unaware of the think version).> Oh yeah, the word also seems to be used quite a bit in your> post too. Not really sure I have learned much from that.> Remember, the FCC recently said it's ok to drop F-bombs on network television, as long as it's not used to refer to anything sexual. So JSH's usage does in fact conform to current FCC policy. =First, do you have a f*ck fetish? Your use of f*ck is extreme. Ihave never seen anybody knowledgealbe in math use f*ck nearly as muchas you have here.> If you f*cking morons think that I will let you get away with not> giving me credit for my f*cking math discoveries then you have another> f*cking thing coming.> What discovery? You can have your fatally §awed factorization. Asfor your PDE, it looks like a rehash of Legendre's Formula. See:http://mathworld.wolfram.com/LegendresFormula.html> What the f*ck??!!!> Where's pure math now, huh? Where's loving math for the f*cking> beauty of it now you f*cking s??!!!> > LOOK AT IT!!!> Here is the partial difference equation and instructions for> integrating.> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))],> S(x,1) = 0.> And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dS> from dS(x,2) to dS(x,y).> http://mathforpro't.blogspot.com/> You f*cking s. I will get credit for my discovery and get f*cking> paid, and you best believe that I will not f*cking let you stupidChange ïpaid' to ïlaid', it would be more consistent with your ïf*ck'rampage.> f*cks get away with your f*cking stupid bull--pure math my> ass--without me coming at you f*ckers with some f*cking PURE f*cking> PURE AS F*CK math that you stupid s have been ting on for over> a f*cking YEAR!!!> You goddamn F*CKS!!!> What the f*ck is wrong with you s??!!! Don't you even believe in> your own stupid ? Where's pure math now?> Where is it?> James Harris =be will have to read them. In thefuture, kindly do not post any text to these newsgroups. later this evening> so that no one will have to read them.what, exactly, gives you the authority to do such thing?> In the future, kindly do not post any text to these> newsgroups.> that no one will have to read them.||what, exactly, gives you the authority to do such thing?i said he could.-- If you ing morons think that I will let you get away with not>giving me credit for my ing math discoveries then you have another>ing thing coming.>What the ??!!!>Where's pure math now, huh? Where's loving math for the ing>beauty of it now you ing s??!!!You got Tourette's syndrome now, Jimmy? That's a pretty serious issue.Doug If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming.snip...Forgive me for posting to the original (I can read usenet from mynewsreader, but not write -- so I post on Google) as this is reallyaddressed to the far off-topic (but far more interesting) sub-threadabout ... another thing/k coming. I heard a phrase on /. the person THEDEVIL'S GOING TO NEED A WIG. Has anyone heard that phrase before? Forsome reason, that cracked me up. I interpret it to mean that the guyis going to do something so wicked that the devil will need to wear adisguise out of shame. Can anyone clarify?> James Harris- Brett If you ing morons think that I will let you get away with > not giving me credit for my ing math discoveries then you > have another ing thing coming.> snip...> Forgive me for posting to the original (I can read usenet from my> newsreader, but not write -- so I post on Google) as this is > really addressed to the far off-topic (but far more interesting) > sub-thread about ... another thing/k coming. I heard a phrase > on /. the other person THE DEVIL'S GOING TO NEED A WIG. Has anyone > heard that phrase before? For some reason, that cracked me up. I> interpret it to mean that the guy is going to do something so > wicked that the devil will need to wear a disguise out of shame. > Can anyone clarify?I've never heard that phrase before, so you'll have to take thiswith a grain of salt, but, irregardless, I would guess thatthe devil refers to the on a (somewhat) mild threatthat I _have_ heard, something along the lines of I'm goingto snatch (you, him, whoever) bald-headed. I think snatching bald-headed is intended to symbolize a seriousbut not life-threatening punishment, such as a mother mightoffer to her child (well, me, really ).Hopefully, I'm right, but, well, whatever, you know?Jim Burns If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming.> > What the ??!!!> You ing s. I will get credit for my discovery and get ing> paid, and you best believe that I will not ing let you stupid> s get away with your ing stupid bull--pure math my> ass--without me coming at you ers with some ing PURE ing> PURE AS math that you stupid s have been ting on for over> a ing YEAR!!!> You goddamn S!!!> What the is wrong with you s??!!! Don't you even believe in> your own stupid ? Where's pure math now?> Where is it?> > James Harris ____ /(( )) ( )6 6( ) (_) l (_) DR. QUINN, DISPOSABLE COCKSUCKER <> ) ____) (_____ ( ____/ ) ) ( )( ) ( / / / / / / )==( / / / / / ï/ //' ï|` ï|` / / ) ( / / / / / / `-...., ,..-' `-..-' =fuffy> If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming.> What the ??!!!> Where's pure math now, huh? Where's loving math for the ing> beauty of it now you ing s??!!!> LOOK AT IT!!!> Here is the partial difference equation and instructions for> integrating.> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))],> S(x,1) = 0.> And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dS> from dS(x,2) to dS(x,y).> http://mathforpro't.blogspot.com/> You ing s. I will get credit for my discovery and get ing> paid, and you best believe that I will not ing let you stupid> s get away with your ing stupid bull--pure math my> ass--without me coming at you ers with some ing PURE ing> PURE AS math that you stupid s have been ting on for over> a ing YEAR!!!> You goddamn S!!!> What the is wrong with you s??!!! Don't you even believe in> your own stupid ? Where's pure math now?> Where is it?> James Harris If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming. What the ??!!! Where's pure math now, huh? Where's loving math for the ing> beauty of it now you ing s??!!! LOOK AT IT!!! Here is the partial difference equation and instructions for> integrating.> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))], S(x,1) = 0. And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dS> from dS(x,2) to dS(x,y). http://mathforpro't.blogspot.com/> You ing s. I will get credit for my discovery and get ing> paid, and you best believe that I will not ing let you stupid> s get away with your ing stupid bull--pure math my> ass--without me coming at you ers with some ing PURE ing> PURE AS math that you stupid s have been ting on for over> a ing YEAR!!! You goddamn S!!! What the is wrong with you s??!!! Don't you even believe in> your own stupid ? Where's pure math now?It's behiiiind yooou - over THERE....![pure math exits stage left, runs round back of stage and appears stageright...] Where is it?No, OVER THERE...! IT'S BEHIND YOUUU!!!!> James Harris =Mike Terry was kind enoughto quote the latest JSH eruption in its entirety so some of us couldsee the poetic phrasing to which Mr Harris has descended:>> You ing s. I will get credit for my discovery and get ing>> paid,Um, one needs to get into an older profession than mathematician for that.(Who, exactly, does he think pays for mathematical discoveries?) Mike Terry was kind enough> to quote the latest JSH eruption in its entirety so some of us could> see the poetic phrasing to which Mr Harris has descended:> You ing s. I will get credit for my discovery and get ing>> paid, Um, one needs to get into an older profession than mathematician forthat. (Who, exactly, does he think pays for mathematical discoveries?)The math fairy. See you leave your proofs under the pillow before you go tobed, and in the morning you'll 'nd in place of said proof a direct depositreceipt.HTHMB Mike Terry was kind enough>to quote the latest JSH eruption in its entirety so some of us could>see the poetic phrasing to which Mr Harris has descended:>> You ing s. I will get credit for my discovery and get ing> paid,Um, one needs to get into an older profession than mathematician for that.(Who, exactly, does he think pays for mathematical discoveries?)People have been wondering that ever since he 'rst said he was inthis for the money.Of course the fact that none of us has ever been paid for discoveringa bit of mathematics doesn't really prove anything, since nobody inhistory has ever made the _sort_ of mathematical discovery he has.Presumably there are plenty of people out there with millions ofdollars, just waiting to hear about it when someone 'nally provessomething non-trivial.David C. Ullrich You ing s. I will get credit for my discovery and get ing> paid, and you best believe that I will not ing let you stupid> s get away with your ing stupid bull--pure math my> ass--without me coming at you ers with some ing PURE ing> PURE AS math that you stupid s have been ting on for over> a ing YEAR!!!> You goddamn S!!! edy .Okay, now we've both got that out of our systems, at least for awhile...Who has made a commitment to pay you? If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming.That should be: ï... you have another THINK coming.' How appropriate.> What the ??!!!> Where's pure math now, huh? Where's loving math for the ing> beauty of it now you ing s??!!!> LOOK AT IT!!!> Here is the partial difference equation and instructions for> integrating.> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))],> S(x,1) = 0.> And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dS> from dS(x,2) to dS(x,y).> http://mathforpro't.blogspot.com/> You ing s. I will get credit for my discovery and get ing> paid, Who is going to pay you? Hmmm? Surely no one here. Well..., someonemight pay you to shut up.> and you best believe that I will not ing let you stupid> s get away with your ing stupid bull--pure math my> ass--without me coming at you ers with some ing PURE ing> PURE AS math that you stupid s have been ting on for over> a ing YEAR!!!> You goddamn S!!!> What the is wrong with you s??!!! Don't you even believe in> your own stupid ? Where's pure math now?> Where is it?> James HarrisYou should include this message with your journal submissions. Surelythe editors wouldn't resist your eloquence, despite all the phone If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing morons think X, you haveanother ing *think* coming, not another ing *thing* coming,which makes no sense.I hope this calms you down a bit. Cordially --Jeremy > If you ing morons think that I will let you get away with not>> giving me credit for my ing math discoveries then you have expression is If you ing morons think X, you have> another ing *think* coming, not another ing *thing* coming,> which makes no sense.You're starting a completely unrelated 'ght here, but more than oneor two of us might claim that another thing coming makes rather moresense than another think coming.Fortunately, some of us also know that another think coming iscommon enough in certain circles not to correct your ignorance.-- Jesse HughesHow lucky we are to be able to hear how miserable Willie Nelson couldimagine himself to be. -- Ken Tucker on Fresh Air If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing ing morons think X, you have>> another ing *think* coming, not another ing *thing* coming,>> which makes no sense.> You're starting a completely unrelated 'ght here, but more than one> or two of us might claim that another thing coming makes rather more> sense than another think coming.> Fortunately, some of us also know that another think coming is> common enough in certain circles not to correct your ignorance.Ignorance! Ha! Believe me I've heard another thing from enough people bynow, especially in America, that I'm far from ignorant that some people sayit--I'm merely pointing out what the expression was, before people with noear for an effective and mildly humorous expression mangled it intosomething less interesting.Let me explain, since you say it makes little sense to you.You've got another think coming suggests that if you think X, you will'nd that you are mistaken, for my actions will disabuse you of the notionX. At that point you will no longer think X, and in fact you shall 'ndanother thought in its place--probably !X, to make this slightly on-topicfor sci.math--hence you've got another think coming.Perhaps you are bothered by think used as a noun, where you would prefer the plainer thought. Have a think about it and get back to us.And perhaps you could also tell us what the other thing is that we might have coming?Cordially -Jeremy Ignorance! Ha! Believe me I've heard another thing from enough people by> now, especially in America, that I'm far from ignorant that some people say> it--I'm merely pointing out what the expression was, before people with no> ear for an effective and mildly humorous expression mangled it into> something less interesting.> Let me explain, since you say it makes little sense to you.> You've got another think coming suggests that if you think X, you will> 'nd that you are mistaken, for my actions will disabuse you of the notion> X. At that point you will no longer think X, and in fact you shall 'nd> another thought in its place--probably !X, to make this slightly on-topic> for sci.math--hence you've got another think coming.> Perhaps you are bothered by think used as a noun, where you would prefer > the plainer thought. Have a think about it and get back to us.> And perhaps you could also tell us what the other thing is that we might > have coming?Right on. The people who say another thing coming are probably the same ones who say I could care less.Gib Right on. The people who say another thing coming are probably the > same ones who say I could care less.> GibI could care less is just sarcasm isn't it? Makes perfect sense then.David Collier >> Right on. The people who say another thing coming are probably the >> same ones who say I could care less.>>> GibI could care less is just sarcasm isn't it? No, it's a corruption of I couldn't care less thatarose from people not realizing why that phrase meanswhat it does: I couldn't care less, because I care solittle that there's simply no such thing as caring less.>Makes perfect sense then.David CollierDavid C. Ullrich > Right on. The people who say another thing coming are probably the > same ones who say I could care less.> Gib>>I could care less is just sarcasm isn't it? No, it's a corruption of I couldn't care less that>arose from people not realizing why that phrase means>what it does: I couldn't care less, because I care so>little that there's simply no such thing as caring less.Yes; though when I use I could care less, I mean I could care less,but it would take so much effort to actually reduce the amount thatit's not worth it... But always as a joke, of course. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.edu> >>Right on. The people who say another thing coming are probably the >>same ones who say I could care less.>>Gib> I could care less is just sarcasm isn't it? Makes perfect sense then.I don't know. What do you mean by sarcasm in this context?Gib Right on. The people who say another thing coming are probably the > same ones who say I could care less.Or this, which totally blew my mind the 'rst time I heard it come out of someone's mouth:For all intensive purposes, you're actually in charge... > Perhaps you are bothered by think used as a noun, where you would prefer >> the plainer thought. Have a think about it and get back to us.>>> And perhaps you could also tell us what the other thing is that we might >> have coming?Right on. The people who say another thing coming are probably the >same ones who say I could care less.I think your point is mute. (-:This is probably a spoken version of Mondegreenshttp://www.sfgate.com/columnists/carroll/ mondegreens.shtml == ==It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.edu>> Perhaps you are bothered by think used as a noun, where you would prefer > the plainer thought. Have a think about it and get back to us.> And perhaps you could also tell us what the other thing is that we might > have coming?>>Right on. The people who say another thing coming are probably the >>same ones who say I could care less.I think your point is mute. (-:Gegch.>This is probably a spoken version of Mondegreenshttp://www.sfgate.com/columnists/carroll/ mondegreens.shtml == =It's not denial. I'm just very selective about> what I accept as reality.> --- Calvin (Calvin and Hobbes)> =Arturo Magidin>magidin@math.berkeley.edu>David C. Ullrich <87k75qmyqh.fsf@phiwumbda.org> Fortunately, some of us also know that another think coming is> common enough in certain circles not to correct your ignorance.Sadly, alt.usage.english agrees that your ignorance is older than thecorrect usage. According to, another thinkis older than another thing.Nonetheless, it's damned silly to say that another thing comingmakes no sense.-- Jesse F. HughesEven I, who know beyond doubt that my death will be caused by a sillygirl, will not hesitate when that girl passes by. -- Merlin, asreported by John Steinbeck. > Fortunately, some of us also know that another think coming is>> common enough in certain circles not to correct your ignorance.> Sadly, alt.usage.english agrees that your ignorance is older than the> correct usage. According to> , another think> is older than another thing.> Nonetheless, it's damned silly to say that another thing coming> makes no sense.In your version--If you think X, you've got another thing coming--what was the *'rst* thing? (I mean the thing implied by the word another)It really seems absurd to believe think/thing is superior. But if it comforts you to use the same expression as James...I could imagine someone thinking the expression should be: If your thingis X, you've got another thing coming. But how could you believe thatthing in the second half of James's sentence balances think in the 'rst half?Wow, I always thought people who said it your way would just need to have the correct version pointed out to them, and they'd say ah yes, how funny that I misheard it all these years. But you defend it!Cordially (really, if anything sounds snotty I didn't mean it that way) --Jeremy <87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org> > Fortunately, some of us also know that another think coming is> common enough in certain circles not to correct your ignorance.>>> Sadly, alt.usage.english agrees that your ignorance is older than the>> correct usage. According to>> , another think>> is older than another thing.>>> Nonetheless, it's damned silly to say that another thing coming>> makes no sense. In your version--If you think X, you've got another thing coming--what > was the *'rst* thing? (I mean the thing implied by the word> another)When one says, If you think X, you've got another thing coming, X istypically a future event or a regularity. The word think here isused synonymously with expect or believe, but I want to stress theexpect meaning.So the 'rst thing is not the thought about X, but X itself (veryloosely). If you expect X, you have another thing coming. If youthink I'll give you $5, you have another thing coming (an experiencecontrary to the actual giving of the $5). This is the interpretationthat I have always had for the expression.Notice that your version equivocates on the use of the word think.In the 'rst case, it is synonymous for believes while in thesecond, it indicates the act of deliberation that leads to belief.This equivocation plus the bad grammar is all part of the humor. Thething version is not overtly humorous.But thing sounds so much more natural to my ears (admittedly, partlythrough repetition) that I have always assumed that think is aback-formation along the lines of the other examples in this thread(the worst of which, I think, is the one you mention: intensivepurposes). In any case, if it's good enough for Judas Priest, then by God, it'sgood enough for me. Next, you will complain that heavy metallyricists are not modern day poets, twentieth century[1] counterpartsto the great bard *and* the traveling troubadours that spread the'nely crafted words.> It really seems absurd to believe think/thing is superior. But if it > comforts you to use the same expression as James...Doesn't everyone here strive to be more like James? The man is equalparts Henry Ford, James Dean and Johann Carl Friedrich Gauss, with alittle Thomas Paine sprinkled on top. You want to be James too, butare too prideful to admit this plain fact.Footnotes: [1] Well, Judas Priest's song *was* last century, so perhaps modernday should be interpreted loosely.-- Jesse HughesWell, if I can get [my proof of FLT accepted], then I hopefully get abook deal down the road, and maybe I get to go on ïOprah'. James Harris, on the rewards of mathematical endeavours. In any case, if it's good enough for Judas Priest, then by God, it's> good enough for me. Next, you will complain that heavy metal> lyricists are not modern day poets, twentieth century[1] counterparts> to the great bard *and* the traveling troubadours that spread the> 'nely crafted words.:-)> You want to be James too, but are too prideful to admit this plain fact.:-) :-) When one says, If you think X, you've got another thing coming, X is> typically a future event or a regularity. The word think here is> used synonymously with expect or believe, but I want to stress the> expect meaning.> So the 'rst thing is not the thought about X, but X itself (very> loosely). If you expect X, you have another thing coming. If you> think I'll give you $5, you have another thing coming (an experience> contrary to the actual giving of the $5). This is the interpretation> that I have always had for the expression.I'll grant that this is plausible. I'm with the thinkcrowd. To me the expression can be shortened to: If that'swhat you think, think again.. Think again = another thinkcoming.> Notice that your version equivocates on the use of the word think.> In the 'rst case, it is synonymous for believes while in the> second, it indicates the act of deliberation that leads to belief.> This equivocation plus the bad grammar is all part of the humor. The> thing version is not overtly humorous.Right. It's that deliberately bad grammar that sounds likethink is the word intended there, in an attempt to beaffectedly folksy. It sounds like one of those made-upTexas proverbs in other words.Now I'm gonna hear from more Texans than you can shake astick at. - Randy When one says, If you think X, you've got another thing coming, X is> Right. It's that deliberately bad grammar that sounds like> think is the word intended there, in an attempt to be> affectedly folksy. It sounds like one of those made-up> Texas proverbs in other words. Now I'm gonna hear from more Texans than you can shake a> stick at. - RandyWhat is the absolute upper limit on the number of Texans thatone can shake a stick at? Does the size of the stick matter? When one says, If you think X, you've got another thing coming, X is> Right. It's that deliberately bad grammar that sounds like> think is the word intended there, in an attempt to be> affectedly folksy. It sounds like one of those made-up> Texas proverbs in other words. Now I'm gonna hear from more Texans than you can shake a> stick at. - Randy> What is the absolute upper limit on the number of Texans that> one can shake a stick at? Does the size of the stick matter?Size always matters. <87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org> <8765hatedb.fsf@phiwumbda.org> > When one says, If you think X, you've got another thing coming, X is>> typically a future event or a regularity. The word think here is>> used synonymously with expect or believe, but I want to stress the>> expect meaning.>>> So the 'rst thing is not the thought about X, but X itself (very>> loosely). If you expect X, you have another thing coming. If you>> think I'll give you $5, you have another thing coming (an experience>> contrary to the actual giving of the $5). This is the interpretation>> that I have always had for the expression. I'll grant that this is plausible. I'm with the think> crowd. To me the expression can be shortened to: If that's> what you think, think again.. Think again = another think> coming.One reason that thing is not so odd: Whenever one says, You've gotanother thing coming, he means that the outcome will not be good.Thing stresses the actual coming occurrence, at least how Iinterpret it. One never says, you've got another thing coming, whenthey mean that good times are ahead and that the listener will bepleasantly surprised. Instead, the thing that is promised is alwaysa bit menacing.In any case, history is on your side. Think predates thing. Butit shouldn't.>> Notice that your version equivocates on the use of the word think.>> In the 'rst case, it is synonymous for believes while in the>> second, it indicates the act of deliberation that leads to belief.>> This equivocation plus the bad grammar is all part of the humor. The>> thing version is not overtly humorous. Right. It's that deliberately bad grammar that sounds like> think is the word intended there, in an attempt to be> affectedly folksy. It sounds like one of those made-up> Texas proverbs in other words. Now I'm gonna hear from more Texans than you can shake a> stick at.As a technical Texan[1], I claim that thing is the commonest Texasversion. At least in my family. - RandyFootnotes: [1] Born in Texas to Texan parents, but raised in Oklahoma. Whichreally isn't very different, except that Okies have an inferioritycomplex that Kansans should understand.-- Jesse HughesBy de'nition m is a variable. By de'nition all then (sic) numbersrepresented by letters are variables--that's algebra[,] Magidin. -- James Harris shows deep understanding of algebra Footnotes: > [1] Born in Texas to Texan parents, but raised in Oklahoma. Which> really isn't very different, except that Okies have an inferiority> complex that Kansans should understand.Say no more.Gib I'll grant that this is plausible. I'm with the think>crowd. To me the expression can be shortened to: If that's>what you think, think again.. Think again = another think>coming.Exactly.Here's what the OED says under think(n): b. to have another think coming: to be greatly mistaken. 1937 Amer. Speech XII. 317/1 Several different statements used for the same idea - that of some one's making a mistake...[e.g.] you have another think coming. 1942 T. BAILEY Pink Camellia xxvii. 199 If you think you can get me out of Gaywood, you have another think coming. 1979 Jrnl. R. Soc. Arts CXXVII. 221/2 Any design consultant who thinks he is going to get British Leyland right by himself on his own has got another think coming.-- Richard-- FreeBSD rules! Fortunately, some of us also know that another think coming is>common enough in certain circles not to correct your ignorance.>>Sadly, alt.usage.english agrees that your ignorance is older than the>>correct usage. According to>>, another think>>is older than another thing.>>Nonetheless, it's damned silly to say that another thing coming>>makes no sense.> > In your version--If you think X, you've got another thing coming--what > was the *'rst* thing? (I mean the thing implied by the word another)> It really seems absurd to believe think/thing is superior. But if it > comforts you to use the same expression as James...> I could imagine someone thinking the expression should be: If your thing> is X, you've got another thing coming. But how could you believe that> thing in the second half of James's sentence balances think in the > 'rst half?> Wow, I always thought people who said it your way would just need to have > the correct version pointed out to them, and they'd say ah yes, how funny > that I misheard it all these years. But you defend it!He doesn't just defend, he attacks you as ignorant! De'nite echoes of JSH here.Gib <87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org> He doesn't just defend, he attacks you as ignorant! De'nite echoes > of JSH here.Evidently, some readers cannot grasp tongue-in-cheek humor. -- Jesse Hughes How come there's still apes running around loose and there arehumans? Why did some of them decide to evolve and some did not? Didthey choose to stay as a monkey or what? -Kans. Board of Ed member <87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org> <878ym6tg75.fsf@phiwumbda.org> =What is this crap? <87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org> <878ym6tg75.fsf@phiwumbda.org> What is this crap?It is the crap what I posted. What is *this* crap?-- Jesse HughesYes, I'm one of those arrogant people who tries to be quotable.There is actually at least one person who quotes me often. -- James Harris If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming. What the ??!!! Where's pure math now, huh? Where's loving math for the ing> beauty of it now you ing s??!!! LOOK AT IT!!! Here is the partial difference equation and instructions for> integrating. dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))], S(x,1) = 0. And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dS> from dS(x,2) to dS(x,y). http://mathforpro't.blogspot.com/ You ing s. I will get credit for my discovery and get ing> paid, and you best believe that I will not ing let you stupid> s get away with your ing stupid bull--pure math my> ass--without me coming at you ers with some ing PURE ing> PURE AS math that you stupid s have been ting on for over> a ing YEAR!!! You goddamn S!!! What the is wrong with you s??!!! Don't you even believe in> your own stupid ? Where's pure math now? Where is it? James HarrisWith all due contempt, James, you have now §unked out of Sandbox 101.Your inexcusable social behavior and immature language are eloquenttestimony that you must return to the Playpen for a course in remedialcounting.If you fail again, you will be sent to crash dummy training school.--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 If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming.> What the ??!!!> That's a lot of f-bombs dude, what's up? If you ing morons think that I will let you get away with not> giving me credit for my ing math discoveries then you have another> ing thing coming. What the ??!!! Where's pure math now, huh? Where's loving math for the ing> beauty of it now you ing s??!!! LOOK AT IT!!! Here is the partial difference equation and instructions for> integrating.> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))], S(x,1) = 0. And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dS> from dS(x,2) to dS(x,y). http://mathforpro't.blogspot.com/> You ing s. I will get credit for my discovery and get ing> paid, and you best believe that I will not ing let you stupid> s get away with your ing stupid bull--pure math my> ass--without me coming at you ers with some ing PURE ing> PURE AS math that you stupid s have been ting on for over> a ing YEAR!!! You goddamn S!!! What the is wrong with you s??!!! Don't you even believe in> your own stupid ? Where's pure math now? Where is it?> James HarrisMAYBE IF YOU LEARNED SOME MATH, YOU'D REALIZE THAT WE WERE TELLING THETRUTH. You've been drinking again, haven't you? You are way too immature forbeing an adult.David Moran =I thought some sci.math readers would be interested in the publisher'sblurb for Arthur C. best-seller (not sure how they knowthat, but anyway):> The Last recent sensational proof of Fermat's Theorem 350 years > later by a young British mathematician, Andrew Wiles, THE LAST THEOREM > charts the story of Ranjit Subramanian, a man fascinated by Fermat's > Last Theorem > - so simple that anyone can understand it, yet not proved for more than > three centuries. Ranjit learns about the Indian mathematical genius > Ramanujan (1887-1920) and discovers a three-page proof of the Last > Theorem: this might even be Fermat's own proof. The discovery of the > Theorem wins Ranjit the Fields Medal - and the attention of the NSA > cryptography branch. However, Ranjit soon 'nds himself drawn by physics > rather than cryptography, as there have been some spectacular advances > in fusion technology. And these in turn lead to a plasma drive that can > open up the Solar System ... Actually I thought one sci.math reader in particular would be interested. James, do you think ACC has been reading your work? He is a visionary, youknow, and many of his science-'ction dreams have become reality. Itcould happen! The connection with the NSA cryptography branch isintriguing--have you sent them your research on prime counting? Becareful! Any plans to turn your attention to fusion technology? Perhapsthat would be better for mankind. =The Last Theorom by A. C. Clark ?I am sci.math readers would be interested in the publisher's> blurb Arthur C Clarke Based on the recent sensational proof of Fermat's Theorem 350 years> later by a young British mathematician, Andrew Wiles, THE LAST THEOREM> charts the story of Ranjit Subramanian, a man fascinated by Fermat's> Last Theorem> - so simple that anyone can understand it, yet not proved for more than> three centuries. Ranjit learns about the Indian mathematical genius> Ramanujan (1887-1920) and discovers a three-page proof of the Last> Theorem: this might even be Fermat's own proof. The discovery of the> Theorem wins Ranjit the Fields Medal - and the attention of the NSA> cryptography branch. However, Ranjit soon 'nds himself drawn by physics> rather than cryptography, as there have been some spectacular advances> in fusion technology. And these in turn lead to a plasma drive that can> open up the Solar System ... Actually I thought one sci.math reader in particular would be interested.> James, do you think ACC has been reading your work? He is a visionary, you> know, and many of his science-'ction dreams have become reality. It> could happen! The connection with the NSA cryptography branch is> intriguing--have you sent them your research on prime counting? Be> careful! Any plans to turn your attention to fusion technology? Perhaps> that would be better for Message-id: I thought some sci.math readers would be interested in the publisher's>blurb for by>> Arthur C Clarke>>> Based on the recent sensational proof of Fermat's Theorem 350 years >> later by a young British mathematician, Andrew Wiles, THE LAST THEOREM >> charts the story of Ranjit Subramanian, a man fascinated by Fermat's >> Last Theorem >> - so simple that anyone can understand it, yet not proved for more than >> three centuries. Ranjit learns about the Indian mathematical genius >> Ramanujan (1887-1920) and discovers a three-page proof of the Last >> Theorem: this might even be Fermat's own proof. The discovery of the >> Theorem wins Ranjit the Fields Medal - and the attention of the NSA >> cryptography branch. However, Ranjit soon 'nds himself drawn by physics >> rather than cryptography, as there have been some spectacular advances >> in fusion technology. And these in turn lead to a plasma drive that can >> open up the Solar System ... Actually I thought one sci.math reader in particular would be interested. >James, do you think ACC has been reading your work? He is a visionary, you>know, and many of his science-'ction dreams have become reality. It>could happen! The connection with the NSA cryptography branch is>intriguing--have you sent them your research on prime counting? Be>careful! Any plans to turn your attention to fusion technology? Perhaps>that would be better for mankind.I told James that he should become a script writer if he wants to make money.Alas, looks someone has beaten him to the punch.--MensanatorAce of Clubs I'm attempting to connect two 3D squares together. They are of 16> points each and I?ve run them through the Bernstein equation.> Now, I?m quite new to 3 dimensional mathematics and new to the> Bernstein formula, but its proving to be very interesting.> I've been trying for days now to get the squares to match up smoothly.> There is a curvature in the surface of the one square/patch and I?m> attempting to keep the same curvature through the second square (to> make half a §at oval). I've heard about the tangents method, about> having tangents from the control points on the one square to the> control points on the other square, which should create a smooth> surface. And here in is the problem, there is a de'nite line, an> inward slope as the two squares/patches meet. As far I can tell the> tangents line up along the join, I just don't have a clue why the join> isn't perfectly smooth.> Many people must have run into this problem. If you have any thoughts,> ideas or possible directions to pursue, they would all be very much> appreciated.> T. OvertonAnyone? Any ideas? =First, let me reassure ya'll I'm not soliciting for a homeworkassignment.There's a puddle of water in front of my house where the main waterline runs about 6 below grade. In that area is a pipe around theshut-off valve that straddles the inlet/outlet pipes on the valve andthe bottom is resting on a couple 're bricks.The puddle covers about 5 square feet and didn't get smaller lastnight, but no larger either. I don't know the relative humidity orhow well water will perk, and the water meter isn't moving witheverything off. There is also some plastic sheeting on the groundunder the water, but it is not complete.It is possible water was spilled there by local children getting adrink, and water under our home (during a previous leak) took MONTHSto evaporate with the vents wide open.Here are some §ow test results pouring water into a quart measuringcup while timing it:A) §ow rates that were not suf'cient to move the water meter 1) *1.24 gph 30 gpd 2) 1.77 gph 42.5 gpdB) §ow rates that make the water meter move very slowly 1) 2.63 gph 63 gpd 2) 1.11 gph 27 gpd* This one got messed up a little when someone forgot to leave thewater alone and §ushed a toilet water was slowed, then increasedover average. Flush time was 46 s. and I hope the net effect wasminimal (self correcting somewhat as there is a water regulatorvalve).It seems there is some hysteresis effect between readings, the meterbeing a mechanical device .What I hope to get help with here is that there is a rate of leakingbelow the threshold that moves the meter in addition to all the above§ow rates (if there is indeed a leak which the math may help todetermine as well?). Since that leakage rate is before the pressurereducing valve which is under the house, it should have a pretty muchconstant rate and effects of such small §ow rates as the above shouldnot change it signi'cantly.Is it calculable what the rate is of the leak (or if there is one) infront of the house give the above sample §ow rates?By the way, one thing that leaves me suspicious about there maybe notbeing a leak is that the water shut-off valve is about 6 inches plusbelow the water puddle and the well surrounding the valve is dampbut there is no standing water. Perhaps it perked with help from thebricks below, but if the water leak was at the pipe below ground leveland saturated the ground enough (won't perk well enough) to breaksurface, you might be led to think the water should also be inside thepipe surrounding the valveI replaced these pipes about a month ago could be if it is a leak,its been leaking this long (or getting worse too) the shrub plantedthere died, perhaps from being water logged but I know it was drythere for some weeks after the work.Anybody feel like messing with the math and trying to apply it to mylittle plumbing problem? I know the easier (at least for someonerusty/math impaired) solution is with wet ground and all, just diggingup the pipe area, but I've been sick and didn't want to do that especially if there is a reasonable (mathematical) chance it's juststanding water and the effort will be wasted.Stephen Kurzban =This ain't math, it's plumbing psychology detective work.At bed time, securely tape all the toilet and faucet handles with duck tape. Take §ashlight, read water meter and note time. Go to bed and don't have any wet dreams. ;-) Upon waking or whenever you get around to it, such as after work, read water meter and note time.If in the mean time the water meter has been spinning its wheels, you know its your leak to entertain or aggravate you plumber who 'xed the privious leak. Otherwise its water company's leak, who before letting you pin the leak on them, will repeat the test.Furthermore, be there any doubt about the source, such as being ground water, the water company can take water samples and test for clorine, §orine or whatever other locally popular water pollutants the water company is found of making you drink and bath in. >There's a puddle of water in front of my house where the main water >line runs about 6?? below grade. In that area is a pipe around the >shut-off valve that straddles the inlet/outlet pipes on the valve >and the bottom is resting on a couple 're bricks.6 what? Inches, feet, meters? >The puddle covers about 5 square feet and didn't get smaller last >night, but no larger either. I don't know the relative humidity or >how well water will perk, and the water meter isn't moving with >everything off. There is also some plastic sheeting on the ground >under the water, but it is not complete. >It is possible water was ^spilled^ there by local children getting >a drink, and water under our home (during a previous leak) took >MONTHS to evaporate with the vents wide open.You could bail the water and see what happens.Is the water tap for public use? If not, get a locking faucet.Otherwise complain to the water department or other lible agency.---- =The Catbert matrix is the Hankel matrix of reciprocals of Catalan numbers. Let C be the Catbert matrix, and let c_ibe the i_th Catalan number, so the (i, j) entry of C isC_{i, j} = 1/c_(i+j).We conjecture that the inverse of the Catbert matrix hasinteger entries.The sequences of Patalan numbers generalize the Catalannumbers [2, A025748-A025755, A034171, A034255, A034687].The Patalan numbers p_n may be de'ned by p_0 = 1,p_n = p_(n-1)*(p^2*n - p)/(n+1) for n > 0. (Our indexingof the Patalans is off by one from the OEIS indexing.) The Catalannumbers are the Patalan numbers for p = 2.The matrices formed from the Catbert matrix by replacingCatalan numbers with corresponding Patalan numbers donot have the inverse integer property, but some closelyrelated matrices do.De'ne the matrix P by letting its (i, j) entry beP_{i, j} = 1/(p_n*(p+1)).I am tempted to call P the Ratbert matrix, but I will not succumb to that temptation just yet.We conjecture that the inverse of P has integer entrieswhen p is an integer greater than 1 or an integer less than -1.Previously[1] we introduced the Dilbert matrix D(t, s),and proved that the entries of the inverse of the Dilbertmatrix are integers when t and s are nonnegative integers. The (i, j) entry of D(t, s) is D(t,s)_{i,j} = 1/( (i+j+t+s+1)*binomial(i+j+t+s, t) ).The matrix P is related to the Dilbert matrix byP = G*D(-2 - 1/p, 1)*G,where G is a diagonal matrix with G_{i, i} = 1/p^(2*i + 1).The relationship between the Catbert and Dilbert matricesis left as an exercise.Using the formula for the entries of the inverse of the Dilbert matrix, we 'nd the the (i, j) entry of the inverse of the matrix P isP^(-1)_{i, j} = p^(2*i + 2*j + 2)*D(-2 - 1/p, 1)^(-1)_{i, j}= *p^(2*i + 2*j + 2) *sum_(l=j)^(n-1) (-1)^j*binomial(l, j) *sum_(k=l)^(n-1) (-1)^(i+k+l) *multinomial(n+i-1-1/p, n-k-1, k-l, i+1, l-2-1/p, 1) *binomial(n + k - 1 - 1/p, n - i - 1) *binomial(i + k - 1 - 1/p, i)= sum_(l=j)^(n-1) sum_(k=l)^(n-1) p^(2*i + 2*j + 2) *(-1)^(i + k + l + j) *binomial(l, j) *binomial(n + i - 1 - 1/p, n - k - 1) *binomial(i + k - 1/p, k - l) *binomial(i + l - 1/p, i + 1) *(l - 1 - 1/p) *binomial(n + k - 1 - 1/p, n - i - 1) *binomial(i + k - 1 - 1/p, i). Although the terms of this sum are not integers, numericalevidence suggests that the sum is an integer.References[1] Richardson, Thomas M. The Dilbert Matrix. Sequences, http://www.research.att.com/~njas/sequences/. =The Catbert matrix is the Hankel matrix of reciprocals of Catalan numbers. Let C be the Catbert matrix, and let c_ibe the i_th Catalan number, so the (i, j) entry of C isC_{i, j} = 1/c_(i+j).We conjecture that the inverse of the Catbert matrix hasinteger entries.The sequences of Patalan numbers generalize the Catalannumbers [2, A025748-A025755, A034171, A034255, A034687].The Patalan numbers p_n may be de'ned by p_0 = 1,p_n = p_(n-1)*(p^2*n - p)/(n+1) for n > 0. (Our indexingof the Patalans is off by one from the OEIS indexing.) The Catalannumbers are the Patalan numbers for p = 2.The matrices formed from the Catbert matrix by replacingCatalan numbers with corresponding Patalan numbers donot have the inverse integer property, but some closelyrelated matrices do.De'ne the matrix P by letting its (i, j) entry beP_{i, j} = 1/(p_n*(p+1)).I am tempted to call P the Ratbert matrix, but I will not succumb to that temptation just yet.We conjecture that the inverse of P has integer entrieswhen p is an integer greater than 1 or an integer less than -1.Previously[1] we introduced the Dilbert matrix D(t, s),and proved that the entries of the inverse of the Dilbertmatrix are integers when t and s are nonnegative integers. The (i, j) entry of D(t, s) is D(t,s)_{i,j} = 1/( (i+j+t+s+1)*binomial(i+j+t+s, t) ).The matrix P is related to the Dilbert matrix byP = G*D(-2 - 1/p, 1)*G,where G is a diagonal matrix with G_{i, i} = 1/p^(2*i + 1).The relationship between the Catbert and Dilbert matricesis left as an exercise.Using the formula for the entries of the inverse of the Dilbert matrix, we 'nd the the (i, j) entry of the inverse of the matrix P isP^(-1)_{i, j} = p^(2*i + 2*j + 2)*D(-2 - 1/p, 1)^(-1)_{i, j}= *p^(2*i + 2*j + 2) *sum_(l=j)^(n-1) (-1)^j*binomial(l, j) *sum_(k=l)^(n-1) (-1)^(i+k+l) *multinomial(n+i-1-1/p, n-k-1, k-l, i+1, l-2-1/p, 1) *binomial(n + k - 1 - 1/p, n - i - 1) *binomial(i + k - 1 - 1/p, i)= sum_(l=j)^(n-1) sum_(k=l)^(n-1) p^(2*i + 2*j + 2) *(-1)^(i + k + l + j) *binomial(l, j) *binomial(n + i - 1 - 1/p, n - k - 1) *binomial(i + k - 1/p, k - l) *binomial(i + l - 1/p, i + 1) *(l - 1 - 1/p) *binomial(n + k - 1 - 1/p, n - i - 1) *binomial(i + k - 1 - 1/p, i). Although the terms of this sum are not integers, numericalevidence suggests that the sum is an integer.References[1] Richardson, Thomas M. The Dilbert Matrix. Sequences, http://www.research.att.com/~njas/sequences/. I learned a lot, not the least of which was that>> sin(1) + sin(2) + ... + sin(n) is bounded independent of n.>>> That statement intrigued me. I searched throughout this thread, to no avail,>> for information about that. Hm, I also thought JB's statement appeared without prior posted context.As another poster noted, this sum can be computed as a geometric series.One should further note the reason the boundedness applies here: thisis now integration by parts (if I have the temerity to refer to a 'nitesum as integration and thereby risk the participation by asci.math troll). If we let F(n) = sum_{1 le k le n} sin(k), whichis this bounded sum, then sin(n) = F(n)-F(n-1) and so the originalin'nite sum is sum( sin(n)/n ) = sum( [F(n)-F(n-1)]/n ) = sum( F(n)[1/n - 1/(n+1)] )which then converges, being bounded by a multiple of the telescoping sum sum( [1/n - 1/(n+1)] ) = 1Indeed, one can then obtain a precise value for the sum since F(n) isitself the sum of a couple of exponentials, and since the in'nite sum sum_{nge 1} x^n/[n(n+1)] = 1 + (1/x-1) ln(1-x)when convergent.dave = [.snip.]> I hereby invite Arturo back into the fray, to respond at least to >posts of others, if not to your own.I am reading the threads, but I do not want to reply even indirectlyto any of James's arguments. =It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.edu =http:// mathforpro't.blogspot.com/Was this really pro'table, and did you claim it as income ?Has anyone here ever thought of Math as a martial art ? I've been away from maths for a while and never really thought it> would be useful when I had the best opportunity to learn (ie. at> school). I now 'nd that I was wrong... to cut a long story short, I'm> short of the tools required to solve a speci'c problem, and I need> nudged in the right direction so I can 'gure it out for myself.> Basically, it's a chemical reaction of the variety:> c+d ---> e+f> It also produces a byproduct (I'll call the byproduct b1), formed in> a quantity dependent on various parameters, which I've given arbitrary> names:> b1=1-(b/m(1+a/q)+(2b^2)/mn)*(1+a/p+b/m(1+a/q+b/n))^-1> (the letters have replaced equilibrium constants and ion> concentrations... doesn't really matter what they are as this is> applicable for a number of reactions.)> one of the parameters, b, that b1 is dependent on is of the same> substance as b1.> My problem is that b1 gets smaller the larger b gets... and I> would like to be able to calculate what the end b is if a certain> number of c+d is converted to e+f where all other parameters are known> Does that make sense?> I suspect I need to use some sort of calculus...I don't believe calculus is necessary -- just algebra. If I knew morespeci'cs about the reaction, I might be able to help. For example:what do you mean by b is of the same substance as b1? Is it anacid:anion pair?If I read your equation right, it can be set up in the form Ab^2 + Bb+ C = 0, where A,B,C depend on b1. If you know a relationship betweenb and b1, then you get a polynomial equation in b alone(therelationship being something like b1/b = k). Then, you can solve byapproximations. Typically, if you know that certain concentrations aresmall, you may simplify the equation by eliminating those tiny terms.You then get an answer, and test it to see how far off it is.You might also check sci.chem-Brett =I am a high school student in Greece and I would like to learnsomething more about logarithms than what the students' book covers.To be more speci'c, I have a question on e, the base of naturallogarithms. Why do we use such a number? What is so important aboutit? Not only is it a never-ending number (2,718281828459045... ),which makes calculations dif'cult, but it is also much more practicalto use simpler bases e.g. 2 or 10, which makes calculations andconvertions from one base to another much easier. What is more, theimportance of a number equal to 1 n e = lim (1+ - ) n-->inf ndoesn't make much sense to me. I would be very glad if someone could I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone couldWhy do we have the number e and why is the above limit important, you ask?Well, the limit generalizes to the following:e^x = lim (1 + x/n) ^ n. This shows that the right hand side, considered asa function of x, evaluates to some constant e raised to the power x. Theactual value of e is obtained simply by choosing the special case x=1.-Michael. I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),irrational, actually transedental irrational means can not be represented as the ratio of two integers transcendental means not the solution of a polynomial with integer coef'cients (all transendentals are irrational, which is obvious, right?).> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations andIt depends upon what you are doing.> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone couldFor calculus, base e is really the way to go. This is becauseif f(x)=e^x, then the instantaneous rate of change of f with respectto x is actually just e^x again (that is, the derivative is e^x is e^x).For any other base, when taking the derivative one would geta funny factor involving, actually, the natural log of the base.Again, it depends what you are trying to do. In computer applications,base 2 shows up a great deal. Base 10 is often used for a varietyof real life purposes: pH (acid/base) measurements are measurementsof concentration log base 10; so called neutral density 'lterswhich allow a 'xed fraction of incident light to pass through themare rated in terms of optical density, which is log base 10 of theratio of the output to the input light; and lots of things getmeasured in decibles.But e^x really is *natural* in a lot of ways. For example,how can one compute a^x? Well, for e^x one has: e^x = 1/0! + x/1! + x^2/2! + x^3/3! + x^4/4! + x^5/5! + ...Here, ! represents the factorial operation, which is de'ned by 0! = 1 (special case) 1! = 1 2! = 2 * 1 3! = 3 * 2 * 1 4! = 4 * 3 * 2 * 1 5! = 5 * 4 * 3 * 2 * 1 ... etc ...The ... after x^5/5! in the formula for e^x represents that thisis an in'nite series. But the point is that the terms get smallvery quickly (if you have a computer, try graphing f(x) = e^x, g(x) = 1/0! + x/1! + x^2/2! + x^3/3! + x^4/4! h(x) = 1/0! + x/1! + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6!)So, the claim is that while the series is in'nite, one can approximatee^x arbitrarily well (for a given range of x) by using an initialpolynomial from the series. Not only that, but the terms reallydo get small fast, so the approximation is actually practical inmany situations.The corresponding series for 10^x and 2^x don't have nicecoef'cients--indeed, the coef'ecents would require calculatingthe natural log of 10 or 2!As often happens in school, things aren't making sense becauseyou see them out of their natural context and order. Once onestart thinking about calculus, e really jumps out at you.Best wishes, Mike of concentration log base 10; so called neutral density 'lters> which allow a 'xed fraction of incident light to pass through them> are rated in terms of optical density, which is log base 10 of the> ratio of the output to the input light; and lots of things getErr, make that the log base 10 of the ration of the input to the output.-Mike of concentration log base 10; so called neutral density 'lters> which allow a 'xed fraction of incident light to pass through them> are rated in terms of optical density, which is log base 10 of the> ratio of the output to the input light; and lots of things get> Err, make that the log base 10 of the ration of the input to the output.> -MikeI once travelled to view an eclipse with some people who had been(correctly) advised to use ND 6 'lters on their cameras forphotographing the partial phases. A camea store clerk, certain thatno one could take pictures through such a 'lter, instead convincedthem to buy 6X 'lters!Cameras and retinas survived the eclipse intact... I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone couldTero,you can also look at:http://mathforum.org/dr.math/faq/faq.e.htmlfor some very interesting results on applications of e, including logs.There are also wonderful web sites on the topic, like:http://en.wikipedia.org/wiki/Natural_logarithmhttp:// mathworld.wolfram.com/e.htmlhttp://numbers.computation.free.fr /Constants/constants.htmlhttp://www.mathcad.com/library/ Constants/e.htmhttp://thesapps.com/Doug/exp/http:// mathforum.org/library/drmath/view/53418.htmlEnjoy and explore!Flip I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone couldNatural logs are used in calculus, which makes certain tasks easier.Actually, you can use any base logarithm, but when you get to calculus,you'll 'nd that natural logs have some nice properties that make them easyto work with.David Moran I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone couldI can think of a couple of examples where e makes a lot of sense.In calculus, if y = e^x then dy/dx = e^x........handy dandy don't youthink.Also, for an exponential annual growth of say 5%, P = Po*1.05^t which isa discrete accumulation (adds 5% to the new principal every year). Forcontinuous growth P = Po*e^(.05t).Phil Holman I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone could'rst, you are well on your way to being a mathematician as it is wonderfulto see you questioning prior results.Anyway, please read the following web page and see if it helps you tounderstand e and its history better.There is also a wonderful book on the topic.HTH, Flip I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number?For exactly the same reason that we use radians, which have an irrationalnumber of units correponding to a whole rotation, rather than degrees:because it screws up results like d(cos x)/dx = -sin(x) andd(ln x)/dx = 1/x, not too mention the vital property that d(e^x)/dx = e^x.> What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier.Now that we have advanced beyond the need for slide rules and log tables,it's better to use the base in which the theoretical analysis is simplestand leave the computers to work out numerical values.-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will 'nd these people and we will bring them to justice. =Tero scribbled the following:> I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone couldWell, one thing that the natural logarithm has going for it, is that itsinverse (e^x) is its own derivative. You don't get that for bases 2 or10. The only functions that are their own derivatives are e^x and thatfunction multiplied by a constant.-- /-- Joona Palaste (palaste@cc.helsinki.') ------------- Finland ---------- http://www.helsinki.'/~palaste --------------------- rules! --------/How can we possibly use sex to get what we want? Sex IS what we want. - Dr. Frasier Crane I am a high school student in Greece and I would like to learn> something more about logarithms than what the students' book covers.> To be more speci'c, I have a question on e, the base of natural> logarithms. Why do we use such a number? What is so important about> it? Not only is it a never-ending number (2,718281828459045... ),> which makes calculations dif'cult, but it is also much more practical> to use simpler bases e.g. 2 or 10, which makes calculations and> convertions from one base to another much easier. What is more, the> importance of a number equal to> 1 n> e = lim (1+ - )> n-->inf n> doesn't make much sense to me. I would be very glad if someone coulde has some very interesting properties.e^1x = cosx + isinx so the trig functions can be derived from ethe hyperbolic functions are de'ned using ed(e^x)/dx = e^x so e^x is a 'xed point of the operation d/dx.e is used to de'ne the fourier transformsthe exponential function e^x is used as integrating factors for differential equations.Bob Kolker =It is a well known theorem that for a 'nite group G the order of Gequals the sum of the squares of the dimensions of the irreduciblerepresentations of G.A famous theorem of Langrange states that any positive integer can berepresented as the sum of four squares.For which groups are the dimensions of the irreducible representationsthe same as the integers that represent the order of G It is a well known theorem that for a 'nite group G the order of G> equals the sum of the squares of the dimensions of the irreducible> representations of G.> A famous theorem of Langrange states that any positive integer can be> represented as the sum of four squares.It doesn't say that the representation is unique. It also permits zeros, e.g., 2 = 1^2 + 1^2 + 0^2 + 0^2. > For which groups are the dimensions of the irreducible representations> the same as the integers that represent the order of GAny group that has exactly 4 conjugacy classes will do. I believe papers have been written 'nding all the ('nite) groups with small numbers of conjugacy classes, where small maybe goes up to 6.-- =I'm aware of bijections from [0,1] to (0,1), but none is continuous. Canthere be? That is, can we shrink [0,1] by an in'nitesimally small amountto yield (0,1)?L I'm aware of bijections from [0,1] to (0,1), but none is continuous. Can> there be? Not with respect to the usual topology -- indeed there can beno continuous surjection: [0,1] is compact; (0,1) is not. I'm aware of bijections from [0,1] to (0,1), but none is continuous. Can> there be? > > Not with respect to the usual topology -- indeed there can be> no continuous surjection: [0,1] is compact; (0,1) is not.It is, however, possible to exhibit a continuous bijection (actually ahomeomorphism) from (0,1) intersect Q to [0,1] intersect Q. But it'sa fairly dif'cult exercise. I'm aware of bijections from [0,1] to (0,1)Hmmm... After spending a few moments trying to come up withone, I can't.In other words, you picked my curiosity. Please do show usone of those bijections!Carlos-- I'm aware of bijections from [0,1] to (0,1)> Hmmm... After spending a few moments trying to come up with> one, I can't.> In other words, you picked my curiosity. Please do show us> one of those bijections!> Carlos> --> One such bijection is f:[0,1] -> (0,1) de'ned as f(0) = 1/2, f(1/2^n) = 1/2(n+2), for n = 0,1,2, ..., f(x) = x otherwise.Bijective but not even close to being continuous. = grava .88 la saucisse et au marteau:> I'm aware of bijections from [0,1] to (0,1)> Hmmm... After spending a few moments trying to come up with> one, I can't.> In other words, you picked my curiosity. Please do show us> one of those bijections!For any real x not of the form 1/n and different from 0, you have f(x) =xFor any n, f(1/n) = 1/(n+2)f(0) = 1/2I've done this quite quick, so it may be wrong, but I think I'm right.-- Nicolas =Leonard M. Wapner grava .88 la saucisse et au marteau:> I'm aware of bijections from [0,1] to (0,1), but none is continuous. Can> there be? That is, can we shrink [0,1] by an in'nitesimally small amount> to yield (0,1)?No. The image of a compact set by a continuous function is compact.-- Nicolas =--www.StealthHostiing.com Webmasters for the TRUman JOIN www.theBanner.netYou rule Truman. http://tinyurl.com/iky4--------------------------------------- -----------------------------------------The Truman show is TRUE, its based on Adam of the Bible living in todays times, me!Eve is his fate, that is why the government keeps them apart like in the movie,to test how the quantum 'eld controlling us all into allowing destiny works.(Didn't you always wonder why *Nic* *Cage* does most of the *prison* movies? True!)http://tinyurl.com/iky5 http://tinyurl.com/iky8 http://tinyurl.com/iky9http://tinyurl.com/iky4 http://tinyurl.com/rv5f http://tinyurl.com/v1yfHey Trueman...love the show. YOU ARE the Truman I heard him.Very spooky! We're sick of you You rule Truman.>Is the truman is living in Townsville? I've been hearing stuff, yeah.and watch here live in Google as I predict back in 2000 Jim Carrey's next romance costarLaurie Holden with http://tinyurl.com/fuf8 she looks exactly like Laurie Holden of the x-'leswith bright green eyes.X-'les little green EYED men, the 1st children of Adam & EveHerchow long does this have to go on for people? You want ALL my hair gray before I see Eve again? =Because Khinchtine's constant exists and is 'nite, we know that thedistribution of the integers among the terms of the simple continuedfraction of a suf'cently random (whatever is meant by this) positivereal is not normal.In other words, 1 is much more likely to occur than, say, 1000 in thecontinued fraction expansion of most positive reals.So, let us say we have a real x where the geometric average of x'scontinued fraction terms approaches Khinchtine's constant, where theseterms are {a(k)}.So, what can be said about the number theoretical aspects of {a(k)},such as:1) What is the likelyhood that any a(k) will be prime, as k -> oo.ie. If pcf(m) is the number of primes among a(k) for 1 <= k <= m, thenwhat is pcf(m) asymptotical towards?2) Same question as (1), but replace ïprimes' with ïsquarefreeintegers'.3) What is likelihood a(k) and a(k+1) will be coprime?4) What are the expected number of divisors for all a(k) where 1 <= k<= m?Etc etc etc =In the spirit of the game 7 Degrees of Kevin Bacon and of Erdosnumbers, etc, I ask....Which branches of mathematics are the most distant?Of course, this is a subjective question. So I am posting this as acompletely plagerized game.Say one of you posts that, as an example, number theory and calculusare distant.This ïmove' is easily countered with ïanalytical number theory', orwith speci'c math results or speci'c topics that connect the 2topics, such as the Riemann zeta function.(An easily-countered example)I am hoping we will post much better, less general, more esoteric, andmore- dif'cult-to-counter examples here... The goal is to create the longest chain of math branches/ resultswhich cannot be shortened by any of us here.(And, obviously, this whole game is subjective and unrigorous andmostly a mindless Which branches of mathematics are the most distant?My guess is that you would be hard-pressed to 'nd many peoplewho are well versed in both K-Theory and Control Theory.> Of course, this is a subjective question.Well, you could quantify it in some ways. I attempted to do thisonce and collected data about the research literature to investigatethis kind of question; see http://www.math-atlas.org/collection/mathmap.htmldave =We observed an asymptotic expression for the density of sequences withequal numbers of zeros and ones among all in'nite sequences.This was somewhat less than 1/ sqrt(n), as n->oo. There are countablymany sequences with equal numbers of ones and zeros.Borel tells us that almost all numbers are normal to any base,absolutely normal. The Borel measure of normal numbers in the unitinterval is one. There are uncountably many absolutely normalnumbers.All absolutely normal numbers are normal to base two. All numbersnormal to base two have equal amounts of ones and zeros in theirin'nite binary expansion. There are uncountably numbers normal tobase two, with equal zero and one densities in their binary expansion.The count of in'nite binary sequences with equal zero and onedensities is countable, yet uncountable. It's uncountable, yetcountable.Is this not so? Why or why not?1. There are countably many sequences with equal zero- andone-density.2. There are uncountably many sequences with equal zero- andone-density.Explain why one or the other of the above statements is false.Use mathematics to solve problems.Half of the sequences have equal zero and one densities.Ross We observed an asymptotic expression for the density of sequences with> equal numbers of zeros and ones among all in'nite sequences.> This was somewhat less than 1/ sqrt(n), as n->oo. There are countably> many sequences with equal numbers of ones and zeros.By almost any reasonable de'nition of equal numbers of ones and zeros, it is easy to show that there are uncountably many. What is your particular, and probably unreasonable, de'nition?