mm-3479 === Subject: topologies on IR^2 let O_1 be the product topology on IR^2 and O_2 be the topology induced by the norm ||(x_1,x_2)|| = a |x_1| + b |x_2| , where a,b > 0 . Then my question is whether O_1 = O_2 ? I guess yes, because (x_n)_nin IN converges to zero in O_1 if and only if it converges to zero in O_2. If the answer to the above question is in the affirmative, does || . || induce the product topology? I guess this is not the case, but am not sure ... === Subject: Re: topologies on IR^2 days. My association with the Department is that of an alumnus. >let O_1 be the product topology on IR^2 and O_2 be the topology induced >by the norm >||(x_1,x_2)|| = a |x_1| + b |x_2| , where a,b > 0 . Then my question is whether O_1 = O_2 ? The set of open boxes (r,s) x (u,v) form a basis for the product topology on R^2, i.e., O_1. Let B1 be the set of all of these open boxes. Tthe set of open balls B(p,r) = { q in R^2 | ||p-q|| < r}, with p in R^2, r>0. Let B2 be the set of all of these open balls. One way to prove that O_1 = O_2 is to show that given any S in B1, and p in S, there exists T in B2 such that p is in T, and T is contained in S; and conversely that given any T in B2 and any q in T, there exists S in B1 such that q is in S and S is contained in T. (The first one will show that for any subset A of R^2, the interior of A with respect to O_1 is contained in the interior of A with respect to O_2; the second will prove the converse. Since a topology is completely determined by its interior operator, this will prove that O_1 is equal to O_2). >I guess yes, because (x_n)_nin IN converges to zero in O_1 if and only >if it converges to zero in O_2. This would not in general be enough; however, since the topology of R is itself induced by a metric which is translation invariant, and the product topology is then also metric, and the metric you introduced for R^2 is also translation invariant, you can get away with that. >If the answer to the above question is in the affirmative, does || . || >induce the product topology? I guess this is not the case, but am not >sure ... Well, yes. If the topology induced by || || is ->identical<- to the product topology, then of =>course<= || || induces the product topology. === Subject: Re: topologies on IR^2 let O_1 be the product topology on IR^2 and O_2 be the topology induced > by the norm > ||(x_1,x_2)|| = a |x_1| + b |x_2| , where a,b > 0 . Then my question is whether O_1 = O_2 ? I guess yes, because (x_n)_nin IN converges to zero in O_1 if and only > if it converges to zero in O_2. > If the answer to the above question is in the affirmative, does || . || > induce the product topology? I guess this is not the case, but am not > sure ... Note that any norm on R^2 induces the product topology. === Subject: Re: Composite functions, surjective, injective I am new to mathematics and have the following difficulty: > I should find functions f, g with Z -> Z so that f o g is surjective, > but g o f is not surjective (the same also with injective). > However, I do not have an idea how to proceed. I do not want a solution; a short hint how to proceed further is > sufficient for me. === Subject: Re: Composite functions, surjective, injective I am new to mathematics and have the following difficulty: > I should find functions f, g with Z -> Z so that f o g is surjective, > but g o f is not surjective (the same also with injective). > However, I do not have an idea how to proceed. I do not want a solution; a short hint how to proceed further is > sufficient for me. That's funny, I am not new to mathematics and have no idea what you're > talking about. > What don't you understand: surjection, injection or composition f o g? === Subject: JSH: Grooming behavior Human beings are primates, so it's not surprising to see primate behavior in the way things go with postings and I find it fascinating!!! Like look over arguments I have with posters, and you'll find an interesting pattern: I make an argument, some poster replies some way or another, praising them for their supposed superior knowledge. It's just primate grooming behavior. Of course, at times I come back in, destroy the reply and may even show it reflects a very poor knowledge of basic logic or mathematics, but it doesn't matter as the purpose of the behavior on Usenet is the same as in the wilds with other primates: to reinforce an accepted dominance hierarchy. Neat!!! In many ways, human beings are more primate--that is they show more in common with other primates like gorillas and chimpanzees--than they are human. === Subject: Re: JSH: Grooming behavior > Human beings are primates, so it's not surprising to see primate > behavior in the way things go with postings and I find it > fascinating!!! That's why we're so interested in primes, I guess. === Subject: Re: JSH: Grooming behavior > Human beings are primates, so it's not surprising to see primate > behavior in the way things go with postings and I find it > fascinating!!! Like look over arguments I have with posters, and you'll find an > interesting pattern: I make an argument, some poster replies > some way or another, praising them for their supposed superior > knowledge. It's just primate grooming behavior. Of course, at times I come back in, destroy the reply and may even show > it reflects a very poor knowledge of basic logic or mathematics, but it > doesn't matter as the purpose of the behavior on Usenet is the same as > in the wilds with other primates: to reinforce an accepted dominance > hierarchy. Neat!!! In many ways, human beings are more primate--that is they show more in > common with other primates like gorillas and chimpanzees--than they are > human. We all know your mathematical knowledge is lacking, that's why you end up resorting to this psychological crap. I'd bet if I gave you a calculus problem, for example, you couldn't do it. === Subject: Re: JSH: Grooming behavior Human beings are primates, so it's not surprising to see primate > behavior in the way things go with postings and I find it > fascinating!!! Like look over arguments I have with posters, and you'll find an > interesting pattern: I make an argument, some poster replies > some way or another, praising them for their supposed superior > knowledge. It's just primate grooming behavior. Of course, at times I come back in, destroy the reply and may even show > it reflects a very poor knowledge of basic logic or mathematics, but it > doesn't matter as the purpose of the behavior on Usenet is the same as > in the wilds with other primates: to reinforce an accepted dominance > hierarchy. Neat!!! In many ways, human beings are more primate--that is they show more in > common with other primates like gorillas and chimpanzees--than they are > human. > We all know your mathematical knowledge is lacking, that's why you end > up resorting to this psychological crap. I'd bet if I gave you a > calculus problem, for example, you couldn't do it. You all know? You keep making the same mistake, but I think you do because some part of you--irrational as it may sound--actually believes that you are part of some large group where there is total consensus. I make political posts because they are necessary, and posts about human psychology as markers, in my possibly futile hope that scientists may get extra value from such postings as they consider group behavior. What I pointed out is real and it does correlate well with primates acting within a dominance hierarchy--grooming each other--as primates have done for millions of years. It'd be stranger not to see the behavior than for it to exist. === Subject: Re: JSH: Grooming behavior > Human beings are primates, so it's not surprising to see primate > behavior in the way things go with postings and I find it > fascinating!!! Like look over arguments I have with posters, and you'll find an > interesting pattern: I make an argument, some poster replies > some way or another, praising them for their supposed superior > knowledge. It's just primate grooming behavior. Of course, at times I come back in, destroy the reply and may even show > it reflects a very poor knowledge of basic logic or mathematics, but it > doesn't matter as the purpose of the behavior on Usenet is the same as > in the wilds with other primates: to reinforce an accepted dominance > hierarchy. Neat!!! In many ways, human beings are more primate--that is they show more in > common with other primates like gorillas and chimpanzees--than they are > human. We all know your mathematical knowledge is lacking, that's why you end >> up resorting to this psychological crap. I'd bet if I gave you a >> calculus problem, for example, you couldn't do it. > > You all know? You keep making the same mistake, but I think you do because some part > of you--irrational as it may sound--actually believes that you are part > of some large group where there is total consensus. I make political posts because they are necessary, and posts about > human psychology as markers, in my possibly futile hope that scientists > may get extra value from such postings as they consider group behavior. What I pointed out is real and it does correlate well with primates > acting within a dominance hierarchy--grooming each other--as primates > have done for millions of years. It'd be stranger not to see the behavior than for it to exist. Math isn't about consensus. If another mathematician tells me something is true, chances are that I would want to see a proof that it's true. Why should I take anything anyone says at face value? You refuse to learn stuff every good mathematician knows, and that is one thing that makes you a crackpot. You're dealing with college educated mathematicians who know the field better than anyone with a physics degree does, especially in the area of pure math. === Subject: Re: JSH: Grooming behavior behavior in the way things go with postings and I find it > fascinating!!! Like look over arguments I have with posters, and you'll find an > interesting pattern: I make an argument, some poster replies > some way or another, praising them for their supposed superior > knowledge. It's just primate grooming behavior. Of course, at times I come back in, destroy the reply and may even show > it reflects a very poor knowledge of basic logic or mathematics, but it > doesn't matter as the purpose of the behavior on Usenet is the same as > in the wilds with other primates: to reinforce an accepted dominance > hierarchy. Neat!!! In many ways, human beings are more primate--that is they show more in > common with other primates like gorillas and chimpanzees--than they are > human. > We all know your mathematical knowledge is lacking, that's why you end > up resorting to this psychological crap. I'd bet if I gave you a > calculus problem, for example, you couldn't do it. You all know? You keep making the same mistake, but I think you do because some part > of you--irrational as it may sound--actually believes that you are part > of some large group where there is total consensus. I make political posts because they are necessary, and posts about > human psychology as markers, in my possibly futile hope that scientists > may get extra value from such postings as they consider group behavior. What I pointed out is real and it does correlate well with primates > acting within a dominance hierarchy--grooming each other--as primates > have done for millions of years. It'd be stranger not to see the behavior than for it to exist. Does any one else watch South Park? Does anyone see any parallels with Cartman, the self-centered, unapologetic liar? === Subject: Re: JSH: Grooming behavior > I make political posts because they are necessary, and posts about >> human psychology as markers, in my possibly futile hope that scientists >> may get extra value from such postings as they consider group behavior. >> What I pointed out is real and it does correlate well with primates >> acting within a dominance hierarchy--grooming each other--as primates >> have done for millions of years. >> It'd be stranger not to see the behavior than for it to exist. Does any one else watch South Park? Does anyone see any parallels with > Cartman, the self-centered, unapologetic liar? > absolutely, but he is talking in a barrel, nobody takes him seriously, he is preaching in a graveyard, nobody listening. He knows it and he knows he cannot do any better than this. (not Math, not Physics) === Subject: Re: JSH: Grooming behavior behavior in the way things go with postings and I find it > fascinating!!! Like look over arguments I have with posters, and you'll find an > interesting pattern: I make an argument, some poster replies > some way or another, praising them for their supposed superior > knowledge. It's just primate grooming behavior. Of course, at times I come back in, destroy the reply and may even show > it reflects a very poor knowledge of basic logic or mathematics, but it > doesn't matter as the purpose of the behavior on Usenet is the same as > in the wilds with other primates: to reinforce an accepted dominance > hierarchy. Neat!!! In many ways, human beings are more primate--that is they show more in > common with other primates like gorillas and chimpanzees--than they are > human. > We all know your mathematical knowledge is lacking, that's why you end > up resorting to this psychological crap. I'd bet if I gave you a > calculus problem, for example, you couldn't do it. You all know? You keep making the same mistake, but I think you do because some part > of you--irrational as it may sound--actually believes that you are part > of some large group where there is total consensus. I make political posts because they are necessary, and posts about > human psychology as markers, in my possibly futile hope that scientists > may get extra value from such postings as they consider group behavior. What I pointed out is real and it does correlate well with primates > acting within a dominance hierarchy--grooming each other--as primates > have done for millions of years. So it's ok for me to start refering to you as a baboon? It'd be stranger not to see the behavior than for it to exist. > === Subject: Re: The Forebears Paradox <44fb7f70$0$11968$afc38c87@news.optusnet.com.au> <1157369985.6518.9.camel@localhost.localdomain> <87ac5fn3lw.fsf@phiwumbda.org > ... > If your wife is Caucasian (as I assume you are), you can pretty much bet on >> having common ancestors as a certainty. Most Europeans are related to >> Europen royalty (which in turn is inter-related), basically because many of >> the European Kings had lots of children. > ... I see. > Whilst the plebs all had only 2.4 children, the Kings had a lot of > children? > How did they do this? It is my understanding that kings are usually > male, whilst it actually requires the presence of a female for child > production. Right. But a king can have 30 kids while thirty peasant women have > only one kid each. Relatively speaking, that king had lots of kids. In days of yore... Poor families had lots of children due to high mortality rates. There were no pension plans available at this time. Henry VIII go through 6 wives but only one(??) child was produced. -- Jeremy Boden === Subject: Re: The Forebears Paradox > Poor families had lots of children due to high mortality rates. There > were no pension plans available at this time. Yes, but they didn't generally reach child bearing age. If poor people (which was most of the population) had an average of four children reaching child bearing age per couple, then the population would have doubled every 20 years or so. That they didn't is obvious, or the world's population would have reached many billions very quickly. The population of poor people was effectively limitted by the resources available. This was not true of Royalty, who tended to have far more children, and far more of these survived to child bearing age. === Subject: Re: The Forebears Paradox Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >Henry VIII go through 6 wives but only one(??) child was produced. Henry had many children - perhaps 15 or more. Three went on to become kings and queens of England: Mary I, Edward VI, and Elizabeth I. His problem was not having children, it was having a legitimate male child who would outlive him. -- Richard === Subject: Re: Russia to Conduct First Flight to Moon in 2011-2012 - purpose > http://www.mosnews.com/news/2006/08/31/moonmission.shtml What is the point of going back to the moon? Is someone planning to develop new materials which can only be produced in low gravity? I'm thinking of larger diamonds, gems, or things which have an engineering purpose. For example I remember reading Arthur C. Clarke about a lift into orbit on a single diamond fibre. If they could produce ultra strong thin transparent fibres they may give furniture designers something good to work with, so it looks like you're floating on your chair. Also architects may like building tension structures. Also where is nanotechnology going? Would it benefit from low gravity? How about energy? Is there sense in solar panels in space? A guess is that it would take more energy to put the system into orbit than would be gained compared to solar on earth, and power transmission back to ground would be dangerous. Is there benefit to fundamental science? All this moon stuff sounds like a lot of rework to me. === Subject: Re: Russia to Conduct First Flight to Moon in 2011-2012 - purpose What's so funny about peace, love and posting the following on 4 Sep 2006 13:58:43 -0700 iin alt.conspiracy? > http://www.mosnews.com/news/2006/08/31/moonmission.shtml >What is the point of going back to the moon? Is someone planning to >develop new materials which can only be produced in low gravity? I'm >thinking of larger diamonds, gems, or things which have an engineering >purpose. That can be done in orbit, or at the Lagrange points. Problem is, we need to improve our ability to live and work in space. Going to the Moon is a goal for engineers to design for. >For example I remember reading Arthur C. Clarke about a lift into orbit >on a single diamond fibre. If they could produce ultra strong thin >transparent fibres they may give furniture designers something good to >work with, so it looks like you're floating on your chair. Also >architects may like building tension structures. Carbon nanotubes are the material of choice for the space elevator, and these guys plan to build one by 2020. http://www.liftport.com/ >Also where is nanotechnology going? Would it benefit from low gravity? Nano is improving every year. > How about energy? Is there sense in solar panels in space? A guess >is that it would take more energy to put the system into orbit than >would be gained compared to solar on earth, and power transmission back >to ground would be dangerous. Short term cost over long term benefit. Right now, the cost of starting a good solar power sattelite system is high. If we get the elevator working, the cost to orbit drops immensely. As for the danger, the power would come down in the form of a microwave beam sent to a receiver in a remote area. It would take something phsically moving the solar power sat to cause the beam to shift off focus, and you can easily build a cut off switch into the sattelite's computer for anything like that. We'll cook and occasional goose, but that's alright. >Is there benefit to fundamental science? All this moon stuff sounds >like a lot of rework to me. We're doing it with forty years of advances in computers, material science, medicine, etc. This time we'll be able to take more people, stay longer, and learn more both about the Moon and how to live and work in space. -- Douglas Berry Do the OBVIOUS thing to send e-mail Atheist #2147, Atheist Vet #5 Jason Gastrich is praying for me on 8 January 2011 The most beautiful thing we can experience is the mysterious. It is the source of all true art and all science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed. - Albert Einstein === Subject: Re: Russia to Conduct First Flight to Moon in 2011-2012 - purpose The most beautiful thing we can experience is the mysterious. It is the > source of all true art and all science. He to whom this emotion is a > stranger, who can no longer pause to wonder and stand rapt in awe, is as > good as dead: his eyes are closed. - Albert Einstein from alt.conspiracy. My point was that people may only support the program is there are tangible benefits at home, like from the original moon mission, non-stick frying pans (OK, they may be carcinogenic), and perhaps fabrics. Meanwhile my autumn project should be to look for fractal dimensions in share prices ... === Subject: Re: Russia to Conduct First Flight to Moon in 2011-2012 - purpose > http://www.mosnews.com/news/2006/08/31/moonmission.shtml > What is the point of going back to the moon? Is someone planning to > develop new materials which can only be produced in low gravity? I'm > thinking of larger diamonds, gems, or things which have an engineering > purpose. For example I remember reading Arthur C. Clarke about a lift into orbit > on a single diamond fibre. If they could produce ultra strong thin > transparent fibres they may give furniture designers something good to > work with, so it looks like you're floating on your chair. Also > architects may like building tension structures. Also where is nanotechnology going? Would it benefit from low gravity? > How about energy? Is there sense in solar panels in space? A guess > is that it would take more energy to put the system into orbit than > would be gained compared to solar on earth, and power transmission back > to ground would be dangerous. Is there benefit to fundamental science? All this moon stuff sounds > like a lot of rework to me. Sorry should have read the original post. Helium 3 may be useful, but it is a long way to transport material. === Subject: Re: Russia to Conduct First Flight to Moon in 2011-2012 Well, finally after 37 years someone has acknowledged what Neal The Wheel said. Unless you can read minds, you don't --know --what Armstrong meant; it stands as an all-time stupid moment. Of course he said it in a studio but no one picked up on it then. > Oh, they haven't gone back 6 x' per year so that means they did > get there twice when the world was young. One small step for Man(kind) {sic}, one giant leap for Manknd Actually, Armstrong claimed he ment to say; One small step for transmission. === Subject: Re: Russia to Conduct First Flight to Moon in 2011-2012 <44f9372c$0$95183$dbd45001@news.wanadoo.nl> Wheel said. Unless you can read minds, you don't --know --what Armstrong meant; it > stands as an all-time stupid moment. Of course he said it in a studio > but no one picked up on it then. I never said I could read minds. They'd asked Armstrong what his statement ment and he explained he ment to say One small step for man, and somehow it got clipped. And no one has managed to show me how they could pump a large enough studio free of air so the flag wouldn't move and the dust wouldn't hang. (And if they been able to, that would have given the lie to Daniel Min's insistance that the astronauts were bouncing around in deflated suits!) Oh, they haven't gone back 6 x' per year so that means they did > get there twice when the world was young. One small step for Man(kind) {sic}, one giant leap for Manknd Actually, Armstrong claimed he ment to say; One small step for transmission. === Subject: Re: Russia to Conduct First Flight to Moon in 2011-2012 <44f9372c$0$95183$dbd45001@news.wanadoo.nl> It's another JFK, another Big Bang vs Goddi/Gotti,-----------no one KNOW. No one will EVER know. > Well, finally after 37 years someone has acknowledged what Neal The > Wheel said. Unless you can read minds, you don't --know --what Armstrong meant; it > stands as an all-time stupid moment. Of course he said it in a studio > but no one picked up on it then. I never said I could read minds. They'd asked Armstrong what his > statement ment and he explained he ment to say One small step for studio free of air so the flag wouldn't move and the dust wouldn't > hang. (And if they been able to, that would have given the lie to > Daniel Min's insistance that the astronauts were bouncing around in > deflated suits!) > Oh, they haven't gone back 6 x' per year so that means they did > get there twice when the world was young. One small step for Man(kind) {sic}, one giant leap for Manknd Actually, Armstrong claimed he ment to say; One small step for transmission. === Subject: Re: basic math for beginners > I have 3 children about to enter school and I was wondering if there > are any good children based sites for teaching math? Empress2454 #124457 Try http://www.bbc.co.uk/schools/ -- Jeremy Boden === Subject: Re: Linearity question, is the proffesor wrong ? > >I've taught a Mathematical Modelling course in the School of >Engineering. In this course we classify ODEs using the definition of >non-linear that you use, in other words it refers only to the way that y >and its derivatives appear in the equations. Another element of >classification is whether the coefficients are constant or functions of >t. The treatment is not mathematically advanced, being limited to >solving 2nd order systems by means of the characteristic equation, but >it may be at a similar level to the course you're teaching.'ve test >>Hi all, the proffessor answered me and suggested me to perform a test, >>which I did. >>I've tested letting u(t)= unit step function as well as u(t)=2*unit >>step function >>The steady state value in the first test is 5, and in the second 7. It >>should have been 10 according to linearity test. >>I am confused now .... >>Linearity definition is with respect to an operator, if a1 =D(b1) then >>c*a1=D(c*b1), where c is a constant, and if a2=D(b2), then >>a1+a2=D(b1+b2). >>In the case of D.E. I should put in the solution y(t) into the >>linearity definition. >>y(t)=D(y(t)), c*y(t)=D(c*y(t)) etc. >>With respect to this definition the D.E is linear. >>Changing the the input signal u=unit step function to u=2*unitstep >>function is actually changing the D.E. and a different linearity >>definition should be used..... >>Freinds, am I getting it now, or am I way off...? > Linearity, as mathematicians apply it to a differential equation, > means linearity in the dependent variable, not linearity in an input > signal that affects coefficients. However, it's possible that > practitioners of control theory use the term differently. I'd say the professor's use of the term linear with respect to ODEs is unconventional. === Subject: Convexity and Connection for a generic function f:R^m -> R^n, we consider the following properties (X denotes a generic subset of R^m): (I) f(X) is connected if X is connected; (I*) if f(X) is connected then X is connected; (II) f(X) is convex if X is convex; (II*) if f(X) is convex then X is convex. Do you know a function that satisfies (II) and (II*) but neither (I) nor (I*)? I honestly couldn't even find a function that satisfies (II) but not (I). Maury === Subject: Convexity and Connection > for a generic function f:R^m -> R^n, we consider the > following properties (X denotes a generic subset of R^m): > (I) f(X) is connected if X is connected; > (I*) if f(X) is connected then X is connected; > (II) f(X) is convex if X is convex; > (II*) if f(X) is convex then X is convex. > Do you know a function that satisfies (II) and (II*) but > neither (I) nor (I*)? > I honestly couldn't even find a function that satisfies > (II) but not (I). Is there a not continuous convexity preserving function? -- f:R^m -> R^n; for all A subset R^m 1. connected A ==> f(A) connected 1a connected f(A) ==> A connected 2. convex A ==> f(A) convex 2a convex f(A) ==> A convex 1a or 2a implies f is injection. Within R^k, convex A ==> A connected. ---- === Subject: Re: Convexity and Connection === > Subject: Convexity and Connection for a generic function f:R^m -> R^n, we consider > the > following properties (X denotes a generic subset of > R^m): (I) f(X) is connected if X is connected; > (I*) if f(X) is connected then X is connected; (II) f(X) is convex if X is convex; > (II*) if f(X) is convex then X is convex. Do you know a function that satisfies (II) and > (II*) but > neither (I) nor (I*)? I honestly couldn't even find a function that > satisfies > (II) but not (I). Is there a not continuous convexity preserving > function? > Surely yes in the case m=n=1: see my post A Class of Special Functions in sci.math and the reply of Robert Isreal there. Maybe a similar construction can be used to find a function f:R^2->R^2 such that for every segment AB, with distinct A,B, we have f(AB)=R^2. Obviously this function would be a not continous function which satisfy (II). > -- > f:R^m -> R^n; for all A subset R^m 1. connected A ==> f(A) connected > 1a connected f(A) ==> A connected 2. convex A ==> f(A) convex > 2a convex f(A) ==> A convex 1a or 2a implies f is injection. Within R^k, convex A ==> A connected. ---- > Maury === Subject: Re: Convexity and Connection > === >> Subject: Convexity and Connection >> >> for a generic function f:R^m -> R^n, we consider >> the >> following properties (X denotes a generic subset of >> R^m): >> >> (I) f(X) is connected if X is connected; >> (I*) if f(X) is connected then X is connected; >> >> (II) f(X) is convex if X is convex; >> (II*) if f(X) is convex then X is convex. >> >> Do you know a function that satisfies (II) and >> (II*) but >> neither (I) nor (I*)? >> >> I honestly couldn't even find a function that >> satisfies >> (II) but not (I). >> >> Is there a not continuous convexity preserving >> function? >> Surely yes in the case m=n=1: see my post A Class of >Special Functions in sci.math and the reply of Robert >Isreal there. Maybe a similar construction can be used >to find a function f:R^2->R^2 such that for every segment >AB, with distinct A,B, we have f(AB)=R^2. Obviously this >function would be a not continous function which satisfy >(II). One can do something like that. I've got a construction here that seems to work. It's a bit complicated, but here goes: First, by a 'segment' I mean a finite line segment in R^2, of positive length, with endpoints. Two segments 'overlap' if their intersection is a segment. We'll let Seg be the set of all segments. The existence of a map that preserves convexity but not connectedness is immediate from the following lemma. Lemma: Suppose A and B are disjoint subsets of R^2 such that for any segment J, |J cap A| < c and |J cap B| < c. Further suppose that a:A -> R^2, b:B -> R^2 are any maps whatsoever. Then there exists a map f:R^2 -> R^2 such that 1. f(J) = R^2 for every segment J. 2. f|A = a, f|B = b. (And then we get a map that preserves convexity but not connectedness by for example letting A and B be two halves of a circle, and a and b constant maps that send the two halves to distict points.) Proof of lemma: For each J in Seg fix a map h_J:J -> R^2 with the following properties: 1. if x in (J cap A) then h_J(x) = a(x), and if x in (J cap B) then h_J(x)=b(x). 2. h_J is c-to-1 and surjective on each subsegment of J. (In other words, if K is any subsegment of J and y any element of R^2 then |K cap h_J^-1(y)| = c.) Note that 2. easily implies: 3. If K is a subsegment of J and |X| < c, then h_J(KX) = R^2. I'll omit the (easy) proof that such maps exist. Now impose a well-ordering of order type c on Seg (so |{K : K < J}| < c for all J). And for x in R^2 let L(x) = min {J in (Seg,<) : x in J}. And finally let f(x) = h_{L(x)}(x) It's immediate from the definitions that f|A = a and f|B = b, so we need only show f(J) = R^2 for segments J. We'll do that by proving the following statement by induction on elements J of (Seg,<): (*) For every subsegment K of J, f(K) = R^2. First, if J is the minimal element of (Seg,<) then L(x) = J for all x in J, and (*) is true by property (3) of h_J. Now suppose (*) is true for all Ky and L(x)=L(y) then L(x) overlaps I, contradicting our assumption that no such overlap exists. Therefore |X| < c. But notice that if z is in IX then L(z) = J. So for y in IX, f(y) = h_J(y). But then property 3. of h_J tells us that f(I) = h_J(IX) = R^2. QED. Robert Sheskey === Subject: Re: Is there a better way to solve this boundary=------------070407010505070806030605 --------------------------------------------------------------------- > I am trying to solve the following seemingly simple problem. > Lets say I take 10 numbers at random in the range (0,100). > e.g. > 10,20,30,40,50,60,70,80,90,90 > Total = 540 > Based on a certain criteria, I divide these numbers into smaller > subsets and compute the totals for each subset. > 10+20+90 = 120 > 30+40+50 = 120 > 60+70 = 130 > 80+90 = 170 Based on another criteria, I divide these numbers into different > subsets and compute the totals again > 10+30+40 = 80 > 20+50+60+70+80 = 280 > 90+90 = 180 Question: > Given only the totals: 540,120,120,130,170,80,280,180.. > 1) How many possible combinations of 10 numbers generate these totals > 2) what is the best way to find all those combinations > 3) Is there a better simpler way to solve this problem. One way is to model it as a Integer Linear Program > sum {i in X} X[i] = 540; > sum {i in X} a[i] * X[i] = 120; > sum {i in X} b[i] * X[i] = 120; > sum {i in X} c[i] * X[i] = 130; > sum {i in X} d[i] * X[i] = 170; > sum {i in X} e[i] * X[i] = 80; > sum {i in X} f[i] * X[i] = 280; > sum {i in X} g[i] * X[i] = 180; > a[i] + b[i] + c[i] + d[i] = 1; (Mutually exclusive) > e[i] + f[i] + g[i] = 1; (Mutually exclusive) > where a,b,c ...g are boolean variables > a[i] * X[i] is linearized as suggested by Paul in However, when I try to solve this using cplex, it complains that the > problem is integer infeasible/ unbounded. along with the data file for your example. The solution it gave me for X is 80, 80, 80, 80, 50, 40, 40, 40, 30, 20. The splits were: collection, subset, i, X[i]: fours first 4 80 fours first 8 40 fours second 1 80 fours second 6 40 fours third 3 80 fours third 9 30 fours third 10 20 fours fourth 2 80 fours fourth 5 50 fours fourth 7 40 threes first 6 40 threes first 7 40 threes second 1 80 threes second 3 80 threes second 5 50 threes second 8 40 threes second 9 30 threes third 2 80 threes third 4 80 threes third 10 20 It could give a different solution if solved again, especially if the model is tweaked. You might be wondering about the last set of constraints, which force the X variables to be enumerated in nonincreasing order. They're not essential for an initial solution, but if you are going to enumerate all solutions, you'll want something like this to mitigate the number of repetitions. (Note that this does not eliminate indistinguishable solutions. With the solution above, since X[1] = ... = X[4] = 80, obviously there are a lot of permutations of the Y values that swap an 80 for another 80.) There are a couple of ways you could modify this to find all solutions, neither of which really excites me. One would be to write custom code (C++ or Java) to solve it, using CPLEX as a library. CPLEX will let you add a callback function to validate new incumbents. You just use the callback to record the new solution (if it is in fact new, not a permutation of a known solution) and then always reject it. That way CPLEX thinks it still does not have an incumbent and keeps on chugging. The other is to replace the integer X variables with binary variables, in effect writing each X[i] as a binary expansion X[i,1] + 2*X[i,2] + 4*X[i,3] + ... Then, having found a solution, you write an expression which is the sum of all X variables that were zero in the solution plus the complements (1-X[i,j]) of all X variables that were ones, and add a constraint forcing that sum to be >= 1. In other words, the constraint says you have to flip at least one bit in the next solution. All that said, my vote would be to use constraint programming to enumerate the solutions. /Paul --------------------------------------------------------------------- name=integers.mod filename=integers.mod option solver cplexamp; param nInts integer > 0; # number of integers needed param grandTotal integer > 0; # grand total of integers set COLLECTIONS; # set of indices of subset constraints set SUBSETS {COLLECTIONS}; # indices of subsets within a collection param total {c in COLLECTIONS, s in SUBSETS[c]} integer > 0; # totals for each subset # make sure subset totals balance check {c in COLLECTIONS}: sum {s in SUBSETS[c]} total[c,s] == grandTotal; var X {1 .. nInts} integer >= 0; # X = integers sought var Y {1 .. nInts, c in COLLECTIONS, s in SUBSETS[c]} binary; # Y[i,j,k] = 1 iff X[i] contributes to total for subset k of collection j var Z {1 .. nInts, c in COLLECTIONS, s in SUBSETS[c]} >= 0; # Z[i,j,k] = contribution of X[i] to subset k of collection j minimize Nothing: 0; # objective is irrelevant s.t. GrandTotal: sum {i in 1 .. nInts} X[i] == grandTotal; # grand total must be right s.t. SubTotal {c in COLLECTIONS, s in SUBSETS[c]}: sum {i in 1 .. nInts} Z[i,c,s] == total[c,s]; # subsets must some to correct totals s.t. Usage {c in COLLECTIONS, i in 1 .. nInts}: sum {s in SUBSETS[c]} Y[i,c,s] == 1; # every integer contributes to exactly one subset of each collection s.t. ZUpper {i in 1 .. nInts, c in COLLECTIONS, s in SUBSETS[c]}: Z[i,c,s] <= grandTotal*Y[i,c,s]; # y = 0 => z = 0 s.t. ZLower {i in 1 .. nInts, c in COLLECTIONS, s in SUBSETS[c]}: Z[i,c,s] >= X[i] - grandTotal*(1 - Y[i,c,s]); # y = 1 => z >= x s.t. ZMax {i in 1 .. nInts, c in COLLECTIONS, s in SUBSETS[c]}: Z[i,c,s] <= X[i]; # so y = 1 => z = x s.t. Monotonicity {i in 2 .. nInts}: X[i] <= X[i-1]; # reduce number of times a permutation of the same solution turns up --------------------------------------------------------------------- name=integers.dat filename=integers.dat param nInts := 10; param grandTotal := 540; set COLLECTIONS := fours threes; set SUBSETS[fours] := first second third fourth; set SUBSETS[threes] := first second third; param total := fours first 120 fours second 120 fours third 130 fours fourth 170 threes first 80 threes second 280 threes third 180; === Subject: group theory question In a finite group the number of elements of order 2 is even. can somebody help me how to show that. === Subject: Re: group theory question >In a finite group the number of elements of order 2 is even. This is false. Take the cyclic group of order 2; how many elements of order 2 does it have? >can somebody help me how to show that. Given that it is false, I hope nobody does. The question is: what was it you were really supposed to prove? One possibility is that you were supposed to prove that in a finite group of EVEN order, the number of elements of EXPONENT 2 is even. This can be done by considering the following equivalence relation on the elements of the group: x~y if and only if x=y or x=y^{-1}. This will partition the group into equivalence classes. Each equivalence class will have either 1 element, or two elements. Figure out what each of those two things mean in terms of the order of the element, and then see what does that tell you given the fact that we are assuming the group has even order. -- === Subject: Re: How to calculate these Expected Value >3 random variable X, Y, Z >>ln(X) ~ Normal (0, Var(X)) >>ln(Y) ~ Normal (0, Var(Y)) >>ln(Z) ~ Normal (0, Var(Z)) >> > >I doubt that you mean that. I suspect it is impossible that >Var(ln X) >= Var(X) except in the case where these variances are 0. In general, it's quite possible. If you replace X by Y = c X for any c > 0, then Var(ln(Y)) = Var(ln(X)) while Var(Y) = c^2 Var(X). So Var(Y) = Var(ln(Y)) if c = sqrt(Var(ln(X))/Var(X)) [assuming, of course, that the variances exist and are nonzero]. However, it's impossible for a random variable where ln(X) has distribution symmetric about 0: if X = exp(L) = cosh(L) + sinh(L), Var(X) = Var(cosh(L)) + Var(sinh(L)) (since by symmetry, cosh(L) and sinh(L) are uncorrelated) >= E[sinh(L)^2] > E[L^2] = Var(L) (noting that |sinh(L)| > |L| whenever L <> 0) Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Triangles Inscribed in a Circle > consider an isosceles triangle ABC, with AC=BC, inscribed > in a circle C. Then move the vertex A (or B) on the > circle to obtain A'C=A'B (respectively B'A= B'C). The new > triangle A'BC (or AB'C, but these two triangles are > equal) has a greater area and and a longer perimeter than > ABC. Now you can repeat the procedure, moving B or C > (respectively A or C), to obtain a new isosceles > triangle, and so on. The sequence of inscribed triangles > has increasing area and perimeter, and I think it > converges to an equilateral triangle inscribed in the > circle. What do you think? > This result would allow to give a pure geometric proof > of the well-known fact that the equilateral triangle > is the triangle of maximum area and perimeter among all > the triangle inscribed in a given circle. schrieb The Qurqirish Dragon : > My previous post, which asserted the conclusion actually > proved that the equilateral triangle has the maximum area. No. You only showed that to every inscribed triangle that is not equilateral, there exists an inscribed triangle with larger area. But you did not mention that one also has to assert that there *is* an inscribed triangle for which the area takes on a maximum value. Otherwise, you could apply your style of proof to show that 0 is the maximum number in the half-open interval [0,1): For every value x in (0,1), the number x*(2-x) also lies in (0,1), and is larger than x itself - so it can't be the maximal value; on the other hand for the value x = 0, x*(2-x) stays at 0. Would you conclude that 0 will be the maximum in the interval [0,1)? [ ... ] > There were two questions to be answered: > 1) what triangle has the maximum area > (and perimeter, which as you noted I did not show)? That's another misunderstanding: I wanted to point out to *Maury* that finding the inscribed triangle of maximal perimeter is slightly more complicated than finding the inscribed triangle of maximal area. - The reason being, that for the second problem you want to find the parallel line farthest away from a given chord of the given circle which still intersects the circle, while for the first problem you want to find the largest ellipse amongst those with focal points in the endpoints of a given chord which still intersects the given circle... > 2) Is there a procedure based on the OP idea of moving a point along > the circumfrence that will generate this triangle? > Once you have answered 1, you can use that result to answer 2- which is > what I did. If you think that my answer/proof for part 1 is incorrect, > then that is a seperate matter- obviously a flawed statement cannot be > used in the proof of another (unless you are doing a proof by > contradiction, where the flaw is the manner of proof). Maybe I missed out one of your messages, but Maury asked for a proof of 2), and wanted to deduce 1) from 2); and in the messages I have seen, you also seemed to argue along these lines. So, what was your proof of 1)? Sorry for the late answer. Thomas === Subject: Re: Triangles Inscribed in a Circle <8002406.1156437735030.JavaMail.jakarta@nitrogen.mathforum.org> <44f1b525@news1.ethz.ch> <44fcaf32@news1.ethz.ch consider an isosceles triangle ABC, with AC=BC, inscribed > in a circle C. Then move the vertex A (or B) on the > circle to obtain A'C=A'B (respectively B'A= B'C). The new > triangle A'BC (or AB'C, but these two triangles are > equal) has a greater area and and a longer perimeter than > ABC. Now you can repeat the procedure, moving B or C > (respectively A or C), to obtain a new isosceles > triangle, and so on. The sequence of inscribed triangles > has increasing area and perimeter, and I think it > converges to an equilateral triangle inscribed in the > circle. What do you think? > This result would allow to give a pure geometric proof > of the well-known fact that the equilateral triangle > is the triangle of maximum area and perimeter among all > the triangle inscribed in a given circle. schrieb The Qurqirish Dragon : > My previous post, which asserted the conclusion actually > proved that the equilateral triangle has the maximum area. No. You only showed > that to every inscribed triangle that is not equilateral, > there exists an inscribed triangle with larger area. > But you did not mention that one also has to assert > that there *is* an inscribed triangle > for which the area takes on a maximum value. Otherwise, you could apply your style of proof > to show that 0 is the maximum number in the half-open interval [0,1): For every value x in (0,1), the number x*(2-x) also lies in (0,1), > and is larger than x itself - so it can't be the maximal value; > on the other hand for the value x = 0, x*(2-x) stays at 0. Would you conclude that 0 will be the maximum in the interval [0,1)? No, since your proof only considers the open interval. Nowhere did you state anything about 0 itself. My triangle statement, now that you use this example so I can see the flaw, does miss the degenerate case where all 3 vertices of the original triangle are the same point. The area of this triangle can only be increased by moving 2 points. For any other case, my proof works. (any non-equilateral triangle can have its area increased by moving a vertex not on the perpendicular bisector of the opposite side to such a point. There are two such points, but my algorithm chooses the one such that the center of the circle lies inside the triangle.) === Subject: Re: Triangles Inscribed in a Circle schrieb The Qurqirish Dragon : >> consider an isosceles triangle ABC, with AC=BC, inscribed >> in a circle C. Then move the vertex A (or B) on the >> circle to obtain A'C=A'B (respectively B'A= B'C). The new >> triangle A'BC (or AB'C, but these two triangles are >> equal) has a greater area and and a longer perimeter than >> ABC. Now you can repeat the procedure, moving B or C >> (respectively A or C), to obtain a new isosceles >> triangle, and so on. The sequence of inscribed triangles >> has increasing area and perimeter, and I think it >> converges to an equilateral triangle inscribed in the >> circle. What do you think? > This result would allow to give a pure geometric proof >> of the well-known fact that the equilateral triangle >> is the triangle of maximum area and perimeter among all >> the triangle inscribed in a given circle. >> My previous post, which asserted the conclusion actually >> proved that the equilateral triangle has the maximum area. >> No. You only showed >> that to every inscribed triangle that is not equilateral, >> there exists an inscribed triangle with larger area. >> But you did not mention that one also has to assert >> that there *is* an inscribed triangle >> for which the area takes on a maximum value. >> Otherwise, you could apply your style of proof >> to show that 0 is the maximum number in the half-open interval [0,1): >> For every value x in (0,1), the number x*(2-x) also lies in (0,1), >> and is larger than x itself - so it can't be the maximal value; >> on the other hand for the value x = 0, x*(2-x) stays at 0. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ >> Would you conclude that 0 will be the maximum in the interval [0,1)? No, since your proof only considers the open interval. Nowhere did you > state anything about 0 itself. You'd better learn how to read! I *did* write: for the value x = 0, x*(2-x) stays at 0. So, I did *exactly* as you did in your proof: I showed that my procedure - replacing x by x*(2-x) - leaves the maximum 0 fixed, while it increases every other number x in the interval [0,1). You showed that your procedure - replacing the triangle T by the largest amongst those which have one chord in common with T - keeps the equilateral triangle fixed, while it enlarges every other inscribed triangle. Correct or not?! > My triangle statement, now that you use > this example so I can see the flaw, does miss the degenerate case where > all 3 vertices of the original triangle are the same point. You seem to have some compactification of the space of real inscribed triangles in mind here. - This will be necessary to turn your pseudo-proof into a real proof, but I don't see where it is important in your current proof. > The area of > this triangle can only be increased by moving 2 points. If you don't consider degenerate triangles, you don't have worry about this case. > For any other > case, my proof works. (any non-equilateral triangle can have its area > increased by moving a vertex not on the perpendicular bisector of the > opposite side to such a point. There are two such points, but my > algorithm chooses the one such that the center of the circle lies > inside the triangle.) This proof goes *exactly* along the same lines as my pseudo-proof of max [0,1) = 0 above. - It is simply wrong (i.e., incomplete). === Subject: How to recognize elliptic integrals from quite a long way away There are many problems which require elliptic integrals in their solutions: * calculating the arc-length of ellipses; * solving the pendulum equation x''(t) = sin(x(t)); * various formulas for the volume of the bodies in three-space which are bounded by two quadrics, plus a number of planes (e.g. the intersection of two cylinders, or of a cone and a cylinder); * ... . elliptic integrals are just line integrals of meromorphic 1-forms on 1-dimensional complex tori. Are there examples where you can recognize that an elliptic integrals arise in the solution of a problem, just by recognizing that the problem depends somehow on two cyclic parameters - in some kind of complex-analytic way? E.g., could one tell that for the volume of bodies in R^3 that are bounded by more than two quadrics, line integrals of meromorphic forms on complex curves of genus > 1 are required? === Subject: Re: How to recognize elliptic integrals from quite a long way away > There are many problems with elliptic integrals in their solutions: * calculating the arc-length of ellipses; > * solving the pendulum equation x''(t) = sin(x(t)); > * various formulas for the volume of the bodies in three-space > which are bounded by two quadrics, plus a number of planes > (e.g. the intersection of two cylinders, or of a cone and a cylinder); > * ... . elliptic integrals are just line integrals > of meromorphic 1-forms on 1-dimensional complex tori. Are there examples where you can recognize > that an elliptic integrals arise in the solution of a problem, > just by recognizing that the problem depends somehow > on two cyclic parameters - in some kind of complex-analytic way? No comments? Wrong subject line? Too esoteric? Too obvious? Complete nonsense? Any ideas? === Subject: Re: do odd perfect numbers exist >can anyone give me the proof why odd perfect numbers have not been >found? The proof that odd perfect numbers have not been found is the fact > that nobody has found an odd perfect number. What sort of proof are > you expecting? > >or atleast why >anyone's not able to prove the non existence or existence of odd >perfect numbers? Again, what sort of proof are you expecting? Nobody has been able to > prove that odd perfect numbers do not exist. That's simply a fact > observable by noting that no such proof is known. Correct on all coounts. Perhaps what OP wants to know is that it has been proved that if there is an odd perfect number then it must satisfy a long list of onerous conditions, and it has been proved that if there is a number satisfying all those conditions, then it must be enormous, so enormous that there's no hope of finding it by just sending a computer out to look for it. So, proving the existence of an odd perfect number is hard partly because, if there is one, it's way, way out there, and proving the non-existence of an odd pefect number is hard partly because there are so many numbers way, way out there that it's hard to show that there aren't any odd perfects among them. -- === Subject: Re: do odd perfect numbers exist >can anyone give me the proof why odd perfect numbers have not been >found? The proof that odd perfect numbers have not been found is the fact > that nobody has found an odd perfect number. What sort of proof are > you expecting? or atleast why >anyone's not able to prove the non existence or existence of odd >perfect numbers? Again, what sort of proof are you expecting? Nobody has been able to > prove that odd perfect numbers do not exist. That's simply a fact > observable by noting that no such proof is known. Correct on all coounts. Perhaps what OP wants to know > is that it has been proved that if there is an odd perfect number > then it must satisfy a long list of onerous conditions, > and it has been proved that if there is a number satisfying > all those conditions, > then it must be enormous, > so enormous that there's no hope of finding it > by just sending a computer out to look for it. Check out the paper at http://lanl.arxiv.org/abs/math.NT/0602485 for a list of these conditions (known as of February 2006). > So, proving the existence of an odd perfect number is hard > partly because, if there is one, it's way, way out there, > and proving the non-existence of an odd pefect number is hard > partly because there are so many numbers way, way out there > that it's hard to show that there aren't any odd perfects among them. -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Powers of 5 studying for a computer science final, that I still haven't come up with an answer to a question that was offered as extra credit on our first quiz in the course. We were given the following Pascal code. function power(n: integer): integer; begin if n <= 1 then return (5) else return (5 * power(n-1)) end; {power} We were then asked: Can you modify the function in such a way that for arguments of the form 2^k it computes the result with just k+1 calls? I'm not especially adept with math, and I have no idea how to approach the question. Please point me in the right direction; I would really like to understand this. -- Josh Zenker === Subject: Re: Powers of 5 > studying for a computer science final, that I still haven't come up > with an answer to a question that was offered as extra credit on our > first quiz in the course. We were given the following Pascal code. function power(n: integer): integer; > begin > if n <= 1 then > return (5) > else > return (5 * power(n-1)) > end; {power} We were then asked: Can you modify the function in such a way that for > arguments of the form 2^k it computes the result with just k+1 calls? > I'm not especially adept with math, and I have no idea how to approach > the question. Please point me in the right direction; I would really > like to understand this. If you still don't know the russian peasant algorithm for multiplying, something that most competant people here probably became aware of under one of several names or rediscovered themselves while their age was still in single digits, then either you're going to fail your degree horribly, or your degree won't be worth the paper it's written on. However, I'm glad to see that there are people feeding my prejudice that most modern CS degrees are possibly the best indicator of someone being _unsuitable_ for a programming job. The answer's so trivially simple I'm sure you'll get it from others in the thread. Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. === Subject: Re: Powers of 5 If you still don't know the russian peasant algorithm for multiplying, > something that most competant people here probably became aware of > under one of several names or rediscovered themselves while their age > was still in single digits, then either you're going to fail your degree > horribly, or your degree won't be worth the paper it's written on. However, I'm glad to see that there are people feeding my prejudice > that most modern CS degrees are possibly the best indicator of someone > being _unsuitable_ for a programming job. Woo hoo, Phil! You totally bitch-slapped that undergrad! Now, let's all retire to the faculty lounge to smoke cigars and swap stories about totally retarded answers students have given on final exams. -- Kevin === Subject: Re: Powers of 5 <87psea4zam.fsf@nonospaz.fatphil.org> <87slj6e2ox.fsf@kube.mad.asaurus.net > If you still don't know the russian peasant algorithm for multiplying, > something that most competant people here probably became aware of > under one of several names or rediscovered themselves while their age > was still in single digits, then either you're going to fail your degree > horribly, or your degree won't be worth the paper it's written on. Perhaps I should have asked a Russian peasant. > However, I'm glad to see that there are people feeding my prejudice > that most modern CS degrees are possibly the best indicator of someone > being _unsuitable_ for a programming job. I suppose only time will tell. > Woo hoo, Phil! You totally bitch-slapped that undergrad! Now, let's > all retire to the faculty lounge to smoke cigars and swap stories > about totally retarded answers students have given on final exams. The data structures exam I took yesterday should give you plenty of material better than I do, or I would despair for the future of computer science. === Subject: Re: Powers of 5 If you still don't know the russian peasant algorithm for multiplying, > something that most competant people here probably became aware of > under one of several names or rediscovered themselves while their age > was still in single digits, then either you're going to fail your degree > horribly, or your degree won't be worth the paper it's written on. However, I'm glad to see that there are people feeding my prejudice > that most modern CS degrees are possibly the best indicator of someone > being _unsuitable_ for a programming job. Woo hoo, Phil! You totally bitch-slapped that undergrad! Now, let's > all retire to the faculty lounge to smoke cigars and swap stories > about totally retarded answers students have given on final exams. Sounds great, except that I don't smoke. Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. === Subject: Re: Powers of 5 > studying for a computer science final, that I still haven't come up > with an answer to a question that was offered as extra credit on our > first quiz in the course. We were given the following Pascal code. function power(n: integer): integer; > begin > if n <= 1 then > return (5) > else > return (5 * power(n-1)) > end; {power} We were then asked: Can you modify the function in such a way that for > arguments of the form 2^k it computes the result with just k+1 calls? > I'm not especially adept with math, and I have no idea how to approach > the question. Please point me in the right direction; I would really > like to understand this. > Powers of 5 of the form 2^k (i.e. 5^(2^k)) can be computed quickly by successive squaring. For example, 5^16 = (5^8)^2 5^8 = (5^4)^2 5^4 = (5^2)^2 5^2 = (5^1)^2 This should give you the modified algorithm almost immediately. If not, since you indicated this isn't homework just ask for clarification. Rick BTW, the argument doesn't have to be a power of 2 for a modification to work in time less than a multiple of log n. It's a well-known way of speeding up exponentiation. Also, I'm sort of curious what school is still using Pascal as an introductory language. Not that I'm complaining--I think Pascal could still be a righteous choice for an intro vehicle; I'd just like to know what school still has the stones to resist jumping on the C#/C++/C/Java bandwagons. === Subject: Re: Powers of 5 by successive squaring. For example, 5^16 = (5^8)^2 5^8 = (5^4)^2 5^4 = (5^2)^2 5^2 = (5^1)^2 This should give you the modified algorithm almost immediately. in Python for the sake of being concise, but now the function takes k rather than 2^k as its argument. So I guess I haven't really answered the question. def power(k): if (k <= 1): return 5 else: return power(k-1)**2 introductory language. Not that I'm complaining--I think Pascal > could still be a righteous choice for an intro vehicle; I'd just > like to know what school still has the stones to resist jumping > on the C#/C++/C/Java bandwagons. That would be Drexel University. The course I've been taking covers data structures and algorithms, and therefore the programming component is small. Courses with greater emphasis on programming have entered into a sordid love affair with OO languages, especially C++ and Java. At least, that's how it seems to an outsider. (My home institution is the University of Chicago. They started us with Scheme, which made me loathe nested parantheses for a while.) At any rate, I digress. I'm just missing how to get the function to take 2^k as its argument. If you don't mind helping a little more, I would be grateful. -- Josh Zenker === Subject: Re: Powers of 5 >Powers of 5 of the form 2^k (i.e. 5^(2^k)) can be computed quickly >>by successive squaring. For example, >>5^16 = (5^8)^2 >> 5^8 = (5^4)^2 >> 5^4 = (5^2)^2 >> 5^2 = (5^1)^2 >>This should give you the modified algorithm almost immediately. > in Python for the sake of being concise, but now the function takes k > rather than 2^k as its argument. So I guess I haven't really answered > the question. def power(k): > if (k <= 1): > return 5 > else: > return power(k-1)**2 >Also, I'm sort of curious what school is still using Pascal as an >>introductory language. Not that I'm complaining--I think Pascal >>could still be a righteous choice for an intro vehicle; I'd just >>like to know what school still has the stones to resist jumping >>on the C#/C++/C/Java bandwagons. > That would be Drexel University. The course I've been taking covers > data structures and algorithms, and therefore the programming component > is small. Courses with greater emphasis on programming have entered > into a sordid love affair with OO languages, especially C++ and Java. > At least, that's how it seems to an outsider. (My home institution is > the University of Chicago. They started us with Scheme, which made me > loathe nested parantheses for a while.) At any rate, I digress. I'm just missing how to get the function to > take 2^k as its argument. If you don't mind helping a little more, I > would be grateful. > I'll do more than that. Here's the algorithm in a form that doesn't require the argument to be a power of 2. It's been a while since I programmed in Pascal, so bear with me. function Power(x, n: integer) : integer; {Returns x^n for n >= 0.} begin if n = 0 then Power := 1 else if odd(n) then Power := x * Power(sqr(x), n div 2)) else Power := Power(sqr(x), n div 2) end; This relies on the binary representation of n, so for example if n = 13 we have x^13 = x^{8 + 4 + 1} = x^8 * x^4 * x^1 and the observation that if n is even then x^n = (x^2)^{n/2} and if n is odd then x^n = x * (x^2)^{floor(n/2)}. Trace the action of the function for n = 13 and you'll see what's going on. Rick === Subject: Re: Powers of 5 ... > Powers of 5 of the form 2^k (i.e. 5^(2^k)) can be computed quickly > by successive squaring. For example, ... > in Python for the sake of being concise, but now the function takes k > rather than 2^k as its argument. So I guess I haven't really answered > the question. def power(k): > if (k <= 1): > return 5 > else: > return power(k-1)**2 ... > At any rate, I digress. I'm just missing how to get the function to > take 2^k as its argument. If you don't mind helping a little more, I > would be grateful. ... You merely replaced return (5 * power(n-1)) with return power(k-1)**2. Instead, make the function split three ways - if n <= 1 return 5, else if n is even return the square of something, else return power(n-1)*5. [Helpful example: x^14 = (x^7)^2.] -jiw === Subject: Re: Powers of 5 > You merely replaced return (5 * power(n-1)) with > return power(k-1)**2. Instead, make the function > split three ways - > if n <= 1 return 5, > else if n is even return the square of something, > else return power(n-1)*5. Why are people mindlessly propagating the complete falsity that 5^0 = 5? You, James, should know better. Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. === Subject: Re: Powers of 5 <44FCEACB.ECF0F49B@pat7.com You merely replaced return (5 * power(n-1)) with > return power(k-1)**2. Ah, you're right. > Instead, make the function > split three ways - > if n <= 1 return 5, > else if n is even return the square of something, > else return power(n-1)*5. I like this idea, but that second conditional statement doesn't seem quite correct. If we do that, then we get the powers of 2, like 2, 4 and 8, as well as other even numbers that don't follow the form 2^k--like 6, for example. -- Josh Zenker === Subject: Re: Powers of 5 <44FCEACB.ECF0F49B@pat7.com You merely replaced return (5 * power(n-1)) with > return power(k-1)**2. Ah, you're right. Instead, make the function > split three ways - > if n <= 1 return 5, > else if n is even return the square of something, > else return power(n-1)*5. I like this idea, but that second conditional statement doesn't seem > quite correct. If we do that, then we get the powers of 2, like 2, 4 > and 8, as well as other even numbers that don't follow the form > 2^k--like 6, for example. > basic algorithm: If k<0, return 1/(power(-k)) If k=0, return 1 If k is even, return (power (k/2))^2 If k is odd, return 5*(power(k-1)) === Subject: Re: Powers of 5 In sci.math, Josh Zenker on 4 Sep 2006 20:45:17 -0700 >> You merely replaced return (5 * power(n-1)) with >> return power(k-1)**2. Ah, you're right. > Instead, make the function >> split three ways - >> if n <= 1 return 5, >> else if n is even return the square of something, >> else return power(n-1)*5. I like this idea, but that second conditional statement doesn't seem > quite correct. If we do that, then we get the powers of 2, like 2, 4 > and 8, as well as other even numbers that don't follow the form > 2^k--like 6, for example. > As a rather dumb observation, one implementation is fairly simple (assuming indeterminate precision of integers): 1 power(int k,unsigned int n) 2 { 3 int prod = k; 4 int res = 1; 5 unsigned int mask = 1; 6 7 while(n >= mask) 8 { 9 if( (n & mask) ) // bitwise AND 10 { res *= prod; } 11 mask *= 2; // or <<= 1 12 prod *= prod; 13 } 14 15 return res; 16 } so power(5,6) might run as follows: 5: k=5 n=6 res=1 prod=5 mask=1 7: succeeds 9: fails 11: mask=2 12: prod=25 7: succeeds 9: succeeds 10: res = 25 11: mask=4 12: prod=625 7: succeeds 9: succeeds 10: res = 15625 11: mask=8 12: prod=390625 7: fails 15: return 15625 An alternative implementation might search for the leftmost bit, then shift n leftward through that bit: power(int k, unsigned int n) { int res = 1; unsigned int mask = 1; while(mask < n) mask *= 2; while(n != 0) { res *= res; if(n & mask) // bitwise AND again { res *= k; n -= mask; } n *= 2; } return res; } The proof that this works is left to the inquisitive reader. It is worth noting, however, that the last multiplication this particular one does is 125*125, as opposed to the one above which did 625*25. Not that it matters, as multiplication is commutative and associative. (In a real 32-bit system one might run into the problem that 2146473648*2 = 0. However, anyone wanting to compute powers over n^2billion is probably going to want to use slightly different and more efficient methods anyway. :-) ) One can also recursivise it: power(int k, unsigned n) { if(n == 0) return 1; if(n odd) return k * power(k, n-1); return power(k,n/2) * power(k,n/2); // or power(k*k, n/2); } If one takes power(5, 6), one gets the following sequence: power(5,6) = power(5,3) * power(5,3) = (5 * power(5,2)) * (5 * power(5,2)) = (5 * (power(5,1) * power(5,1))) * (5 * (power(5,1) * power(5,1))) = (5 * ((5*power(5,0)) * (5*power(5,0)))) * (5 * ((5*power(5,0)) * (5*power(5,0)))) = (5 * ((5*1) * (5*1))) * (5 * ((5*1) * (5*1))) -- #191, ewill3@earthlink.net Windows Vista. Because it's time to refresh your hardware. Trust us. === Subject: Re: prime theorems Nntp-Posting-Host: apps.cwi.nl >M.J.T. Guy escribi.97: > arunloboforever@gmail.com escribi.97: >> prove this: >> if 1+a+a^2+a^3+....a^2k has a prime factor greater than a.a is an >> odd prime. I think you has made an incredibly bad wording ... I guess, very aventuradamente, that really the statement is If a is an odd prime, then 1+a+a^2+a^3+....+a^(2k) has a prime > factor greater > than a This statement seems more feasible. >> It may be more feasible, but you still dont have to look far for a >> counterexample: >> 1 + 67 + 67^2 = 3 * 7^2 * 31 >> That was found by looking for cases where 3 and 7 divide 1 + a + a^2 >> and hence for primes of the form 42n + 25 or 42n + 37; the second >> prime did the trick. > >Yes, there are many counter-examples. For k = 1, there are 31 primes less >than 1000. And for k = 2, a = 7307 and a = 9769 up to 10000. But I think that the OP meant this, wrongly of course. For k = 3 and a = 7144363, the largest prime factor of (a^7 - 1)/(a - 1) is 200117, about a/35. This example was found using a sieve to search for values of a such that all prime factors of (a^7 - 1)/(a - 1) are below 10^6. -- Expel the plutarchs from Washington, DC this November. Put us on a sensible orbit amongst the solar powers. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: prime theorems <4m36c3F43m7gU1@individual.net> Well yeah.I had written it in a hurry. And thats false now. please check this: if a is an odd prime, 1+a^2+a^4+a^6+....a^4k has a prime factor greater than a. a is a prime of the form 4j+1. >M.J.T. Guy escribi.97: > arunloboforever@gmail.com escribi.97: >> prove this: >> if 1+a+a^2+a^3+....a^2k has a prime factor greater than a.a is an >> odd prime. I think you has made an incredibly bad wording ... I guess, very aventuradamente, that really the statement is If a is an odd prime, then 1+a+a^2+a^3+....+a^(2k) has a prime > factor greater > than a This statement seems more feasible. >> It may be more feasible, but you still dont have to look far for a >> counterexample: >> 1 + 67 + 67^2 = 3 * 7^2 * 31 >> That was found by looking for cases where 3 and 7 divide 1 + a + a^2 >> and hence for primes of the form 42n + 25 or 42n + 37; the second >> prime did the trick. > >Yes, there are many counter-examples. For k = 1, there are 31 primes less >than 1000. And for k = 2, a = 7307 and a = 9769 up to 10000. But I think that the OP meant this, wrongly of course. For k = 3 and a = 7144363, the largest prime factor > of (a^7 - 1)/(a - 1) is 200117, about a/35. > This example was found using a sieve to search > for values of a such that all prime factors > of (a^7 - 1)/(a - 1) are below 10^6. > -- > Expel the plutarchs from Washington, DC this November. > Put us on a sensible orbit amongst the solar powers. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: prime theorems > Well yeah.I had written it in a hurry. And thats false now. please > check this: if a is an odd prime, 1+a^2+a^4+a^6+....a^4k has a prime > factor greater than a. a is a prime of the form 4j+1. Your wording is really obscure ... The statement a is a prime of the form 4j+1, Àis the thesis or the hypothesis part of your claim? If it is the hypothesis, consider k = 1, a = 373: 1 + 373^2 + 373^4 = 3*7^2*13*19*67*73*109 If it is the thesis .... Consider k = 1, a = 3 -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: prime theorems <4m5srrF4ldm6U1@individual.net> Ah well im losing everywhere! please give me the next two counterexamples. And also check this: If a is a prime of the form 4j+1, 1+a+a^2+a^3+.....a^2k has a prime factor greater than a.Please give the 3 least counterexamples. > arunloboforever@gmail.com escribi.97: > Well yeah.I had written it in a hurry. And thats false now. please > check this: if a is an odd prime, 1+a^2+a^4+a^6+....a^4k has a prime > factor greater than a. a is a prime of the form 4j+1. Your wording is really obscure ... The statement a is a prime of the form > 4j+1, Àis the thesis or the hypothesis part of your claim? If it is the hypothesis, consider k = 1, a = 373: 1 + 373^2 + 373^4 = 3*7^2*13*19*67*73*109 If it is the thesis .... Consider k = 1, a = 3 > -- Ignacio Larrosa Ca.96estro > A Coru.96a (Espa.96a) > ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: prime theorems <4m5srrF4ldm6U1@individual.net> Rather than having the community running around looking for counterexamples for you (as you can do that yourself), why not take a couple of moments to collect your thoughts and tell us just what it is you are looking for? Are you trying to find some sequence that always generates a prime or what? cheers Eric > Ah well im losing everywhere! please give me the next two > counterexamples. And also check this: > If a is a prime of the form 4j+1, 1+a+a^2+a^3+.....a^2k has a prime > factor greater than a.Please give the 3 least counterexamples. > arunloboforever@gmail.com escribi.97: > Well yeah.I had written it in a hurry. And thats false now. please > check this: if a is an odd prime, 1+a^2+a^4+a^6+....a^4k has a prime > factor greater than a. a is a prime of the form 4j+1. Your wording is really obscure ... The statement a is a prime of the form > 4j+1, Àis the thesis or the hypothesis part of your claim? If it is the hypothesis, consider k = 1, a = 373: 1 + 373^2 + 373^4 = 3*7^2*13*19*67*73*109 If it is the thesis .... Consider k = 1, a = 3 > -- Ignacio Larrosa Ca.96estro > A Coru.96a (Espa.96a) > ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: prime theorems > Well yeah.I had written it in a hurry. And thats false now. please > check this: if a is an odd prime, 1+a^2+a^4+a^6+....a^4k has a prime > factor greater than a. a is a prime of the form 4j+1. Don't top post, it makes you look like someone who doesn't understand that responses come after the thing to which they respond. Anyway, your question is still malformed. You do not specify what k is. Do you mean a) If a is a prime of the form 4j+1 then 1+a^2+a^4+a^6+....a^4k has a prime factor greater than a for some k>0. b) If a is a prime of the form 4j+1 then 1+a^2+a^4+a^6+....a^4k has a prime factor greater than a for all k>0. . === Subject: Re: prime theorems And in the previous theorem too a is of the form 4j+1 > Well yeah.I had written it in a hurry. And thats false now. please > check this: if a is an odd prime, 1+a^2+a^4+a^6+....a^4k has a prime > factor greater than a. a is a prime of the form 4j+1. > arunloboforever@gmail.com escribi.97: >> prove this: >> if 1+a+a^2+a^3+....a^2k has a prime factor greater than a.a is an >> odd prime. I think you has made an incredibly bad wording ... I guess, very aventuradamente, that really the statement is If a is an odd prime, then 1+a+a^2+a^3+....+a^(2k) has a prime > factor greater > than a This statement seems more feasible. >> It may be more feasible, but you still dont have to look far for a >> counterexample: >> 1 + 67 + 67^2 = 3 * 7^2 * 31 >> That was found by looking for cases where 3 and 7 divide 1 + a + a^2 >> and hence for primes of the form 42n + 25 or 42n + 37; the second >> prime did the trick. > >Yes, there are many counter-examples. For k = 1, there are 31 primes less >than 1000. And for k = 2, a = 7307 and a = 9769 up to 10000. But I think that the OP meant this, wrongly of course. For k = 3 and a = 7144363, the largest prime factor > of (a^7 - 1)/(a - 1) is 200117, about a/35. > This example was found using a sieve to search > for values of a such that all prime factors > of (a^7 - 1)/(a - 1) are below 10^6. > -- > Expel the plutarchs from Washington, DC this November. > Put us on a sensible orbit amongst the solar powers. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: 100 Mbits of PI needed [Tim Peters, suggests trying to alter http://www.swox.com/gmp/pi-with-gmp.html to produce base 16 output to get lots of pi digits quickly ] [Cristiano] > I already used that program and it is very fast, but it works only for > small d (I use mpf_get_str because mpf_out_str doesn't work). Peculiar! > When I use d= 1,000,000 (which is still small), the program terminates > with an unhandled exception in the 'div' routine (stack overflow). Ditto! > I have the version 4.1.4 of GMP buil as dll because under Windows Ah, that explains everything <0.3 wink>. > I don't know how to use the source code directly (I use MinGW/MSYS > to build the dll). > I'd like to try the new version 4.2.1, but 'make check' fails and the > resulting dll is really snarffed. Upgrade to Linux ;-) Just for the heck of it, I tried building a current GMP on Windows under a full Cygwin installation, and it crapped out late in the compilation phase. Not motivated to pursue it, and sorry for wasting your time. ... I see from a later post that everything worked out for you without GMP. Great! Use your hex digits only for good, never for evil purposes :-) === Subject: Re: 100 Mbits of PI needed > Upgrade to Linux ;-) I tried Knoppix, but ./configure gives Permission denied with the gmp folder on the ram disk and read/write permission enabled (!?). > ... I see from a later post that everything worked out for you > without GMP. Great! Use your hex digits only for good, never for > evil purposes :-) I'll use it for Gioco del lotto (an Italian lottery). :-) Not true, I use it for testing purposes. Cristiano === Subject: Re: testing... > i'm using slrn and i need to test... sorry.... I never received it. Dale. === Subject: a question about matrix variate normal distribution Let X (pxn) be a a random matrix that has a matrix variate normal distribution with mean matrix M (pxn), and covariace matrix Sigma otimes Psi, where Sigma is a pxp matrix, Psi is an nxn matrix, and Sigma otimes Psi denotes the kronecker product of two matrices. For the sake of simplicity, we can assume the mean matrix M is 0, and covariance is an identity matrix. My question is: What is the distribution of the matrix XC for a constant matrix C (nxq)? Here XC is the regular matrix product of X and C. Does XC also have a matrix variate normal distribution? Comments: For n-dimensional normal random vector, the answer is Yes under centain conditions. I am not sure what is gonna happen for matrix variate normal distribution. Roy === Subject: Re: a question about matrix variate normal distribution >Let X (pxn) be a a random matrix that has a matrix variate normal >distribution >with mean matrix M (pxn), and covariace matrix Sigma otimes Psi, >where > Sigma is a pxp matrix, Psi is an nxn matrix, and Sigma otimes >Psi denotes >the kronecker product of two matrices. >For the sake of simplicity, we can assume the mean matrix M is 0, and >covariance is an identity matrix. >My question is: What is the distribution of the matrix XC for a >constant matrix C (nxq)? >Here XC is the regular matrix product of X and C. >Does XC also have a matrix variate normal distribution? Any set of linear combinations of jointly normal random variables is jointly normal. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: a question about matrix variate normal distribution distribution >with mean matrix M (pxn), and covariace matrix Sigma otimes Psi, >where > Sigma is a pxp matrix, Psi is an nxn matrix, and Sigma otimes >Psi denotes >the kronecker product of two matrices. For the sake of simplicity, we can assume the mean matrix M is 0, and >covariance is an identity matrix. My question is: What is the distribution of the matrix XC for a >constant matrix C (nxq)? >Here XC is the regular matrix product of X and C. Does XC also have a matrix variate normal distribution? Any set of linear combinations of jointly normal > random variables is jointly normal. > -- Let us consider an example. Let X be an n-dimensional normal random vector. Let A be a 2xn constant matrix. And assume the two row vectors in A are the same, i.e., rank(A) = 1. Is AX a multivariate normal random vector? I saw in some statistics books that only if A has full row rank, can we guarantee that AX is multivariate normal. Roy === Subject: Re: a question about matrix variate normal distribution >>Let X (pxn) be a a random matrix that has a matrix variate normal >>distribution >>with mean matrix M (pxn), and covariace matrix Sigma otimes Psi, >>where >> Sigma is a pxp matrix, Psi is an nxn matrix, and Sigma otimes >>Psi denotes >>the kronecker product of two matrices. >>For the sake of simplicity, we can assume the mean matrix M is 0, and >>covariance is an identity matrix. >>My question is: What is the distribution of the matrix XC for a >>constant matrix C (nxq)? >>Here XC is the regular matrix product of X and C. >>Does XC also have a matrix variate normal distribution? >> Any set of linear combinations of jointly normal >> random variables is jointly normal. >> -- >Let us consider an example. >Let X be an n-dimensional normal random vector. Let A be a 2xn constant >matrix. And assume the two row vectors in A are the same, i.e., rank(A) >= 1. >Is AX a multivariate normal random vector? >I saw in some statistics books that only if A has full row rank, can we >guarantee that AX is multivariate normal. AX will not have a density, but it is still multivariate normal. Not requiring a density has many advantages, including going over (properly defined) to infinite dimension. In finite dimensions, if guarantees that the limit of sequence of multivariate normal distributions with no probability going off to infinity is multivariate normal, which would not be the case otherwise. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Complex Variables - Analytic Where is Log( z + z^(-1) - 2 ) analytic? where Log is the real log and z is complex. For the real log( A ), then A >= 0. Is it true in this case? What's next? === Subject: Re: Complex Variables - Analytic > Where is Log( z + z^(-1) - 2 ) analytic? > where Log is the real log and z is complex. For the real log( A ), then A >= 0. Is it true in this case? What's > next? > How are you defining the real log? Do you mean the natural logarithm instead? Dale. === Subject: Re: Complex Variables - Analytic W. Dale Hall .9b.8d.93.b9.81F > Where is Log( z + z^(-1) - 2 ) analytic? > where Log is the real log and z is complex. For the real log( A ), then A >= 0. Is it true in this case? What's > next? > How are you defining the real log? > Do you mean the natural logarithm instead? Oh, yes. I mean ln. Dale. === Subject: Re: Complex Variables - Analytic > W. Dale Hall .9b.8d.93.b9.81F Where is Log( z + z^(-1) - 2 ) analytic? > where Log is the real log and z is complex. For the real log( A ), then A >= 0. Is it true in this case? What's > next? > How are you defining the real log? >> Do you mean the natural logarithm instead? Oh, yes. I mean ln. > >> Dale. > Well, then, the logarithm is analytic where the argument is nonzero. The argument of your function is also analytic where z is nonzero, so your function will be analytic where both conditions coincide: z != 0 and z + z^(-1) - 2 != 0 The second condition can be turned into a quadratic inequality z^2 - 2z + 1 != 0 i.e., (z - 1)^2 != 0. Looks like we find that your function is analytic for z != 0,1. I'd check the arithmetic, but I think that's it. Dale === Subject: Re: Complex Variables - Analytic where Log is the real log and z is complex. For the real log( A ), then A >= 0. Is it true in this case? What's > next? > How are you defining the real log? >> Do you mean the natural logarithm instead? Oh, yes. I mean ln. > Dale. > Well, then, the logarithm is analytic where the argument is nonzero. > The argument of your function is also analytic where z is nonzero, > so your function will be analytic where both conditions coincide: z != 0 and z + z^(-1) - 2 != 0 The second condition can be turned into a quadratic > inequality z^2 - 2z + 1 != 0 i.e., (z - 1)^2 != 0. Looks like we find that your function is analytic > for z != 0,1. I'd check the arithmetic, but I think that's it. This is true for a multivalued logarithm. There is no analytic function on C Ä {0,1} that is a logarithm of z + z^(-1) - 2. If you want a single-valued function, you need to choose a branch of the logarithm. This will introduce branch cuts where your function is not analytic (in fact discontinuous), joining the three branch points 0, 1 and infinity. For example, you could take the branch cut to be the interval (-infinity, 1] of the real line. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Complex Variables - Analytic where Log is the real log and z is complex. For the real log( A ), then A >= 0. Is it true in this case? What's > next? > How are you defining the real log? >> Do you mean the natural logarithm instead? Oh, yes. I mean ln. > Dale. > Well, then, the logarithm is analytic where the argument is nonzero. > The argument of your function is also analytic where z is nonzero, > so your function will be analytic where both conditions coincide: z != 0 and z + z^(-1) - 2 != 0 The second condition can be turned into a quadratic > inequality z^2 - 2z + 1 != 0 i.e., (z - 1)^2 != 0. Looks like we find that your function is analytic > for z != 0,1. I don't understand.. This is my first undergrad complex variable course. I'd check the arithmetic, but I think that's it. This is true for a multivalued logarithm. There is no analytic > function > on C Ä {0,1} that is a logarithm of z + z^(-1) - 2. > If you want a single-valued function, you need to choose a branch of > the logarithm. This will introduce branch cuts where your function is > not analytic (in fact discontinuous), joining the three branch points > 0, 1 > and infinity. For example, you could take the branch cut to be the > interval (-infinity, 1] of the real line. I get what you mean, but the answer given at the back is: the complement of the set { z : | z | = 1 } or { z : z = x, x is real, x <= 0 } Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada === Subject: Re: Complex Variables - Analytic > W. Dale Hall .8cøÇ.8eªã.95.b9û >> Where is Log( z + z^(-1) - 2 ) analytic? >> where Log is the real log and z is complex. >>1;2c > For the real log( A ), then A >= 0. Is it true in this case? What's >> next? > How are you defining the real log? > Do you mean the natural logarithm instead? >> Oh, yes. I mean ln. > Dale. >> Well, then, the logarithm is analytic where the argument is nonzero. >> The argument of your function is also analytic where z is nonzero, >> so your function will be analytic where both conditions coincide: >> z != 0 >> and >> z + z^(-1) - 2 != 0 >> The second condition can be turned into a quadratic >> inequality >> z^2 - 2z + 1 != 0 >> i.e., >> (z - 1)^2 != 0. >> Looks like we find that your function is analytic >> for z != 0,1. I don't understand.. >This is my first undergrad complex variable course. > I'd check the arithmetic, but I think that's it. >> This is true for a multivalued logarithm. There is no analytic >> function >> on C {0,1} that is a logarithm of z + z^(-1) - 2. >> If you want a single-valued function, you need to choose a branch of >> the logarithm. This will introduce branch cuts where your function is >> not analytic (in fact discontinuous), joining the three branch points >> 0, 1 >> and infinity. For example, you could take the branch cut to be the >> interval (-infinity, 1] of the real line. I get what you mean, but the answer given at the back is: >the complement of the set { z : | z | = 1 } or { z : z = x, x is real, >x <= 0 } Ah... It's using the principal branch of the logarithm. That is, Log(w) is analytic in the complement of the real interval (-infty, 0]. So then the question is, when is z + 1/z - 2 not in (-infty, 0], and the answer is what your book gives. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Complex Variables - Analytic Robert Israel .9b.8d.93.b9.81F > W. Dale Hall .9b.8d.93.b9.81F >> Where is Log( z + z^(-1) - 2 ) analytic? >> where Log is the real log and z is complex. >>1;2c > For the real log( A ), then A >= 0. Is it true in this case? What's >> next? > How are you defining the real log? > Do you mean the natural logarithm instead? >> Oh, yes. I mean ln. > Dale. >> Well, then, the logarithm is analytic where the argument is nonzero. >> The argument of your function is also analytic where z is nonzero, >> so your function will be analytic where both conditions coincide: >> tz != 0 >> and >> tz + z^(-1) - 2 != 0 >> The second condition can be turned into a quadratic >> inequality >> tz^2 - 2z + 1 != 0 >> i.e., >> t(z - 1)^2 != 0. >> Looks like we find that your function is analytic >> for z != 0,1. I don't understand.. >This is my first undergrad complex variable course. > I'd check the arithmetic, but I think that's it. >> This is true for a multivalued logarithm. There is no analytic >> function >> on C Ä {0,1} that is a logarithm of z + z^(-1) - 2. >> If you want a single-valued function, you need to choose a branch of >> the logarithm. This will introduce branch cuts where your function is >> not analytic (in fact discontinuous), joining the three branch points >> 0, 1 >> and infinity. For example, you could take the branch cut to be the >> interval (-infinity, 1] of the real line. I get what you mean, but the answer given at the back is: >the complement of the set { z : | z | = 1 } or { z : z = x, x is real, >x <= 0 } Ah... It's using the principal branch of the logarithm. That is, > Log(w) is analytic in the complement of the real interval > (-infty, 0]. So then the question is, when is z + 1/z - 2 not in > (-infty, 0], and the answer is what your book gives. All right, I understand the function isn't analytic in the set { z : | z | = 1 } or { z : z = x, x is real, x <= 0 }. But how do we obtain this set? Is z + 1/z - 2 equivalent to [ sqrt(z) - 1/sqrt(z) ]^2 ? > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada === Subject: Re: ctrl msg rmgroup sci.math > because it's filled with crap and endless replies to crap 90% of everything is crap === Subject: Re: ctrl msg rmgroup sci.math In sci.math, the enemies of god <44FC7D15.10500@verizon.net>: > because it's filled with crap and endless replies to crap > Erm...did I miss the RFD? :-) Followups. -- #191, ewill3@earthlink.net Windows Vista. Because it's time to refresh your hardware. Trust us. === Subject: Re: ctrl msg rmgroup sci.math > because it's filled with crap and endless replies to crap ooooohhhhh you're a clever man. === Subject: Re: Usefulness of calculating length <1157370232.6518.11.camel@localhost.localdomain Working backwards... Time = INT [ f'(x)^2 + 1]^0.5 dx > dx is length, f' is dimensionaless. Thus integral(a,b) sqr(f'(x)^2 + 1) is length and your equation equating time to length is nonsense. > dt/dx = [ f'(x)^2 + 1]^0.5 > (dt/dx)^2 - 1 = f'(x)^2 > f'(x) = [ (dt/dx)^2 - 1 ] ^0.5 > f(t) = INT [ (dt/dx)^2 - 1 ] ^0.5 dx Now what would X represent? > Obviously the same thing as Y. === Subject: Masiv combinatoric I have 100 slots. A identical things need to placed in the first 20 slots and B identical things need to be placed in the remaining (of the 100) slots. How many ways can this be done? I think it is: 20! / A! * (100-A)! / B! / (100-A-B)! Is that correct? WE === Subject: Re: Masiv combinatoric I am very sorry for posting an ambiguous problem, please allow me to clarify: There are 100 slots, A red balls and B blue balls. The red balls may only be placed in the first 20 slots. How many configurations meet these considerations? My guess was that you place the red balls in the 20 slots: 20!/A!/(20-A)! And then you are left with 100-A slots with which to place the blue balls: (100-A)!/B!/(100-A-B)! Giving: 20!/A!/(20-A)! * (100-A)!/B!/(100-A-B)! Is that correct? WE === Subject: Re: Masiv combinatoric > I am very sorry for posting an ambiguous problem, please allow me to > clarify: There are 100 slots, A red balls and B blue balls. The red balls may > only be placed in the first 20 slots. How many configurations meet > these considerations? My guess was that you place the red balls in the 20 slots: 20!/A!/(20-A)! Well, let's try a much smaller exampple and see whether your answer is reasonable. If the red balls were restricted to the 1st 2 slots, you would get 2 / ( A! (2 - A)! ). If A = 2, this formula gives 1. But in fact if A = 2 you could put both balls in the first slot, or one in each of the two slots, or both in the 2nd slot, so the correct answer is 3. Unless, of course, no slot can hold more than one ball. But you never said that. Did you mean that? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Masiv combinatoric > I am very sorry for posting an ambiguous problem, please allow me to > clarify: There are 100 slots, A red balls and B blue balls. The red balls may > only be placed in the first 20 slots. How many configurations meet > these considerations? My guess was that you place the red balls in the 20 slots: 20!/A!/(20-A)! And then you are left with 100-A slots with which to place the blue > balls: (100-A)!/B!/(100-A-B)! Giving: 20!/A!/(20-A)! * (100-A)!/B!/(100-A-B)! Is that correct? Looks like it. Ignore my 'show me 2' comment. Mensanator's post had persuaded me that B=100-A, which is not a fair assumption. How's that for blame-shifting! :-) Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. === Subject: Re: Masiv combinatoric > I have 100 slots. A identical things need to placed in the first 20 slots > and > B identical things need to be placed in the remaining (of the 100) > slots. How many ways can this be done? So let's look at the question, how many ways can you put n identical objects into m distinct boxes. Line up n + m - 1 identical objects, and then paint m - 1 of them blue. You're left with n identical objects, separated into m groups. How many ways can you choose from the original n + m - 1 objects the m - 1 objects to be painted blue? Can you take it from here? Note - the method needs some simple modification if the original objects were already blue. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Masiv combinatoric Now I think it is: 20! / A! / (20-A)! * (100-A)! / B! / (100-A-B)! === Subject: Re: Masiv combinatoric > Now I think it is: > 20! / A! / (20-A)! * (100-A)! / B! / (100-A-B)! I think it's 1. === Subject: Re: Masiv combinatoric > Now I think it is: > 20! / A! / (20-A)! * (100-A)! / B! / (100-A-B)! I think it's 1. Whilst you could put forward an argument supporting that, I think it's a bogus one. Your post is a rec.puzzles post, not a sci.math post. It's only the case if the 'A' items are indistinguishable from the 'B' items. (Or A=20 or 0, but that de-parameterises the problem, which is bogus. Or the slots are indistinguishable, which they aren't, so that's bogus too.) Given that the first 'A' items are _explicitly_ indicated as being identical to each other, and that the 'B' items are also explicitly indicated as being identical to each other, there is precisely no justification for assuming that any items not _explicitly_ indicated as being identical to each other would so be. But it didn't say they weren't identical. is not a counter argument. It didn't say that the boxes aren't rotated by one in order after the placement (thus making box 20 out of bounds for an 'A', and changing the answer). It didn't say that the boxes are too small for the objects, making the answer 0. Your assumtion that all things unless otherwise stated is one that is counterindicated by the wording of the problem. And to the OP - once the 'A' items are placed, how many ways are there to fill all the remaining slots with identical balls? Can you describe 2 such ways, for example? Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. === Subject: Re: Masiv combinatoric <87ejuq4xn5.fsf@nonospaz.fatphil.org > Now I think it is: > 20! / A! / (20-A)! * (100-A)! / B! / (100-A-B)! I think it's 1. Whilst you could put forward an argument supporting that, > I think it's a bogus one. Your post is a rec.puzzles post, not a sci.math post. No, it's an alt.sci.math.combinatorics post. > It's only the case if the 'A' items are indistinguishable > from the 'B' items. (Or A=20 or 0, but that de-parameterises > the problem, which is bogus. Or the slots are indistinguishable, > which they aren't, so that's bogus too.) Given that the first 'A' items are _explicitly_ indicated as > being identical to each other, and that the 'B' items are > also explicitly indicated as being identical to each other, > there is precisely no justification for assuming that any > items not _explicitly_ indicated as being identical to each > other would so be. But it didn't say they weren't identical. is not a counter > argument. It didn't say that the boxes aren't rotated by > one in order after the placement (thus making box 20 out of > bounds for an 'A', and changing the answer). It didn't say > that the boxes are too small for the objects, making the > answer 0. Your assumtion that all things unless otherwise stated is > one that is counterindicated by the wording of the problem. > And to the OP - once the 'A' items are placed, how many ways > are there to fill all the remaining slots with identical > balls? Can you describe 2 such ways, for example? Phil > -- > Home taping is killing big business profits. We left this side blank > so you can help. -- Dead Kennedys, written upon the B-side of tapes of > /In God We Trust, Inc./. === Subject: Re: Masiv combinatoric <87ejuq4xn5.fsf@nonospaz.fatphil.org > Now I think it is: > 20! / A! / (20-A)! * (100-A)! / B! / (100-A-B)! I think it's 1. Whilst you could put forward an argument supporting that, > I think it's a bogus one. It depends how you interpret the question. There is a valid interpretation that results in an answer of 1, so it's not bogus. Your post is a rec.puzzles post, not a sci.math post. Are you saying that the question how many combinations are there of 100 items taken 100 at a time? isn't valid mathematics? And doesn't 100choose100 = 1? It's only the case if the 'A' items are indistinguishable > from the 'B' items. (Or A=20 or 0, but that de-parameterises > the problem, which is bogus. Or the slots are indistinguishable, > which they aren't, so that's bogus too.) But my answer doesn't depend on A and B being identical or that the slots are indistinguishable. Given that the first 'A' items are _explicitly_ indicated as > being identical to each other, and that the 'B' items are > also explicitly indicated as being identical to each other, > there is precisely no justification for assuming that any > items not _explicitly_ indicated as being identical to each > other would so be. But it didn't say they weren't identical. is not a counter > argument. And I'm not using that argument. > It didn't say that the boxes aren't rotated by > one in order after the placement (thus making box 20 out of > bounds for an 'A', and changing the answer). It didn't say > that the boxes are too small for the objects, making the > answer 0. Your assumtion that all things unless otherwise stated is > one that is counterindicated by the wording of the problem. Take a tall graduated cylinder. Into that cylinder drop 20 red marbles (whose diameter matches the I.D. of the cylinder). Now drop in 80 blue marbles. How many different possible outcomes are there? Answer: 1. All the red marbles are identical. All the blue marbles are identical. The red marbles are not identicle to the blue marbles. The slots (vertical position within the cylinder) are distinguishable. The red marbles are in the first 20 slots. The blue marbles are in the rest of the slots. Now, in what way does this model violate the original question? > And to the OP - once the 'A' items are placed, how many ways > are there to fill all the remaining slots with identical > balls? Can you describe 2 such ways, for example? Phil > -- > Home taping is killing big business profits. We left this side blank > so you can help. -- Dead Kennedys, written upon the B-side of tapes of > /In God We Trust, Inc./. === Subject: Re: Masiv combinatoric *Please note that Masiv is a incredible typo of Basic. === Subject: shift columns > statdat<- rnorm(1000, mean=0, sd=1) > dat.matrix <- matrix (data=statdat, ncol=4) > dat.matrix I need to shift the 1st set of 250 rows all of the cols by 3, and then shift the 4th 250 rows all of the cols by -3. What is the function for this? thx in advance. === Subject: Re: Review of Lee Smolin's THE TROUBLE WITH PHYSICS Here is my succinct interpretation of the author's major complaints. The field is full of phonies who teach physical science. They often act like they understand what reality is, no matter. The possible dimensions of reality are infinte, damne 'em all to helle forever. Einstein couldn't tie his shoe w/o Mom's help until 8 or 9 or 10 (like me). Werner von Braun utilized slave-labor Oppy wore a porkpie hat while building the bomb, rather than a protective pithy helmet. Jeanne Dixon predicted JFK's fate: She is one of the better ones, I suppose. > http://physicsmathforums.com/ The first step to recovery is admitting that there is a problem. Houston, we have a problem. Pomo-hipster String Theory. Books like Smolin's give me faith that Reason, Science, and Physics > will again be celebrated in our lifetimes. Smolin provides an excellent review of the Standard Model, and a > well-balanced overview of where we can go from here. 'Tis the dawn of a new era. This book, together with Woit's Not Even Wrong signifies the beginning > of the end of a dark, postmodern age in physics. Postmodern philosophies brought us all the Enrons, all the plagiarism > scandals that plague the literary industry, and string theory. All were > all based on the same pomo-hipster trick--an elite group of snarky > insiders deconstructing classical ideals and instituting secret > handhsakes and jumbled jargon in their stead, corrupting those very > entities that supported them. Truth, beauty, and physics were sacrificed--indeed--today string > theorists would rather get rid of science than admit that string > theology has not only failed, but it has created a dependent class of > thousands of physicists who depend on string theory hoaxes for salaries > and benefits. Search nsf.org and arxiv.org for String, or Strings, or Stringy, > and you will see the sad, embarrassing results of massive egoes gone > wild. Self-referential, meaningless paper after self-referential, > meaningless paper--all penned by those with ambitions overshadowning > their talents. And no physics in sight--indeed, physics became the > cause of expulsion from the physics department. Logic, reason, and Reality must be held over all else in physics. For physics is not math--it is physics. It is not politics--it is physics. It is not jargon nor cutesy jokes--it is physics. Einstein, Bohr, Wheeler, Feynman, and Dirac were all very humble > men--humble before Reality. We would be wise to follow their lead, to not deconstruct these giants > and all they stood for, but to stand upon their shoulders so as to see > further. Physics must first and foremost be rooted in physical reality. Now and > forever. Hopefully LQG and Moving Dimensions Theory can pave the way to a brand > new day where one is not exlcuded from scientific conferences and > funding because of one's love of logic, reason, and science. http://physicsmathforums.com/ === Subject: Re: Review of Lee Smolin's THE TROUBLE WITH PHYSICS http://physicsmathforums.com/showthread.php?t=1174 Moving Dimensions Theory T-shirts! Fundraiser to hire postdocs!! A great research project for grad students and postdocs would be to show how both the space-time metric and the Schr.9adinger equation arise from the first postulate of MDT: The Fourth Dimension is expanding relative to the Three Spatial Dimensions. While String Theory is absorbing the tens of millions of NSF funds and the accompanying billions of local tax and tuition dollars: http://nsf.gov/awardsearch/progSearch.do;jsessionid=B6AE5664DB11367AE... The best way to raise funds for alternative theories is t-shirts: http://www.cafepress.com/autumnrangers.72464949 If at first the idea is not absurd, then there is no hope for it. -Albert Einstein Every great advance in science has issued from a new audacity of the imagination. -John Dewey, The Quest For Certainty There ain't no rules around here! We're trying to accomplish something! -Thomas Alva Edison (1847-1931) U. S. inventor. All religions, arts and sciences are branches of the same tree. All these aspirations are directed toward ennobling man's life, lifting it from the sphere of mere physical existence and leading the individual towards freedom. -Albert Einstein http://physicsmathforums.com/showthread.php?t=1174 FREEEEEEEEEEEEEEEEEEDOOOOOOOOOOOM!!!!!!-William Wallace in Braveheart INTRODUCTION In this day and age of unprecedented hype, hand-waving, and hoaxes in theoretical physics, Dynamic Dimensions Theory rocks the universe with a simple postulate. The object of science is to unify formerly disparate phenomena within a deeper framework, and Dynamic Dimensions Theory unifies both Quantum Mechanics and Relativity with a simple framework. For instance, both the results of the double-slit experiment and length contraction of relativity are suddenly understood in the profound context of Dynamic Dimensions Theory. The famous equation E=mc2 is presented in a deeper context, finally accounting for the deeper physical reality that gives massive energies to small masses seemingly at rest. Godel's paradox of block-time is resolved, and action at a distance and quantum entanglement are explained within the simple framework of Dynamic Dimensions Theory. The arrow of time and entropy-the second law of thermodynamics-are accounted for, and time dialation is explained. Dynamic dimensions theory is a theory so revolutionary that it may be banned from the academy for years, as health benefits and tenure are far more important than truth, logic, and reason in today's postmodern academies. And at last time is explained as a phenomenon that emerges from a deeper physical reality. THE GENERAL POSTULATE The fourth dimension is expanding relative to the three spatial dimensions. THE SPECIFIC POSTULATE The fourth dimension is expanding relative to the three spatial dimensions at the rate of c in units of the Planck length, giving rise to time and all classical, quantum mechanical, and relativistic phenomena. MDT IN BRIEF Without further adieu, allow me to present the beauty and elegance of MDT by showing both its simplicity and far-reaching ability to account for and answer fundamental questions. All of the below will be elaborated on throughout the book. Questions Addressed by MDT: Why does light have a maximum, constant speed independent of the source? The fourth dimension is expanding relative to the three spatial dimensions. A photon is momenergy that exists orthogonal to the three spatial dimensions. It is carried along by the expanding fourth dimension. So no matter how fast the source is moving when the photon is emitted, the photon travels at the rate with which the fourth dimension is expanding relative to the three spatial dimensions. Thus c is always independent of the movement of the source. Why are light and energy quantized? The fourth dimension is expanding in a quantized manner relative to the three spatial dimensions. Light and energy are matter rotated completely into the fourth expanding dimension, and as it expands in a quantized manner, light and energy are thus quantized. Why is the velocity of light constant in all frames? Time is an emergent phenomena that arises because the fourth dimension is expanding relative to the three spatial dimensions. The flow of time is inextricably wed to the emission and propagation of photons. In all biological, mechanical, and electronic clocks, the emission and propagation of photons is what determines time. The velocity of light is always measured with respect to time, which is inextricably linked to the velocity of light. This tautology ensures that the velocity of light, measured relative to the velocity of light, will always be the same. fundamental photon propagates as a spherical wave-front, surfing the fourth expanding dimension. This is because the fourth expanding dimension appears as a spherical wavefront as it expands through the three spatial dimensions. The act of measurement localizes the photon's momenergy, taking it out of the expanding fourth dimension and trapping it in the three stationary spatial dimensions, and it grain on a photographic plate. fundamental electron is abuzz with photons. Photons are continually being emitted into the fourth expanding dimension and reabsorbed by the electron. The continual dance with these photons gives the electron its wave properties. Nothing moves without photons which up the net probability that the combine momenergy will be in the expanding fourth dimension. The more photons one adds to an object, the greater the chance it has of existing in the expanding fourth dimension, and thus it moves. Why are there non-local effects in quantum mechanics? The fourth dimension is expanding relative to the three spatial dimensions. That means that what begins as a point in the fourth dimension is a sphere with a 186,000 mile radius one second later. So it is that the entire spherical wavefront of the photon exists in the exact same place in time. Hence the non-locality observed in double slit experiments, the EPR effect, and quantum entanglement. Take two interacting spin ¸ photons and let them propagate at the speed of c in opposite directions. They are yet at the exact same place in time! And too, they are yet in the exact same place of the fourth expanding dimension. Why does time stop at the speed of light? Time depends on the emission and propagation of photons. If no photons are emitted, time does not occur. This holds true whether the clock is an unwinding copper spring, a biological system such as a heart, or an oscillating quartz crystal. No photom emission=no time! As an object approaches the speed of light, its ability to emit photons without reabsorbing them diminishes. An object traveling at the speed of light cannot emit a photon. How come a photon does not age? A photon represents momenergy rotated entirely into the fourth expanding dimension. A photon stays the exact same place in the fourth dimension, no matter how far it travels. A photon stays the exact same place in time, no matter how far it travels. Again, time is not the fourth dimension, but in inherits properties of the fourth dimension. Why are inertial mass and gravitational mass the same thing? Why do moving bodies exhibit length contraction? Movement is always accompanied by a shortening in length. This is because the only way for a body to move is for it to undergo a rotation into the forth dimension, which is expanding relative to the three spatial dimensions. The more energy an electron has, the more photons it possesses, and the higher probability it exists in the expanding fourth dimension. Hence its length appears contracted as perceived from the three spatial dimensions. Why are mass and energy equivalent? The fourth dimension is expanding relative to the three spatial dimensions. That means that a baseball sitting on a lab table stationary in our three-dimensional inertial reference frame, is yet moving at a fantastic velocity relative to the fourth dimension. Hence every seemingly stationary mass has a vast energy, as given by E=mc2. In a nuclear reaction matter is rotated into the expanding fourth dimension, appearing as high-enegry photons (gamma rays) propagating at the same velocity of the fourth expanding dimension-c. Why does time's arrow point in the direction it points in? The fourth dimension is expanding relative to the three spatial dimensions. Hence every photon naturally expands in a spherically symmetric manner. Hence every electron, or piece of matter that interacts with photons, is naturally carried outward from a central point in a spherically pool dissipate in a spherically symmetric manner, and are never reunited. Hence time's arrow and entropy. Why do photons appear as spherically-symmetric wavefronts traveling at a velocity c? The fourth dimension is expanding relative to the three spatial dimensions at the velocity c. Hence photons, which are tiny packets of momenergy rotated entirely into the fourth dimension, appear as spherically-symmetric wavefronts propagating at the velocity c. Why is there a minus sign in the following metric? x^2+y^2+z^2-c^2t^2=s^2 The fourth dimension is expanding relative to the three spatial dimensions at the velocity c. Hence the only way to stay still in the space-time continuum, and to achieve a 0 interval, is to move with the velocity of light. What deeper reality underlies Einstein's postulates of relativity? The fourth dimension is expanding relative to the three spatial dimensions at the velocity c. This single postulate assures that the speed of light is constant for all observers and that the laws of physics are the same in all inertial frames. What deeper reality underlies Newton's laws? Newton's laws are an approximation of relativity and quantum mechanics, and as MDT underlies QM & relativity, it underlies Newton's laws. Why is an increase in velocity always accompanied by a decrease in length as measured by an external observer? All increases in velocity are accompanied by rotations into the fourth dimension. All momenrgy component in the expanding fourth dimension, the greater the It never changes. It prefers the three spatial dimensions. In order for it to move, one must gain energy in the form of photons. These photons prefer the fourth expanding dimension. The more photons one adds, the greater the component of the momenergy 4-vector that appears the shorter it appears, and the faster it moves. How MDT Is Aiding Fellow Physicists The conclusions from Bell's theorem are philosophically startling; either one must totally abandon the realistic philosophy of most working scientists or dramatically revise our concept of space-time. -Abner Shimony and John Clauser Moving Dimensions Theory provides this new concept of space-time. The vast ambitions of most tenure-track physicists, including string theorists and LQG hypers, causes them to focus on irrelevant, minute questions, and thus, though funded by millions for over thirty years, have not yet been able to string the bow. Deeper, true physicists, such as Abner Shimony and John Clauser are alert to the fact that physics need news ideas. The expanding fourth dimension gives rise to non-local phenomena and quantum entanglement, as the expanding fourth dimension means that two events separated in the three spatial dimensions can yet appear to be at the exact same place in the fourth dimension. MDT thus provides the new concept of space-time. For me, then, this is the real problem with quantum theory: the apparently essential conflict between any sharp formulation and fundamental relativity. It may be that a real synthesis of quantum and relativity theories requires not just technical developments but radical conceptual renewal. -John Bell Moving Dimensions Theory provides this radical conceptual renewal. The expanding fourth dimension gives rise to non-local phenomena and quantum entanglement, as the expanding fourth dimension means that two events separated in the three spatial dimensions can yet appear to be at the exact same place in the fourth dimension. MDT thus provides the new concept of space-time. Entanglement is not one but rather the characteristic trait of quantum mechanics. -Erwin Schrodinger The expanding fourth dimension gives rise to non-local phenomena and quantum entanglement, as the expanding fourth dimension means that two events separated in the three spatial dimensions can yet be at the exact same place in the fourth dimension. MDT thus provides the new concept of space-time. The discovery of the quantum of action shows us not only the natural limitation of classical physics, but, by throwing a new light upon the old philsophical problem of the objective existence of phenomena indepedently of our observations, confronts us with a situation hitherto unknown in natural science. -Niels Bohr The expanding fourth dimension gives rise to non-local phenomena and quantum entanglement, as the expanding fourth dimension means that two events separated in the three spatial dimensions can yet appear to be at the exact same place in the fourth dimension. MDT thus provides the new concept of space-time. I think we need a new way to look at time, not either Quantum Mechanics or Relativity. -Roger Penrose Time is an emergent property of a fourth dimension that is expanding relative to the three spatial dimensions. Thus it inherits properties of the fourth expanding dimension, but time is not the fourth expanding dimension. MDT underlies time and explains time on a more fundamental level. Should we be prepared to see some day a new structure for the foundations of physics that does away with time? . . . Yes, because 'time' is in trouble. -John Wheeler Time is an emergent property of a fourth dimension that is expanding relative to the three spatial dimensions. Thus it inherits properties of the fourth expanding dimension, but time is not the fourth expanding dimension. MDT underlies time and explains time on a more fundamental level. Time is clothed in a different garment for each role it plays in our thinking. -John Wheeler Time is an emergent property of a fourth dimension that is expanding relative to the three spatial dimensions. Thus it inherits properties of the fourth expanding dimension, but time is not the fourth expanding dimension. MDT underlies time and explains time on a more fundamental level. The word time came not from heaven but form the mouth of man. -John Wheeler Time is an emergent property of a fourth dimension that is expanding relative to the three spatial dimensions. Thus it inherits properties of the fourth expanding dimension, but time is not the fourth expanding dimension. MDT underlies time and explains time on a more fundamental level. My ideas about time all developed from the realization that if nothing were to change we could not say that times passes. Change is primary, time, if it exists at all, is something we deduce from it. My Italian collaborator Bruno Bertotti and I found that the deep structure of Einstein's general theory of relativity does correspond to this truth. It is telling us that time does not exist as an independent thing and that change is indeed primary. However, this is in the framework of so-called classical physics, the form of physics that developed before quantum mechanics was discovered. When the idea that time has no independent existence is combined with the basic facts of quantum mechanics in the simplest possible way, the implications are startling. . .The quantum universe is static. Only timeless Nows exist. The quantum rules give them different probabilities. We experience the most probable Nows as individual instants of time. The appearance of motion and a flow of time are both illusions created by very special structure of the instants that we experience. -Julian Barbour, http://www.platonia.com/ideas.html Time is an emergent property of a fourth dimension that is expanding relative to the three spatial dimensions. Thus it inherits properties of the fourth expanding dimension, but time is not the fourth expanding dimension. MDT underlies time and explains time on a more fundamental level. The mystery of time's arrow is the oldest problem in science concerning the nature of time, predating even the theory of relativity. -Paul Davies, About Time MDT accounts for Time's Arrow: The fourth dimension is expanding relative to the three spatial dimensions. Photons are momenergy rotated entirely into this expanding fourth dimension. Hence any group of photons originating from a central point will be found distant to one-another. Hence every photon naturally expands in a spherically symmetric manner. Hence every electron, or piece of matter, is naturally carried outward from a central point in a spherically symmetric manner. Hence a drop of dye in a swimming pool dissipates in a spherically symmetric manner, and is never reunited. Hence time's arrow and entropy. Moving Dimensions Theory & On The Advancement Of Physics Physics has been furthered far more often by a rugged individual acknowledging the simple and obvious in a pursuit of the truth than book-keepers-in-training playing games in the abstruse in pursuit of tenure. The advancement of physics has ever depended far more on logic, reason, and Truth than government grants, tenure, group think, peer-reviewed journals, and aging bureaucracies. That is the way things are because that is the way things are, has lead to far more physics than the contemporary, things can't be that way because the math dictates that we live in thirty-three dimensions and four are curled up, and that is what NSF is funding. When nobody could measure nor detect the supposed ether, Einstein proclaimed, there is no ether. When experiments showed that light existed only in quantized packets, Einstein proclaimed that light only existed in quantized packets, and he won the Nobel Prize. When spectra from atoms showed discreet energies, Niels Bohr proclaimed that electrons orbits were quantized, and he received a Nobel Prize. When Maxwell's Equations had a recurring constant, Maxwell used c to denote it, and Einstein proclaimed that the speed of light must be constant for all observers-and so Special Relativity was born. When Einstein juxtaposed objects falling towards the earth getting closer together with the fact that two people starting at the equator, walking on originally parallel lines of longitude towards the North Pole, would come together because they were walking on a curve surface, Einstein proclaimed that the space-time around a massive object must also be curved. This along with Einstein's realization that the force of gravity would be rendered null in free-fall, lead to General Relativity. And so it is that in the above paragraph you have the roots of the greatest achievements of physics in the past 100+ years, dwarfing String Theory, Loop Quantum Gravity, and thousands of their variatons, which deal in the abstruse, complicated, muddled, and mythological worlds which are safe from physics simple rigor. > Here is my succinct interpretation > of the author's major complaints. The field is full of phonies who teach physical science. > They often act like they understand what reality is, no matter. > The possible dimensions of reality are infinte, damne 'em all to helle > forever. > Einstein couldn't tie his shoe w/o Mom's help until 8 or 9 or 10 (like > me). > Werner von Braun utilized slave-labor > Oppy wore a porkpie hat while building the bomb, rather than a > protective pithy helmet. > Jeanne Dixon predicted JFK's fate: She is one of the better ones, I > suppose. > http://physicsmathforums.com/ The first step to recovery is admitting that there is a problem. Houston, we have a problem. Pomo-hipster String Theory. Books like Smolin's give me faith that Reason, Science, and Physics > will again be celebrated in our lifetimes. Smolin provides an excellent review of the Standard Model, and a > well-balanced overview of where we can go from here. 'Tis the dawn of a new era. This book, together with Woit's Not Even Wrong signifies the beginning > of the end of a dark, postmodern age in physics. Postmodern philosophies brought us all the Enrons, all the plagiarism > scandals that plague the literary industry, and string theory. All were > all based on the same pomo-hipster trick--an elite group of snarky > insiders deconstructing classical ideals and instituting secret > handhsakes and jumbled jargon in their stead, corrupting those very > entities that supported them. Truth, beauty, and physics were sacrificed--indeed--today string > theorists would rather get rid of science than admit that string > theology has not only failed, but it has created a dependent class of > thousands of physicists who depend on string theory hoaxes for salaries > and benefits. Search nsf.org and arxiv.org for String, or Strings, or Stringy, > and you will see the sad, embarrassing results of massive egoes gone > wild. Self-referential, meaningless paper after self-referential, > meaningless paper--all penned by those with ambitions overshadowning > their talents. And no physics in sight--indeed, physics became the > cause of expulsion from the physics department. Logic, reason, and Reality must be held over all else in physics. For physics is not math--it is physics. It is not politics--it is physics. It is not jargon nor cutesy jokes--it is physics. Einstein, Bohr, Wheeler, Feynman, and Dirac were all very humble > men--humble before Reality. We would be wise to follow their lead, to not deconstruct these giants > and all they stood for, but to stand upon their shoulders so as to see > further. Physics must first and foremost be rooted in physical reality. Now and > forever. Hopefully LQG and Moving Dimensions Theory can pave the way to a brand > new day where one is not exlcuded from scientific conferences and > funding because of one's love of logic, reason, and science. http://physicsmathforums.com/ http://physicsmathforums.com/showthread.php?t=1174 === Subject: Re: Review of Lee Smolin's THE TROUBLE WITH PHYSICS > >> http://physicsmathforums.com/ >> The first step to recovery is admitting that there is a problem. >> Houston, we have a problem. Pomo-hipster String Theory. >> Books like Smolin's give me faith that Reason, Science, and Physics >> will again be celebrated in our lifetimes. >> Smolin provides an excellent review of the Standard Model, and a >> well-balanced overview of where we can go from here. >> 'Tis the dawn of a new era. >> This book, together with Woit's Not Even Wrong signifies the beginning >> of the end of a dark, postmodern age in physics. >> Postmodern philosophies brought us all the Enrons, Name these Postmodern philosophies. > >> all the plagiarism >> scandals that plague the literary industry, and string theory. All were >> all based on the same pomo-hipster trick--an elite group of snarky >> insiders deconstructing classical ideals and instituting secret >> handhsakes and jumbled jargon in their stead, corrupting those very >> entities that supported them. >> Truth, beauty, and physics were sacrificed--indeed--today string What is truth? Define physics. > >> theorists would rather get rid of science than admit that string Define science. > >> theology has not only failed, but it has created a dependent class of >> thousands of physicists who depend on string theory hoaxes for salaries >> and benefits. >> Search nsf.org and arxiv.org for String, or Strings, or Stringy, >> and you will see the sad, embarrassing results of massive egoes gone >> wild. Self-referential, meaningless paper after self-referential, >> meaningless paper--all penned by those with ambitions overshadowning >> their talents. And no physics in sight--indeed, physics became the >> cause of expulsion from the physics department. >> Logic, reason, and Reality must be held over all else in physics. >> For physics is not math--it is physics. What's the exact difference between physics and math? Could there be a > physics without math? > >> It is not politics--it is physics. >> It is not jargon nor cutesy jokes--it is physics. >> Einstein, Bohr, Wheeler, Feynman, and Dirac were all very humble >> men--humble before Reality. >> We would be wise to follow their lead, to not deconstruct these giants >> and all they stood for, but to stand upon their shoulders so as to see >> further. >> Physics must first and foremost be rooted in physical reality. Now and >> forever. Define physical reality, Mr. self proclaimed expert. SD: Physical reality? It's not WHAT , but who defines physical reality. Here's how it works, palsy. You and me , let's say, we're standing on a cliff. A big one. You jump off -- I keep my feet planted. A few seconds later the person who's alive gets to define physical reality. Repeat pairwise: with one smarmy butthead and one mechanic. Reality gets defined pretty quick then, cause for the first pair-of-mechanics gets paired ... and thereafter, neither jumps. Instead they negotiate a reality, and trot off for a beer. Any questions palsy? nss ********** === Subject: Re: Linear Algebra 2 [...] The problem was: Let P(F) be the space of all polynomials with coefficients in field F; let U be the subset of P(F) consisting of all polynomials of the form p(z) = az^5 + bz^2. Show U is a subspace and find a subspace W such that P(F) = U + W. (Presumably, the sum is direct; did the + have a circle around it?). > I actually worked this one out and defined W in general as the subspace > of P(F)consisting of polynomials p of degree n not equal to 2 or 5. > but z^6 + z^5 + (-x^6) has degree 5. Your W isn't a subspace. I. Describe W precisely. II. Show W is a subspace i. Show that the zero polynomial belongs to W; ii. Show that if p(z) and q(z) belong to W then so does p(z) + q(z); iii. Show that if p(z) is in W and r is in F then rp(z) is in W. [You must do all 3 in excruciating detail using I.] III. Show that U + W = P(F). [Show that if q(z) is in P(F) and p(z) is in U then q(z) - p(z) is in W.] IV. Show that U and W have only the zero polynomial in common. [Or else W = P(F) works.] -- Paul Sperry Columbia, SC (USA) === Subject: Re: Randomness debate, some ground rules > My credibility is not an issue. Because you have none, except for being a crackpot. Math people have already marked me as a crackpot across the web. Hmm, I wonder why you're marked as a crackpot. LEARN SOME MATH!