mm-261 === Subject: Re: Borel measurable trying to write up a proof, and I'm using the fact if a function> f: R -> R> is differentiable everywhere, then the derivative f' is Borel measurable.> Is there an easy way to do this?f'(x) is the limit of n (f(x+1/n) - f(x)).--Ron === I'm trying to write up a proof, and I'm using the fact if a function>> f: R -> R>> is differentiable everywhere, then the derivative f' is Borel>> measurable. >> Is there an easy way to do this?> f'(x) is the limit of n (f(x+1/n) - f(x)).> --Ron === another question about circular sectors :)continued...There may be two solutions:For example take Area= .7854 and L =.7071 , input data for a quartercircle.In Mathematica, Plot[ { (1-Cos[th])/th - L^2/Area,L/Sin[th/2]},{th,1.5,Pi }];Apart from the expec quarter circle of radius 1, you also get === question about circular sectors :)A math wiz I am not, but when I run your numbers backwards, I get L =1.4142.Using your first answer where theta = 90 deg (or Pi radians) and radius = 1: L = 2 (r * sin(theta/2)) = 1.4142And, your second where theta = 180 deg (Pi radians) and radius = 1/sqrt(2) =0.7071: L = 2 (r * sin(theta/2)) = 1.4142By the way, I agree that there are two solutions...> continued...> There may be two solutions:> For example take Area= .7854 and L =.7071 , input data for a quarter> circle.> In Mathematica, Plot[ { (1-Cos[th])/th - L^2/Area,L/Sin[th/2]> },{th,1.5,Pi }];> Apart from the expec quarter circle of radius 1, you also get a> === question about circular sectors :)> When I run your numbers backwards, I get L => 1.4142.> Using your first answer where theta = 90 deg (or Pi[/2 you meant] radians) and radius = 1: L = 2 (r * sin(theta/2)) = 1.4142> And, your second where theta = 180 deg (Pi radians) and radius = 1/sqrt(2) => 0.7071: L = 2 (r * sin(theta/2)) = 1.4142Please see my above post again ; L is === another question about circular sectors :)Oops!> When I run your numbers backwards, I get L => 1.4142.> Using your first answer where theta = 90 deg (or Pi[/2 you meant]radians) and radius = 1: L = 2 (r * sin(theta/2)) = 1.4142> And, your second where theta = 180 deg (Pi radians) and radius =1/sqrt(2) => 0.7071: L = 2 (r * sin(theta/2)) = 1.4142> Please see my above post again ; L is the SEMI- chord, not === easy to lie in future!Here's 5 peoples posts to verify that mind reading technology is already here.When you think, you are not silent, a radar can pick up your thoughtsJUST LIKE SPEACH and sound them out.Because I'm the truman I get it constantly.100,000 people in townsville australia know for certain that 100% clear mind readingis possible, all my neighbours listen to my every thought every day.its the most hideous torture possible to constantly have you're thoughts playedback to you audibly and be FORCED to answer truthfully every passing remark,like I do every time I go out.Your voice box gets a trace stimulus of every thought you think, makes a smallnoise just like speaking, it can be picked up. They can play with the timing,they can hear compressed phonetics of whole sentences you are about tothink, and tell you your thought a second before you are aware of it.HercI CANNOT LIEEEEor another Jim Carrey, Majestic costarring Laurie HoldenExhibit A: http://tinyurl.com/fuf8 she looks exactly like Laurie HoldenExhibit B: http://tinyurl.com/fuf2 dates on these 2 posts place them clearlyin between the release dates ofThe Truman Show : 1998 : Jim CarreyMajestic : 2002 : Jim Carrey and Laurie HoldenI'm from Townsville and YOU ARE the Truman!http://tinyurl.com/iky5I was in Townsville over the weekend, and I heard him.Very spooky!http://tinyurl.com/iky8>phone someone in Townsville, half of you must know someone there,>every day I go out people say THERES THE TRUMANI'm in Townsville. We're sick of you.http://tinyurl.com/iky9http://tinyurl.com/iky4You rule Truman!>Do you know if the truman is living in Townsville?I've === Hey, look! It will not as easy to lie in future!>Here's 5 peoples posts to verify that mind reading technology is already> here.When you think, you are not silent, a radar can pick up your thoughtsJUST LIKE SPEACH and sound them out.Because I'm the truman I get it constantly.100,000 people in townsville australia know for certain that 100% clear> mind reading> is possible, all my neighbours listen to my every thought every day.its the most hideous torture possible to constantly have you're thoughts> played> back to you audibly and be FORCED to answer truthfully every passing> remark,> like I do every time I go out.Your voice box gets a trace stimulus of every thought you think, makes a> small> noise just like speaking, it can be picked up. They can play with the> timing,> they can hear compressed phonetics of whole sentences you are about to> think, and tell you your thought a second before you are aware of it.Herc> I CANNOT LIEEEEor another Jim Carrey, Majestic costarring Laurie Holden> Exhibit A: http://tinyurl.com/fuf8 she looks exactly like Laurie Holden> Exhibit B: http://tinyurl.com/fuf2 government has posts place them clearly> in between the release dates ofThe Truman Show : 1998 : Jim Carrey> Majestic : 2002 : Jim Carrey and Laurie Holden>>I'm from Townsville and YOU ARE the Truman!> http://tinyurl.com/iky5>I was in Townsville over the weekend, and I heard him.> Very spooky!> http://tinyurl.com/iky8>phone someone in Townsville, half of you must know someone there,>every day I go out people say THERES THE TRUMAN> I'm in Townsville. We're sick of you.> http://tinyurl.com/iky9>http://tinyurl.com/iky4> You rule Truman!>Do you know if the truman is living in Townsville?> I've been hearing stuff, yeah> http://tinyurl.com/p0w3>> I can remember listening to Amazing Randi's radio show in the '60's .> He was aware of the phenomenon.> It's called Subvocalization and apparently, some people can detect the> acoustial or electro/acoustical energy from you saying what you're thinking> under your breath .That's how I assume its done, but its done remotely with machines, a sateliteI think, you can't hear it yourself.Popular Science had a May issue with I was shot by the army's pain beam onthe cover, which was a clue to the 'weather' satelite story, that usetrains of satelites pumping quote 94 GHZ radar beam and pulsed laser array.> Somehow perple claiming to to mind reading suffer a stunning loss of> accuracy when isola acoustically from their victim ( oops..) client. (> oops!), I mean subject.> the Scientific study of this is slim, but Magick is not an acceptable> alternative at this time, when this is at least in the hypothesis stage.> Drifting off topic for sci.math??yes, but I'm not writing about esp, I'm submitting data that requires a simple statisticalanalysis for 6 months now as to whether it occurs naturally.I need sci.math to stamp my stat analysis so other groups won't dismiss it.Its not unlike a 5 mark question from second year stats. H0 the correlation isevident.... H1 Herc is rambling about nothing :H0 : ...Spend 10 seconds checking each tiny url I gave. Why would a man yell hisheart out he's the truman, then get numerous responses like this :>Do you know if the truman is living in Townsville?> I've been hearing stuff, yeah> http://tinyurl.com/p0w3>Do you want Duggy to post in sci.math and tell you to check my empiricial datashowing who I am? He's quite conversant and he'll tell you just what my inner thoughtssound like. I found his James Cook University email address and he's usedit since before I was in Townsville.I put the truman verifying posts there to show that lots of people know thetechnology is already here is complete form. You can listen to my thoughtstoday if you come to Townsville.Honestly, media cover up + internet global communications = apathy.HercNow if you don't mind, I have a dole form to submit and I have to stand in queuefor half an hour with 50 people around me all interrogating my thoughts, thenI'll thaw my last half loaf of frozen bread for lunch and then return into the mysteriesof the internet where the whole morning of EVERYONE in sight knowingwho I am never happened, like I have been for 2 years now.and don't set the group === lie in future!> and don't set the group header on memy mistake, cutting a group is fine, I jumped to the conclusion you set the forward,I often find myself posting just to alt.kibble once day i'll go there and it'll beall posts from === in future!and don't set the group header on me>> my mistake, cutting a group is fine, I jumped to the conclusion you set the forward,> I often find myself posting just to alt.kibble once day i'll go there and it'll be> all posts from me.dole office took nearly 2 hours btw, had to murmur a sermon to everyone to keep themquiet, then next month you can all tune into it word for word on everyone loves raymondor nic cages new prison scene.Hercaka the very poor star who gets abused === future!> Here's 5 peoples posts to verify that mind reading technology is already here.> When you think, you are not silent, a radar can pick up your thoughts> JUST LIKE SPEACH and sound them out.> Because I'm the truman I get it constantly.> 100,000 people in townsville australia know for certain that 100% clear mind reading> is possible, all my neighbours listen to my every thought every day.> its the most hideous torture possible to constantly have you're thoughts played> back to you audibly and be FORCED to answer truthfully every passing remark,> like I do every time I go out.> Your voice box gets a trace stimulus of every thought you think, makes a small> noise just like speaking, it can be picked up. They can play with the timing,> they can hear compressed phonetics of whole sentences you are about to> think, and tell you your thought a second before you are aware of it.> Herc> I CANNOT LIEEEE> or another Jim Carrey, Majestic costarring Laurie Holden> Exhibit A: http://tinyurl.com/fuf8 she looks exactly like Laurie Holden> Exhibit B: http://tinyurl.com/fuf2 dates on these 2 posts place them clearly> in between the release dates of> The Truman Show : 1998 : Jim Carrey> Majestic : 2002 : Jim Carrey and Laurie Holden> I'm from Townsville and YOU ARE the Truman!> http://tinyurl.com/iky5> I was in Townsville over the weekend, and I heard him.> Very spooky!> http://tinyurl.com/iky8>phone someone in Townsville, half of you must know someone there,>every day I go out people say THERES THE TRUMAN> I'm in Townsville. We're sick of you.> http://tinyurl.com/iky9> http://tinyurl.com/iky4> You rule Truman!>Do you know if the truman is living in Townsville?> I've been hearing stuff, yeah> http://tinyurl.com/p0w3An infinite number of exhibits constitutes merely evidence -- not proof.- -The second greatest error in reasoning is mistaking evidence for proof. The greatest ismistaking testimony for === It will not as easy to lie in future!Here's 5 peoples posts to verify that mind reading technology is already here.When you think, you are not silent, a radar can pick up your thoughtsJUST LIKE SPEACH and sound them out.Because I'm the truman I get it constantly.100,000 people in townsville australia know for certain that 100% clear mind reading> is possible, all my neighbours listen to my every thought every day.its the most hideous torture possible to constantly have you're thoughts played> back to you audibly and be FORCED to answer truthfully every passing remark,> like I do every time I go out.Your voice box gets a trace stimulus of every thought you think, makes a small> noise just like speaking, it can be picked up. They can play with the timing,> they can hear compressed phonetics of whole sentences you are about to> think, and tell you your thought a second before you are aware of it.Herc> I CANNOT LIEEEEor another Jim Carrey, Majestic costarring Laurie Holden> Exhibit A: http://tinyurl.com/fuf8 she looks exactly like Laurie Holden> Exhibit B: http://tinyurl.com/fuf2 government has spied on me so longNote in between the release dates ofThe Truman Show : 1998 : Jim Carrey> Majestic : 2002 : Jim Carrey and Laurie HoldenI'm from Townsville and YOU ARE the Truman!> http://tinyurl.com/iky5I was in Townsville over the weekend, and I heard him.> Very spooky!> http://tinyurl.com/iky8>>phone someone in Townsville, half of you must know someone there,>every day I go out people say THERES THE TRUMAN> I'm in Townsville. We're sick of you.> http://tinyurl.com/iky9http://tinyurl.com/iky4> You rule Truman!>>Do you know if the truman is living in Townsville?> I've been hearing stuff, yeah> http://tinyurl.com/p0w3> An infinite number of exhibits constitutes merely evidence -- not proof.that's why we use limits to reason what point it makesHerc> - -> The second greatest error in reasoning is mistaking evidence for proof. The greatest is> mistaking testimony for evidence.> --> --> === coverage area of a few circles?In my simulation, N circles with the same radius r are randomlyplaced. Let P_i denote the center of circle i. For any i, p_i lieswithin the coverage range of at least one other clicle, i.e. at leastone other circle contains p_i. How to calculate the total coveragearea of the N overlaped circle? The method should be easy to beimplemen by programming for simulation.Any comments is === calculate the total coverage area of a few circles?> In my simulation, N circles with the same radius r are randomly> placed. Let P_i denote the center of circle i. For any i, p_i lies> within the coverage range of at least one other clicle, i.e. at least> one other circle contains p_i. How to calculate the total coverage> area of the N overlaped circle? The method should be easy to be> implemen by programming for simulation.You can calculate the intersection of any two circlesanalytically:http://mathworld.wolfram.com/ Circle-CircleIntersection.htmlHowever, your problem might involve a large number of these,and in addition you need to calculate overlaps of 3 circles,4 circles, etc.It seems to me your best bet (reasonable accuracy in reasonabletime) is Monte Carlo integration: Generate some large number (M) of points uniformly in an area (A) that covers all your circles. Count how many (Mc) are covered by your circles.Estimate the coverage area as A*(Mc/M).This should be pretty quick if you === calculate the total coverage area of a few circles? 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21:> In my simulation, N circles with the same radius r are randomly> placed. Let P_i denote the center of circle i. For any i, p_i lies> within the coverage range of at least one other clicle, i.e. at least> one other circle contains p_i. How to calculate the total coverage> area of the N overlaped circle? The method should be easy to be> implemen by programming for simulation.> You can calculate the intersection of any two circles> analytically:> http://mathworld.wolfram.com/Circle-CircleIntersection.html> However, your problem might involve a large number of these,> and in addition you need to calculate overlaps of 3 circles,> 4 circles, etc.Edelsbrunner has inclusion-exclusion formulas that depend only on overlaps of at most three circles: The union of balls and its dual shape, http://portal.acm.org/citation.cfm?id=161139-- David Eppstein http://www.ics.uci.edu/~eppstein/Univ. of California, Irvine, === very much, Randy and Prof. Eppstein. The number of circles isabout 10. I think that the calculation time of Monte Carlo integration mightbe too long. Is it possible to use numerical intergration, i.e calculatingthe area enclosed by the envelope of those circles? But how can I get theexpression of this envelop easily in my simulation? I also need to considerthe sunk parts while integrating the area, right? The calculation is relato computer graphics. Could you please give me more with the same radius r are randomly> placed. Let P_i denote the center of circle i. For any i, p_i lies> within the coverage range of at least one other clicle, i.e. at least> one other circle contains p_i. How to calculate the total coverage> area of the N overlaped circle? The method should be easy to be> implemen by programming for simulation.You can calculate the intersection of any two circles> analytically:http://mathworld.wolfram.com/ Circle-CircleIntersection.htmlHowever, your problem might involve a large number of these,> and in addition you need to calculate overlaps of 3 circles,> 4 circles, etc.> Edelsbrunner has inclusion-exclusion formulas that depend only on> overlaps of at most three circles: The union of balls and its dual> shape, http://portal.acm.org/citation.cfm?id=161139> -- > David Eppstein http://www.ics.uci.edu/~eppstein/> Univ. of California, Irvine, School of Information & Computer === Eppstein. The number of circles is> about 10. I think that the calculation time of Monte Carlo integration might> be too long. Is it possible to use numerical intergration, i.e calculating> the area enclosed by the envelope of those circles? But how can I get the> expression of this envelop easily in my simulation? I also need to consider> the sunk parts while integrating the area, right? The calculation is rela> to again.> Leng Supengquo, but it sounds like it contains an efficient algorithmto do exactly what you want to do. Note that it's from acomputer graphics conference.Monte Carlo integration is a quick way of estimating theintegral you're talking about doing explicitly. On my Solarismachine here's the result of a quick run with 10 circlesand a half million points. That took 9 seconds of real time,6.6 seconds of CPU time, and converged to 4 decimal places.By the way, another method that occurs to me is to actuallyrender the circles in some pixela medium and then countcolored pixels. - Randy------------------------------------------Circle centersC = 0.1942 0.1138 0.0846 0.9897 0.9635 0.5098 0.4557 0.0639 0.6524 0.0272 0.0005 0.0413 0.3786 0.4947 0.0858 0.8082 0.5010 0.4129 0.3872 0.9048Circle radiians = Columns 1 through 8 0.9391 0.4621 0.9122 0.2243 0.6262 0.2088 0.4072 0.7326 Columns 9 through 10 0.3542 0.2420N A(est) clocktime CPUtime20000 4.791736 0.3326 0.270040000 4.768794 0.6383 0.540060000 4.775305 0.9956 0.810080000 4.774684 1.3191 1.0800100000 4.772452 1.6082 1.3500120000 4.772566 2.0246 1.6300140000 4.770211 2.4158 1.9000160000 4.768678 2.9425 2.1700180000 4.769965 3.2977 2.4400200000 4.769445 3.5792 2.7000220000 4.769358 3.9876 2.9700240000 4.767554 4.4163 3.2400260000 4.769772 4.7735 3.4900280000 4.769901 5.1083 3.7600300000 4.769807 5.4628 4.0100320000 4.768561 5.8629 4.2600340000 4.766387 6.1869 4.5300360000 4.768725 6.5527 4.7900380000 4.769789 7.0027 5.0500400000 4.768623 7.3632 5.3100420000 4.767096 7.7723 5.5800440000 4.767117 8.0896 5.8400460000 4.765990 8.4463 6.1000480000 4.764828 8.8470 === calculate the total coverage area of a few circles?> ...Randy Poe...:> In my simulation, N circles with the same radius r are randomly> placed. Let P_i denote the center of circle i. For any i, p_i lies> within the coverage range of at least one other clicle, i.e. at least> one other circle contains p_i. How to calculate the total coverage> area of the N overlaped circle? The method should be easy to be> implemen by programming for simulation....> http://mathworld.wolfram.com/Circle-CircleIntersection.html> However, your problem might involve a large number of these,> and in addition you need to calculate overlaps of 3 circles,> 4 circles, etc.Edelsbrunner has inclusion-exclusion formulas that depend only on> overlaps of at most three circles: The union of balls and its dual> shape, http://portal.acm.org/citation.cfm?id=161139> [...] The number of circles is about 10. I think that the calculation > time of Monte Carlo integration might be too long. Is it possible to > use numerical integration [...] need to consider the sunk parts while> integrating the area [...] The sunk parts presumably are accoun for by the inclusion-exclusionformulae of Edelsbrunner. Here is an alternative plane-sweep method thatwould be ok for ~ 10 circles (its complexity is probably O(N^3) as sta):Make a list of the N*(N-1) circle intersection points and the 2Npoints p_i with x +/- r that are not in the interior of anothercircle, and sort into ascending order by x coordinate, then sum theareas of vertical zones bounded by these critical points. Thereare no arc intersections within a zone. A zone may contain disjointsegments but each segment is bounded above and below by an arc of acircle, and left and right by straight lines.For example, if we have 3 circles of radius 5 and centers at (5,12),(8,9), and (9,5), the event-points list for the plane sweep is: 0.0 12.0 3.3 7.3 4.0 5.0 4.1 5.9 9.7 13.7 12.9 8.1 13.0 9.0 14.0 5.0The following edges or intersections are interior and not relevant: 3.0 9.0 4.4 7.0 9.6 10.0 10.0 === mass is an aggregation - via gravitation and other centripetal> forces - of the substances comprising any object; body, and or mass of> material matter; which causes these accumulations to have inertia, and/or> heft; the property of matter whereby it becomes more obvious that it> requires greater> (net) force to change the velocity of an accumulation as it becomes> larger:Here on Earth Galileo found that the rate of change in the velocity of any> body free falling at Earth's surface was about [s/t = 16'/sec] - half of> 'Newton's' acceleration of free fall [g] - and furthermore - in effect -> he> found that the force restraining this change in velocity of free fall from> continuing toward Earth's center is the mutual force exer between the> body and Earth's terra firma; which force is commonly measured with> weight-scales: On Earth, the ratio of this weight-force [w], divided by> the> rate of change in velocity [s/t = 16'/sec] that it restrains, is a> constant [wt/16' = wt/s]; to be known hereafter, as one half of the> body'sgravitational mass [g/2], and/or inertia.On any similar planet, such as the moon, this _ratio_ will still be equal> to> half of any body's mass, and/or inertia!>In everyday use, a body's mass [m] is commonly confused with its weight> [w]; A practice which must cease immediately; for the sake of physics!> According to newton's second law weight is the product of mass [m] and the> gravitational acceleration of free fall [g]: This erroneous formula w = ma> is a special case of f = ma, and like f = wa/g, must be written as w = fg/a:> All because inertial mass m = f/a, and gravitational mass m = w/g; where it> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a => w/g.> Cheeze, what have I got to do, write a book for youse people(:^?I doubt if it is necessary for you to distinguish between 'mass' and 'weight'for any physicist. What's your problem?--There are two things you must never attempt to prove: the unprovable -- and === quantity of matterPhysical mass is an aggregation - via gravitation and othercentripetal> forces - of the substances comprising any object; body, and or mass of> material matter; which causes these accumulations to have inertia,and/or> heft; the property of matter whereby it becomes more obvious that it> requires greater> (net) force to change the velocity of an accumulation as it becomes> larger:Here on Earth Galileo found that the rate of change in the velocity ofany> body free falling at Earth's surface was about [s/t = 16'/sec] -half of> 'Newton's' acceleration of free fall [g] - and furthermore - ineffect -> he> found that the force restraining this change in velocity of free fallfrom> continuing toward Earth's center is the mutual force exer betweenthe> body and Earth's terra firma; which force is commonly measured with> weight-scales: On Earth, the ratio of this weight-force [w], dividedby> the> rate of change in velocity [s/t = 16'/sec] that it restrains, is a> constant [wt/16' = wt/s]; to be known hereafter, as one half of the> body's> gravitational mass [g/2], and/or inertia.On any similar planet, such as the moon, this _ratio_ will still beequal> to> half of any body's mass, and/or inertia!>In everyday use, a body's mass [m] is commonly confused with itsweight> [w]; A practice which must cease immediately; for the sake of physics!According to newton's second law weight is the product of mass [m] andthe> gravitational acceleration of free fall [g]: This erroneous formula w =ma> is a special case of f = ma, and like f = wa/g, must be written as w =fg/a:> All because inertial mass m = f/a, and gravitational mass m = w/g; whereit> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a=> w/g.Cheeze, what have I got to do, write a book for youse people(:^? I'm sure that people would stand in line for blocks === matterCut<> I'm sure that people would stand in line for blocks to get a signedcopy!!> RJ PAs a matter of fact RJ, I have already written a couple, and can't even give'em away! As long as the gravy train keeps running, nobody wants to rock thegravy === people would stand in line for blocks to get a signed>copy!!>As a matter of fact RJ, I have already written a couple, and can't even give>'em away! As long as the gravy train keeps running, nobody wants to rock the>gravy boat.Your metaphors are as mixed as your understanding of === I'm sure that people would stand in line for blocks to get a signed>copy!!>As a matter of fact RJ, I have already written a couple, and can't evengive>'em away! As long as the gravy train keeps running, nobody wants to rockthe>gravy boat.> Your metaphors are as mixed as your understanding of elementary> physics.I think his metaphors are funny!But the idea is that THE ESTABLISHMENT (shiver, shiver)controls the TROOTH (tm) and suppresses publication of HERESY, especiallybecause they're making lotsa money is certainly neither novel nor humorous.You'd think that he could at least GIVE them away to social friends whowould say Gee, I'll get around to giving this some serious study.(Nudge,nudge, wink, wink) Bob === is that THE ESTABLISHMENT (shiver, shiver)>controls the TROOTH (tm) and suppresses publication of HERESY, especially>because they're making lotsa money is certainly neither novel nor humorous.Well that's the standard crackpot conspiracy theory, but somehow it'seven more ludicrous when adap to the scientific community.>You'd think that he could at least GIVE them away to social friends who>would say Gee, I'll get around to giving this some serious study.>(Nudge,nudge, wink, wink) The trouble is, most people have learned elementary physics in === Re: Mass is a quantity of matterCut<> I doubt if it is necessary for you to distinguish between 'mass' and'weight'> for any physicist. What's your problem?The problem isn't mine: It's that any physicist doesn't think that there_is_ a problem.Here I am attempting to prove the obvious, and all the while democracy isallowing crooked popularity to triump over === mass is an aggregation - via gravitation and other centripetal>> forces - of the substances comprising any object; body, and or mass of>> material matter; which causes these accumulations to have inertia, and/or>> heft; the property of matter whereby it becomes more obvious that it>> requires greater>> (net) force to change the velocity of an accumulation as it becomes>> larger:Here on Earth Galileo found that the rate of change in the velocity of any>> body free falling at Earth's surface was about [s/t = 16'/sec] - half of>> 'Newton's' acceleration of free fall [g] - and furthermore - in effect ->> he>> found that the force restraining this change in velocity of free fall from>> continuing toward Earth's center is the mutual force exer between the>> body and Earth's terra firma; which force is commonly measured with>> weight-scales: On Earth, the ratio of this weight-force [w], divided by>> the>> rate of change in velocity [s/t = 16'/sec] that it restrains, is a>> constant [wt/16' = wt/s]; to be known hereafter, as one half of the>> body's>> gravitational mass [g/2], and/or inertia.On any similar planet, such as the moon, this _ratio_ will still be equal>> to>> half of any body's mass, and/or inertia!>>In everyday use, a body's mass [m] is commonly confused with its weight>> [w]; A practice which must cease immediately; for the sake of physics!According to newton's second law weight is the product of mass [m] and the>> gravitational acceleration of free fall [g]: This erroneous formula w = ma>> is a special case of f = ma, and like f = wa/g, must be written as w = fg/a:>> All because inertial mass m = f/a, and gravitational mass m = w/g; where it>> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a =>> w/g.Cheeze, what have I got to do, write a book for youse people(:^?>I doubt if it is necessary for you to distinguish between 'mass' and 'weight'>for any physicist. What's your problem?With physicists, it's often the opposite problem--imaginingdifferences when they don't exist. Both of those words, of course,are ambiguous words with several different meanings.What Dense Donny is talking about is things like the net weight ofmy bag of sugar, 4 lb (1.81 oz) right on the label. Of course, netweight is not a physics concept in the first place. Second, whenevernet weight is used this is always the very same thing as mass inphysics jargon--but that synonym is not legal on those labels, the lawrequires either the spelled out word weight or the particularabbreviation wt on the labels.If you are a physicist, of course, you can choose not to call thisquantity weight. But what often happens instead is that they stillcall it weight but misapply a definition of weight which isinappropriate for the context. That simply is not an acceptablealternative.Just remember that your choosing not to call it weight does not meanthat someone else is making some error if they do call it weight--wehave a prior claim to this word by over 750 years over the physicistswho recently borrowed it and often use it with a different meaning.I really have a hard time understanding how so many people can be soGod-awful stupid as to think that when we buy and sell goods byweight, we'd want to measure some quantity that varies with location.We should not do so; we do not do so; we have never done so.I'm sure you wouldn't think twice about somebody calling a troy ouncea unit of weight. However, those troy units of weight, unlike theiravoirdupois cousins, and unlike grams and kilograms, have neverspawned a unit of force of the same name. They are always units ofmass.-- Gene Nygaardhttp://ourworld.compuserve.com/homepages/Gene_Nygaard/ It's not the things you don't know what gets you into trouble. It's the things you do know that just ain't so. Will === mass is an aggregation - via gravitation and other centripetal> forces - of the substances comprising any object; body, and or mass of> material matter; which causes these accumulations to have inertia, and/or> heft; the property of matter whereby it becomes more obvious that it> requires greater> (net) force to change the velocity of an accumulation as it becomes> larger:Here on Earth Galileo found that the rate of change in the velocity of any> body free falling at Earth's surface was about [s/t = 16'/sec] - half of> 'Newton's' acceleration of free fall [g] - and furthermore - in effect -> he> found that the force restraining this change in velocity of free fall from> continuing toward Earth's center is the mutual force exer between the> body and Earth's terra firma; which force is commonly measured with> weight-scales: On Earth, the ratio of this weight-force [w], divided by> the> rate of change in velocity [s/t = 16'/sec] that it restrains, is a> constant [wt/16' = wt/s]; to be known hereafter, as one half of the> body's> gravitational mass [g/2], and/or inertia.On any similar planet, such as the moon, this _ratio_ will still be equal> to> half of any body's mass, and/or inertia!>In everyday use, a body's mass [m] is commonly confused with its weight> [w]; A practice which must cease immediately; for the sake of physics!According to newton's second law weight is the product of mass [m] and the> gravitational acceleration of free fall [g]: This erroneous formula w = ma> is a special case of f = ma, and like f = wa/g, must be written as w = fg/a:> All because inertial mass m = f/a, and gravitational mass m = w/g; where it> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a => w/g.Cheeze, what have I got to do, write a book for youse people(:^?>>I doubt if it is necessary for you to distinguish between 'mass' and 'weight'>>for any physicist. What's your problem?>With physicists, it's often the opposite problem--imagining>differences when they don't exist. Both of those words, of course,>are ambiguous words with several different meanings.>What Dense Donny is talking about is things like the net weight of>my bag of sugar, 4 lb (1.81 oz) right on the label. Of course, netcorrection--that's a typo; make that 4 lb (1.81 kg)Those pounds, of course, are legally defined as 0.45359237 kg exactly.Neither pounds force nor kilograms force are legal for use in themarketplace in selling goods by weight.>weight is not a physics concept in the first place. Second, whenever>net weight is used this is always the very same thing as mass in>physics jargon--but that synonym is not legal on those labels, the law>requires either the spelled out word weight or the particular>abbreviation wt on the labels.>If you are a physicist, of course, you can choose not to call this>quantity weight. But what often happens instead is that they still>call it weight but misapply a definition of weight which is>inappropriate for the context. That simply is not an acceptable>alternative.>Just remember that your choosing not to call it weight does not mean>that someone else is making some error if they do call it weight--we>have a prior claim to this word by over 750 years over the physicists>who recently borrowed it and often use it with a different meaning.>I really have a hard time understanding how so many people can be so>God-awful stupid as to think that when we buy and sell goods by>weight, we'd want to measure some quantity that varies with location.>We should not do so; we do not do so; we have never done so.>I'm sure you wouldn't think twice about somebody calling a troy ounce>a unit of weight. However, those troy units of weight, unlike their>avoirdupois cousins, and unlike grams and kilograms, have never>spawned a unit of force of the same name. They are always units === get the question (fractional iteration?), but> folks should know that the universality of the M-set,> that is the recurrence of mini-bugs or cardioids,> at every level of magnification, is just an artifact> of the floating-point ops (IEEE-755, -855, I think).> this was (really/partially) confirmed by monsieur M,> when he glroriously begged my (only) technical question> at a talk for a general audience.It's quite simple to disprove your claim by setting the roundingmethod, which you can do in hardware on the Pentium (and manyother processors as well) to all it's values and seeing what changes,or doesn't change, in calculations. FP doubles are good for severaltens of thousand of iterations, down to an area of 10E-10 or so,before the precision gives out. The cartoids are visible severalorders of magnitude above === you might be the first. after all,I said that it was quite trivial. seriously, when I was passing though Santa Cruz,I stopped at Otto-Pagan's office to pursue this, buthe was on his European half of academia. when I sugges that,changing the hardware setting on the machine from double-precisionto single, he pooh-poohed it -- the grad student that I found. anyway, as I said, monsieur M. had already done it, orat least he made that inference at Young Hall.correction to what I typed:the mini-Ms do not appear at *every* magnification, sincethe rounding-errors are tied to the lengths of the registers,which is enough for a few iterations. as your statement impliese.g. the specification is inherently chaotic,as the term of art goes, no matter how variously implimen. > that is the recurrence of mini-bugs or cardioids,> at every level of magnification, is just an artifact> of the floating-point ops (IEEE-755, -855, I think).> this was (really/partially) confirmed by monsieur M,> when he glroriously begged my (only) technical question> at a talk for a general audience. > It's quite simple to disprove your claim by setting the rounding> method, which you can do in hardware on the Pentium (and many> other processors as well) to all it's values and seeing what changes,> or doesn't change, in calculations. FP doubles are good for several> tens of thousand of iterations, down to an area of 10E-10 or so,> before the precision gives out. The cartoids are visible several> orders of magnitude above this.--les ducs de Buffet;vote NONE OF THE BELOWon Trickier Dick Cheney's California === length of Gamma function>But I also wonder though, for the family of function where the arc length>from 0 to x *is* the same as its function value in x-1, so :>f(y-1) = int,0_y(sqrt(1+Df(x)^2) dx)Differentiating, you getf'(y-1) = sqrt(1 + f'(y)^2)or f'(y-1)^2 = 1 + f'(y)^2If g(y) = f'(y)^2, this has general solutiong(y) = h(y) - y where h is periodic with period 1.and thusf(x) = int_0^{x+1} sqrt(1 + h(t) - t) dt = int_{-1}^x sqrt(h(t)-t) dtwhich satisfies the original equation. For example, with h(t) = 1you get f(x) = (4 sqrt(2) - 2 (1-x)^(3/2))/3.Of course things get interesting when g(t) becomes negative...Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of === === Combination/Permutation Question...>Subject: Re: Combination/Permutation Question...>Image that you have 10 balls.>Permutation : The way(order) you can take them out from a box.>Example:> 1,2,3,4,5,6,7,8,9,0> 0,1,2,3,4,5,6,7,8,9> ..................> 4,3,2,1,5,8,6,7,9,0>Combination with no repeating: The way(order) you can take only m of them out>and m<10.>Example: m=3 > 1,2,3> 5,3,7> 0,8,2> .....>Combination with repeating: The way(order) you can take only m of them out>and m<10 and you have to return this ball back before take out another one .>Example: m=3 > 1,2,3> 4,7,7> 1,1,1> === finding an equationi know the roots of an equation, how do i go about finding the equation ifit is unknown (and the equation must not contain any irrational numbers)?i've used the quad.formula and worked backwords on some but what if === finding an equationSLH escribi.97 en finding the equation if> it is unknown (and the equation must not contain any irrational numbers)?> i've used the quad.formula and worked backwords on some but what if the> roots are sqrt(2) + sqrt(3)....any hints?Let x = sqrt(2) + sqrt(3)x^2 = 5 + 2*sqrt(6)x^2 - 5 = 2*sqrt(6)(x^2 - 5)^2 = x^4 - 10x^2 + 25 = 24 ==>x^4 - 10x^2 + 1 = 0-- Ignacio Larrosa Ca.96estroA Coru.96a === do i go about finding the equation if> it is unknown (and the equation must not contain any irrational numbers)?> i've used the quad.formula and worked backwords on some but what if the> roots are sqrt(2) + sqrt(3)....any hints?You form the polynomial whose roots are the conjugates of the given root eg.(x - sqrt(2) - sqrt(3)) . (x - sqrt(2) + sqrt(3)) . (x + sqrt(2) - sqrt(3)). (x + sqrt(2) + sqrt(3))= x^4 - 10.x^2 + === equation, how do i go about finding the equation if> it is unknown (and the equation must not contain any irrational numbers)?> i've used the quad.formula and worked backwords on some but what if the> roots are sqrt(2) + sqrt(3)....any hints?There may not be any equation having only the roots you want and still have all rational coefficients, but if you are allowed to have extra roots to achieve rational coefficients, you can take any algebraic number and all its conjugates as roots of a polynomial.For example, the conjugates of r1 = sqrt(2) + sqrt(3) are r2 = -sqrt(2) + sqrt(3), and r3 = sqrt(2) - sqrt(3), and r4 = -sqrt(2) - sqrt(3), so that (x-r1)(x-r2)(x-r3)(x-r4) = x^4 - 10*x^1 +1 is a polynomial with integer coefficients.This can always be done in theory, but can be difficult in practice, especially where roots other than square roots are involved or more than 2 square roots are === of an equation, how do i go about finding the equation if>it is unknown (and the equation must not contain any irrational numbers)?>i've used the quad.formula and worked backwords on some but what if the>roots are sqrt(2) + sqrt(3)....any hints?If r_1, r_2, ..., r_n are all the (complex) roots of a polynomial in x, then the polynomial is a constant multiple of product_{j=1}^n (x - r_j).Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === students?> I will have to give a short talk (15-20 minutes) to our> new mathematics students, and I find myself at a bit of> a loss as to what I am going to say. I should talk about> something mathematical, maybe some application, but the> technical part should be minimal. Ideally, it should> demonstrate some mathematical point or somehow prepare> and motivate them for the courses. It would also be> nice if there were some story attached to it to grip> their attention. However, anything entertaining and> mathematical would be acceptable.> Any ideas?a couple of years ago my brother an I both studying as maths teachers would havehad 20 emergency fun maths puzzles to fill up half lessons.The bird flying between two trains has a good anecdote, some students ofVon Neumon asked him after a lecture one day the puzzle how far doesthe bird fly, and after a couple seconds thought he said '20 miles'. Thestudents said oh you know the trick and he answered, 'no I just summedthe infinite series'.And / or you could talk about historic mathematicians, notably that most ofthem lead tragic lives. Pythagorus, Newton, Turing, ... there's many othersgetting all sorts of terrible things === happen, its a jinxed profession.HercSubject: Re: Length of go around a circumference.> Eg> If I wan to go around a piece of pipe 3 1/2 in diameter, with 1/4 flatbar, how long should I cut the bar. This is considering the fact that I can form it completely roundC = 2 * pi * r = pi * d, where C = circumference of the circle,pi = 3.14159..., r = radius, d = === for length of stock required to go around a circumference.Eg>> If I wan to go around a piece of pipe 3 1/2 in diameter, with 1/4 flatbar, how long should I cut the bar. This is considering the fact that I can form it completely round>C = 2 * pi * r = pi * d, where C = circumference of the circle,>pi = 3.14159..., r = radius, d = diameter.The diameter of the neutral axis is to be used in this formula:http://archive.metalformingmagazine.com/1999/12/ DieD.pdfAnd note that the outside diameter of a nominal 3.5-inch pipe is not3.5 inches:http://mdmetric.com/tech/pipe0010.htmHTHJoe === construction (doubling a point):Let X be a compact Hausdorff space and p be a non-isola point of X.> Take some q not in X and define X' = X+{q}, where the base in q is the> set of all {q}+U{p}, U is a neighbourhood of p; the topology of X is> the original one. Then any two neighbourhoods of p and q in X' have> non-empty intersection, but they have Hausdorff compact neighborhoods.What's the compact Hausdorff nhood of q?If U is a compact Hausdorff neighbourhood of p, then {q}+U{p} ishomeomorphic to U in the topology of X'. In fact we may take U=X.Simeon> Could anyone give me an example of a topological space X such that every> point of X has a Hausdorff compact === neighborhood, but X is not Hausdorff?>Subject: Re: Topology question === Re: Topology question > A general construction (doubling a point): > Let X be a compact Hausdorff space and p be a non-isola point > of X. Take some q not in X and define X' = X+{q}, where the base > in q is the set of all {q}+U{p}, U is a neighbourhood of p; the > topology of X is the original one. Then any two neighbourhoods of > p and q in X' have non-empty intersection, but they have Hausdorff > compact neighborhoods. >> What's the compact Hausdorff nhood of q? >If U is a compact Hausdorff neighbourhood of p, then {q}+U{p} is >homeomorphic to U in the topology of X'. In fact we may take U=X.Ok, I agree.>> Could anyone give me an example of a topological space X such that>> every point of X has a Hausdorff compact === Test |-|erc (was Re: A test proposal for Herc)> ie, you post. Someone responds. The content of their post > can be matched to their name.Lets try some reverse engineering here, what would someone whose nameis Hung Too Long post? > pretty good, the magic is inducing the response, Got news for you, Herky, magic is just a pretty synonymn for parlortrick.> so there's no divining necessaryThere is no divining parlor tricks.> its just analytical work to figure out the name, > anyone can do it once the posts are available, but since> you've set it up so I have to deduce them myself divine is apt. > for this claim.Yeah, well deduce MY name, k00k. As usual, you will either not respond or will reply evasively andoff-topic.--Grand Inspector of the K00kfinder GeneralSkepticult Member 518-27581-876For Entertainment Purposes Only. - DisclaimerWe're not laughing with you. - Skepticult CreedIn my life, I have prayed only one prayer in asking for divine favor: 'O Lord, make my enemies ridiculous.'And God gran it. - VoltaireAgainst stupidity, the Gods themselves contend invain. - The Postman Syndrome, Volume === Herc)>Herc,>Just so we all know what we are attempting here, can you please>clarify your claimed powers. How about we just drop the whole thing down a well, and then fill thewell with concrete?--V.G.People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper StickerSarcasm is my sword, Apathy === proposal for Herc)>Herc,>>Just so we all know what we are attempting here, can you please>clarify your claimed powers.> How about we just drop the whole thing down a well, and then fill the> well with concrete?> --that's no way to vanilla a === gorillaHerc electron-dot-cloud are galaxiesSubject: Superconductivity/Electronegativity Experiment Re: true theory of Superconductivity; differentiating Classical Physics from news item in SCIENCE NEWS of 11May02 of a inverse> proximity effect of lead with silver films. Superconductivity> temperature increased from 1.6 K to that of 1.8 K.That experimental datum suggests several things.(i) all materials at a cold enough temperature are superconductive and ifso, then superconductivity is a Classical physics phenomenon(ii) the difference between pure lead superconductive at 1.6 K and that oflead mixed with silver films is a huge difference even though the rise inTc is a small difference suggesting again that all materials aresuperconductive and thus Classical physics involved(iii) Superconductivity must be a geometrical phenomenon such asdiffraction gratings in that lead at 1.6 K and then another geometricaladdition of silver film to enhance the geometry, again suggestive ofClassical physicsClassical physics is Conduction Band theory and no-one has really exploredthe Maximization of Conduction Bands such that normal-conductivity becomessuperconductivity.Classical physics is Electronegativity/Electropositivity and no-one hasreally explored the Maximization of Electronegativity where we take sayfluorine and cesium, cool them, and where the cesium wants to push aelectron and the fluorine wants to pull a electron and between the push andpull is a electric current self-initia and self-built. Superconductivityis not a spontaneous current creation but when maximization ofelectronegativity and electropositivity is crea then superconduction isa cinch.So if we combine Conduction Band Maximization withElectronegativity/Electropositivity then Superconduction is a cinch.It seems to me, at this time, that Superconductivity as evidenced by leadand silver film can all be explained by Classical Physics of ConductionBand and Electronegativity and that the state of superconduction is amaximization of Conduction Band and/or/or both Electronegativity.Troubles with Quantum strangeness as explanation of Superconductivity:My favorite here is to think that photon messengers combine to become someneutrino hybrid as a messenger and we all know that neutrinos travelthrough matter with almost zero resistance. Trouble with that idea is thatif lead is superconductive at 1.6 K by quantum strangeness in convertingphotons into hybrid neutrinos then why should the additional sprinkling ofsilver film enhance this conversion. When we study polarization we do notget better polarization by the tiny adjustment, or do we.Archimedes Plutonium, a_plutonium@hotmail.comwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are === Experiment Re: true theory of Superconductivity; differentiating Classical Physics from Quantum Physics> (snip everything else)> That experimental datum suggests several things.> (i) all materials at a cold enough temperature are superconductive and if> so, then superconductivity is a Classical physics phenomenonPlease think about what you're saying. All is a mighty big word. Does sapphire superconduct or quarts or any insulators at low temperatures and normal === Experiment Re: true theory of Superconductivity; differentiating Classical Physics from Quantum Physics> (snip everything else)> That experimental datum suggests several things.> (i) all materials at a cold enough temperature are superconductive and if> so, then superconductivity is a Classical physics phenomenon> Please think about what you're saying. All is a mighty big word. Does > sapphire superconduct or quarts or any insulators at low temperatures > and normal pressures?Yes, I often choose the wrong word of all when most or some.But I think in this case I have chosen the correct word of all. Ilike to make the analogy of digital versus analog TV in comparisonof Quantum Physics to Classical Physics where the analog is ClassicalPhysics and that we get not discrete bits but a wavelength of varianceand gradation. So that in Superconductivity of pure lead at 1.6 K addsilver and boost Tc to 1.8 K is one of those fine analog gradationswhere if we look hard enough we can get a 1.7 Kby sprinkling in some other ingredient.I believe Classical Physics gives us this variable spectrum ofendresults. Whereas if superconductivity were a Quantum Physicsphenomenon can not get any such desired endresults that we want. If QMthen superconductivity would not be so fine-tunable.Indeed, I have not adequately considered pressures insuperconductivity, but that pressure makes the case of ClassicalPhysics even a stronger case in that we add the variable of pressureand get a whole new spectrum of gradations. Everyone knows thatBoyle's Law of PV = nRT is Classical Physics.How much of an argument can I wager that Fusion Physics is ClassicalPhysics and not Quantum Physics, ie, no quantum strangeness. Ditto forfission energy technology. No-one would say that quantum tunnelling orquantum strangeness occurrs in fission physics. So, if Fission andFusion were smack square in the middle of Classical Physics, then itseems to me that Superconductivity should be purely a ClassicalPhysics phenomenon regardless of the tie with the upper limit offusion-- 2/3 breakeven.Cooper pairs of electrons as per the BCS theory of superconductivityis a purely Quantum Physics phenomenon in that it invokes quantumstrangeness. What could be utterly more strange than 2 electronspairing?? Nothing. In fact, I believe the Cooper pair is the mostabsurd and preposterous modern day assumption and on par with theabsurdity of black holes.Archimedes Plutoniumwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are galaxies electron-dot-cloud are === galaxiesSubject: Self crea current Re: Superconductivity/Electronegativity Experiment Re: true theory of Superconductivity; differentiating Classical Physics from saying. All is a mighty big word. Does> sapphire superconduct or quarts or any insulators at low temperatures> and normal pressures?I did not seem to spell out the Experiment which I had lis into the title.The experiment I had in mind which Joseph seems to have jogged my memory, for it seems as though Ihave so many things on my mind that if I wander off just minorly I run the risk of not completingwhat I had star off in doing.Joseph talks about sapphire or quartz. And I would say they are superconductive but at a temperatureso close to 0 Kelvin.The experiment I wan to discuss and prod someone into doing involves fluorine and cesium. Or anytwo of the most electronegative and electropositive elements. I just picked fluorine and cesium butit could just as well be Cl and Ba or I and Rb. What I want to do is to get an Experiment where Iget the cold temperature to release a Self-Current or Spontaneous Current. Perhaps such a thing hasa different name.I want an experiment where the coldness of temperature of 2 atoms, one electropositive and the otherelectronegative, that the push of the electron and the pull of the electron from these 2 atomscreates a Self Current of electron flow.So that the Coldness creates electric current without resistance. And that Superconductivity is justthe creation of an electric current because the coldness has made the Electropositive atom push anelectron and the Electronegative has caused a pull of an electron to such an extent that Electropositive push + Electronegative pull = current creation = SuperconductivityIf Experiment can show that the push and pull of electrons when combined with very low temperatures,that Superconductivity is the creation of a Self Current and that is why there is no resistancebecause when you add an outside source of current it just ties in with the pushing and pulling ofthe atoms.So, set up an experiment with say Fluorine and Cesium and cool them to very low temperatures andlook for a Spontaneous current flow. If there is one, then I believe this is the ultimateexplanation of Superconductivity in that you have simply Maximized theElectronegativity/Electropositivity and also maximized Conduction Bands. Whether Conduction Bands isseparate from Electronegativity is unclear, but we do remind ourselves that Superconductivity is acomplex process and so it is likely that Superconductivity involves both Electronegativity andConduction Bands.Experiment I request: I am looking for an experiment where no outside source of electric current isapplied but wherein upon very cold temperatures a Spontaneous current appears to arise in the testsample.If lead with silver film superconducts at 1.8 K, then is there a tiny Spontaneous current at 1.5Kelvin??? Where Pb is considered the electronegative element and silver considered theelectropositive element and disregarding Conduction bands.But I still preferr to use Fl and Cs, or I and perhaps K.If I can get a current creation by lowering the temperature to very cold, then I think I cancompletely explain Superconductivity. It would be a tiny current, perhaps milliamperes, but if I canget current creation then the riddle of superconduction will be closer to a finish.Archimedes Plutonium, a_plutonium@hotmail.comwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are a nice coincidence that my return to Superconductivity theory in the last week is alsosuperfluidity and superconductivity. But it seems as though the Nobel Committee is falling into a trapof sorts. Call it a shotgun effect where they give the prize to just about anything insuperconductivity or superfluidity where the long march of history will look back and say what minordetails were given the highest awards or in the case of the BCS theory with its Cooper pairing as anutterly false theory that makes the Nobel Committee look bad. So the shotgun approach to making a pastmistake ameliora is by giving out so many Nobels in the area of superconductivity to drown out theinitial mistake and flaw of the BCS theory.And at the rate of the mistakes of the Nobel prize, it maybe frightening prospect in the future that1/2 of the Nobel awards had reached a point of becoming science mistakes and false science. And shouldthat Committee start awarding for black holes, wormholes and other exotica would accelerate thereaching of 1/2 of the Nobel prizes were given for false science.About the only way that the Nobel Science awards can have a record of being 90% or more of true scienceand the other 10% for falsehoods is if the Committee stuck tenaciously to awarding for Experimentalscience.The amazing thing about the Nobel prizes in physics was that in the early part of the 20th century, theCommittee went out on a limb by awarding for Quantum Mechanics to a large extent, and they got itphysics. Perhaps the Committee became lax with its success of awarding Quantum Physics from1900 to 1950, that this laxness made the Committee feel it could spot theoretical true physics fromwhere BCS and Cooper pairing come to mind. Neutron stars is another falsehood. And although Priontheory is not physics but biology, a case of another falsehood in science being awarded.And the entire Quark theory is at best a mere scaffolding. Not even architects and the general publicwould award a scaffold structure as one of the 7 wonders of the world.The Nobel Committee tries to get it right for 100% of the time, but they are fallible as any humanorganization becomes prone to error. It would behoove the Nobel Committee and for TV to run adocumentary on the historical fallibility and errors that the Nobel has thus far commit. If anorganization becomes so arrogant that they never seem to admit any wrong, then they only increase theirerror rate, rather than reducing it.I should mention Dr. Legget discussing the Nobel on TV. He mentions the fact that high-temperaturesuperconductors seem to have no theory to explain them.As far as I am concerned, the supercold superconductors such as pure lead can be explained byConduction Band theory and to experiment seeing whether a Self Crea current can be gotten bylowering the temperature. As for high temperature Superconductothe idea of Electronegativity versusElectropositivity maximized would explain them.Archimedes Plutonium, a_plutonium@hotmail.comwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are === Experiment Re: true theoryof Superconductivity; differentiating Classical Physics from Quantum Physics> (snip everything else)>>That experimental datum suggests several things.> (i) all materials at a cold enough temperature are superconductive and if> so, then superconductivity is a Classical physics phenomenon> Please think about what you're saying. All is a mighty big word. Does> sapphire superconduct or quarts or any insulators at low temperatures> and normal pressures?You are arguing with a psychotic idiot troll who is triviallygainsaid. 1) Bose-Einstein condensates operate at nanokelvins. No anomaloussuperconductiviy has been observed. The Meissner effect would betrivially detec. 2) Liquid helium will stay liquid right to absolute zero unlesscompressed to at least 25 atmospheres. We can say with completeassurance that liquid helium under its own vapor pressure will not bean electrical superconductor at any low temperature - right down toabsolute zero in the limiting case. There will never be a solidlattice whose phonons promote Cooper-pairing (much less free electronsto be Cooper-paired).http://www.eng.vt.edu/fluids/msc/super/ super-f.htmArchie-Poo is not only a jackass, he is a trolling boring ignorantjackass. -- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The === Experiment Re: true theoryof Superconductivity; differentiating Classical Physics from Quantum Physics> You are arguing with a psychotic idiot troll who is trivially> gainsaid. I know who he is. I normally ignore his posts but I read this one without looking to see who it was from. I just had to comment. Even he must know the difference between insulators and metals. He even tries to make retractions === textbook (Rosen) I'm reading about sequences like this: a_n = sum_{1<=i<=k} c_i * a_{n-i}where the c_i ' s are constants. The sequences that satisfy an equation likethis constitute a k-dimensional vector space V, since the first k terms in anysequence determine all the remaining terms. The equation above can easily befiddled into a polynomial equation whose solutions are exactly the numbers x forwhich a_n=x^n is a solution of the original equation. The book states, withoutproof, that if the polynomial equation has no repea roots, then V is spannedby the set of vectors (1,x,...x^{k-1}) for which x is a solution of thepolynomial equation. A while ago, when I first skimmed through the chapter, Itried to think of a reason why that set of vectors should necessarily beindependent, but I came up empty. Now one of my homework questions is to provethe analogous theorem, but without the hypothesis of no repea roots. Doesanybody have a good hint about how to crack this nut, or, failing that, areference? === Re: recurrence relations> In my discrete math textbook (Rosen) I'm reading about sequences like this:> a_n = sum_{1<=i<=k} c_i * a_{n-i}> where the c_i ' s are constants. The sequences that satisfy an equation like> this constitute a k-dimensional vector space V, since the first k terms in any> sequence determine all the remaining terms. The equation above can easily be> fiddled into a polynomial equation whose solutions are exactly the numbers x> for> which a_n=x^n is a solution of the original equation. The book states,> without> proof, that if the polynomial equation has no repea roots, then V is> spanned> by the set of vectors (1,x,...x^{k-1}) for which x is a solution of the> polynomial equation. A while ago, when I first skimmed through the chapter, I> tried to think of a reason why that set of vectors should necessarily be> independent, but I came up empty. Now one of my homework questions is to prove> the analogous theorem, but without the hypothesis of no repea roots. Does> anybody have a good hint about how to crack this nut, or, failing that, a> reference? Needless to say, all help is mucho appreciado.> Peaceone word: Vandermonde.[I think it's one word. Or is it Van Der Monde === talk>Also, when are the hour, minute and second hands positioned so that>they divide the clockface in three equal sectors?The short answer is never.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === students in a 10-15 minute talk>>Also, when are the hour, minute and second hands positioned so that>>they divide the clockface in three equal sectors?>The short answer is never.I was aware of the result. The interesting bit is thinking ofdifferent mathematical solutions to solving it, which leads us to thetopic of having multiple ways to reach a given === 10-15 minute talk>Also, when are the hour, minute and second hands positioned so that>they divide the clockface in three equal sectors?>>The short answer is never.> I was aware of the result. The interesting bit is thinking of> different mathematical solutions to solving it, which leads us to the> topic of having multiple ways to reach a given mathematical result.So if you think of the three hands as rotating unit vectors,their sum will never be zero. So here's another problem. Assignangular velocities to the three unit vectors so that the mimimumlength of their sum is as large as === minute talk>>Also, when are the hour, minute and second hands positioned so that>>they divide the clockface in three equal sectors?>>The short answer is never.>>I was aware of the result. The interesting bit is thinking of>>different mathematical solutions to solving it, which leads us to the>>topic of having multiple ways to reach a given mathematical result.> So if you think of the three hands as rotating unit vectors,> their sum will never be zero. So here's another problem. Assign> angular velocities to the three unit vectors so that the mimimum> length of their sum is as large as possible.There are various answers depending on what additional restrictionsyou apply. For j = 1,2,3, let Case j allow up to j hands to have thesame angular velocity. Define subcase (a) by requiring all hands tomeet at some time, and subcase (b) by not requiring this.(Applying no additional restrictions yields the trivial case 3b.)Solutions:Cases 3a and 3b: maxmin length is 3. All hands have equal angularvelocities and initial positions.Cases 2a, 2b, and 1b: maxmin length is 1. 2a: two hands have the same angular velocity, and one hand has an angular velocity different from the other two. 2b: The (2a) solution + another. The second solution is the same as (2a) except that the first two hands have opposite initial positions. 1b: The hour and minute hand start at the same initial position; the second hand, at the opposite position. The second hand moves twice as fast as the minute hand in the frame of reference of the hour hand.Case 1a: maxmin length is sqrt((47 - 14 sqrt(7))/27) ~= 0.607346.The second hand moves three times as fast as the minute hand in theframe of reference of the hour hand.I can't swear by these results: it's easy to slip up when casesproliferate. Nor have I proven them.I found it interesting to compute the minimal length for our standardclocks. It turns out to be 0.0025408119679, and occurs at2:54:34.56169 and9:05:25.43831.Note: none of the above results apply to digital clocks.-- | Jim Ferry | Center for Simulation |+------------------------------------+ of Advanced Rockets || http://www.uiuc.edu/ph/www/jferry/ +------------------------+| jferry@[delete_this]uiuc.edu | === in a 10-15 minute talk@vixen.cso.uiuc.edu:>> So if you think of the three hands as rotating unit vectors,>> their sum will never be zero. So here's another problem. Assign>> angular velocities to the three unit vectors so that the mimimum>> length of their sum is as large as possible.> There are various answers depending on what additional restrictions> you apply. For j = 1,2,3, let Case j allow up to j hands to have the> same angular velocity. Define subcase (a) by requiring all hands to> meet at some time, and subcase (b) by not requiring this.> Case 1a: maxmin length is sqrt((47 - 14 sqrt(7))/27) ~= 0.607346.> The second hand moves three times as fast as the minute hand in the> frame of reference of the hour hand.Yeah, that's the interesting case, and that's the right answer.We can restate it as trying to evaluate:sup over {a_1 sup over {a_1 Where the a_k and t are real. > Let f(n) be the maxmin length for n hands. Then f(1)=1, f(2)=0,> and f(3)=sqrt((47 - 14 sqrt(7))/27).> Some questions:> 1. Is f(n) always attainable by specific values of the a_i?> (And if so, is it attainable by integer values as in the case> of f(3)?)> 2. Is f(n) monotonically increasing?> 3. What are some other values for f(n)? (4- or 5-handed clock.)> 4. f(n) has a trivial upper bound of sqrt(n+1), but this is pretty> gross. How about a substantial improvement?> I have a particular interest in question 1.Some comments:* It seems like a good idea to normalize a_1 = 0 (i.e., replace a_k with a_k - a_1 for all k).* The hands all meet at t=0, z=1. Define T to be the minimum positive value of t such that all values of exp(a_k*t*i) are identical. With the normalization a_1 = 0, this is equivalent to requiring that each a_k*t is an integer multiple of 2 pi. This minimum can fail to exist in two ways: (1) because no t exists to make the values identical, in which case let T = infinity, and (2) because a_k = 0 for all k, in which case let T = 0. Let's assume that n > 1 so that the latter case never occurs (because the a_k are unequal).* If T = infinity, the function |sum_{k=1}^{n} exp(ak*t*i)| attains all values between 0 and n, so this case can be ignored. (Exercise for the reader.)* Now we now apply a further normalization (for finite, positive T): let T = 2 pi. I.e., replace each a_k by (2 pi/T) * a_k (making the complete normalization to replace each a_k by (2 pi/T) * (a_k - a_1)). This makes the a_k's co-prime integers (meaning that the GCD of all of them together is 1, not that they're pairwise co-prime).* So the candidate a_k's are n-tuples like (0,3,7,8,11): i.e., n-tuples of increasing integers beginning with 0. I would now apply one final normalization. Replacing each a_k by a_n - a_k results in the same solution (i.e., frame of reference is now the fastest hand, and we look at the clock in the mirror). In the example, this yields (0,3,4,8,11). The final normalization is to take the smaller of the two (in lexicographical order), which is (0,3,4,8,11) in this case.* Now the key to proving things about this problem is to obtain a bound on the minimal sum, showing that for big (a_k) it's small, leaving only a few cases to check. In particular, the answer to 1 should be yes, because the maximal minimum sum will be obtained for small (a_k), not approached by a sequence of large (a_k).* Okay, that's a lot of hot air, and no real proofs, but I it's how I'd approach the problem if I wan to prove things.-- | Jim Ferry | Center for Simulation |+------------------------------------+ of Advanced Rockets || http://www.uiuc.edu/ph/www/jferry/ +------------------------+| jferry@[delete_this]uiuc.edu | University of Illinois === * If T = infinity, the function |sum_{k=1}^{n} exp(ak*t*i)| attains all> values between 0 and n, so this case can be ignored. (Exercise for the> reader.)Oops. Not correct. I was thinking of the case where each pair (a_j,a_k)has an irrational ratio. The T = infinity case should never pertain, butJustifying this is more complica than I thought.-- | Jim Ferry | Center for Simulation |+------------------------------------+ of Advanced Rockets || http://www.uiuc.edu/ph/www/jferry/ +------------------------+| jferry@[delete_this]uiuc.edu | University of Illinois === talkWell, for (1), all you need is a series.> fly's speed and that time, work out the distance.I expect most people here have heard it, but I'll pass on the followinganecdote:Someone presen Feynman with that problem [bug flying between twoapproaching vehicles], and he of course solved it very quickly. The otherguy said, would you believe some people solve it with series! To whichFeynman responded, what's the === minute talkWell, for (1), all you need is a series.fly's speed and that time, work out the distance.> I expect most people here have heard it, but I'll pass on the following> anecdote:> Someone presen Feynman with that problem [bug flying between two> approaching vehicles], and he of course solved it very quickly. The other> guy said, would you believe some people solve it with series! To which> Feynman responded, what's the other way?> BOh Feynman not Neumann ;-) My apologies to both === 10-15 minute talkWell, for (1), all you need is a series.fly's speed and that time, work out the distance.> I expect most people here have heard it, but I'll pass on the following> anecdote:> Someone presen Feynman with that problem [bug flying between two> approaching vehicles], and he of course solved it very quickly. The other> guy said, would you believe some people solve it with series! To which> Feynman responded, what's the other way?:-)Here (USA), it's almost always (i.e., every time I've ever heard it) toldabout Von Neumann himself: he solves the problem very quickly, exclaims,Ah! Yes, it is 150 miles! or whatever, and, when the curious onlookersask him how he did it so quickly, he gives a blank look and replies,I summed the series.I never heard the algebraic approach referred to as the Von Neumannapproach === minute talk>Well, for (1), all you need is a series.fly's speed and that time, work out the distance.I expect most people here have heard it, but I'll pass on the following> anecdote:> Someone presen Feynman with that problem [bug flying between two> approaching vehicles], and he of course solved it very quickly. The other> guy said, would you believe some people solve it with series! To which> Feynman responded, what's the other way?>> Here (USA), it's almost always (i.e., every time I've ever heard it) told> about Von Neumann himself: he solves the problem very quickly, exclaims,> Ah! Yes, it is 150 miles! or whatever, and, when the curious onlookers> ask him how he did it so quickly, he gives a blank look and replies,> I summed the series.> I never heard the algebraic approach referred to as the Von Neumann> approach before.> -ArthurI meant the sum the series approach is the von Neumann approach,only to be told that the clever fellow was Feynman not Neumann. Now I'mutterly confused.Of course it's possible that von Feynmann _didn't_ sum the series andwas pulling his === Chapter 5 is all about colourings. The place where I'm getting the feeling> that there is a disconnect in terminology is that when we say a graph is> 5-chromatic, it just means that some colouring with 5 colours is possible> (and no fewer than 5). The specific colouring you use to establish this is> unimportant and is not considered to permanently colour the graph and all> its subgraphs (you might call this a graph with colouring to distinguish> it from just a graph).> There could be more than one colouring (and in fact there likely is more> than one). So unlike a bipartite graph where if it is bipartite then the> colouring is basically unique (assuming connecness), a 5-partite graph> doesn't necessarily have a uniquely well-defined partition into 5 classes.Those vertices that do not belong in only one specific partition maybe summarily discarded. Then the partitioning will be uniquelydefined.> That's why it doesn't make too much sense to talk about whether the sizes> of the five partitions must be such-and-such. When we delete a vertex> and ask if the resulting graph is 4-chromatic, the colouring needn't share> any common ground with the 5-colouring in the larger graph.There is a common ground. Vertices that were adjacent before removinga vertex are still adjacent afterwards. So 4-chroma coloring ispartially the same as the 5-chroma coloring.> Certainly it is possible to formulate the idea of a minimal 5-chromatic> graph in a way you might find more pleasing. For instance, we could say> that it's equivalent to a graph that is 5-chromatic in such a way that> for every vertex v there exists a 5-colouring of the graph in which v> is the sole vertex with the colour blue.The only 5-chroma === Re: Minimal Graph, Four Color Theorem>> Certainly it is possible to formulate the idea of a minimal 5-chromatic>> graph in a way you might find more pleasing. For instance, we could say>> that it's equivalent to a graph that is 5-chromatic in such a way that>> for every vertex v there exists a 5-colouring of the graph in which v>> is the sole vertex with the colour blue.>The only 5-chroma graph that I am likely to find pleasing is K5! Fair enough, but there are other minimal 5-chromatic graphs besides K5even if you aren't pleased by them :). For example, glue two regularpentagonal cones together at the base to get a polyhedron with 7 vertices,and form the natural adjacency graph on those vertices (of course, we geta planar graph). Then add one more edge joining the apex vertices.The resulting graph is 5-chromatic, but removing any vertex, no matterwhich one, always gives a graph that is 4-chromatic (and also planar,if I'm not mistaken). Personally === Minimal Graph, Four Color Theorem>> Certainly it is possible to formulate the idea of a minimal 5-chromatic>> graph in a way you might find more pleasing. For instance, we could say>> that it's equivalent to a graph that is 5-chromatic in such a way that>> for every vertex v there exists a 5-colouring of the graph in which v>> is the sole vertex with the colour blue.>The only 5-chroma graph that I am likely to find pleasing is K5! > Fair enough, but there are other minimal 5-chromatic graphs besides K5> even if you aren't pleased by them :). For example, glue two regular> pentagonal cones together at the base to get a polyhedron with 7 vertices,> and form the natural adjacency graph on those vertices (of course, we get> a planar graph). Then add one more edge joining the apex vertices.> The resulting graph is 5-chromatic, but removing any vertex, no matter> which one, always gives a graph that is 4-chromatic (and also planar,> if I'm not mistaken). Personally I find it just as pleasing as K5 :).The 5-chroma graph is non-planar and therefore, cannot be an mc-e === Minimal Graph, Four Color TheoremWhat I have been trying to say is that if chi(G)=5, G cannot beplanar!The only minimal counter-example to the FCT is K5!The conjecture that there exists a 5-chroma graph may be recolored to4-chroma is false.Yes I am confused! I am confused as to why anyone would argue sopassionately and so ineffectually in favor of a false FCT. I amconfused as to why I am considered crazy for wanting the FCT to betrue.Let H be any subgraph of G, where G has n vertices and H has n-1vertices. Then, the description of H seems to imply that the deletionof 'any' vertex from G will make chi(H)<=4.But this interpretation is generally false and is valid only forn=5!!! I intended this to mean that only K5 could be an mc-e to the FCT. Noneof the responses have convinced me to change my mind. Although, someof the responses really confused me!Perhaps, I erred in not offering a more substantial proof for theconjectureG = K5? Probably because I === Graph, Four Color Theorem Visiting Assistant Professor at the University of Montana.>What I have been trying to say is that if chi(G)=5, G cannot be>planar!You know, you ->really<- should drop the stupid exclamation signs. Itmakes you look like a raving loon.The claim that if chi(G)=5 then G cannot be planar IS the 4-colortheorem. You cannot ->assume<- it if you are trying to ->prove<- the 4color theorem.>The only minimal counter-example to the FCT is K5!No, K5 is NOT a counterexample to the Four Color Theorem, because the4 color theorem states that any ->planar<- graph can be colored withat most 4 colors in such a way that no two adjacent vertices share thesame color.>The conjecture that there exists a 5-chroma graph may be recolored to>4-chroma is false.There is no such conjecture.>Yes I am confused! I am confused as to why anyone would argue so>passionately and so ineffectually in favor of a false FCT. Nobody is. > I am>confused as to why I am considered crazy for wanting the FCT to be>true.You are being considered crazy because your posts read like you arecrazy. >Let H be any subgraph of G, where G has n vertices and H has n-1>vertices. Then, the description of H seems to imply that the deletion>of 'any' vertex from G will make chi(H)<=4.This is true if G is a minimal counterexample for the 4 Color Theorem.>But this interpretation is generally false and is valid only for>n=5!!!The triple exclamation points make you look like a raving loon. Sostart by removing them.Then note that the original argument star by ->assuming<- that theFCT is ->false<-, from which we deduce that if this is the case, thenamong them, there is one with the least number of vertices. Call n thenumber of vertices of this HYPOTHETICAL counterexample. Then, by thedefinition of n, any graph with fewer than n vertices must be4-colorable. In particular, if you took this HYPOTHETICAL example G,and removed one vertex, then the resulting graph would be 4-colorable.What exactly are you having trouble understanding about the aboveargument? Try to answer without using a ->single<- exclamation point.>I intended this to mean that only K5 could be an mc-e to the FCT.K5 is NOT even a hypothetical a counterexample to the FCT, because K5is not planar.> None>of the responses have convinced me to change my mind. Although, some>of the responses really confused me!If the responses have been confusing, I must say that it is becauseYOU are a very confusing fellow.>Perhaps, I erred in not offering a more substantial proof for the>conjecture>G = K5? Probably because I don't have one as yet. There is no such conjecture. The existence of a minimal counterexampleG (which must be PLANAR) is assumed as part of a === Four Color Theorem>The only minimal counter-example to the FCT is K5!> No, K5 is NOT a counterexample to the Four Color Theorem, because the> 4 color theorem states that any ->planar<- graph can be colored with> at most 4 colors in such a way that no two adjacent vertices share the> same color.>The conjecture that there exists a 5-chroma graph may be recolored to>4-chroma is false.> There is no such conjecture.>>Let H be any subgraph of G, where G has n vertices and H has n-1>vertices. Then, the description of H seems to imply that the deletion>of 'any' vertex from G will make chi(H)<=4.> This is true if G is a minimal counterexample for the 4 Color Theorem.>But this interpretation is generally false and is valid only for>n=5!!!> The triple exclamation points make you look like a raving loon. So> start by removing them.Point taken, Could you explain why?> Then note that the original argument star by ->assuming<- that the> FCT is ->false<-, from which we deduce that if this is the case, then> among them, there is one with the least number of vertices. Call n the> number of vertices of this HYPOTHETICAL counterexample. Then, by the> definition of n, any graph with fewer than n vertices must be> 4-colorable. In particular, if you took this HYPOTHETICAL example G,> and removed one vertex, then the resulting graph would be 4-colorable.> What exactly are you having trouble understanding about the above> argument? Try to answer without using a ->single<- exclamation point.I understand the argument perfectly. I have given the problem somethought and I have concluded that HYPOTHETICAL G is impossible. Nograph meets all three criteria; ie, G is === Graph, Four Color Theorem Visiting Assistant Professor at the [.snip.]>I understand the argument perfectly. I have given the problem some>thought and I have concluded that HYPOTHETICAL G is impossible. No>graph meets all three criteria; ie, G is 5-chroma, G is planar, H is>4-chroma.As has been no, if by H you mean any graph obtained by removinga single vertex from G, then your statement is exactly equivalent tothe 4 Color Theorem.Reading back through the thread, I ->think<- I'm beginning tounderstand what exactly it is that you are finding problematic. You are arguing that a G which requires 5 coloand with theproperty that removing any vertex results in a planar graph thatrequires just 4 coloand with a minimal number of vertices amongall graphs that satisfy that condition, must be equal to K5.This is true, but note that we have dropped the key property ofplanar from the assumptions of G. Rather, we are expec to take aplanar G which requires 5 coloand with the property that removingany vertex results in a graph that requires just 4 coloand withthe minimal number of vertices among all graphs that satisfy thatcondition. Clearly, any graph G that satisfies the conditions of (a) Being planar; (b) Requiring 5 colors; (c) Removing any vertex results in a planar graph that requires only 4 colors;also satisfies just (b) and (c); so a graph which satisfies (a)-(c)and has a minimal number of vertices among all graphs satisfying thiscondition would necessarily have 5 or more vertices (since a graphthat satisfies just (b) and (c) and has a minimal number of verticesamong all graphs satisfying (b) and (c) has exactly 5 vertices). Butthe only minimal graph that satisfies (b) & (c), namely K5, does notsatisfy (a). So the number of vertices of this hypothetical G will bestrictly greater than 5.But after that, you seem to be arguing that in fact this hypotheticalG would necessarily have no more than 5 vertices. And I think thatthe reason you are making this argument is that you point out that inorder to reduce G to a 4-colorable graph, you must remove all verticesof some color, which means that the vertex you removed was the onlyvertex of the given color; which in turn means that each vertex is theonly vertex of its color, which means G has 5 vertices, which means Gis K5, which is a contradiction.Or something like that.Now, if that is not what you are arguing, then you may ignore thispost and everything that follows.So, assuming I got the gist of your argument correct, the error is inthe step that goes from the vertex you removed was the only vertex ofthe given color to each vertex is the only vertex of its color.The fact that G-{v} can be 4 colored but G cannot means that for eachvertex v, there exists a coloring C(v), which ->depends on v<-, withthe property that v is the only vertex of its color. However, if v andw are two distinct vertices, there is no guarantee that the coloringof G-{v} is compatible with the coloring of G-{w}; that is, there is acoloring C(v) which depends on v and in which v is the only vertex ofits color, and there is a different coloring C(w) which depends on win which w is the only vertex of its color, but there is no reason toassume that w is the only vertex of its color under the coloring C(v),and there is no reason to assume that v is the only vertex of itscolor under the coloring C(w). This was mentioned by Erick Wong 40morgoth.sfu.caAgain, consider the 5-cycle, that is the graph consisting of 5vertices, {1,2,3,4,5}, with adjacencies 1-2-3-4-5-1 (so each n isadjacent to n+1 (mod 5) ).Removing any vertex results in a 2-chromatic graph; the graph itself,however, is not 2-chromatic, it requires 3 colors. For ->each<-vertex, there is a coloring of G in which that vertex is the onlyvertex of its color. However, there is no 3 coloring of the graph inwhich ->each<- vertex is the only vertex of its color, and there is noreason to assume that this is the case from the fact that there is acoloring for each vertex.That is, we are encountering a typical fallacious exchange ofquantifiers. We have:(1) For every vertex v, there exists a coloring C such that v is the only vertex of its color;and you seem to be interpreting this as being equivalent to(2) There exists a coloring C such that for every vertex v, v is the only vertex of its color.The two statements are not equivalent; (2) implies (1), but (1) doesnot imply === understand the argument perfectly. I have given the problem some|thought and I have concluded that HYPOTHETICAL G is impossible. No|graph meets all three criteria; ie, G is 5-chroma, G is planar, H is|4-chroma.I assume by this last clause you mean that all the graphs one obtains bydeleting a single vertex from G are 4-chromatic.This conclusion is equivalent to the four color theorem. If the four colortheorem is true, then no planar graph has chromatic number 5. On the otherhand, on the assumption that your conclusion above is correct, the fourcolor theorem follows by induction on the number of vertices. Once we'veshown it's true for planar graphs of up to n vertices, then it also mustbe true for planar graphs of n+1 vertices, since whatever graph H is, it'salready been shown to have chromatic number <=4 (and its chromatic numberdiffers from that of G by at most 1).So the only way you can reach such a conclusion is by an argument which isat most one short paragraph shorter than a proof of the four color theorem.I would assume that you're just relying upon the existing proof, except thatit wouldn't usually take some thought to conclude that a 5-chromaticplanar graph having a certain kind of subgraph doesn't exist, given thatno 5-chromatic planar graph exists at all.I just hate to see someone go away still confused, so I hope your clarityon the argument has reached the point of recognizing that this conclusionyou state above is very far from trivial, without taking the proof of thefour color theorem for gran. If there's some simple way to showsuch a G doesn't exist, a number of smart people have failed tosee it despite working hard on it === Theorem> |I understand the argument perfectly. I have given the problem some> |thought and I have concluded that HYPOTHETICAL G is impossible. No> |graph meets all three criteria; ie, G is 5-chroma, G is planar, H is> |4-chroma.> I assume by this last clause you mean that all the graphs one obtains by> deleting a single vertex from G are 4-chromatic.> This conclusion is equivalent to the four color theorem. If the four color> theorem is true, then no planar graph has chromatic number 5. On the other> hand, on the assumption that your conclusion above is correct, the four> color theorem follows by induction on the number of vertices. Once we've> shown it's true for planar graphs of up to n vertices, then it also must> be true for planar graphs of n+1 vertices, since whatever graph H is, it's> already been shown to have chromatic number <=4 (and its chromatic number> differs from that of G by at most 1).> So the only way you can reach such a conclusion is by an argument which is> at most one short paragraph shorter than a proof of the four color theorem.> I would assume that you're just relying upon the existing proof, except that> it wouldn't usually take some thought to conclude that a 5-chromatic> planar graph having a certain kind of subgraph doesn't exist, given that> no 5-chromatic planar graph exists at all.> I just hate to see someone go away still confused, so I hope your clarity> on the argument has reached the point of recognizing that this conclusion> you state above is very far from trivial, without taking the proof of the> four color theorem for gran. If there's some simple way to show> such a G doesn't exist, a number of smart people have failed to> see it despite working hard on it for a long time.> I hope you will be gracious and respond to my previous posting Re:Four Color Theorem Simplified.I notice that you have not responded to any of my previous posting rethe FCT.May I inquire as to === Assistant Professor at the University of Montana.>>The only minimal counter-example to the FCT is K5!>> No, K5 is NOT a counterexample to the Four Color Theorem, because the>> 4 color theorem states that any ->planar<- graph can be colored with>> at most 4 colors in such a way that no two adjacent vertices share the>> same color.>>The conjecture that there exists a 5-chroma graph may be recolored to>>4-chroma is false.>> There is no such conjecture.>>Let H be any subgraph of G, where G has n vertices and H has n-1>>vertices. Then, the description of H seems to imply that the deletion>>of 'any' vertex from G will make chi(H)<=4.>> This is true if G is a minimal counterexample for the 4 Color Theorem.>>But this interpretation is generally false and is valid only for>>n=5!!!>> The triple exclamation points make you look like a raving loon. So>> start by removing them.>Point taken, Could you explain why?Can I explain why the triple exclamation points make you look like araving loon? Because they do. It makes it seem like you are jumping upand down, yelling, spitting, and foaming at the mouth. That's themental image they conjure up.>> Then note that the original argument star by ->assuming<- that the>> FCT is ->false<-, from which we deduce that if this is the case, then>> among them, there is one with the least number of vertices. Call n the>> number of vertices of this HYPOTHETICAL counterexample. Then, by the>> definition of n, any graph with fewer than n vertices must be>> 4-colorable. In particular, if you took this HYPOTHETICAL example G,>> and removed one vertex, then the resulting graph would be 4-colorable.>> What exactly are you having trouble understanding about the above>> argument? Try to answer without using a ->single<- exclamation point.>I understand the argument perfectly. Then why did you think somebody was claiming the Four Color Theoremwas ->false<-?>I have given the problem some>thought and I have concluded that HYPOTHETICAL G is impossible.Good for you. > No>graph meets all three criteria; ie, G is 5-chroma, G is planar, H is>4-chroma.Good for you. But your argument seems to be no such G can exist,because then G would be K5, and that does not even begin to makesense. G cannot be K5 if it is assumed to be planar.Indeed, the proof of the Four Color Theorem rests on showing thatthere does not exist any graph G which requires 5 colois planar,and such that the removal of any vertex results in a graph which canbe colored with only 4 colors. But you have not given any coherentargument to establish this proposition that I can see anywhere. Allyou have done is yell === Re: Minimal Graph, Four Color Theorem>>The only minimal counter-example to the FCT is K5! No, K5 is NOT a counterexample to the Four Color Theorem, because the>> 4 color theorem states that any ->planar<- graph can be colored with>> at most 4 colors in such a way that no two adjacent vertices share the>> same color.The conjecture that there exists a 5-chroma graph may be recolored to>>4-chroma is false. There is no such conjecture.>>Let H be any subgraph of G, where G has n vertices and H has n-1>>vertices. Then, the description of H seems to imply that the deletion>>of 'any' vertex from G will make chi(H)<=4. This is true if G is a minimal counterexample for the 4 Color Theorem.But this interpretation is generally false and is valid only for>>n=5!!! The triple exclamation points make you look like a raving loon. So>> start by removing them.>>Point taken, Could you explain why?> Can I explain why the triple exclamation points make you look like a> raving loon? Because they do. It makes it seem like you are jumping up> and down, yelling, spitting, and foaming at the mouth. That's the> mental image they conjure up.>> Then note that the original argument star by ->assuming<- that the>> FCT is ->false<-, from which we deduce that if this is the case, then>> among them, there is one with the least number of vertices. Call n the>> number of vertices of this HYPOTHETICAL counterexample. Then, by the>> definition of n, any graph with fewer than n vertices must be>> 4-colorable. In particular, if you took this HYPOTHETICAL example G,>> and removed one vertex, then the resulting graph would be 4-colorable. What exactly are you having trouble understanding about the above>> argument? Try to answer without using a ->single<- exclamation point.>I understand the argument perfectly. > Then why did you think somebody was claiming the Four Color Theorem> was ->false<-?>I have given the problem some>thought and I have concluded that HYPOTHETICAL G is impossible.> Good for you. > No>graph meets all three criteria; ie, G is 5-chroma, G is planar, H is>4-chroma.> Good for you. But your argument seems to be no such G can exist,> because then G would be K5, and that does not even begin to make> sense. G cannot be K5 if it is assumed to be planar.Assume that G is 5-chroma, then show that no 5-chroma graph can beplanar. This seems to be a make sense.> Indeed, the proof of the Four Color Theorem rests on showing that> there does not exist any graph G which requires 5 colois planar,> and such that the removal of any vertex results in a graph which can> be colored with only 4 colors. But you have not given any coherent> argument to establish this proposition that I can see anywhere. All> you have done is yell like a loon that G would be K5!, which is> nonsense.You have misinterpre my use of the triple explanation marks. I amsuprised that you don't have me 'foaming at the mouth' if I use justone.I agree that it is nonsense to 'yell like a loon'; so I don't.Although, I would be in good company, ie, Archimedes.I think of G as a 5-chroma graph that might be planar; while you thinkof it as a planar graph that is or could be 5-chroma. I am afraid thatI overlooked our differing points of view.By the way, are there any other punctuation marks that you === Graph, Four Color Theorem Visiting Assistant Professor at the University of Montana. [.snip.]>> No>>graph meets all three criteria; ie, G is 5-chroma, G is planar, H is>>4-chroma.>> Good for you. But your argument seems to be no such G can exist,>> because then G would be K5, and that does not even begin to make>> sense. G cannot be K5 if it is assumed to be planar.>Assume that G is 5-chroma, then show that no 5-chroma graph can be>planar. This seems to be a make sense.Then show that no 5-chroma graph can be planar ->IS<- the 4 ColorTheorem. What you are saying is that to prove the 4-color map theorem,you should just prove the 4 color map theorem. Well, duh. >> Indeed, the proof of the Four Color Theorem rests on showing that>> there does not exist any graph G which requires 5 colois planar,>> and such that the removal of any vertex results in a graph which can>> be colored with only 4 colors. But you have not given any coherent>> argument to establish this proposition that I can see anywhere. All>> you have done is yell like a loon that G would be K5!, which is>> nonsense.>You have misinterpre my use of the triple explanation marks.I didn't say it ->made<- you into a raving loon. I said it made you->look<- like a raving loon.> I am>suprised that you don't have me 'foaming at the mouth' if I use just>one.One exclamation mark, if not overused, indicates emphasis, surprise,any number of things. A triple exclamation mark reads likeyelling. Continual use of multiple exclamation marks, joined with(apparently) not reading the responses to your questions, is whatmakes you look like you are foaming at the mouth.>I agree that it is nonsense to 'yell like a loon'; so I don't.>Although, I would be in good company, ie, Archimedes.Yes, and some people were laughed at and turned out to be geniuses. Onthe other hand, most of the people who are laughed at are clowns.>I think of G as a 5-chroma graph that might be planar; while you think>of it as a planar graph that is or could be 5-chroma. I am afraid that>I overlooked our differing points of view.If you think of G as a graph which has chi(G)=5 and may or may not beplanar, then you completely misunderstood the paragraph you quowhen this began, and then that is at least part of the reason for yourarguments/misunderstandings/confusion. The paragraph you quostar as part of a proof by contradiction, by assuming that therewas a graph G which was planar, and which had chi(G)=5, and which hada minimal number of vertices from among all graphs that satisfiedthose properties: being planar, AND having chi(G)=5.If you thought that such a graph might be planar, then you missedthe point entirely. The assumption is that it ->is<- planar.I do not think of this hypothetical planar G as a graph that couldhave or fail to have chi(G)=5. If it is part of a proof bycontradiction, then I ->must<- assume that the graph is planar ANDthat it satisfies chi(G)=5.Now say you are trying to prove The 4 Color Theorem by induction onthe number of vertices, as Keith Ramsey sugges. You have proventhat a planar graph with fewer than 5 vertices is 4 colorable. Now asan induction hypothesis, we assume that a planar graph with fewer thann vertices is necessarily 4-colorable, and consider a graph G which isplanar and has n vertices.At this point, G is a graph that could, indeed, have chi(G)>=5 orchi(G)<=4. Since for any given vertex v we have, by the inductionhypothesis, that G-{v} is 4-colorable, that shows that G is certainly5-colorable, so chi(G)<=5. Thus, at this stage, we have a graph whichis planar and which may or may not satisfy chi(G)=5. Perhaps that iswhat you meant. Note that in this situation, since the assumption isthat ALL planar graphs with fewer than n vertices are 4-colorable,that would mean that if chi(G)=5, then G is a minimal counterexampleto the conjecture. However, I fail to see how assuming that the graph G has n verticesand chi(G)=5, and being unsure as to whether G is or is not planar,would help you in figuring out the situation. Removing a vertex is notenough to be able to apply the induction hypothesis, since the resultmay not be planar. You would not be able to say that G is a minimalcounterexample. Or rather, assuming that G is a graph with thesmallest number of vertices among all graphs with chi(G)=5 is NOT thecorrect assumption to make; the induction hypothesis does notguarantee that chi(G) is minimal with this property, it onlyguarantees that any proper subgraph of G ->which is planar<- is4-colorable. So, for example, for all you know the graph properlycontains K5. There is no minimality property you could apply to thisG, or rather, the minimality hypothesis here is just plain ->wrong<-.>By the way, are there any other punctuation marks that you consider>signs of emotional unbalanceSigh. It was the entirety of your interaction. You star by asking areasonable question. When you encountered replies, your immediatereaction was to ->argue<- about those replies, using multipleexclamation marks. So you were being ->very<- emphatic, at the veryleast. The more exclamation marks you put, the more emphasis/volumeone is expec to read into the statement. Surprise! is not readthe same way as Surprise!! or as Surprise!!! or asSurprise!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Frankly, I felt like you were yelling in my face, and making littlesense to boot. In addition, your statements made it seem like you wereeither not reading, or not understanding, what people were writing. Assuch, we have someone who asks a question, and starts yelling toeveryone who replies, apparently without listening to their answers.->That<- makes you look like a === Theorem|Yes I am confused! I am confused as to why anyone would argue so|passionately and so ineffectually in favor of a false FCT. I am|confused as to why I am considered crazy for wanting the FCT to be|true.No, you are confused because you read badly. Nobody has been arguing thatthe four color theorem is false. Nobody!When in the proof of the four color theorem, it is assumed temporarilythat there exists a planar graph with chromatic number >4, that assumptionis being made in order to do a proof by contradiction. In a proof bycontradiction, one makes a temporary assumption in order to show eventuallythat the assumption must have been false. So the only reason anyone hasbeen considering the assumption that there exists such a graph, has beenin order to explain the proof which eventually shows that such a graphcan't exist after all. and I have both explained this rather key point before, butas far as I can see you just skimmed past both explanations. I urged youto deal with any qualms you might have with proof by contradiction first.If you keep ignoring our explanations, and instead keep pretending thatwe're arguing so passionately and so ineffectually for something thatwe're not arguing for at all, then there's not much point in continuing.It isn't necessary to phrase the proof as a proof by contradiction. Itcould be rephrased to make it a direct proof by induction on the numberof vertices of the graph. But such a rephrasing wouldn't affect the contentof the proof, and isn't needed to make the === Minimal Graph, Four Color Theorem> No, you are confused because you read badly. Nobody has been arguing that> the four color theorem is false. Nobody!I will concede that you are not arguing against the FCT if you willconcede that I have no === Ball selection from mulitple urns question>I have a probability question I could use some help with. I'll use>the urn/ball model to make it more general. Assume that I have n>urns, each with a different proportion of colored balls in it. I know>the probability P_n(c) of selecting each color of ball from each urn. >I will select one ball from each urn, for a total of n unordered balls>in the selec set S. Now, given a particular reference set of>unordered balls T, how do I determine the probability that S will>match it? For example, if I have 10 urns, how might I determine the>probability that I will choose 3 red, 2 blue and 5 green balls?If P_j(c) is the probability of selecting colour c from urn j, the probability of getting n_1 balls of colour 1, ..., n_k balls of colour kis the coefficient of x_1^n_1 x_2^n_2 ... x_k^n_k in product_{j=1}^n (P_j(1) x_1 + P_j(2) x_2 + ... + P_j(k) x_k).Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === of CST,which book of maths should I read first?> I am a freshman of computer science and technology.> I came across a problem.That is I don't know which book should I read> first,which should I read second......> I have three books right here.They are , Maths>,.> If you know,please help me! Change your major to basket weaving. What do any of the newsgroups have to do with CS?-- Memory Hole: The only al Qaeda cell uncovered intact wasopera by Israel. -- The Iron Webmaster, === should I read first?> I am a freshman of computer science and technology.> I came across a problem.That is I don't know which book should I read> first,which should I read second......> I have three books right here.They are , Maths>,.> If you know,please help me!> Change your major to basket weaving. What do any of the newsgroups have todo> with CS?Piss off, racist.-- Felony case 02-CR-0617 9/1/03: Oregon Department ofJustice V. Raymond Ronald Karczewski, Defendant.The === HELPHaving a complex number z = a+ib and its derivates as dz/dt = da/dt + idb/dt.Now, putting r = sqrt(a^2 + b^2), how === HELPdr/dt=(a.da/dt+b.db/dt)/2r> Having a complex number z = a+ib and its derivates as dz/dt = da/dt + i> db/dt.> Now, putting r = sqrt(a^2 + b^2), how to obtain the dr/dt ? Thanxs === dr/dt=(a.da/dt+b.db/dt)/2rwrong> Having a complex number z = a+ib and its derivates as dz/dt = da/dt + i> db/dt.> Now, putting r = sqrt(a^2 + b^2), how to obtain the dr/dt ? Thanxs TL>>an easy way to derivation is to differentiate the expressionr^2 = x^2 + y^2 on both sides, giving:2r dr/dt = 2x dx/dt + 2y dy/dt, or:dr/dt = (x dx/dt + y dy/dt)/r = (x dx/dt === Theorem/Andrew Wiles> There is a new perspective on this question. Please check out the post by> Tomas, http://mathforum.com/discuss/sci.math/m/77228/77277New? That's === Theorem/Andrew Wilescan anyone recommend a good book with regard to Wiles'>attempt to solve the problem over 7 years?> For the sincere and dedica amateurs:> Fermat's Last Theorem for Amateurs> (currently on sale in the Springer Yellow sale) and> Invitation to the Mathematics of Fermat-Wiles> (written by the individual who first realized that> elliptic curves were an essential step towards the solution)> Both these are by what I believe to be respec individuals> writing serious mathematics and are not just popularized near-fiction.The same goes for Alf van der Poorten's book, Notes on Fermat's Last === === sets, relationshipPerhaps I should have stuck to basics...If you go half way between 1 and 2 for example, to one and a half... as in1.5... and then half way again to between 1 and 1.5 to 1.25 and so on...this will give an infinite number of decimal places... and will never findan end result, as there will always be another half way point between anytwo points on the number line...Though I entered this discussion in the osophy newsgroup... and not as amathematician, but I am not completely ignorant of maths and physics. Itwas in regard to the most held belief by people who can really make stuffhappen in a spiritual way (ie. magic, answered prayer, healing powerflowing, and even the most recognised messiah JC) and they believe that Godis within all things, microscopic and beyond. That the building blocks ofall things are obviously within, and this in itself is God. That all thingstherefore are manifestations of various arrangements and organisations ofthese building blocks of energy as structures. Because man cannot yet makeone atom into another atom, does not mean it is not possible by greaterknowledge. In fact all elements are thought to come from hydrogen in starsaren't they? With we ourselves being from the substance of stars?You said that energy quantities were predictable... is this the case with anuclear bomb? For that is the energy potential which we access on aphysical plane. Because the invisible is not seen does not mean it does notexist.So it just seemed relevant to me that if we look for God in all things, bysubdividing them into various parts, there is always more, infinity. AndGod has always been described as the Infinite that holds all thingstogether... So it seemed that maths was backing this up.The if one was to consider the intelligence of mankind as being acollection of structures of mass/energy within the area of the human skullor body if one considers each living cell as well. Then it would not seem sostrange that the greater universe of everything, with the rawest of energybuilding blocks previously called God, would also be intelligent, especiallyconsidering the eternal life span of energy as being indestructable.So it seems to me while many say that God is not provable and is taken as afaith issue, I tend to find that God is actually provable as energy and asintelligence. Certainly we see how an invisible program in a computer isjust electrical charges made into the artificial structure of the hardware.What if it actually was the structure of human intelligence? What if thereis a higher intelligence than human intelligence and that structure couldtake up the whole universe? We would be as thoughts in the mind of such avast space of the universal mind?Have you considered that radio waves for example, can only travel becausethey are in fact in a medium that will let them travel? This in the same wayas waves on the surface of a pond when a rock is thrown in. So where thereis nothing by the human eye, even void of the elements, there still issomething else they would not be able to travel. Man is only discoveringthings that are already there. When mankind can make life from nothing,then we may be able to argue more, before then it would seem that mankindwill only be using what already is... and even then, if man was able to makean element out of nothing (nothing being an abesence of elements, but infact is zero point energy), then he would truly understand.There are those who blindly adhere to faith, I am not one of them. I believethat there is always an explanation which seems more than feasible... and isnot less valid than a big bang. I know my parents had a big bang and had toget married and then I came into being. :p The big band still had to have aprehistory... and that always comes back to energy which always exis andalways will exist... such would form and attract and repel into variousstructures, and has as much chance of evolving into an intelligent invisiblebeing as any Darwinian theory of evolution.So mathematically the infinite is within all things and between all things.So spiritually so it the invisible energy which holds all things together.--Peace is within, and is projec outwardly.Searching... for eternal truths...>Something interesting with the infinite...> Maybe, if one considers nonsense to be interesting.>How many decimal places are possible between the numbers 1 and 2?> Hint: decimal places do not occur *between* numbers; they occur *within*> the decimal representations of numbers.>Just goes to show that mathematics can go on forever.> Meaning that mathematics deals with the notion of infinity? That's> certainly true, and indeed prior to Cantor there really wasn't a coherent> notion of infinity in the first place; infinitary language just meant> big -- really big -- you just won't believe how vastly, hugely> mind-bogglingly big it is [D. Adams]> So to say that there>is the eternal infinity within all things is not such a long shot.> To say that is to spout mush.> Some call this God, others energy, and others again don't even considerit.> If you mean energy as physics uses it then you're completely out to> lunch. The mass-energy of a thing is finite. If you mean energy in> some flakey new-age sense, then goodness knows what, if anything, you> actually mean.> While some>things add up, sometimes the answer is infinite and beyond the reach of>precision.> Hint: precision is an entirely different notion from magnitude.> Is just halving the difference a million times enough? Or just>accepting the figure of infinity?> Or maybe just getting a clue somewhere?>The more two things are compared, the more differences can be found if we>look hard enough.> Not in mathematics (1 + 1 absolutely exactly equals 2 all the time), and> not in physics (two electrons are absolutely indistinguishable, except for> their positions and velocities).> Factually would all things have infinite difference?> No, see above.>Darn now some scientist will get noble peace prize for this...lol> Ever heard of Linus Pauling? Anyway, in addition to peace prizes of> various degrees of nobility, maybe some scientist will also get a Nobel> peace prize for it.>-->Peace is within, and is projec outwardly.>Searching... for eternal truths...> --> ---------------------------> | BBB b barbara minus knox at iname stop com> | B B aa rrr b |> | BBB a a r bbb |> | B B a a r b b |> | BBB aa a r bbb |> === sets, relationship> Though I entered this discussion in the osophy newsgroup... and not as a> mathematician, but I am not completely ignorant of maths and physics. ItIt seems to me that this is by now out of topic in all the groupspos to (Sorry everyone not interes in the thread). In fact youare not completely ignorant of maths and physics but almost. I'm notgoing to point out every single wrong statement you made in are serious aboutsearching for truths I respectfully advise you to do some moreserious reading, you cannot benefit from science without properunderstanding.Now to what concerns me most. You seem to be mixing up God with theinvisible and the not (yet) known. These are not attributes of Godregardless of wether you believe in God or not. You, and many othersare looking for God between the spaces of the aparent world, and inwhat seems to you to be spaces left unexplored by modern science.However, the spaces you identified are not really there, but in yourignorance of science. God is outside the realm of science, even if hemanifests himself in the physical world, it is beyond science tocomprehend full knowledge of God. You seem to be confused by the factthat science has nowadays alternative explanations for most of what inthe past could only be explained in terms of divine origin. Sciencehas given us a model of the world which enables us to interact with itin a much more practical way and is as a system of knowledge mostlytrue. However, science is not the owner of Truth. If you are abeliever you may find Truth in religion, and yet accept the truth ofscience even where they contradict. With wisdom, you can resolve thoseconflicts in the proper manner when they arise.We live in a time when science and technology have become the onlysanctioned ways of knowing and interacting with the world. We havebecome addic to them because they so greatly satisfy our reason,and yet they are poor replacemens for religion, because they cannotsatisfy our spirit.If you are looking for spiritual truths you shouldn't look for themthrough science which might be the currently accep way of lookingat the world, but be brave, and look for them in religion. There is noshame in being irrational, as long as you don't meddle into the realmof the rational. If you are looking for God, don't look for him inhalf-baked pseudo-scientific theories, look for him by faith which isthe proper way to God. God does not need to be explained, and === mathematics, sets, relationship>> Though I entered this discussion in the osophy newsgroup... and not as a>> mathematician, but I am not completely ignorant of maths and physics. It>It seems to me that this is by now out of topic in all the groups>pos to (Sorry everyone not interes in the thread). In fact you>are not completely ignorant of maths and physics but almost. I'm not>going to point out every single there are several, and if you are serious about>searching for truths I respectfully advise you to do some more>serious reading, you cannot benefit from science without proper>understanding.For sure.>Now to what concerns me most. You seem to be mixing up God with the>invisible and the not (yet) known. These are not attributes of God>regardless of wether you believe in God or not. You, and many others>are looking for God between the spaces of the aparent world, and in>what seems to you to be spaces left unexplored by modern science.Yes indeed. The god of the gaps is a shy creature indeed, sincewhenever a gap gets closed she needs to go hide somewhere else.>However, the spaces you identified are not really there, but in your>ignorance of science.Well said. But if the OP takes the trouble to learn enough science thenthey will come across some very interesting current gaps, for example theincompatibility between our current understandings of general relativityand of quantum mechanics (i.e., between gravity and everything else). Soas long as one is willing to shift their god(s) into whereever the currentgaps are, a god of the gaps strategy is at least plausible.> God is outside the realm of science, even if he>manifests himself in the physical world,Not so fast. If something manifests itself in the physical world then itshould be measurable by physical science, at least in principle.>it is beyond science to comprehend full knowledge of God.It's beyond us finite creatures to comprehend full knowledge of *anything*sufficiently complex. For example, even in mathematics, Goedel'sincompleteness theorem shows that we can never have full knowledge of theproperties of even something as seemingly straightforward as the countingnumbers.> You seem to be confused by the fact>that science has nowadays alternative explanations for most of what in>the past could only be explained in terms of divine origin. Science>has given us a model of the world which enables us to interact with it>in a much more practical way and is as a system of knowledge mostly>true. However, science is not the owner of Truth.To paraphrase a famous bad-guy, what is this capital-T Truth?> If you are a>believer you may find Truth in religion, and yet accept the truth of>science even where they contradict.That way lies madness, or at least dysfunction. You had jolly well betterstill look both ways before crossing the street, regardless of yourreligious beliefs. > With wisdom, you can resolve those>conflicts in the proper manner when they arise.Psychologists call this dissociation, which is a psychological means tocope with impossible situations. It is a necessary survival response, butas a preferred life strategy it is surely a Bad Thing.>We live in a time when science and technology have become the only>sanctioned ways of knowing and interacting with the world.Sanctioned by whom? Many professional religious people still make apretty good living, as do astrologepsychics, stock-market touts, andother purveyors of wishful thinking.> We have>become addic to them because they so greatly satisfy our reason,No. Most people don't understand much science or engineering at all, butthey still benefit greatly from its products.>and yet they are poor replacemens for religion, because they cannot>satisfy our spirit.Currently the psychological sciences and technologies are way way behindthe physical ones, but it's entirely plausible that someday there will beeffective and well-understood ways to treat all sorts of emotionaldistress (including ones currently labeled spiritual).>If you are looking for spiritual truths you shouldn't look for them>through science which might be the currently accep way of looking>at the world, but be brave, and look for them in religion.It doesn't take much bravery. It just takes the realisation that ifyou're looking to find non-scientific Truths then science is rather apoor place to search for them.> There is no>shame in being irrational, as long as you don't meddle into the realm>of the rational.As I said, you'd better still look both ways before crossing the street.> If you are looking for God, don't look for him in>half-baked pseudo-scientific theories,Amen!> look for him by faith which is>the proper way to God. God does not need to be explained, and you>cannot find God through explanation.Although, a lot of human experience of gods can be explained. Forexample, have a look at Julian Jaynes's _The Origin of Consciousness inthe Breakdown of the Bicameral Mind_.-- ---------------------------| BBB b barbara minus knox at iname stop com| B B aa rrr b || BBB a a r bbb | | B B a a r b b | | === mathematics, sets, relationshipMcSnip> If you are a>believer you may find Truth in religion, and yet accept the truth of>science even where they contradict.> That way lies madness, or at least dysfunction. You had jolly well better> still look both ways before crossing the street, regardless of your> religious beliefs.Doublethink is OK for a lot of Folks.Unfortunately, the OP has expressed a version of what is called Hiqmat bysome Bahai's. ( I was posting to Soc.religion.bahai but got kicked out forpointing stuff like this out)Actually the meaning put to the word wasTrue science, which agrees with the Scriptures, contras with FalseScience, believed by infidelsthis becomes a marvelous exercise in TautologyThe idea is that God can do miracles by temporarily suspending the Laws ofthe Universe.I might add an Ambrose Bierce here..... for one petitioner, admitly unworthy.Did Not Mark Twain define Faith asBelieving stuff you know ain't === true?RJ PSubject: Re: mathematics, sets, Truth in religion, and yet accept the truth of>>science even where they contradict.That way lies madness, or at least dysfunction. You had jolly well better>> still look both ways before crossing the street, regardless of your>> religious beliefs.>Doublethink is OK for a lot of Folks.>Unfortunately, the OP has expressed a version of what is called Hiqmat by>some Bahai's. ( I was posting to Soc.religion.bahai but got kicked out for>pointing stuff like this out)>Actually the meaning put to the word was>True science, which agrees with the Scriptures, contras with False>Science, believed by infidels>this becomes a marvelous exercise in Tautology>The idea is that God can do miracles by temporarily suspending the Laws of>the Universe.>I might add an Ambrose Bierce here.. for one petitioner, admitly unworthy.>Did Not Mark Twain define Faith as>Believing stuff you know ain't true?>RJ PNo doubt religion is a shady business these days,but those profiteers aren't really serious. You won't findany of them taking their own advice, for instance.Pity the fools who follow and send money.On the other hand, I find the childish belief in 'nuthin-butmaterialism' pretty weak as well. How did the elephantget a long nose? A crocodile stretched it at the water hole.No wonder any more. We know. It's Just so.How did the big bang lead to man at the pinnacle of thefood chain? Well, there's the laws of physics, and evolution,and lots of time, and here we are. Just so.Of course it's just so. That's what happened no doubt.We don't even have to imagine a maker or a plan.All that is gran. Science is correct. No argument.But what does that mean for us?That's where science can't go. Those who imagine thatscience has all the answers don't understand science,and certainly don't understand their own 'being' in theworld. The story of science is of a magnificent creationof everything from nothing. Sort of miraculous, eh?Oh, === relationship>>McSnip> If you are a>>believer you may find Truth in religion, and yet accept the truth of>>science even where they contradict.That way lies madness, or at least dysfunction. You had jolly wellbetter>> still look both ways before crossing the street, regardless of your>> religious beliefs.>Doublethink is OK for a lot of Folks.>Unfortunately, the OP has expressed a version of what is called Hiqmatby>some Bahai's. ( I was posting to Soc.religion.bahai but got kicked outfor>pointing stuff like this out)>>Actually the meaning put to the word was>True science, which agrees with the Scriptures, contras with False>Science, believed by infidels>this becomes a marvelous exercise in Tautology>The idea is that God can do miracles by temporarily suspending the Lawsof>the Universe.>>I might add an Ambrose Bierce here.. for one petitioner, admitly unworthy.>>Did Not Mark Twain define Faith as>Believing stuff you know ain't true?>>RJ P>> No doubt religion is a shady business these days,> but those profiteers aren't really serious. You won't find> any of them taking their own advice, for instance.> Pity the fools who follow and send money.> On the other hand, I find the childish belief in 'nuthin-but> materialism' pretty weak as well. How did the elephant> get a long nose? A crocodile stretched it at the water hole.> No wonder any more. We know. It's Just so.> How did the big bang lead to man at the pinnacle of the> food chain? Well, there's the laws of physics, and evolution,> and lots of time, and here we are. Just so.> Of course it's just so. That's what happened no doubt.> We don't even have to imagine a maker or a plan.> All that is gran. Science is correct. No argument.> But what does that mean for us?> That's where science can't go. Those who imagine that> science has all the answers don't understand science,> and certainly don't understand their own 'being' in the> world. The story of science is of a magnificent creation> of everything from nothing. Sort of miraculous, eh?> Oh, wait. It's just so.You might be surprised that I agree with you to some extent.I would have to characterize myself as agnostic, with the belief that thereis a teleology, but at this stage, we couldn't understand what it is becausewe lack the intelligence and language to do so.The creation of everything Ex nihilo is a peculiar idea, ( unless doneby the Gnostic Deity Shaddai) but the alternatives of Giving politicalsupport to madmen who want to make second class citizens out ofNon-Christians is certainly not the coolest gig I can envision.Blind Faith in Science is certainly not an absolute good.In my worldview, I regard it as a safer alternative that the other extreme,as presen in the movie Wizards (Technology is BAD, and MAGICK isGOOD!!!,and that Science encourages Despotism and suppression of the Human SpiritMORE than Superstition.)The problem is that practically nobody I know knows enough about Science tobe able to justify anything BUT blind faith in it, possible as the PC modeof belief in their peers.Usually, the best answer I can get if someone is trying to justify theCopernican Theory in classical terms is 1. (lowered pince-nez) Ockham's razor, ya know!!.Or The Catholic Church was against it, so it MUST have been correct!!)the AnswerScientists sez so is maybe better than these, but not much.If you have to watch Football and be on the Office Bowling team, you don'thave much time to study Natural osophy or Mathematics. Rave mode approaching.Gotta === relationship> We live in a time when science and technology have become the only> sanctioned ways of knowing and interacting with the world. We have> become addic to them because they so greatly satisfy our reason,> and yet they are poor replacemens for religion, because they cannot> satisfy our spirit.Spirit, schmirit. When you can live a comfortable life, not worry about your next meal, avoid or be cured of diseases that used to kill millions and outlive all but a few of the Patriarchs, what is there to complain about? Don't you know a Good Thing when you see it. While it is true that man does not live by bread alone, there is nothing wrong with having plenty of bread.> If you are looking for spiritual truths you shouldn't look for them> through science which might be the currently accep way of looking> at the world, but be brave, and look for them in religion.Spiritual truths are the work of delusions and self deception. The only truths that matter are those that can be associa with facts in the real world. There is no> shame in being irrational, as long as you don't meddle into the realm> of the rational. If you are looking for God, don't look for him in> half-baked pseudo-scientific theories, look for him by faith which is> the proper way to God. God does not need to be explained, and you> cannot find God through explanation.Faith and one dollar will get you a ride from Alewife Brook station to Park Street Under on the MBTA. Which gives a good indication of what faith === relationship> We live in a time when science and technology have become the only> sanctioned ways of knowing and interacting with the world. We have> become addic to them because they so greatly satisfy our reason,> and yet they are poor replacemens for religion, because they cannot> satisfy our spirit.> Spirit, schmirit. When you can live a comfortable life, not worry about > your next meal, avoid or be cured of diseases that used to kill millions > and outlive all but a few of the Patriarchs, what is there to complain > about? Don't you know a Good Thing when you see it. While it is true > that man does not live by bread alone, there is nothing wrong with > having plenty of bread.Hi Bob why have you deliberately taken this statement out of context?In my posting, I had a few lines above clearly sta that science isbenefitial or a Good Thing as you put it. In this quo paragraph Iam making a point wich you choose not to address. Nor am I complainingat all. The point is, that nowadays it is virtually impossible to talkseriously to common people about anything that lies outside therational, scientific or technically explainable. Raising issues suchas spirituality, miracles, divine intervention etc. will get youmostly contempt and ridicule. That is a clear impoverishment of ourlives, because it imposes a limit on what thoughts we can exchangewith each other. The point is not wether such issues are valid or not,but wether it is valid to adopt an obscurantistic point of view anddenigrate them out of hand because they don't fit into the rationalscheme of things. Is this not exactly the same kind of thing (underinver signs) that scientists had to face in what you would callless enlightened times?> If you are looking for spiritual truths you shouldn't look for them> through science which might be the currently accep way of looking> at the world, but be brave, and look for them in religion.> Spiritual truths are the work of delusions and self deception. The only > truths that matter are those that can be associa with facts in the > real world.Hey Bob, get yourself a life. Even if you don't accept the exsitenceof spiritual truths, the claim that only those truths be which can beassocia with facts in the real world is preposterous. You would bethen declaring the inexistence of the truth of the beauty of poetry,the truth of humour, etc. etc.> There is no> shame in being irrational, as long as you don't meddle into the realm> of the rational. If you are looking for God, don't look for him in> half-baked pseudo-scientific theories, look for him by faith which is> the proper way to God. God does not need to be explained, and you> cannot find God through explanation.> Faith and one dollar will get you a ride from Alewife Brook station to > Park Street Under on the MBTA. Which gives a good indication of what > faith is worth.> Bob KolkerSorry, I dont get this one since I don't know what a ride from AlewifeBrook station to Park Street Under implies, can you enlighten me?Faith is a === mathematics, sets, relationship> The point is, that nowadays it is virtually impossible to talk> seriously to common people about anything that lies outside the> rational, scientific or technically explainable. Raising issues such> as spirituality, miracles, divine intervention etc. will get you> mostly contempt and ridicule.It is very bad when people cannot express their beliefs andconvictions, or cannot act in accordance with them. The beliefs andconvictions may be true or false; it does not matter.I personally believe that the universe we inhabit works according tosimple and beautiful mathematics, and I am temp to say thatmiracles cannot occur. It is possible to experience miracles happening-- it is even possible for several people to experience the samemiracle, but when one experiences a miracle one learns more about onesbrain than about the universe inside which it lives.The word spirituality does not say much to me. It is suggestive, buttoo vague to really mean anything.Divine intervention seems to refer to situations where gods or higherpowers make things happen in a certain way. I do not think one canusefully describe our universe that way.> That is a clear impoverishment of our> lives, because it imposes a limit on what thoughts we can exchange> with each other.I agree.> The point is not wether such issues are valid or not,> but wether it is valid to adopt an obscurantistic point of view and> denigrate them out of hand because they don't fit into the rational> scheme of things.When discussing a rational world it is permissible to dismiss anythingthat is not rational. When discussing an irrational world it is not.The worlds of fiction can be irrational, inner worlds that some peoplehave can be irrational, etc. Both rational and irrational worldsshould be studied.> Spiritual truths are the work of delusions and self deception. The only > truths that matter are those that can be associa with facts in the > real world.> Hey Bob, get yourself a life. Even if you don't accept the exsitence> of spiritual truths, the claim that only those truths be which can be> associa with facts in the real world is preposterous. You would be> then declaring the inexistence of the truth of the beauty of poetry,> the truth of humour, etc. etc.Would he? The truth of the beauty of poetry is a subjective truth --poetry may be beautiful to some and rather meaningless to others --but if a person finds poetry beautiful, the person need not sayPoetry is beautiful, but can say instead I find poetry beautiful,and there is nothing subjective about the latter sentence. I thinksubjective truths are really just objective truths viewed in asubjective way.I think, however, that it is wrong to say that the only truths thatmatter are those that can be associa with facts in the realworld. Mathematical truths matter, but are at least in partindependent of our === relationship boundary=----=_NextPart_000_01A5_01C38D9A.56475020------------ --------------------------------------------------------- read it!!Also attributions would be nice.Bob Pease > The point is, that nowadays it is virtually impossible to talk> seriously to common people about anything that lies outside the> rational, scientific or technically explainable. Raising issues such> as spirituality, miracles, divine intervention etc. will get you> mostly contempt and ridicule.> It is very bad when people cannot express their beliefs and> convictions, or cannot act in accordance with them. The beliefs and> convictions may be true or false; it does not matter.> I personally believe that the universe we inhabit works according to> simple and beautiful mathematics, and I am temp to say that> miracles cannot occur. It is possible to experience miracles happening> -- it is even possible for several people to experience the same> miracle, but when one experiences a miracle one learns more about ones> brain than about the universe inside which it lives.> The word spirituality does not say much to me. It is suggestive, but> too vague to really mean anything.> Divine intervention seems to refer to situations where gods or higher> powers make things happen in a certain way. I do not think one can> usefully describe our universe that way.> That is a clear impoverishment of our> lives, because it imposes a limit on what thoughts we can exchange> with each other.> I agree.> The point is not wether such issues are valid or not,> but wether it is valid to adopt an obscurantistic point of view and> denigrate them out of hand because they don't fit into the rational> scheme of things.> When discussing a rational world it is permissible to dismiss anything> that is not rational. When discussing an irrational world it is not.> The worlds of fiction can be irrational, inner worlds that some people> have can be irrational, etc. Both rational and irrational worlds> should be studied.> Spiritual truths are the work of delusions and self deception. The only > truths that matter are those that can be associa with facts in the > real world.> Hey Bob, get yourself a life. Even if you don't accept the exsitence> of spiritual truths, the claim that only those truths be which can be> associa with facts in the real world is preposterous. You would be> then declaring the inexistence of the truth of the beauty of poetry,> the truth of humour, etc. etc.> Would he? The truth of the beauty of poetry is a subjective truth --> poetry may be beautiful to some and rather meaningless to others --> but if a person finds poetry beautiful, the person need not say> Poetry is beautiful, but can say instead I find poetry beautiful,> and there is nothing subjective about the latter sentence. I think> subjective truths are really just objective truths viewed in a> subjective way.> I think, however, that it is wrong to say that the only truths that> matter are those that can be associa with facts in the real> world. Mathematical truths matter, but are at least in part> === mathematics, sets, relationship> Perhaps I should have stuck to basics...> If you go half way between 1 and 2 for example, to one and a half... as in> 1.5... and then half way again to between 1 and 1.5 to 1.25 and so on...> this will give an infinite number of decimal places...There's an elementary error here which a lot of people make: thenumbers you will generate this way all have finite precision.None of them have an infinite number of decimal places. Youwill never generate a number in this process which is irrational,or a repeating decimal (1/3 for instance). However, thereare infinitely many numbers in this collection.> and will never find> an end result, as there will always be another half way point between any> two points on the number line...> Because man cannot yet make> one atom into another atom,Don't know what you had in mind here, but both nuclearfission and fusion involve making atoms into other atoms.> does not mean it is not possible by greater> knowledge. In fact all elements are thought to come from hydrogen in stars> aren't they?Yes. Stars are fusion engines that get their energy byfusing hydrogen. All the heavier elements are waste products,the results of additional fusions. Star poop as Carl Saganput it.> With we ourselves being from the substance of stars?In the sense that all elements come from stayes.> You said that energy quantities were predictable... is this the case with a> nuclear bomb?Yes.> For that is the energy potential which we access on a> physical plane.I'm reading this in the physics newsgroup. Physics equations dealwith the physical universe.> Because the invisible is not seen does not mean it does not> exist.Physics is the science of measurement. If it has no effect onthe observable universe, it's not part of physics.> Have you considered that radio waves for example, can only travel because> they are in fact in a medium that will let them travel?This was the belief of natural osophers up to a centuryago. It is no longer a belief held by anyone except === an example of a function (R=real numbers) f:R-->Rsuch that for any a,b,c in R, there exists an x in R such that aR>such that for any a,b,c in R, there exists an x in R such that aand f(x)=c?Not so that it's continuous for the whole R, but it can be continuouswithin the open interval ]a, b[: f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))]-oo when x=a, oo when === function>>can anyone show me an example of a function (R=real numbers) f:R-->R>>such that for any a,b,c in R, there exists an x in R such that a>and f(x)=c?>Not so that it's continuous for the whole R, but it can be continuous>within the open interval ]a, b[: >f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))]>-oo when x=a, oo when x=b and gets all values when between them.Another example:g[x_]:=Pi/(b-a)*x-Pi*(a+b)/(2*(b-a))f[x_]:=Cos[g[x]]/g [x]This one is continous everywhere except at x=(a+b)/2, and === anyone show me an example of a function (R=real numbers) f:R-->R>such that for any a,b,c in R, there exists an x in R such that aand f(x)=c?>>Not so that it's continuous for the whole R, but it can be continuous>>within the open interval ]a, b[: >>f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))]>>-oo when x=a, oo when x=b and gets all values when between them.>Another example:>g[x_]:=Pi/(b-a)*x-Pi*(a+b)/(2*(b-a))>f[x_]:=Cos[g[x]] /g[x]>This one is continous everywhere except at x=(a+b)/2, and gets all>values between a and b.You are not reading the question! It says for all a,b,c, soa and b are not constants. And the question does not say anything aboutcontinuity.I think there was one correct solution pos. Here is another possibility.I will define f:R -> [0,1], which is good enough, because there are bijectionsfrom [0,1] to R.Let x in R and look at the decimal expansion of x. Choose m maximalsuch that x = n . a_1 a_2 ... a_m a_1 a_2 ... a_m ...; that is,the first m digits after the decimal point repeat.If there is no maximal m with this property, then define f(x) = 0.Otherwise, let x = n . a_1 a_2 ... a_m a_1 a_2 ... a_m b_1 b_2 b_3 ...and define f(x) = 0.b_1 b_3 b_5 ...(The point of taking only the odd b_i is to enable me to select the even b_iso as to prevent me accidentally getting a larger value of m in x.)Derek === a function (R=real numbers) f:R-->R>such that for any a,b,c in R, there exists an x in R such that aand f(x)=c?Let f(x) === a function (R=real numbers) f:R-->R>such that for any a,b,c in R, there exists an x in R such that aand f(x)=c?> Let f(x) = 5.No, let c=6, there is no x such that 5=6.>