mm-358 === Subject: : Poker Odds question Let me start by saying that i'm *not* a math guy (but I am very interested in learning more about it), but I've been trying to [CapitalThorn]gure this out for quite some time now and, after endless googling, I've yet to [CapitalThorn]nd the answer. My question is this... Most hold'em players know that, before the ßop, a middle pair (6's or 8's, for example) is a 55% favorite over two overcards (A K, etc.) at 45%. If the overcards are suited, the odds are almost exactly 50/50. Now, the number of outs for the overcards before the ßop is 6, which means that out of 48 unseen cards, the overcards have a 12.5% chance of pairing either card. Can somebody explain to me how this works? Again, apologies if this isn't the appropriate forum for a question like this, but I'd have to believe this is the kind of thing you guys eat for breakfast :) Brett === Subject: : Re: Poker Odds question > Let me start by saying that i'm *not* a math guy (but I am very > interested in learning more about it), but I've been trying to [CapitalThorn]gure > this out for quite some time now and, after endless googling, I've yet > to [CapitalThorn]nd the answer. My question is this... > Most hold'em players know that, before the ßop, a middle pair (6's or > 8's, for example) is a 55% favorite over two overcards (A K, etc.) at > 45%. If the overcards are suited, the odds are almost exactly 50/50. > Now, the number of outs for the overcards before the ßop is 6, which > means that out of 48 unseen cards, the overcards have a 12.5% chance of > pairing either card. > Can somebody explain to me how this works? Two resources you may be interested in is the newgroup rec.gambling.poker and the website http://www.twodimes.net Basically, with the overcards you are hoping to match one up for a high pair, straight (if the ranks are close enough together), or a ßush (if suited). With the pair, you are hoping for no help with the [CapitalThorn]ve community cards for either side (pair beats high card), trips, or a full house. So for example, lets say that you hold a J-9 offsuit. You can win a J or a 9 comes up (6 outs), but you can also win if a T comes up along with one of the three combinations KQ, Q8, or 87 (more outs). Plus if four of the [CapitalThorn]ve cards match the suit of the J (more outs) or the 9 (more outs), you win. This is of course assuming their hand does not improve. Because of the variety of possibilities for the over cards (against pocket 6s a JT offsuit is stronger than AQ offsuited, but weaker than AQ suited), this is called a coinßip since the odds are approximately 50:50 and the slight advantage of the pair is offset by the fact that there is only one trial. Saint Cad === Subject: : Re: Poker Odds question > Let me start by saying that i'm *not* a math guy (but I am very > interested in learning more about it), but I've been trying to [CapitalThorn]gure > this out for quite some time now and, after endless googling, I've yet > to [CapitalThorn]nd the answer. My question is this... > Most hold'em players know that, before the ßop, a middle pair (6's or > 8's, for example) is a 55% favorite over two overcards (A K, etc.) at > 45%. If the overcards are suited, the odds are almost exactly 50/50. > Now, the number of outs for the overcards before the ßop is 6, which > means that out of 48 unseen cards, the overcards have a 12.5% chance of > pairing either card. > Can somebody explain to me how this works? > Again, apologies if this isn't the appropriate forum for a question > like this, but I'd have to believe this is the kind of thing you guys > eat for breakfast :) If you have a pair already, you have better odds of trips, another pair, or a full house than if you don't have a pair. However, if you hold any two cards in the same suit, you also have increased chances for a ßush, which is a relatively strong hand. === Subject: : Re: Connect 4 solved For a 15 year old, you sure do have a mouth on you, which shows your > immaturity. Dave > kids my age cuss. it's a fact of life. were you never a kid? let's > not ignore you just said that FOR a 15 year old i sure have a mouth on > me. implying i have more of a mouth on me than a normal 15 year old. > i haven't been shackled down by social oppressions yet so i'm eager to > speak a bit more than my mind from time to time. do the people you > associate with know that you have no en clue how to deal with > kids? that's a sign of maturity you know; putting kids in their > place.. dealing with them. i see you have no skills in that > department you old goat. you're a miser. you eat out the > assholes of your one-night-stands and save their in a jar to > sniff and wiff and lick later to your heart's content. > Excuse me but as a meteorologist, I get to go talk to kids about your age > about my job and what I do at work. These kids have even come up to me at > stores, restaurants, etc. and talked to me even more because they know that > I will talk to them. None of these people have ever said anything bad to my > face which shows their maturity because I am physically disabled and while > others their age might make rude comments to my face, those that have seen > one of my presentations know that though I am disabled, I am not stupid. > Davre ok so you found some stupid kids in some backwater town that look up to you. to them there's no difference between you and einstein. but in the real world. smart kids (like me) know that a meteorologist is one of those low-form scientists. totally scared less of pure mathematics and theoretical physics. as a kid i'm con[CapitalThorn]dent i know more about mathematics or physics or computer programming or pretty much any scienti[CapitalThorn]c subject save meteorology than you. how does that make you feel? my brother is almost as young as me and he knows a lot more! === Subject: : Re: Connect 4 solved For a 15 year old, you sure do have a mouth on you, which shows your > immaturity. Dave kids my age cuss. it's a fact of life. were you never a kid? let's > not ignore you just said that FOR a 15 year old i sure have a mouth on > me. implying i have more of a mouth on me than a normal 15 year old. > i haven't been shackled down by social oppressions yet so i'm eager to > speak a bit more than my mind from time to time. do the people you > associate with know that you have no en clue how to deal with > kids? that's a sign of maturity you know; putting kids in their > place.. dealing with them. i see you have no skills in that > department you old goat. you're a miser. you eat out the > assholes of your one-night-stands and save their in a jar to > sniff and wiff and lick later to your heart's content. > Excuse me but as a meteorologist, I get to go talk to kids about your age > about my job and what I do at work. These kids have even come up to me at > stores, restaurants, etc. and talked to me even more because they know that > I will talk to them. None of these people have ever said anything bad to my > face which shows their maturity because I am physically disabled and while > others their age might make rude comments to my face, those that have seen > one of my presentations know that though I am disabled, I am not stupid. > Davre > ok so you found some stupid kids in some backwater town that look up > to you. to them there's no difference between you and einstein. but > in the real world. smart kids (like me) know that a meteorologist is > one of those low-form scientists. totally scared less of pure > mathematics and theoretical physics. as a kid i'm con[CapitalThorn]dent i know > more about mathematics or physics or computer programming or pretty > much any scienti[CapitalThorn]c subject save meteorology than you. how does that > make you feel? my brother is almost as young as me and he knows a lot > more! That could be true. I remember seeing a math test that one of our former af[CapitalThorn]liates used to give anyone wanting to apply. If you'd like to see it, let me know. I don't know if I still have a copy of it, but if not, I can give you a few of the sample problems. Dave === Subject: : Re: Connect 4 solved For a 15 year old, you sure do have a mouth on you, which shows your > immaturity. Dave kids my age cuss. it's a fact of life. were you never a kid? let's > not ignore you just said that FOR a 15 year old i sure have a mouth on > me. implying i have more of a mouth on me than a normal 15 year old. > i haven't been shackled down by social oppressions yet so i'm eager to > speak a bit more than my mind from time to time. do the people you > associate with know that you have no en clue how to deal with > kids? that's a sign of maturity you know; putting kids in their > place.. dealing with them. i see you have no skills in that > department you old goat. you're a miser. you eat out the > assholes of your one-night-stands and save their in a jar to > sniff and wiff and lick later to your heart's content. Excuse me but as a meteorologist, I get to go talk to kids about your > age > about my job and what I do at work. These kids have even come up to me > at > stores, restaurants, etc. and talked to me even more because they know > that > I will talk to them. None of these people have ever said anything bad to > my > face which shows their maturity because I am physically disabled and > while > others their age might make rude comments to my face, those that have > seen > one of my presentations know that though I am disabled, I am not stupid. Davre > ok so you found some stupid kids in some backwater town that look up > to you. to them there's no difference between you and einstein. but > in the real world. smart kids (like me) know that a meteorologist is > one of those low-form scientists. totally scared less of pure > mathematics and theoretical physics. as a kid i'm con[CapitalThorn]dent i know > more about mathematics or physics or computer programming or pretty > much any scienti[CapitalThorn]c subject save meteorology than you. how does that > make you feel? my brother is almost as young as me and he knows a lot > more! > That could be true. I remember seeing a math test that one of our former > af[CapitalThorn]liates used to give anyone wanting to apply. If you'd like to see it, > let me know. I don't know if I still have a copy of it, but if not, I can > give you a few of the sample problems. > Dave yes i want to see what kind of math test's score would gauge the aptitude someone for this science. and i warn you, if it's just analysis, you should hardly bother i will have answers for you instantly. === Subject: : Re: Connect 4 solved For a 15 year old, you sure do have a mouth on you, which shows your > immaturity. Dave kids my age cuss. it's a fact of life. were you never a kid? let's > not ignore you just said that FOR a 15 year old i sure have a mouth on > me. implying i have more of a mouth on me than a normal 15 year old. > i haven't been shackled down by social oppressions yet so i'm eager to > speak a bit more than my mind from time to time. do the people you > associate with know that you have no en clue how to deal with > kids? that's a sign of maturity you know; putting kids in their > place.. dealing with them. i see you have no skills in that > department you old goat. you're a miser. you eat out the > assholes of your one-night-stands and save their in a jar to > sniff and wiff and lick later to your heart's content. Excuse me but as a meteorologist, I get to go talk to kids about your > age > about my job and what I do at work. These kids have even come up to me > at > stores, restaurants, etc. and talked to me even more because they know > that > I will talk to them. None of these people have ever said anything bad to > my > face which shows their maturity because I am physically disabled and > while > others their age might make rude comments to my face, those that have > seen > one of my presentations know that though I am disabled, I am not stupid. Davre ok so you found some stupid kids in some backwater town that look up > to you. to them there's no difference between you and einstein. but > in the real world. smart kids (like me) know that a meteorologist is > one of those low-form scientists. totally scared less of pure > mathematics and theoretical physics. as a kid i'm con[CapitalThorn]dent i know > more about mathematics or physics or computer programming or pretty > much any scienti[CapitalThorn]c subject save meteorology than you. how does that > make you feel? my brother is almost as young as me and he knows a lot > more! > That could be true. I remember seeing a math test that one of our former > af[CapitalThorn]liates used to give anyone wanting to apply. If you'd like to see it, > let me know. I don't know if I still have a copy of it, but if not, I can > give you a few of the sample problems. > Dave > yes i want to see what kind of math test's score would gauge the > aptitude someone for this science. and i warn you, if it's just > analysis, you should hardly bother i will have answers for you > instantly. I don't remember the exact problems, but here's a few similar ones. 1. You have 1000 square centimeters of material to make a cylindrical container. Find the dimensions of the cylinder that maximizes the volume and the optimal volume. 2. A hemispheric tank is full of water. The tank has a top radius of 4 feet and a hole in the bottom 1 inch in diameter. How long does it take the tank to empty? 3.Assuming a gas behaves ideally, [CapitalThorn]nd the rate of change of the volume of one mole of gas if the pressure of the gas increases at a rate of .05 kPa/sec and the temperature increases at .15 K/s. Granted, these aren't too terribly dif[CapitalThorn]fcult, but I'd expect anyone that I was considering hiring to be able to do these. Let me know if you want my solutions. Dave === Subject: : Re: Poker Odds question Right, but I'm unable to reconcile the fact that a pair is only a 2-4% favorite over two overcards. A more general question I've been struggling with is how odds are calculated using tools like this one -> http://www.cardplayer.com/poker_odds/ I can't imagine the math is all that diffucult, but all I've been able to [CapitalThorn]nd anything other than things like the odds of hitting a third jack, etc. as opposed to the respective odds of two or more hands against one another. > Let me start by saying that i'm *not* a math guy (but I am very > interested in learning more about it), but I've been trying to [CapitalThorn]gure > this out for quite some time now and, after endless googling, I've yet > to [CapitalThorn]nd the answer. My question is this... > Most hold'em players know that, before the ßop, a middle pair (6's or > 8's, for example) is a 55% favorite over two overcards (A K, etc.) at > 45%. If the overcards are suited, the odds are almost exactly 50/50. > Now, the number of outs for the overcards before the ßop is 6, which > means that out of 48 unseen cards, the overcards have a 12.5% chance of > pairing either card. > Can somebody explain to me how this works? > Again, apologies if this isn't the appropriate forum for a question > like this, but I'd have to believe this is the kind of thing you guys > eat for breakfast :) > If you have a pair already, you have better odds of trips, another pair, or > a full house than if you don't have a pair. However, if you hold any two > cards in the same suit, you also have increased chances for a ßush, which > is a relatively strong hand. === Subject: : Re: Poker Odds question > Right, but I'm unable to reconcile the fact that a pair is only a 2-4% > favorite over two overcards. Actually, in general, it is not. If the two overcards have good straight or ßush possibilities than the odds tend to be closer. But if they if they do not then odds favor the pair. a pair a 6s from 2 other suits. But a 9/10 of spades has a 51% chance of winning vs. the same pair. To get a little sense of this, ask what is the probability of no aces or 7s showing up in 5 cards. That would be ((48-6)/48)^5 = .51. So the chance of an ace or a 7 coming up is 49%. But that is not the chance of the holder of Ace/7 winning because another 6 could come up. So you have to subtract the odds of a third 6 coming up given at least Ace/7 came up. But there could be 2 aces or 2 7s to beat the 3 6s. Etc. Also you have to [CapitalThorn]gure in straights and ßushes. Bill === Subject: : Re: Poker Odds question > Right, but I'm unable to reconcile the fact that a pair is only a 2-4% > favorite over two overcards. > Actually, in general, it is not. If the two overcards have good straight or > ßush possibilities than the odds tend to be closer. But if they if they do > not then odds favor the pair. vs. > a pair a 6s from 2 other suits. But a 9/10 of spades has a 51% chance of > winning vs. the same pair. > To get a little sense of this, ask what is the probability of no aces or 7s > showing up in 5 cards. That would be ((48-6)/48)^5 = .51. Should have said approximately. Bill > So the chance of an > ace or a 7 coming up is 49%. > But that is not the chance of the holder of Ace/7 winning because another 6 > could come up. So you have to subtract the odds of a third 6 coming up given > at least Ace/7 came up. But there could be 2 aces or 2 7s to beat the 3 6s. > Etc. Also you have to [CapitalThorn]gure in straights and ßushes. > Bill === Subject: : Re: Poker Odds question > Right, but I'm unable to reconcile the fact that a pair is only a 2-4% > favorite over two overcards. Picking 5 cards out of 7, most hands will be a pair or better. To make a pair from the high cards, 6 cards out of the unknown 48 are good. To improve the middle-card pair to trips, only two of the 48 cards will help. === Subject: : Re: sci.math vs True Believers > WHAT IS YOUR MAJOR MALFUNCTION? > WHAT IS YOUR MAJOR MALFUNCTION? > You are WRONG - totally and without doubt. Maxwells equations are > Lorentz invariant but not Galiliean invariant. > Do you know WHY Galilean transforms are ? No, of course not. >>A True Believer much like you but with the guts to expose himself showed me >>his demonstration that Maxwell was not invariant under the galilean xforms. >>What he demonstrated was that it was indeed invariant if one did not impose >>the corrupt (because anti-theoretical per Newton, etc) time transform. >hehe, really? >Show me, i've had a boring weekend. Seeing Maxwell's equations being >invariant under Galilian transforms would be interesting. Eleaticus's comments here are based on his inadequate appreciation of the Chain Rule for multiple variables. The Galilean Transformation is t' = t, x' = x - vt, y' = y, z' = z, and the inverse transformation is given by t = t', x = x' + vt', y = y', z = z'. Denoting partial differentiation with respect to a variable u by @/@u, the components of the 4-gradient operator transform as @/@t' = @/@t + v @/@x, @/@x' = @/@x, @/@y' = @/@y, @/@z' = @/@z. But that is not good enough for Eleaticus. Eleaticus holds to it that since t = t', then the differential operators @/@t and @/@t' MUST be equal. All arguments about the fact that x, y, z are being held constant when evaluating @f/@t, and that x', y', z' are being held constant when evaluating @f/@t', fell on deaf ears. Eleaticus also ignored the physical example of the difference between C_p (the speci[CapitalThorn]c heat under constant pressure) and C_v (the speci[CapitalThorn]c heat under constant volume), and the fact that these two quantities are de[CapitalThorn]nitely not equal, in spite of the fact that each of them has the form @U/@T, where U is the energy and T is temperature, and the fact that the reason why they differ is *solely* because C_p is evaluated under the conditions of constant pressure and C_v is evaluated under the conditions of constant volume. David ----- === Subject: : The corruption of the Relativity cultist was: sci.math vs True Believers > and the inverse transformation is given by > t = t', > x = x' + vt', > y = y', > z = z'. > Denoting partial differentiation with respect to a variable u by @/@u, > the components of the 4-gradient operator transform as > @/@t' = @/@t + v @/@x, > @/@x' = @/@x, > @/@y' = @/@y, > @/@z' = @/@z. AH HA! I [CapitalThorn]nally get the answer to a question that has bothered me for years: why insist on basing discussion on the Ôinverse' transformation instead of the basic function in which X' is a dependent variable, and X indepenent. An X dependent on time, which is absolutely not true in the galilean transform, but allows you to pretend X and t are dependent on each other. [ Note that in the case of any dependent variable you can solve the equation for the dependent variable(s) in terms of the dependent variable, but it is misleading to do so.] Let's examine this: > @/@t' = @/@t + v @/@x, What that says is that a change in the value of the function due to t' depends on a change in the value of X, which is obvious bull. Neither t' (nor t) change due to a change in X. Remember? t'=t according to the corrupt imposition of the idea there is a time transform. It is obvious bull to say that a change in the value of the function due to t' depends on a change in the value of X. Which is what @/@t' = @/@t + v @/@x says. Moo-plop. So, using the non-corrupt original transforms where X maintains its honest identity as an idependent variable we have @t/@x =0, @v/@x =0, @x/@t =0, @x/@v =0, and especially @t'/@x = 0. In which case it is obvious that @/@t' <> f(@x), and @/@t' <> @/@t + v @/@x. Not that it matters, since the imposition of the time transform is ridiculous to start with. In a Newtonian-concept universe there is not time transform. Further, it doesn't matter because anyone who hasn't forgotten their childhood lessons in how to use a ruler when you don't position it optimally will use some (x1-x0) instead of a bare x, and invariance is guaranteed in such an equation. Everytime I am caused to wonder why the True Believer cultists do or say something strange it turns out to be a corrupt necessity to maintain an underpinning of their anti-galilean stance. eleaticus === Subject: : Re: The corruption of the Relativity cultist was: sci.math vs True Believers >Not that it matters, since the imposition of the time transform is >ridiculous to start with. In a Newtonian-concept universe there is not time >transform. Except that the universe is not Newtonian. It is Lorentzian, which meand a moving frame is not the same as a stationary frame. There are a whole host of experiments demonstrating this phenomenan... === Subject: : Re: The corruption of the Relativity cultist was: sci.math vs True Believers >Not that it matters, since the imposition of the time transform is >ridiculous to start with. In a Newtonian-concept universe there is not time >transform. > Except that the universe is not Newtonian. It is Lorentzian, which > meand a moving frame is not the same as a stationary frame. > There are a whole host of experiments demonstrating this phenomenan... Moral cretins who can't honestly examine the simple math of alternative theories are doubly incapable of honestly examining experimental results vis a vis the alternative theories. It is one thing to set up strawmen to corruptly argue for the sake of ego protection, it is quite another to delude yourself with images of the strawmen. eleaticus === Subject: : Re: The corruption of the Relativity cultist was: sci.math vs True Believers >>Not that it matters, since the imposition of the time transform is >>ridiculous to start with. In a Newtonian-concept universe there is not >time >>transform. >> Except that the universe is not Newtonian. It is Lorentzian, which >> meand a moving frame is not the same as a stationary frame. >> There are a whole host of experiments demonstrating this phenomenan... >Moral cretins who can't honestly examine the simple math of alternative >theories are doubly incapable of honestly examining experimental results vis >a vis the alternative theories. It is one thing to set up strawmen to >corruptly argue for the sake of ego protection, it is quite another to >delude yourself with images of the strawmen. Let me use simple words to explain the dilema. Lets see if this helps. Newtonian gravity does not square against observation. There appears to be a maximum speed for which anything can go, and it is c. There is no such thing in your beloved classical mechanics. go faster. There is no such thing in your beloved classical mechanics. If you still insist Galilean transforms are correct even though there are numerous counterexamples, I am forced to say you are a ing idiot. >eleaticus === Subject: : Re: The corruption of the Relativity cultist was: sci.math vs True Believers > Let me use simple words to explain the dilema. Lets see if this helps. > Newtonian gravity does not square against observation. To bring it up is another amazingly stupid avoidance technique. Do you even realize what you are doing? > There appears to be a maximum speed for which anything can go, and it > is c. There is no such thing in your beloved classical mechanics. That is closer to being a direct response, one that can be addressed - through MMX for example - after you have gathered the courage it takes for a True Believer to admit the mathematical transformation is invariant. > go faster. There is no such thing in your beloved classical mechanics. That is the subject of the riddle: if you call cats dogs you can reasonable make erroneous statements about dogs. meant anything about actual light, without the obvious quali[CapitalThorn]cations, is mindless. > If you still insist Galilean transforms are correct even though there > are numerous counterexamples, I am forced to say you are a ing > idiot. I showed McNally's demonstration to contradict itself on two counts, and you can't refute it. I showed both linear and inverse-square equations in coordinates to be invariant and you can't refute that. So, you obviously are once again ßailing around. How dare you say my pastries suck! exclaimed the angry chef, my soups are perfect! eleaticus === Subject: : Re: The corruption of the Relativity cultist was: sci.math vs True Believers >> Let me use simple words to explain the dilema. Lets see if this helps. >> Newtonian gravity does not square against observation. >To bring it up is another amazingly stupid avoidance technique. Do you even >realize what you are doing? No, I don't. I have no free will. >> There appears to be a maximum speed for which anything can go, and it >> is c. There is no such thing in your beloved classical mechanics. >That is closer to being a direct response, one that can be addressed - >through MMX for example - after you have gathered the courage it takes for a >True Believer to admit the mathematical transformation is invariant. Switching between pure math and applied physics at will, are you? >> go faster. There is no such thing in your beloved classical mechanics. >That is the subject of the riddle: if you call cats dogs you can reasonable >make erroneous statements about dogs. >meant anything about actual light, without the obvious quali[CapitalThorn]cations, is >mindless. Isn't it nice being able to totally misunderstand everything said to you? >> If you still insist Galilean transforms are correct even though there >> are numerous counterexamples, I am forced to say you are a ing >> idiot. >I showed McNally's demonstration to contradict itself on two counts, and you >can't refute it. >I showed both linear and inverse-square equations in coordinates to be >invariant and you can't refute that. Do you think I care what you Ôshow' when no matter what you say, at the end of the day Galilean transforms do NOT work in physical reality? >So, you obviously are once again ßailing around. How dare you say my >pastries suck! exclaimed the angry chef, my soups are perfect! I wish I had a pony. Wouldn't it be nice if I had a pony? >eleaticus === Subject: : Re: The corruption of the Relativity cultist was: sci.math vs True Believers >>and the inverse transformation is given by >>t = t', >>x = x' + vt', >>y = y', >>z = z'. >>Denoting partial differentiation with respect to a variable u by @/@u, >>the components of the 4-gradient operator transform as >>@/@t' = @/@t + v @/@x, >>@/@x' = @/@x, >>@/@y' = @/@y, >>@/@z' = @/@z. > AH HA! > I [CapitalThorn]nally get the answer to a question that has bothered me for years: why > insist on basing discussion on the Ôinverse' transformation instead of the > basic function in which X' is a dependent variable, and X indepenent. > An X dependent on time, which is absolutely not true in the galilean > transform, but allows you to pretend X and t are dependent on each other. > [ Note that in the case of any dependent variable you can solve the > equation for the dependent variable(s) in terms of the dependent variable, > but it is misleading to do so.] > Let's examine this: >>@/@t' = @/@t + v @/@x, > What that says is that a change in the value of the function due to t' > depends on a change in the value of X, which is obvious bull. Neither t' > (nor t) change due to a change in X. Remember? t'=t according to the > corrupt imposition of the idea there is a time transform. > It is obvious bull to say that a change in the value of the function due > to t' depends on a change in the value of X. > Which is what @/@t' = @/@t + v @/@x says. Moo-plop. > So, using the non-corrupt original transforms where X maintains its honest > identity as an idependent variable we have > @t/@x =0, > @v/@x =0, > @x/@t =0, > @x/@v =0, > and especially @t'/@x = 0. > In which case it is obvious that @/@t' <> f(@x), and > @/@t' <> @/@t + v @/@x. > Not that it matters, since the imposition of the time transform is > ridiculous to start with. In a Newtonian-concept universe there is not time > transform. > Further, it doesn't matter because anyone who hasn't forgotten their > childhood lessons in how to use a ruler when you don't position it optimally > will use some (x1-x0) instead of a bare x, and invariance is guaranteed in > such an equation. > Everytime I am caused to wonder why the True Believer cultists do or say > something strange it turns out to be a corrupt necessity to maintain an > underpinning of their anti-galilean stance. > eleaticus Invariance is beyond their abilities to comprehend, as is the fact that reciprocity in itself is suf[CapitalThorn]cient to [CapitalThorn]le-13 the special theory of relativity. What they repeatedly fail to grasp, is that if an observer is free to consider himself at rest, then he is also free to arbitrarily assign the positive and negative halves of the axis. Special relativity cannot deal with such a reversal, since immediate contradictions arise, such as the twin contradiction. Any third grader can perceive the fault. Alas, many adults have spent their lives rationalizing the spew of their peers, losing touch with their earlier sensibility. This theory is no exception. The triplets experiment that I've posted several times is a valid argument that proves that once having settled on a rest frame, no other frames are thereafter valid rest frames, thus contradicting the claim of invariance. Though Dirk and others pretend to provide mathematical proofs that this is incorrect, note that they never once give the alternate frame the option of reversing his axes, thus stating, essentially, that the frame is restricted in its choices of axes and handedness, and thus not a rest frame, but rather a frame bound in its perceptions to the initial rest frame chosen. More importantly, you don't need to prove the Galilean invariance of any set of equations in order to demonstrate the ßaw in the reasoning of special relativity adherents. That's a cheap shot on their part. To hell with them, let them prove that there is no alternate theory of electromagnetism that is Galilean invariant and empirically consistent. Then let them provide a single observed behavior to support the claim that the Galilean transform is disproved daily in the lab. I'm aware of a long list of behaviors that they claim to be proof, and yet I cannot seem to [CapitalThorn]nd a valid contradiction between these behaviors and the Galilean transform. As for clock ticking rates, DOA, no two clocks in my possession tick at the same rate, and so according to their -for-brains logic, time must be passing at different rates in every room in the house. I've taken up sleeping in a room with a broken clock that doesn't tick, thus preventing 8 hr. worth of aging every day. Now as for relativistic energy: Moving charges generate an electromagnetic [CapitalThorn]eld. The faster the charge, the more energy stored in the [CapitalThorn]eld. Where in that observation does the Galilean transform fail?. Moreover, it was Einstein's take on the matter as well, that inertia was just electromagnetic drag. Now the only problem to resolve is this, drag wrt to what?. This is where I part ways with the special theory of relativity, having derived from [CapitalThorn]rst principles that space is just the superposition of the is the entire universe of matter. Due to the inverse square relationship, local matter has a greater inßuence on a charge than do the more distance sources. Thus we derive frame dragging. Contrary to the old thinking that mass is a fundamental dimension, it is nonetheless charge that is the base of all matter, or better the [CapitalThorn]eld associated with the charge. The extension of that [CapitalThorn]eld is space, they are one in the same, and thus the curvature is of an electromagnetic rather than being of gravitational nature. I.e. gravity is a macroscopic delusion. The fact that local matter has a greater inßuence is just an empirical observation, and thus could have served as a premise to the argument. Another way of perceiving this area of the argument is that space is more dense near massive objects, i.e. since massive objects are composed entirely of charge. We can pull all manner of semantic approaches out of our asses at this point to account for the details of the interactions, but in the end it must be noted that math is a logical tool used to sort truths, and not a truth of any sort in and of itself. Thus, for instance, arbitrarily compressing radii verses altering charge in order to calculate force between charges, is nothing more than semantic quibbling. Both radii and charge are free inventions of the human mind. Likewise time and space. In any system of interrelated functions (mathematically speaking), the associated curves of the functions are set by initial ad hoc assumptions about the linearity or nonlinearity of some gradient. I've already shown mathematically that we can arbitrarily posit that gravitational force obeys an inverse power of 1 rule. In order to provide empirical consistency with this rule we need only have mass change as a function of distance. In short, when you have a bunch of brain damaged geeks with their thoughts set on realization of a Star Trek future...well they tend to overlook the underpinnings of the thought process itself. And more oft than not, they will abhor anything that might lead back once more to Newton's clockwork universe. Their mantra is Give me magic or give me death. BTW, Maxwell was wrong because there is no temperature dependence factored into his equations. A high-temp ionized gas will exert more force on an external charge than the same gas at room temp. This can even be derived from the relativistic approach to magnetism, which point I've endlessly repeated to no avail. In any case, it's a moot point whether Heaviside's version of the equations are Galilean invariant, since had we devoted equal man hours to it, deriving Galilean epicycles, they would no doubt have worked suf[CapitalThorn]ciently well with the equations as do the existing special relativistic epicycles. Richard Perry http://www.cswnet.com/~rper/Electromagnetism.html Electromagnetism: First Principles If it can be transformed away, then you had your head up your ass when you gave it a name. === Subject: : McNally illustrates corrupt time xform. was: sci.math vs True Believers Well, I'll be durned. An actual responsive piece except for the atheoretical, corrupt imposition of the idea that there is a time transform. > Eleaticus's comments here are based on his inadequate appreciation of the > Chain Rule for multiple variables. The Galilean Transformation is > t' = t, Nonsense, unless you also impose equally valid v'=v, which would screw up your LET Ôinvariance'. t=t=t=t=t anywhere and everywhere in the universe in Newtonian/Galilean theory. > x' = x - vt, > y' = y, > z' = z, > and the inverse transformation is given by > t = t', But t'=t is no more an honest Newtonian/Glailean transform than is v'=v. These two variables/terms are on exactly the same ground as being transforms that should imposed. The imposition of the time transform - besides being corrupt on theoretical grounds - is also especially corrupt since there is in general no time term in equations in X. The question of df/dX is thus a relevant aspect of the equation in general, but the time term is not Just show us how invariant anything is under the LET if you impose the equally corrupt and anti-theoretical v'=v transform, McNally. Only by imposing the dishonest, anti-theoretical time transform do you Ôshow' non-invariance. > x = x' + vt', > y = y', > z = z'. > Denoting partial differentiation with respect to a variable u by @/@u, > the components of the 4-gradient operator transform as > @/@t' = @/@t + v @/@x, > @/@x' = @/@x, > @/@y' = @/@y, > @/@z' = @/@z. See? The only non-invariance term is the time term imposed contra-theory. So, Gisse, you dishonest buffoon. It does indeed take the corrupt time transform imposition to declare non-invariance under the galielan transforms. So, that answers one question: yes, at least some of the supposed non-invariance demonstrations are based on the corrupt imposition of the atheoretical, anti-theoretical idea that an absolute time undergoes transformation. eleaticus === Subject: : Re: McNally illustrates corrupt time xform. was: sci.math vs True Believers >Well, I'll be durned. An actual responsive piece except for the >atheoretical, corrupt imposition of the idea that there is a time transform. >> Eleaticus's comments here are based on his inadequate appreciation of the >> Chain Rule for multiple variables. The Galilean Transformation is >> t' = t, >Nonsense, unless you also impose equally valid v'=v, which would screw up >your LET Ôinvariance'. >t=t=t=t=t anywhere and everywhere in the universe in Newtonian/Galilean >theory. Really? Explain the experiments that support SR rather than Newton, then. === Subject: : Re: McNally illustrates corrupt time xform. was: sci.math vs True Believers >t=t=t=t=t anywhere and everywhere in the universe in Newtonian/Galilean >theory. > Really? > Explain the experiments that support SR rather than Newton, then. Lordy! You True Believer cretins might as well have been born brain dead. In response to Ôin newtonian/glilean' theory' you say ÔExplain the experiments that support SR rather than Newton, then.' LOL! LOL! LOL! LOL! LOL! LOL! LOL! eleaticus === Subject: : Re: McNally illustrates corrupt time xform. was: sci.math vs True Believers >>t=t=t=t=t anywhere and everywhere in the universe in Newtonian/Galilean >>theory. >> Really? >> Explain the experiments that support SR rather than Newton, then. >Lordy! You True Believer cretins might as well have been born brain dead. >In response to Ôin newtonian/glilean' theory' you say ÔExplain the >experiments that support SR rather than Newton, then.' >LOL! LOL! LOL! LOL! LOL! LOL! LOL! >eleaticus What is so funny? That you lack the understanding of the subject matter to answer my question? === Subject: : Re: sci.math vs True Believers >A True Believer much like you but with the guts to expose himself showed me >his demonstration that Maxwell was not invariant under the galilean xforms. >What he demonstrated was that it was indeed invariant if one did not impose >the corrupt (because anti-theoretical per Newton, etc) time transform. > hehe, really? Yes. > Show me, i've had a boring weekend. Seeing Maxwell's equations being > invariant under Galilian transforms would be interesting. The idea that you would be honest about anything moe complicated than the simple stuff you are already despicably dishonest about is hardly reasonable. However, I'd be delighted to had I the posts involved; I do not recall it all and at the time just followed his material. But I will follow and post my own approach to it just as soon as you answer the simple questions that have been posed to you time and again, and which you continually avoid with irrelevancies and abuse. Answer: a. Couched in difference coordinate form such as (x1-x0) which is valid even if x0 is not at the origin, rather than a bare x which is valid only when and as long as it the one end of the relevant distance is at the origin (which is Ônever' true for x'), is there a non-reductive (non-constant output) equation that is not invariant mathematically under the Galilean transform, X' = X -Vt? No time transform, of course. Demonstrate such an equation. [a'. Are all of the non-Maxwell demonstrations of the Galilean xforms non-invariance based on transforming equations in x,y,z to equations in x',y',z', which is to say, equations not based on difference forms like X1-X0 (vector intended there).] b. Without the assertion that there is a time transform in the proper Newtonian/Galilean transforms, are the Maxwells demonstrably invariant mathematically? If no, show it. Exercise that bored and stinky mind of yours. [b'. Are the Ôproofs' by experiment that the Maxwells as transformed galilean-wise based on forms derived by assertion of the time transform?] > Your simple math is coupled to reality, thus physics. > If the physics is wrong, it is wrong. Given the universe really likes > to behave in a Lorentzian way, guess what my response to Ôthe universe > is Galilian!' is? The nature of the universe is completely immaterial to the basic questions I have posed. On the one hand there are the equations, a math form, and there are math operations on those forms. Honest folk could answer no sweat had they your wonderful skills. So simple, and then you can have an honest laugh, right? Wouldn't that feel much better than the hyena laugh you've been so intensely emitting? eleaticus === Subject: : Re: sci.math vs True Believers >>A True Believer much like you but with the guts to expose himself showed >>his demonstration that Maxwell was not invariant under the galilean >xforms. >>What he demonstrated was that it was indeed invariant if one did not >impose >>the corrupt (because anti-theoretical per Newton, etc) time transform. >> hehe, really? >Yes. >> Show me, i've had a boring weekend. Seeing Maxwell's equations being >> invariant under Galilian transforms would be interesting. >The idea that you would be honest about anything moe complicated than the >simple stuff you are already despicably dishonest about is hardly >reasonable. Whats the matter, big man? Can't do it? >However, I'd be delighted to had I the posts involved; I do not recall it >all and at the time just followed his material. *snicker* So you don't have enough comprehension of the material to express Maxwell's equations in a Galilian-invariant form? I think that says something. No. You are making the assertions, you can back them up. I will not do your homework or legwork for you. >The nature of the universe is completely immaterial to the basic questions I >have posed. On the one hand there are the equations, a math form, and there >are math operations on those forms. Honest folk could answer no sweat had >they your wonderful skills. Since you are only talking about the math of Galilian transforms, you can stop posting to sci.physics. >So simple, and then you can have an honest laugh, right? Wouldn't that feel >much better than the hyena laugh you've been so intensely emitting? I only do that after consuming the souls of the damned. >eleaticus === Subject: : Re: sci.math vs True Believers Nothing. Hey asshole, Lorentz tranformations are hyperbolic rotations in spacetime. Galilean transforms are bull. Newton was wrong. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: : Re: sci.math vs True Believers > Hey asshole, Lorentz tranformations are hyperbolic rotations in > spacetime. Galilean transforms are bull. Newton was wrong. The make perfectly good sense one thinks the speed of light is in[CapitalThorn]nite. It was not known (but it was suspected) that the speed of light is [CapitalThorn]nite prior to 1676 (Ole Roemer). Even then it was not clear that the speed of light is the upper bound on the speed at which information or energy can be moved from Here to There. That took a much longer time. But now we know better, so there is no excuse for believing the Galilean transform is either true or fundemental. In one of my physics classes where we were studying waves in elastic continua we assumed the Galilean transform held, which was o.k. because elastic wave propagation in most material is slow compared to light. In the theory of aerodynmaics the physics is pure classical. Quantum theory does not enter into the study of turbulence of airßows over an airfoil. The only places where the quantum matters is in the construction of ultra light ultra strong airfoils. That is the materials science, no the aerodynamics. Aerodynamics is a world is which Mach 10 or Mach 20 is considered fast. Bob Kolker === Subject: : Re: sci.math vs True Believers >>> WHAT IS YOUR MAJOR MALFUNCTION? >> WHAT IS YOUR MAJOR MALFUNCTION? >> You are WRONG - totally and without doubt. Maxwells equations are >> Lorentz invariant but not Galiliean invariant. >> Do you know WHY Galilean transforms are ? No, of course not. >A True Believer much like you but with the guts to expose himself showed me >his demonstration that Maxwell was not invariant under the galilean xforms. >What he demonstrated was that it was indeed invariant if one did not impose >the corrupt (because anti-theoretical per Newton, etc) time transform. > hehe, really? > Show me, i've had a boring weekend. Seeing Maxwell's equations being > invariant under Galilian transforms would be interesting. What Eleaticus actually backing his rubbish up? First law of the crank is assert you want as if it was fact beyond question and never actually detail what you claim. Bill >I suppose your failure to also do so lies in the fact that you know that is >a fault in the demonstration. > Of course. Your demonstration... You are yet to demonstrate what you > speak about. >Either that are you don't know your own cult demonstration. >(I left in a bad typo below in honor of the spell checker that didn't catch >it.) >Gee. Such a simple thing to demonstrate for such a knowlegeaboe person as >you, but do you do it? > Only an undergrad. Then again, what education do you have or are in > the process of getting? >Oh no! You'd rather rave. Not to mention rather not expose what you say to >the basic theses I have been posting. > If I'd rather rave, guess what i would be doing? >To expose your Ôwhy' against the material you have been avoiding on the >pretext that experiment disproves simple math could not be done without >shame even by such a near-shameless sham as you have been. > *snicker* > Your simple math is coupled to reality, thus physics. > If the physics is wrong, it is wrong. Given the universe really likes > to behave in a Lorentzian way, guess what my response to Ôthe universe > is Galilian!' is? >eleaticus === Subject: : Re: sci.math vs True Believers > What Eleaticus actually backing his rubbish up? First law of the crank is > assert you want as if it was fact beyond question and never actually detail > what you claim. I've demonstrated the invariance of zillions of equations under the Galilean xforms, X'=X-Vt. Just use a general differnce form such as (x1-x0) instead of the moronic bare x in equations where one end of the relevant distance is not at the origin or when the equation will be tranformed to a system where the end's location is not at the origin. Back up your abusive rubbish and show us an equation where that isn't true (except for nonsense like f=x^0). (Well, that one would also be invariant.) Back up your rubbish, hobbled. I've demonstrated that the equivalence principle is wrong, that in fact what an inertial system sees as linear motion is also seen as such in an accelerated system. Under the Galilean X'=X-Vt. of course. The proof is exceptionally simple in concept and very simple in execution. (Apologies if the poe doesn't actually say/imply a curved trajectory.) Wanna [CapitalThorn]ght about it, hobbled? Back up your rubbish, hobbled. Here's another one for you. Show even one actual (use-this-to-calculate) EM equation in coordinate form that is invariant under the LE coordinate transforms. I bet you don't even understand the catch in that challenge, do you? Certainly the other cretin, Gisse, won't. And I doubt you have half his marbles, as few as his are. eleaticus === Subject: : Re: sci.math vs True Believers > I've demonstrated the invariance of zillions of equations under the Galilean > xforms, X'=X-Vt. The electromagnetic wave equation is not Galilean invariant. transform is corroberated and Galilean transform is falsi[CapitalThorn]ed. Bob Kolker === Subject: : duality of fourier transform This really bothered me for some time. For continuous fourier transform pairs as de[CapitalThorn]ned usually (in terms of f, not omega), suppose F x(t) <-----> X(f) The duality property tells us that F X(t) <-----> x(-f) Now one of the fourier transform pair is the time delay: x(t-t0) <-----> X(f)exp(-j2(pi)f(t0)) I cannot [CapitalThorn]gure out how I can prove the other pair (modulation theorem) by the duality property. The modulation theorem should be like this: x(t)exp(j2(pi)(f0)t) <-----> X(f-f0) This can be proved by actually doing the integration, but I am confused how this can be proved by duality. Any help is appreciated. wil. === Subject: : System of equations If 100 bushels of corn are distributed to 100 people in such a manner that each man gets 3 bushels, each woman gets 2 bushels and each child gets 1/2 of a bushel, how many men, women and children are there? I let m= # of men, w= # of woman and c= # of children. But then I only got 2 equations. m+w+c=100 and 6m + 4w+c= 200. Is there a 3rd equation that I missed or can C be any even integer between 68 and 70?? === Subject: : Re: System of equations says... > If 100 bushels of corn are distributed to 100 people in such a manner that > each man gets 3 bushels, each woman gets 2 bushels and each child gets 1/2 > of a bushel, how many men, women and children are there? > I let m= # of men, w= # of woman and c= # of children. But then I only got 2 > equations. > m+w+c=100 and 6m + 4w+c= 200. Is there a 3rd equation that I missed or can C > be any even integer between 68 and 70?? You only have unknowns. c=100-(m+w) Peter === Subject: : Re: System of equations > If 100 bushels of corn are distributed to 100 people in such a manner that > each man gets 3 bushels, each woman gets 2 bushels and each child gets 1/2 > of a bushel, how many men, women and children are there? > I let m= # of men, w= # of woman and c= # of children. But then I only got 2 > equations. > m+w+c=100 and 6m + 4w+c= 200. Is there a 3rd equation that I missed or can C > be any even integer between 68 and 70?? Or between 68 and 80? 2 men, 30 women, 68 children. 5 men, 25 women, 70 children. . . . 17 men, 5 women, 78 children. 20 men, 0 women, 80 children. === Subject: : Re: System of equations > If 100 bushels of corn are distributed to 100 people in such a manner that > each man gets 3 bushels, each woman gets 2 bushels and each child gets 1/2 > of a bushel, how many men, women and children are there? > I let m= # of men, w= # of woman and c= # of children. But then I only got 2 > equations. > m+w+c=100 and 6m + 4w+c= 200. Is there a 3rd equation that I missed or can C > be any even integer between 68 and 70?? Combining your two equations yields 5m + 3w = 100. This may be called a Diophantine equation since m and w are integers. Your problem now is to [CapitalThorn]nd m and w (integers) which satisfy this equation. Alex === Subject: : Re: System of equations > If 100 bushels of corn are distributed to 100 people in such a manner that > each man gets 3 bushels, each woman gets 2 bushels and each child gets 1/2 > of a bushel, how many men, women and children are there? > I let m= # of men, w= # of woman and c= # of children. But then I only got > equations. > m+w+c=100 and 6m + 4w+c= 200. Is there a 3rd equation that I missed or can > be any even integer between 68 and 70?? > Combining your two equations yields 5m + 3w = 100. > This may be called a Diophantine equation since m and w are integers. > Your problem now is to [CapitalThorn]nd m and w (integers) which satisfy this equation. > Alex It is also required that m and w be non-negative integers. Then one may deduce that w is a multiple of 5 between 0 and 30 inclusive, and each of these 7 values of w produces a solution. === Subject: : Re: System of equations Typo: ...can C be any even integer between 68 and 80?? > If 100 bushels of corn are distributed to 100 people in such a manner that > each man gets 3 bushels, each woman gets 2 bushels and each child gets 1/2 > of a bushel, how many men, women and children are there? > I let m= # of men, w= # of woman and c= # of children. But then I only got 2 > equations. > m+w+c=100 and 6m + 4w+c= 200. Is there a 3rd equation that I missed or can C > be any even integer between 68 and 70?? === Subject: : Re: nitpicking in the complex plane? there isn't such a [CapitalThorn]gment of our lazy imaginations! > In sci.math, David Bandel > mathematical circles, scribbling i's and a+bi's on classroom > chalkboards. it's time to open our eyes ladies and gentlemen. if you > can equate the square root of -1 to i. why not equate the meaning of > life to x? then x can be whatever we want. so if we say x is not > the meaning life, then we have a contradiction. mathematicians have > become lazy as of late, when running into dead ends, all too eager to > just invent a whole new category of existence containing an element > that proposes a solution. let me iterate: > ^ > |3 > |2 > |1 > -3i 2i -i | i 2i 3i > <-----------+----------- |-1 > |-2 > |-3 > | > v > WHERE INGENUITY COULD NOT PREVAIL. show me a negative anything and > i'll show you a crook.. and that crook would be YOU > Ingenuity? > I'm not sure what you're getting at here. Of course, one reason > why complex numbers are often used in such [CapitalThorn]elds as electrical > small-signal analysis is that they're *useful* as a shorthand. > But we can go simpler than that. I have two apples. You want three. > Do I: > [1] give you the two apples and owe you one later? > [2] magically conjure up an anti-apple and apple pair, > and give you the apple and keep the anti-apple somehow? > level, although for apples it's fairly ridiculous.) > [3] Slice one of the apples in half and give you three pieces? > [4] Wait until next growing season? > Presumably, negative numbers were initially used to represent > bar tabs. youc annot give me 3 apples. it's stupid to say you have -1 apples after giving me 2 and owing me 1. you simply gave me 2. owe me +1.. and you have 0. there is no negative value.. that's like dealing with in[CapitalThorn]nity as some kind of real number. === Subject: : Re: Weird group James Harris > It amazes me how weird sci.math'ers are and it's clearly some weird > social thing where many of you are simply abnormal when it comes to > even basic social function. If you want to see something really weird, examine the Google archives to see how long Harris has been behaving exactly like this, in the same group, against the same people, on the same two topics (factorization to vaporize FLT, and a bit of computer code which counts primes) without learning anything. === Subject: : Re: E. W. Dijkstra VS. John McCarthy. A rebuttal to Paul Graham's web writings. > Anyway, I don't see how Emacs/vi would lend itself to a Lisp/C war: > vi does not have a C-like extension language. In fact, no serious > editor I know of has a C-like extension language, since you would > not want to crash your editor too easily. In the context of > user-accessible extension languages, C basically has lost > completely. There's Epsilon (http://www.lugaru.com/). -- http://hertzlinger.blogspot.com === Subject: : Re: E. W. Dijkstra VS. John McCarthy. A rebuttal to Paul Graham's web writings. > Anyway, I don't see how Emacs/vi would lend itself to a Lisp/C war: > vi does not have a C-like extension language. In fact, no serious > editor I know of has a C-like extension language, since you would > not want to crash your editor too easily. In the context of > user-accessible extension languages, C basically has lost > completely. > There's Epsilon (http://www.lugaru.com/). Note that lugaru is homophone of loup-garrou which means werewolf, and which is a name that indeed matches perfectly that creepy beast. -- __Pascal Bourguignon__ http://www.informatimago.com/ Our enemies are innovative and resourceful, and so are we. They never stop thinking about new ways to harm our country and our people, and neither do we. === Subject: : Re: JSH: Your fantasy world problem > James Often in error, but never in doubt! Harris This is why James is so often in error. For most of the rest of us, error does cause doubt, and to remove that doubt, we go and learn more. That reduces the error rate. -- --Tim Smith === Subject: : Re: JSH: Your fantasy world problem > James Often in error, but never in doubt! Harris > This is why James is so often in error. For most of the rest of us, error > does cause doubt, and to remove that doubt, we go and learn more. That > reduces the error rate. Like I noted in my original post, sci.math'ers live in their own fantasy world where they apparently just make up what they decide is reality! One sci.math poster *claims* that I'm never in doubt, and another poster replies as if its true!!! To sci.math'ers that's all you need--group conformity as if simply seeing one sci.math'er that you like making a claim means it must be true! What's odd about this following behavior among sci.math'ers is that it continues against mathematics, so if a *LIKED* sci.math'er makes claims that are mathematically false, even attacking basic algebra, other sci.math'ers will follow along as the group and the group fantasy is more important to them than mathematics. As for my error rate, it has been VERY high, which has caused me to often doubt, question and go over my work repeatedly. Doubt goes hand in hand with real discovery, while so does the courage to make those leaps, and make mistakes!!! Failure is simply a part of putting yourself out there and trying. But it takes certain kind of people to lie in such a way when they see someone trying so hard and admitting their mistakes. And sci.math'ers have shown themselves to be those kind of people over a period of years. There's something wrong with many of you--something basic inside your heads that's just messed up. James Harris === Subject: : Re: JSH: Your fantasy world problem > Like I noted in my original post, sci.math'ers live in their own > fantasy world where they apparently just make up what they decide is > reality! This from someone who hasn't been in touch with reality for 7 plus years! === Subject: : Re: JSH: Your fantasy world problem Discussion, linux) >> James Often in error, but never in doubt! Harris > This is why James is so often in error. For most of the rest of us, error > does cause doubt, and to remove that doubt, we go and learn more. That > reduces the error rate. Altered signatures aren't funny. If you're quoting someone then quote them. If you want to say something about someone then say it. Don't put words in their mouths, facetiously or not. There's more than enough forgery of quotes on other groups. Don't drag it in here, even if it's a transparent joke. If it's marked as a quotation, then let it be a quotation. -- If you see math knowledge as a tool--as a hammer--with which you can attack other people then ... you defeat rational discourse. I get to call my proof the Hammer. It's more powerful than *any* physical object. It is overwhelming force. -- Two JSH quotes === Subject: : Binomial Theorem for X^n + Y^n x+y = A x-y = B [A+B]/2 = x [A-B]/2 = y x^2+y^2 = [A^2 + B^2]/2 x^3+y^3 = [A^3 + 3AB^2]/4 x^4+y^4 = [A^4 + 6A^2 B^2 + B^4]/8 x^5+y^5 = [A^5 + 10A^3 B^2 + 5AB^4]/16 x^6+y^6 = [A^6 + 15A^4 B^2 + 15A^2 B^4 + B^6]/32 x^7+y^7 = [A^7 + 21A^5 B^2 + 35A^3 B^4 + 7AB^6]/64 etc... etc... etc... === Subject: : Re: Binomial Theorem for X^n + Y^n > x+y = A > x-y = B > [A+B]/2 = x > [A-B]/2 = y > x^2+y^2 = [A^2 + B^2]/2 > x^3+y^3 = [A^3 + 3AB^2]/4 > x^4+y^4 = [A^4 + 6A^2 B^2 + B^4]/8 > x^5+y^5 = [A^5 + 10A^3 B^2 + 5AB^4]/16 > x^6+y^6 = [A^6 + 15A^4 B^2 + 15A^2 B^4 + B^6]/32 Very interesting observations. It can further be simpli[CapitalThorn]ed. As an example, write x^5 + y^5 = Q5/16 Then x^5 + y^5 = Q5/2^(5-1) So in general, x^m + y^m = Qm/2^(m-1) Now, the challenge is to write Qm in more compact form so that it can be recognized that Qm/(2^m-1) cannot be an m-th power of an integer without applying FLT. > x^7+y^7 = [A^7 + 21A^5 B^2 + 35A^3 B^4 + 7AB^6]/64 > etc... > etc... > etc... === Subject: : Re: Intuitive argument for this simple problem? > If you take the parabola de[CapitalThorn]ned by y = x^2 and then consider the effect of > adding the term kx, many people expect the curve to be skewed by this and to > lose its symmetry. A nice, if vague, question. It is nicely vague. In a way, the answer is merely: Coz it's hard to *damage* a parabola; but that's not very helpful. Probably you need to have an overall idea of conics. And thus know that adding lower-order terms doesn't damage quadraticity, or, therefore, conic-ness. And brief thought shows that skewing the graph isn't going to turn it into a circle, ellipse or hyperbola, for obvious topological reasons, so it has to stay a parabola. And skewing, or rather *shearing*, which is what you're really doing, isn't going to change the direction of the major axis, when the shear is also in that direction. So it stays in orientation. Well! That's a long-winded explanation invoking many higher-order concepts than just doing the basic co-ordinate geometry. But maybe it contains some sort of answer for you? Put simply, shearing the graph in a vertical direction can't damage a parabola! This is not the only thing that can't! Neither can expanding/contracting either axis. Everyone knows that expanding an axis will damage a circle into an ellipse. Though expanding both axes by the same amount won't, if you regard all circles as similar. Similarly, (pun!), all parabolas are similar! This is not so widely known. There is only one shape of parabola just as there is only one shape of circle. Any parabola can be turned into any other co-vertical at the origin, by merely changing the scale of the graph. Like getting to America by sailing ship, obvious enough once you've heard it's been done before... ------------------------------------------------------------- --------------- -- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------------- --------------- -- The [CapitalThorn]rst European words spoken in the new world... They don't look very Chinese, Christopher. ------------------------------------------------------------- --------------- -- === Subject: : Re: Question: SPACE of super-complex numbers SNIP > There are no 3 D division algebras over the reals. You have to go from > complex number to quaternions. > All of these manipulations were tried back in the middle of the 19-th > century. Google on the history of quaternions and octernians. Also > Google division algebras. Having a quotient is a strong constraint. Moufang loops (all groups and Octonions) provide that constraint. They are mxm Cayley multiplication tables with left and right multiplicative inverses for every element. They multiply and divide sets of unsigned coef[CapitalThorn]cients A, and have Frobenius conservation, with Detm[A]Detm[B]=Detm[AB], where Detm is the determinant of the multiplication table mapped with the coef[CapitalThorn]cients. They become algebras when another operation (generalized negation) collapses an mxm table (iff it has r-fold symmetry) to an (m/r)x(m/r) table, and the m unsigned coef[CapitalThorn]cients to m/r signed coef[CapitalThorn]cients. Their multiplicative inverses Ai have Detm[A] as divisors; if this factorises (into conserved sizes) the inverse splits into partial fractions. Division by zero occurs if any size becomes zero; it can be avoided by working in a constrained sub-algebra (renormalization). Real algebras have r=2; the four real division algebras without divisors of zero (R, C, H, O) conserve the sum of their squared elements. As they are monosized, they cannot renormalize. They do not have (non-trivial) real divisors of zero because the sum of squares is only zero in the {0,0...} case. Every Group and Octonion de[CapitalThorn]nes a Hoop algebra [1] over the real (r=2), terplex (r=3), complex (r=4), (etc.) numbers. Most of these are partial fraction division algebras; Clifford, Davenport, Pauli-sigma, etc, algebras are Hoops; Wedge (exterior) and Lie algebras are obtained by constraining particular hoops. There are many ways to go to multiple dimensional algebras, and some of them are interesting, despite the mathematicians horror of division by zero. Roger Beresford. [1]http://library.Wolfram.com/infocenter/Mathsource/4894 It was the secrets of heaven and earth that I desired to learn. (Mary Shelley). === Subject: : LaTeX resources (of interest to mathematicians?) Only slightly off topic, I hope! Some here might be interested in LaTeX for Logicians @ www.phil.cam.ac.uk/teaching_staff/Smith/LaTeX/ which has (surprise, surprise!) links to LaTeX resources intended to be of interest to Logicians, but also probably some of interest to mathematicians more generally. Peter S. === Subject: : Re: LaTeX resources (of interest to mathematicians?) > Only slightly off topic, I hope! > Some here might be interested in LaTeX for Logicians @ > www.phil.cam.ac.uk/teaching_staff/Smith/LaTeX/ > which has (surprise, surprise!) links to LaTeX resources intended to be of > interest to Logicians, but also probably some of interest to > mathematicians more generally. > Peter S. Ôcid Ôooh === Subject: : Problems from Hungerford's Algebra I recall that Hungerford, Algebra (the Grad Texts in Math--yellow one), had some good problems on 1) the groups Z*_p^k for both 2 and odd p, basically looking at the generators of Z*_p^k, and showing that for p = 2, Z*_2^k =~ Z_2 x Z_2^(k-2). Also some good problems on cyclotomic polynomials for various cases; like phi_pn(x) = phi_p(x^n) or something like that. If someone has this text and the time, would he post some of these problems? I can't recall them all. Van === Subject: : Re: Uncountable sets in CZF? > Loewenheim is remembered for the Loewenheim-Skolem paradox (which > Skolem pointed out is not a paradox!) which produces non-standard > models, for example a denumerable model of the reals. > Translation: Loewenheim says the reals are countable. I wish I'd > known about this before, I could have invoked Loewenheim-Skolem > instead of independently developing appropriate notions myself, for > several of the contentious issues we discuss. Now I won't deny that > they say so, unless necessary. You have misunderstood Lowenheim-Skolem. It says that any consistent [CapitalThorn]rst-order theory with a countable number of constants and axioms has a countable model. This means that the model is countable in the >meta<-theory. A Ôcountable' model of ZFC just means that there is some relation R on natural numbers, which when interpreted as membership, satis[CapitalThorn]es the ZFC axioms. But no function (in the meta-theory) from the Ôreals' (of the model) to the naturals of the meta-theory will be a set of the model (obviously). You cannot count the reals of the model >within< the model. The relation R satis[CapitalThorn]es no useful property, and its existence is quite mysterious as it is proved by contradiction. Anyway, the de[CapitalThorn]nition of reals implicitly used by mathematicians is second order. === Subject: : Re: Uncountable sets in CZF? > Loewenheim is remembered for the Loewenheim-Skolem paradox (which > Skolem pointed out is not a paradox!) which produces non-standard > models, for example a denumerable model of the reals. Translation: Loewenheim says the reals are countable. I wish I'd > known about this before, I could have invoked Loewenheim-Skolem > instead of independently developing appropriate notions myself, for > several of the contentious issues we discuss. Now I won't deny that > they say so, unless necessary. > You have misunderstood Lowenheim-Skolem. > It says that any consistent [CapitalThorn]rst-order theory with a countable number > of constants and axioms has a countable model. This means that the model > is countable in the >meta<-theory. A Ôcountable' model of ZFC just means > that there is some relation R on natural numbers, which when interpreted > as membership, satis[CapitalThorn]es the ZFC axioms. But no function (in the > meta-theory) from the Ôreals' (of the model) to the naturals of the > meta-theory will be a set of the model (obviously). You cannot count the > reals of the model >within< the model. The relation R satis[CapitalThorn]es no > useful property, and its existence is quite mysterious as it is proved > by contradiction. > Anyway, the de[CapitalThorn]nition of reals implicitly used by mathematicians is > second order. Hi Nath, I necessarily disagree. Instead, I theorize that ordinals are ubiquitous, and any function between the set of natural numbers and the set that is the set of real numbers is an implicit composition with EF. That's the problem with demanding a metatheory, instead of just having a theory, that as the variable of the n'th order logic n diverges, it's still subject to the same problems of considering an in[CapitalThorn]nite ordinal. That reminds me of hearing about epsilon chains of length sixteen or eleven dimensions. The universe is n-dimensional. If these are nonstandard, let's consider a nonstandard model of the reals: the hyperreals. The reals are just the hyperreals, and vice versa, and the integers are just the hyperintegers, and vice versa. They're just called nonstandard so they can be discussed rationally. The second order logic, and n'th order logic, can only be within a [CapitalThorn]rst order logic. EF is a function between the set N and the unit interval of the set R. No classes in set theory, and no models in theory. Ross F. === Subject: : Re: Uncountable sets in CZF? >> | Does the construction of L require AC to begin with? >> | >> | To go back to a previous example, AC -> there is a non-measurable >> | subset of R. >> | >> | If V=L -> AC, then L must contain such a set. >> | >> | Clearly, adding more sets to V (and making AC false) can't eliminate >> | anything, so there must be a non-measurable regardless of AC, if L is >> | contained in V, right? >> `---- >> I don't know diddly about the de[CapitalThorn]nitions, but I imagine that the >> de[CapitalThorn]nition of non-measurable includes some quanti[CapitalThorn]ers. If we enlarge >> the universe of discourse, then a set which was previously >> non-measurable may well become measurable. Someone that knows a >> de[CapitalThorn]nition or two can con[CapitalThorn]rm or deny this, but I'd reckon that a set >> is not measurable/non-measurable all by its lonesome, but only in some >> context, namely the universe. > The universe, in this case, is R. This is contained in L, and adding > more sets can't change the properties of R. > Don't be too hasty here. The properties of R can certainly change, > depending on what one means by properties. If by property, you mean > all of the true statements involving R, then *of course* this may > change by adding more sets, because *some* of these statements involve > quanti[CapitalThorn]ers. The real numbers are a de[CapitalThorn]nite entity. If two theories predict differing properties of R, one must be wrong. If Lesbesgue measure has quanti[CapitalThorn]ers in this sense, then it is inconsistent. >> In any case, it is just painfully obvious that V >= L. L is the >> collection of all sets which provably exist. Any model of ZF must >> contain every set which provably exists (well, perhaps with some of >> these sets identi[CapitalThorn]ed). Let's call a model M of ZF /good/ if, >> whenever M |= s = t for any terms s and t, then also ZF |- s = t. >> Some model theorist can tell me what the right terminology is, but >> good will do for now. Clearly, if M is good, then L <= M. That is, >> then L is a submodel of M. > You have passed the limits of my knowledge here; this notation is > gibberish to me. > Darn shame. Probably doesn't bode well for your ability to converse > sensibly on the relationship between V and L. I am not a crank! Perhaps direct me to an explanation I could understand of how L is constructed; I have examined texts on set theory but [CapitalThorn]nd them either to be too simple to discuss this, or incomprehensible to me. >> V is clearly a good model of ZF. Therefore L is a submodel (not >> necessarily proper) of V. >> This is more detail than ought to be necessary. Every constructible >> set is a *set*. Therefore, L is contained in V. Duh. > L is not Ôconstructible' by the ordinary de[CapitalThorn]nition constructible = > determined by a [CapitalThorn]nite proof. > I didn't say L is a set. I said that every element of L is a > constructible set. Sorry, I meant Ôevery element of L'. There are only a countable number of constructible sets, therefore some elements of L are not constructible. > Since L contains all the ordinals, it must be larger than any > cardinal. This is much larger than the (countable) set of > constructible objects. Also, the construction of L requires > uncountably many steps, since it uses trans[CapitalThorn]nite induction. Hence I > can't accept L is a valid concept anyway. > I see no reason why it can't be said that L is Ôtoo big'. It's just > obvious that any universe containing AC will be larger than an > equivalent one that does not. > What does equivalent mean? How is V equivalent to L? Since AC does not imply V=L, this is a non sequitur. > Every constructible set is a set. L cannot be larger than V. This is > just obvious. No, that is meaningless. For an arbitrary element of L, you can't give me a construction for it. The sense in which L is Ôconstructible' is a different meaning of the word Ôconstructible' than the one used in stating that it is Ôobvious' that constructible sets all exist. There are two kinds of mathematics: the useful, meaningful part dealing with [CapitalThorn]nite and countable sets, computable numbers, and constructible objects; and the division dealing with uncountable sets, unde[CapitalThorn]nable objects, and unintelligible concepts - nonsense upon nonsense upon nonsense. You apparently belong to the latter school. Andrew Usher === Subject: : Re: Uncountable sets in CZF? <87llgd3prz.fsf@phiwumbda.org> <87ekm212vo.fsf@phiwumbda.org> Discussion, linux) >> Don't be too hasty here. The properties of R can certainly change, >> depending on what one means by properties. If by property, you mean >> all of the true statements involving R, then *of course* this may >> change by adding more sets, because *some* of these statements involve >> quanti[CapitalThorn]ers. > The real numbers are a de[CapitalThorn]nite entity. If two theories predict > differing properties of R, one must be wrong. If Lesbesgue measure has > quanti[CapitalThorn]ers in this sense, then it is inconsistent. The properties are different only if we understand property to be synonymous with formal statement, and our interpretation of certain formal statements has changed. Consider a statement about R of the form For all sets X, blah(R,X). If we call such a statement a property, then *of course* whether R satis[CapitalThorn]es that statement can depend on what sets we have in our universe. Statements like R is Cauchy-complete are exactly this kind of statement. What Cauchy-completeness means depends on what Cauchy sequences are available. Given L c V, it's not obvious that the set R in L is the same set as the set R in V. If there are more sets in V, then there may be more Cauchy sequences in V than in L and so the unique set satisfying the de[CapitalThorn]nition of R in V may be a superset of the unique set satisfying the de[CapitalThorn]nition of R in L. I don't know if that's the case. I don't study these issues. However, the idea that the reals are a de[CapitalThorn]nite entity independent of what set constructions are available to us seems a bit naive. -- Jesse F. Hughes Yesterday was Judgment Day. How'd you do? -- The Flatlanders === Subject: : Re: Uncountable sets in CZF? > Don't be too hasty here. The properties of R can certainly change, > depending on what one means by properties. If by property, you mean > all of the true statements involving R, then *of course* this may > change by adding more sets, because *some* of these statements involve > quanti[CapitalThorn]ers. >> The real numbers are a de[CapitalThorn]nite entity. If two theories predict >> differing properties of R, one must be wrong. If Lesbesgue measure has >> quanti[CapitalThorn]ers in this sense, then it is inconsistent. >The properties are different only if we understand property to be >synonymous with formal statement, and our interpretation of certain >formal statements has changed. >Consider a statement about R of the form For all sets X, blah(R,X). >If we call such a statement a property, then *of course* whether R >satis[CapitalThorn]es that statement can depend on what sets we have in our >universe. >Statements like R is Cauchy-complete are exactly this kind of >statement. What Cauchy-completeness means depends on what Cauchy >sequences are available. >Given L c V, it's not obvious that the set R in L is the same set as >the set R in V. Typically, it isn't. >If there are more sets in V, then there may be more >Cauchy sequences in V than in L and so the unique set satisfying the >de[CapitalThorn]nition of R in V may be a superset of the unique set satisfying >the de[CapitalThorn]nition of R in L. >I don't know if that's the case. I don't study these issues. It is the case. >However, the idea that the reals are a de[CapitalThorn]nite entity independent of >what set constructions are available to us seems a bit naive. In any model of ZFC, R is completely determined up to isomorphism. On the other hand, given any model V of ZFC, and any two in[CapitalThorn]nite sets A and B in V, there exists a generic extension V[G] of V in which A and B are bijective. Speci[CapitalThorn]cally, there is a generic extension V[G] in which N and R^V (the realization of R in V) are bijective, i.e. there is a generic extension V[G] of V in which R^V is countable. Obviously, in such an extension, R^V cannot be regarded as a model for the real numbers in V[G], even though it is a model for the real numbers in V. In the generic extension, there are Cauchy sequences which do not occur in V (there are, in V[G], Cauchy sequences in R^V which do not have a limit in R^V - such Cauchy sequences are elements of V{G}, but not elements of V). It follows that there are generic extensions of L in which there are real numbers which are not constructible. David ----- === Subject: : Re: Uncountable sets in CZF? <87llgd3prz.fsf@phiwumbda.org> <87ekm212vo.fsf@phiwumbda.org> <87wtzrpq6r.fsf@phiwumbda.org> Discussion, linux) >>I don't know if that's the case. I don't study these issues. > It is the case. -- Jesse Hughes Like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as it might seem. -- Alan MacKay === Subject: : Re: Uncountable sets in CZF? <87smamyuzm.fsf@phiwumbda.org> <2oeo5fF9p595U1@uni-berlin.de> <87657hzhdr.fsf@phiwumbda.org> <2ogm7kFag9cfU1@uni-berlin.de> <4124f4fa$3$fuzhry+tra$mr2ice@news.patriot.net> <41265af9$22$fuzhry+tra$mr2ice@news.patriot.net> X-CompuServe-Customer: Yes X-Coriate: interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: George Cox X-Punge: Micro$oft X-Sanguinate: The MVS Guy X-Terminate: SPA(GIS) X-Tinguish: Mark Grif[CapitalThorn]th X-Treme: C&C,DWS at 02:09 PM, raf@tiki-lounge.com (Ross A. Finlayson) said: >It's nice of you to resort to a word that doesn't exist except of >your own construction if you don't want it to be understood. Well, most people who failed to get the joke would have assumed that it was a typo and proceded from there. >Janos Bolyai, who independently developed some aspects of hyperbolic >(non-Euclidean) geometry There you go again. >So, by your pun, are you trying to imply that you've heard of >someone else much farther along in this line of argument, No, I was trying to humorously convey the basic concept that Hyperbolic Geometry is not the same as Non-Euclidean Geometry, and that Bolyai worked on a non-Euclidean Geometry that was *NOT* Hyperbolic. >rabid camp of Skolemites, Not relevant to the difference between elliptical and hyperbolic. >I want to tell you that you have some spelling errors on your web >page, abd and expereience. >I generally don't use sarcasm But you did in this case. Inappropriately, since you had missed, and continued to miss, the entire point. >Shmuel, 2+2=4, and in a model of ubiquitous ordinals, in[CapitalThorn]nite sets >are equivalent. What does that have to do with the point at issue? >Loewenheim-Skolem says what I said it does above. What does that have to do with the point at issue? >So, if we're talking about Loewenheim and Skolem, No. The text that I quoted and commented on had nothing to do with Loewenheim and Skolem, as I already pointed out to you. But perhaps you're only pretending to be dense as a form of humor and I'm simply failing to understand how funny it is. Or maybe Non-Euclidean geometry is also called extra- or super-Euclidean, or Minkowskian. was intended as a joke. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: : Equalization Does this belong on sci.math or sci.crypto? Equalization curves: I think Europe used the CCIR equalization curve or the new name ITU or something like that instead of the US standard RIAA curve for records. I ought to explain myself: I won a [CapitalThorn]fty dollar jackpot on the nickel slot machines at the EM club in Bamberg Germany in 1966 and put ten dollars back in and hit again. I used the ninety dollars and some more to buy a Philips stereo record player with two small speakers. It had a wooden case. The three Beatles albums Rubber Soul, Revolver, and Sgt. Pepper's sounded very very good. Make that (very)^3. I can't get any music system in the USA to sound that good or even in tune. I was in Germany again in 1975 and heard a song (Indian Love Call by Ray Stevens) from the radio in a Mercedes-Benz bus and the announcer said ausgezeitnich as if she were saying the word delicious. I agreed and I bought the album when I got back to the States, but, it is no good on any system I've tried here. Everything you plug into or out of can change the equalization, but I think it can be made to sound good with a graphic equalizer. There was no standard equalization curve for records in the USA until about 1956, so, try to guess what Bix Beiderbecke's cornet really sounded like in 1927. I can change the word Borneo to Corneo or Porneo in the lyrics to one of Bix's songs with an equalizer. http://www.kspc.org/ Forward Into The Past from 2 PM to 5 PM Sundays California time plays a lot of good music from 1925-1945. It is very good if it is equalized correctly, but how can it be done? Cliff Nelson Dry your tears, there's more fun for your ears, Forward Into The Past 2 PM to 5 PM, Sundays, California time, at: http://www.kspc.org/ Don't be a square or a blockhead; see: http://users.adelphia.net/~cnelson9/ === Subject: : cos(x) >= 1 - x^2/2! + x^4/4! - x^6/6! f_0(x) = 1 f_1(x) = x^2/2! f_2(x) = x^4/4! f_3(x) = x^6/6! f_n(x) = x^(2n) / (2n)! We know that cos(x) = f_0(x) - f_1(x) + f_2(x) - f_3(x) + ... + (-1)^n*f_n(x) + ... for all x. I wonder if it is true that: cos(x) >= f_0(x) - f_1(x) + .... + (-1)^(2n+1)*f_(2n+1)(x) for all x. This inequality is true for very small x, because the sequence ( f_0(x), f_1(x), f_2(x), ... ) is a decreasing sequence when -1= f_(n+1)(x) for all x and for all nonnegative integer n. I failed to prove or disprove the inequality. Can somebody help me on this? === Subject: : Re: cos(x) >= 1 - x^2/2! + x^4/4! - x^6/6! > f_0(x) = 1 > f_1(x) = x^2/2! > f_2(x) = x^4/4! > f_3(x) = x^6/6! > f_n(x) = x^(2n) / (2n)! > We know that cos(x) = f_0(x) - f_1(x) + f_2(x) - f_3(x) + ... + > (-1)^n*f_n(x) + ... for all x. > I wonder if it is true that: > cos(x) >= f_0(x) - f_1(x) + .... + (-1)^(2n+1)*f_(2n+1)(x) for all x. > This inequality is true for very small x, because the sequence ( f_0(x), > f_1(x), f_2(x), ... ) is a decreasing sequence when -1 This inequality is true for very large x, because then > f_0(x) - f_1(x) + .... + (-1)^(2n+1)*f_(2n+1)(x) > is a negative number far away from 0 while cos(x) is bounded between -1 and > 1. > Therefore we can guess that this inequality may be also true for other > values of x. > This inequality comes from a book called Problem-Solving Through Problems. > The author of the book assumes the inequality to be true, for the only > reason that the sequence f_0(x), -f_1(x), f_2(x), -f_3(x), ... is an > alternating sequence. ( That the tayler series for cos(x) is an alternating > series.) But the absolut values of the sequence is not steadily decreasing. > it is not true that f_n(x) >= f_(n+1)(x) for all x and for all nonnegative > integer n. > I failed to prove or disprove the inequality. Can somebody help me on this? Since cos is an even function you will need to prove this for positive x only. Start with the inequality sin(x) <= x (for all x >= 0). Integrate both sides from 0 to x to get 1 - cos(x) <= x^2/2, that is, cos(x) >= 1 - x^2/2 (for all x >= 0). Integrate both sides from 0 to x again, and again. __________________________________________ Eric J. Wingler (wingler@math.ysu.edu) Dept. of Mathematics and Statistics Youngstown State University One University Plaza Youngstown, OH 44555-0001 330-941-1817 === Subject: : Re: cos(x) >= 1 - x^2/2! + x^4/4! - x^6/6! > f_0(x) = 1 > f_1(x) = x^2/2! > f_2(x) = x^4/4! > f_3(x) = x^6/6! > f_n(x) = x^(2n) / (2n)! > We know that cos(x) = f_0(x) - f_1(x) + f_2(x) - f_3(x) + ... + > (-1)^n*f_n(x) + ... for all x. > I wonder if it is true that: > cos(x) >= f_0(x) - f_1(x) + .... + (-1)^(2n+1)*f_(2n+1)(x) for all > x. It is certainly true whenever f_(2n+2)(x) < f_(2n+1)(x). === Subject: : Re: cos(x) >= 1 - x^2/2! + x^4/4! - x^6/6! >f_0(x) = 1 >f_1(x) = x^2/2! >f_2(x) = x^4/4! >f_3(x) = x^6/6! >f_n(x) = x^(2n) / (2n)! >We know that cos(x) = f_0(x) - f_1(x) + f_2(x) - f_3(x) + ... + >(-1)^n*f_n(x) + ... for all x. >I wonder if it is true that: > cos(x) >= f_0(x) - f_1(x) + .... + (-1)^(2n+1)*f_(2n+1)(x) for all x. Hint: Let P_n(x) = 1 - x^2/2! +...+(-1)^n x^(2n)/(2n)! and Q_n(x) = x - x^3/3! +...+(-1)^n x^(2n+1)/(2n+1)! Note that Q_n(x) = int_0^x P_n(t) dt and P_{n+1}(x) = 1 - int_0^x Q_n(t) dt and use mathematical induction. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: : Re: 1+i > i He/she/it believes, or is made to believe in an ordering. Third son can give orders to [CapitalThorn]rst daughter, even if he is much younger. In school, doctor(=teacher) can punish pupil, beating on the [CapitalThorn]ngers. That's not a Korean speciality, even between nations you have a ranking like this, for example money : state of USA lowest =greatest depts. Or education : North Korea highest (see Ermanno Furlanis http://homepage.mac.com/nealemvf/iblog/C821734565/E930395014/ and so Kim must be the most educated. Is he still bachelor, mina_? He is in love with Anne Frank, but that is a different area). mina_ , Your example translated: (1 gold + 1 silver)- 1 silver = 1 gold > no medals in Olympia 1 gold + 1 silver > 1 silver True, not paradox. The question is 2 silver > or < or = 1 gold ? The same with a house: upstairs 1 ßoor + along a corridor to the [CapitalThorn]rst room and back = just upstairs 1 ßoor the value is here in how far from ground, which level You are, but on a given level it doesn't matter, with people it's called, they are peers. The paradox is arising, when one believes in only this one kind of ordering. Other orderings can be seen on the left hand of this page (the postings ordered by who replies to whom - and by date). In sci.math one has an ordering of, who gives the best answers, who helps another, (sometimes who has the best insulting words). So in who is asking interesting questions You are not low, mina_. But order is only half of life, the structure is more and i hope You have done Your homework on who is captain in this miracolous world wide web. Have fun Hero === Subject: : JSH: So what's the point? So why do I keep posting when sci.math'ers have demonstrated that they're not willing to agree for years now? Because what is happening here is a drama for more than you who are sci.math'ers and I think it's a peculiarity that so many of you live in a tiny little world where you're the only important people. So to you, if I can't get acknowledgement of what I call facts, and people call me names, etc., you think I should just go away as I can't win. But you're not the only people reading these posts. There are other people out there who get them, for instance, on the Internet, and I try to keep up through various means with the discussions that occur elsewhere. My prime counting function is also on my blog, and people doing Internet searches on prime counting are likely to be lead to Usenet, where they will see sci.math'ers lying about my research, and learn what I learned a few years ago, which is that sad reality that mathematicians and math groupies seem comfortable lying to the public about mathematics. In a way it makes sense, as so many people are afraid of mathematics, and for so many years mathematicians have gotten away with pure math, which they chortle is completely useless to the real world, and the more useless the better!!! But, if only a few mathematicians can understand something, and then that few is also saying what's great or proven, then why can't that few reward themselves for work that is not correct? Like Andrew Wiles has such a simple logical error in his work that it's explainable to a child, but mathematicians STILL champion him as having proven Fermat's Last Theorem, when most mathematicans can't even go over his work in detail and don't bother!!! The [CapitalThorn]eld is corrupted in the way that many human [CapitalThorn]elds or areas where people both police and reward themselves become. People when given the power to cheat, cheat. Mathematicians have been given the job of both policing and rewarding each other, so they have reached a point where mathematics is secondary to the bene[CapitalThorn]ts. So the point is that by talking about my research in a public forum, and so thoroughly refuting lies from sci.math'ers--but it not mattering to the mathematicians--the public at large can begin to understand that the [CapitalThorn]eld of mathematics has become corrupted. Eventually, maybe, the world will react and begin policing mathematicians or forcing them to do more to prove that a proof is actually a proof. A good start would be requiring computer proof checking!!! So then as a start, you can consider my position to be one that will *require* that mathematical works over a certain length like 10 pages must be checked by a computer before they will be accepted as being correct because human beings are too error prone when it comes to long works. That is just a reality that most people accept, but for some reason lately they've been acting as if mathematicians can be perfect over hundreds of pages, when the reality is that many long mathematical works are probably just ßawed, like Wiles, but mathematicians feel good saying they're perfect, whether they are or not. James Harris === Subject: : Re: JSH: So what's the point? Discussion, linux) > Like Andrew Wiles has such a simple logical error in his work that > it's explainable to a child, but mathematicians STILL champion him as > having proven Fermat's Last Theorem, when most mathematicans can't > even go over his work in detail and don't bother!!! As recently as January, 2002, you didn't know whether Wiles's work was correct or not[1]. Now, there's a simple logical error explainable to a child. Could you explain it please? So far all you've said is that it involves an informal fallacy that applies only to causal arguments. Since mathematical justi[CapitalThorn]cations are never causal arguments (as far as I can tell), this is puzzling. But I have a three-year-old on hand who will help me out, if you'll just give me that explanation. Footnotes: ,---- | I do hope Wiles' work is correct. It's just at this time I don't | know. | | Still I think his story is kind of suspicious, but hey, truth is | stranger than [CapitalThorn]ction. `---- -- Jesse F. Hughes If the car stops and you're not getting out, then you have to start it again. -- Quincy P. Hughes on his father's skills with a manual transmission. === Subject: : Re: JSH: So what's the point? >So why do I keep posting when sci.math'ers have demonstrated that >they're not willing to agree for years now? that's a good question, especially in light of the fact that sci.math doesn't matter. >[...] >Like Andrew Wiles has such a simple logical error in his work that >it's explainable to a child, but mathematicians STILL champion him as >having proven Fermat's Last Theorem, when most mathematicans can't >even go over his work in detail and don't bother!!! this is the second time recently you've said his proof was wrong. your fans wish you'd explain what the error is. >[...] >A good start would be requiring computer proof checking!!! when are you going to get around to this for your own work, btw? >So then as a start, you can consider my position to be one that will >*require* that mathematical works over a certain length like 10 pages >must be checked by a computer before they will be accepted as being >correct because human beings are too error prone when it comes to long >works. >That is just a reality that most people accept, but for some reason >lately they've been acting as if mathematicians can be perfect over >hundreds of pages, when the reality is that many long mathematical >works are probably just ßawed, like Wiles, but mathematicians feel >good saying they're perfect, whether they are or not. what's your position on the fact that an -error- was found in the [CapitalThorn]rst version of wiles' proof? i mean why didn't we all just agree that -that- one was right? >James Harris ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: JSH: So what's the point? >[...] >Like Andrew Wiles has such a simple logical error in his work that >it's explainable to a child, but mathematicians STILL champion him as >having proven Fermat's Last Theorem, when most mathematicans can't >even go over his work in detail and don't bother!!! > this is the second time recently you've said his proof was wrong. > your fans wish you'd explain what the error is. JSH said somewhere that Wiles's fallacy was cum hoc ergo propter hoc. That seems to mean with this, so because of this. It's a variant, new to me, on the more common tag, post hoc ergo propter hoc which means after this, so because of this. Either way, it's a fallacy all right. But... where does Wiles commit the fallacy? -- Chris Henrich The total lack of evidence is the surest sign that the conspiracy is working. === Subject: : Re: JSH: So what's the point? >A good start would be requiring computer proof checking!!! That's a good start you can make on your own. You can lead by example by using computer proof checking on your own claims. Please do so and report the results (in such a way that they are repeatable by others). === Subject: : Re: JSH: So what's the point? Discussion, linux) >>A good start would be requiring computer proof checking!!! > That's a good start you can make on your own. You can lead by example > by using computer proof checking on your own claims. Please do so and > report the results (in such a way that they are repeatable by others). Oh, but he only wants to require it for papers over ten pages. Maybe (just maybe) his are short enough to escape his tentative requirement. -- This confused and outraged many Matrix fans, who'd already spent hours on the web explaining that man and computers could never really live in such a state of harmony and mutual bene[CapitalThorn]t. -- http://www.pointlesswasteoftime.com === Subject: : Re: division by zero > In the ancient times, one could not calculate 1-2. Then came the negative > numbers. > In the middle agres, one struggled with the square root of positive numbers. > Then came the irrational numbers. > In the middle agres, one struggled with the square root of negative numbers. > Then came the complex numbers. > However, we still haven't solved how to divide by zero. When will we see the > in[CapitalThorn]nite numbers? > A priori, it seems just to be a matter of de[CapitalThorn]nition: One just needs to > invent a new number [CapitalThorn]eld, show that it contains solutions to x*0=1, and > that it contains the complex numbers as a sub [CapitalThorn]eld. In particular, it > should be possible to invert a singular matrix. > Has it already been proven that such a construction is impossible? > Or is this really just projective geometry? Notice that for any y, y * 0 = y * (0 + 0) = y * 0 + y * 0 This uses the de[CapitalThorn]nition of 0 to get 0 = 0 + 0, and distributivity. Then adding the additive inverse of y * 0 to both sides gives y * 0 + [-(y * 0)] = (y * 0 + y * 0) + [-(y * 0)] 0 = y * 0 + (y * 0 + [-(y * 0)]) 0 = y * 0 + 0 0 = y * 0 This uses associativity of addition, the de[CapitalThorn]nition of 0, and the de[CapitalThorn]nition of additive inverses. In other words, anything times 0 is 0. Then if x * 0 = 1, the left side is 0 (by the last sentence), so 0 = 1. (And now if I multiply both sides by 42, I get 0 = 42. And so on. So everything is equal to 0. This makes arithmetic pretty easy, but it doesn't make for very interesting mathematics. :-) ) In other words, you can't have x * 0 = 1 unless you're willing to give up something else. But the properties I used above --- distributivity, the de[CapitalThorn]nition of 0, associativity of addition, the de[CapitalThorn]nition of additive inverses --- are not things that most people would give up. They're some of the axioms for a ring, and most interesting number systems are rings. Negative numbers and complex numbers arise naturally, in trying to solve certain problems (e.g. x + 5 = 3, or x^2 + 1 = 0). It's not clear what would be gained by de[CapitalThorn]ning a multiplicative inverse of 0, particularly when one would be forced to give up something that is more important. This isn't related to de[CapitalThorn]ning in[CapitalThorn]nite quantities. For example, one can construct nonstandard extensions of the reals which contain numbers which are larger than any real number. However, you still can't divide by 0. I don't think this has anything to do with projective geometry. I hope this helps. Bruce I. === Subject: : Re: A functional measure of roughness The measure of roughness based on the Bezier which is like a Lyapunov exponent average, but more sensative: It turns out to be like a second derivative: Measure[n]=Sum[Log[1+Abs[f''(x(i))]/4],{i,1,n} ]/n Limit[Measure[n],n-> In[CapitalThorn]nity]=rho The roughness measure is relate to the Lyapunov exponent average by k=Sum[Log[f'(x(i))],{i,1,n}]/n/Sum[Log[1+Abs[f[CapitalOTilde ]'(x(i))]/4],{i ,1,n}]/n k is close to 2 for the primes. So I have actually found two measures. > In thinking of a way to get a better than Lyapunov , Hausdorff or Kolmogorov > measure of dimension , I thought of this: > F(curve)=0 if smooth and continuous > F(curve)<>0 if rough or discontinuous > The best measure of dimensional roughness (Mandelbrot's way of > expressing it) is the > Lyapunov exponent (or maybe the Hurst exponent?). > Box counting or capacity/ entropy dimension of the Kolmogorov type > is too big most of the time > while Hausdorff being very cut-off measure like > is usually too small. > The trouble with Lyapunov is that it depends on a derivative > and unless you are talking about a fractional derivative, > many fractal functions are of the Weierstrass fractal type > where the classical derivative doesn't exist. > I did some work on Bezier functions in IFS in the past > and fractional partial derivatives of an angular sort as well. > I came to realize that the three point Bezier function of an iterative > sequence in n: > Bezier[p,n]=p^2*f(n+2+2*p*(1-p)*f(n+1)+(1-p)^2*f(n) > is such that if smooth and continuous: > f(n+1)=Bezier[1/2,n]=f(n+2)/4+f(n+1)/2+f(n)/4 > So that the function : > delta[n]=f(n+2)/4+f(n+1)/2+f(n)/4-f(n+1) > is a measure of the roughness. > Putting this measure in an Lyapunov average type function: > Measure[n]=Sum[Log[1+delta[i]],{i,1,n}]/n > I tried this out by comparing it to a known rough set, the primes > and it's Lyapunov integer difference average. > In this experiment the new Bezier roughness measure performs better than the > Lyapunov equivalent over the same range in detecting roughness. > Respectfully, Roger L. Bagula > tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: > 619-5610814 : > URL : http://home.earthlink.net/~tftn > URL : http://victorian.fortunecity.com/carmelita/435/ > ------------------------------------------------------------- ----------- -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: : Re: Partial difference equation, primes > There is no other known in recorded history used to count prime > numbers besides my dS(x,y) and yes, I'm talking about all of human > history here. > To me that's a simple enough claim that it should either be refutable > by you sci.math'ers, or if you're sane, you'll quit trying to > insinuate that it's not a big deal. reference. There are thousands of novel results published yearly -- not all of which are big deals. > James Harris -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: : Re: Partial difference equation, primes >> Posters continually use the world algorithm in a derisive manner, >> when actually the prime counting function is just a formula. >> There is nothing derisive about the term algorithm, the derision you sense arises from the fact that you do >> not understand the term. >Nope. Posters say I just have an algorithm for counting primes like >any other and that the best way to evaluate an algorithm is by speed. >Then they point out that the direct *algorithmic* implementation of my >prime counting function is slow, and they say it's not worth >discussing. >It's all part of a rather steady campaign to try and dismiss my work. >So yes, posters use the word algorithm derisively in a direct attack >on my prime counting research to try and support arguments for >dismissing it. no, people point out that it's your algorithm, not your Ôfunction', because they think that you -might- be interested in using the terminology correctly instead of making a fool of yourself. what you [claim to have] invented is -an algorithm for calculating- the prime counting function - the -function- pi(n) is the same function regardless of what algorithm is used to calculate it. at least people used to point this out, eventually they gave up. yes, people are dismissive of your algorithm, but that has nothing to do with the word Ôalgorithm'. >> The compressed explicit prime counting function exists as I've shown. >> Notice it too is a formula and not an algorithm. >> Here's a de[CapitalThorn]nition of algorithm: >> SYLLABICATION: >> algorithm >> PRONUNCIATION: >> AUDIO: lg-rthm KEY >> NOUN: >> A step-by-step problem-solving procedure, >> especially an established, recursive >> computational procedure for solving a problem >> in a [CapitalThorn]nite number of steps. >> ETYMOLOGY: >> Variant (probably inßuenced by arithmetic) of >> algorism. >> OTHER FORMS: >> algorithmic (-rthmk) ?ADJECTIVE >> algorithmically ?ADVERB >> How does your implementation of a prime counting function differ from an algorithm? >By that de[CapitalThorn]nition ANY [CapitalThorn]nite summation is an algorithm. >That is, whenever you see the summation sign in mathematics for a >[CapitalThorn]nite sum then by the position of this poster it's an algorithm. not quite. a [CapitalThorn]nite sum, like sum[n=1..42] a[n], is a -number-. the obvious method for [CapitalThorn]nding this number, namely s = 0 for n = 1 to 42 s = s + a[n] return s is an algorithm. so what? ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: Partial difference equation, primes >> oops; >> I'm sure that Harris can [CapitalThorn]nd your little mistake, if >> even I can [CapitalThorn]gure it out. >I wouldn't bet on it! Many of his replies to posts remind me of an >improved Eliza program: He parses the structure of a post, just enough >to delete things and insert his rantings in suitable places, but without >making any attempt to understand the actual contents. For a computer >program, this would actually be quite clever. why do you say that he parses the structure of a post, just enough to delete things and insert his rantings in suitable places, but without making any attempt to understand the actual contents? >> However, the straightforward de[CapitalThorn]nition of prime numbers gives an even >> less complex de[CapitalThorn]nition: >> Let f (x) = product (x mod i) for 2 <= i < x >> Let g (x) = 0 if f (x) = 0 and g (x) = 1 otherwise. >> Then pi (n) = sum (f (x)) for 2 <= x <= n. ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: tossing coins >I have a question regarding the distribution of a coin tossing >experiment. Somebody actually simulated the following experiment and >claims it is exponentially distributed. I would like to know if this is >common knowledge, or if it is false. So I describe the experiment: >I continously toss a coin with the outcome H or T. Assume also I have a >subset of the integers I = {1,...,N} for some large integer constant N. >I also have a set K = {N+1,..,2*N}. >Each time I toss my coin repeatedly with 1 seconds inter-arrival time I >do the following: >- If I get a H I randomly (with uniform distribution) remove one element >in K and add it to I. >- If I get a T, I do the reverse, ie. I remove an element randomly (with >uniform distribution} from I and add it to K. >I do this for suf[CapitalThorn]ciently long time (in[CapitalThorn]nitely). If we count the >number of seconds each integer was inside K before being removed, and >plot a frequency diagram over the lifetime of each integer in K, will >it be exponentially distributed? If so why? Okay, I've simply recreated the simulation (see Matlab code below). The result is not an exponential distribution. Hope this helps (I also hope I did it correctly ;) ). ---begin code--- close all; clear all; N = 1000; pH = 0.5; % probability heads T = 500; pT = 1 - pH; % probability tails. for x = 1:(2*N) A(x) = 1/x; P(1,x) = pH*A(x); end for t = 1:T for x = 1:(2*N) if x == 1 P(t+1,x) = pT*P(t, x+1); elseif x == (2*N) P(t+1,x) = pH*(1-A(x))*P(t,x-1) + (pT*P(t, x)); else P(t+1,x) = (pH*(1-A(x))*P(t,x-1)) + (pT*P(t, x+1)); end end end plot(1:(T+1), P(:,N)); y = log(P(:,N)); U = poly[CapitalThorn]t((1:(T+1)), y', 1); u = U(1)*(1:(T+1)) + U(2); d = y'-u; [CapitalThorn]gure; plot(d); % d is the difference between an exponential distribution % and the found Ôcurve' --- end code --- === Subject: : Re: tossing coins > I have a question regarding the distribution of a coin tossing > experiment. Somebody actually simulated the following experiment and > claims it is exponentially distributed. I would like to know if this > is common knowledge, or if it is false. So I describe the experiment: > I continously toss a coin with the outcome H or T. Assume also I have > a subset of the integers I = {1,...,N} for some large integer constant > N. I also have a set K = {N+1,..,2*N}. > Each time I toss my coin repeatedly with 1 seconds inter-arrival time > I do the following: > - If I get a H I randomly (with uniform distribution) remove one > element in K and add it to I. > - If I get a T, I do the reverse, ie. I remove an element randomly > (with uniform distribution} from I and add it to K. > I do this for suf[CapitalThorn]ciently long time (in[CapitalThorn]nitely). If we count the > number of seconds each integer was inside K before being removed, and > plot a frequency diagram over the lifetime of each integer in K, > will it be exponentially distributed? If so why? I can believe this is true. Let p be the probability of heads. Fix an integer k originally in K and let X be the time until it is [CapitalThorn]rst moved . It is not hard to show that for any positive integer x, P{X <= x} <= p x / (N-x). Hence, X tends to in[CapitalThorn]nity (in distribution) as N grows arbitrarily large. Thus, to talk about an exponential distribution, let us [CapitalThorn]rst scale the problem: Say you ßip a coin every 1/N seconds. For N >> n, the probability that k is chosen at time n/N given that it has not been chosen before is roughly p/N. That is, the probability that k is picked in any given time interval of small length (e.g., 1/N), given that it has not been picked before, is approximately p times the length of the interval. We say that X has constant hazard rate; a constant hazard rate is unique to exponential distributions. By this informal argument, in this scaled situation, the distribution of X tends towards the exponential distribution with parameter p. If you wish, now rescale the problem. Then, for large N, X has roughly exponential distribution with parameter p/N. If probability interests you, you might pick up a basic book ont he subject. Ross SM, A First Course in Probability is an excellent elementary introduction. Drake AW, Fundamentals of Applied Probability Theory provides an introduction which, though less mathematically precise, might also impart insights similar to the one used in the explanation above. Now some exercises for those who already know probability. 1) Let k be a [CapitalThorn]xed interger in I. For large N, what can you say about the distribution of the time k is [CapitalThorn]rst removed from K? (Yes, K.) 2). Fix k in K, and let X be the time until k is [CapitalThorn]rst removed from K. Develop equations which determine the expectation, variance, and moment generating function of X. Use these to determine the expectations, variances, and moment generating functions for a few small values of N. 3) Consider the M/GI/infty queue were service times follow a uniform distribution. What is the density of the time until the departure of the [CapitalThorn]rst customer whose service time exceeds x? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: : Re: tossing coins ask a few things. > I can believe this is true. > Let p be the probability of heads. Fix an integer k originally in > K and let X be the time until it is [CapitalThorn]rst moved . It is not hard to > show that for any positive integer x, P{X <= x} <= p x / (N-x). > Hence, X tends to in[CapitalThorn]nity (in distribution) as N grows arbitrarily > large. Please regard N as a constant, I should have maybe used a small letter. Also, p is 0.5 as there is equal probability of H and T. I understand your sentence Hence, X tends to in[CapitalThorn]nity as N grows arbitrarily large. The probability of picking a number in a set which is ini[CapitalThorn]nitely large is 1/infty, which limits 0. But I do not understand the formula P{X <= x} <= p x / (N-x). P(X <= x) is the CDF for the life-times X right, why should it be less than or equal to p*x/(N-x) ? P(X = x) = 0.5 / N, right ? Then the CDF P(X <= x) = 0.5x/N, right? > Thus, to talk about an exponential distribution, let us [CapitalThorn]rst scale the > problem: Say you ßip a coin every 1/N seconds. For N >> n, the > probability that k is chosen at time n/N given that it has not been > chosen before is roughly p/N. That is, the probability that k is Why is it p/N, why isn't it 0.5n/N. The [CapitalThorn]rst ßip is done at time 0, and the probability of picking k is 0.5/N . As time advances to n/N where you have ßipped the coin n times, the probability should be 0.5n/N not 0.5/N (I assume everywhere p=0.5). picked in any given time interval of small length (e.g., 1/N), given > that it has not been picked before, is approximately p times the > length of the interval. We say that X has constant hazard rate; a > constant hazard rate is unique to exponential distributions. By this > informal argument, in this scaled situation, the distribution of X > tends towards the exponential distribution with parameter p. I am lost :-(. > If you wish, now rescale the problem. Then, for large N, X has > roughly exponential distribution with parameter p/N. > If probability interests you, you might pick up a basic book ont he > subject. Ross SM, A First Course in Probability is an excellent > elementary introduction. Drake AW, Fundamentals of Applied Probability > Theory provides an introduction which, though less mathematically > precise, might also impart insights similar to the one used in the > explanation above. Andersen === Subject: : Re: tossing coins > I continously toss a coin with the outcome H or T. Assume also I have > a subset of the integers I = {1,...,N} for some large integer constant > N. I also have a set K = {N+1,..,2*N}. > Each time I toss my coin repeatedly with 1 seconds inter-arrival time > I do the following: > - If I get a H I randomly (with uniform distribution) remove one > element in K and add it to I. > - If I get a T, I do the reverse, ie. I remove an element randomly > (with uniform distribution} from I and add it to K. > I do this for suf[CapitalThorn]ciently long time (in[CapitalThorn]nitely). If we count the > number of seconds each integer was inside K before being removed, and > plot a frequency diagram over the lifetime of each integer in K, > will it be exponentially distributed? If so why? >> I can believe this is true. >> Let p be the probability of heads. Fix an integer k originally >> in K and let X be the time until it is [CapitalThorn]rst moved . It is not >> hard to show that for any positive integer x, P{X <= x} <= p x / >> (N-x). Hence, X tends to in[CapitalThorn]nity (in distribution) as N grows >> arbitrarily large. > Please regard N as a constant, I should have maybe used a small > letter. Also, p is 0.5 as there is equal probability of H and T. > I understand your sentence Hence, X tends to in[CapitalThorn]nity as N grows > arbitrarily large. The probability of picking a number in a set which > is ini[CapitalThorn]nitely large is 1/infty, which limits 0. > But I do not understand the formula P{X <= x} <= p x / (N-x). > P(X <= x) is the CDF for the life-times X right, why should it be > less than or equal to p*x/(N-x) ? > P(X = x) = 0.5 / N, right ? Then the CDF P(X <= x) = 0.5x/N, right? No. X=x iff k is not picked the [CapitalThorn]rst x-1 times and then picked the xth time. The exact probability is complicated, since you need to take into consideration whether coins are added to or taken away from K at each of the x-1 previous trials. To get the upper bound, for integer x, X <= x if and only if k is picked on one of the [CapitalThorn]rst x trials. The probability of k being picked on the nth trial is at most p/(N-n) <= p/(N-x). It is a basic theorem of probaility that the probability of a disjunction of events is at most the sum of the probability of events. >> Thus, to talk about an exponential distribution, let us [CapitalThorn]rst scale >> the problem: Say you ßip a coin every 1/N seconds. For N >> n, >> the probability that k is chosen at time n/N given that it has >> not been chosen before is roughly p/N. That is, the probability >> that k is > Why is it p/N, why isn't it 0.5n/N. The [CapitalThorn]rst ßip is done at time 0, > and the probability of picking k is 0.5/N . As time advances to n/N > where you have ßipped the coin n times, the probability should be > 0.5n/N not 0.5/N (I assume everywhere p=0.5). We are talking about the *conditional* probability that k is picked at time n, *given* that it has not been picked before. At time n, there are between N-n and N+n integers in K; since N >> n, there are approximately N integers in K. > picked in any given time interval of small length (e.g., 1/N), given >> that it has not been picked before, is approximately p times the >> length of the interval. We say that X has constant hazard rate; a >> constant hazard rate is unique to exponential distributions. By this >> informal argument, in this scaled situation, the distribution of X >> tends towards the exponential distribution with parameter p. > I am lost :-(. Here is how the exponential distribution arises: Suppose you are the entry of a customer. You model this process as follows: The probability of an event occuring to a small time inverval (t, t+dt] is roughly proportional to its length dt, regardless of what has happened previously, with a [CapitalThorn]xed constant of proportionality. I.e., there is some constant c such that the probability of an emission (or arrival) in (t, t+dt] is approximately c dt, regardless of t.. (c is called the hazard rate.) Then the time until the [CapitalThorn]rst emission has an exponential distribution with parameter c. (For simplicity, I have omitted some details in this informal explanation.) -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: : Re: abstract algebra..... === Subject: : Re: abstract algebra..... >If g: F --> R is onto, actually R must be a [CapitalThorn]eld if F is. If f ring homomorphism from [CapitalThorn]eld F onto non-trivial ring R, then R is [CapitalThorn]eld and isomorphic to F Corollary. A surjective homomorphism between [CapitalThorn]elds is an isomorphism. The [CapitalThorn]rst step to show is R is commutative ring with identity. If r,s in R, then some x,y in F with f(x) = r, f(y) = s rs = f(x).f(y) = f(xy) = f(yx) = f(y).f(x) = sr Some nonzero a in R. Some x in F with f(x) = a. f(1) nonzero. Otherwise a = f(x) = f(x1) = f(x).f(1) = a0 = 0 For all r in R, some y in F with f(y) = r. r.f(1) = f(y)f(1) = f(y1) = f(y) = r f(1) identity 1 /= 0 So far F may as well be a ring. Thus when f:R -> R' is ring homomorphism onto a non-trivial ring R' if R is commutative, then so is R' if R has (left or right) identity, then so does R' (Note, (left or right) identity nonzero iff ring non-trivial.) For all nonzero r in R, some nonzero x in F with f(x) = r 1 = f(1) = f(xx^-1) = f(x).f(x^-1) = r.f(x^-1) f(x^-1) multiplicative inverse r. If f(x) = 0, x /= 0, then 1 = f(1) = f(xx^-1) = f(x).f(x^-1) = 0. But 1 /= 0. Thus ker f = { 0 }, f [CapitalThorn]eld ismorphism. ---- === Subject: : Re: abstract algebra..... > hello....doctor~ > suppose that > F is [CapitalThorn]eld. > R is ring. > g : F -> R is surjective ring homomorphism. > show that g is isomorphism. > *** since g is surjective, g(1) is unity of R. *** > --------------------------------------- > um......i can't understand this part ***. As several people pointed out, you have to suppose that R is not the 0 ring. Although some writers ban that ring, that is wrong. On the other hand, a [CapitalThorn]eld is usually assumed not the 0 ring. (Why the difference? Well the category of rings is equational if you allow 0, but [CapitalThorn]elds are more like primes and 1 is not a prime.) Anyway, although it is certainly true that the only ideals in a [CapitalThorn]eld are {0} and the whole [CapitalThorn]eld, let me answer the question directly. Since g is surjective, let r in R and write r = g(x). Then r.g(1) = g(x).g(1) = g(1.x) = g(x) = r, so g(1) is indeed the identity of R. === Subject: : Re: Automorphisms of groups === Subject: : Re: Automorphisms of groups >In general, I recall that Z*_n = A(Z_n), and Z_*2 = (1). >Z*_3 = (-1,1) =~ Z_2 ; Z*_4 = Z_2 = (1,3) = (-1,1) >> Interesting enuf. >> When m,n coprime, Z_mn = Z_m x Z_n; Thus >> Z_mn^* = (Z_m x Z_n)^* =??= Z_m^* x Z_n^* >> Whence voile, phi(mn) = phi(m) phi(n) >> What about phi(p^n) = p^n - p^(n-1) ? >> Back to the old fashion way of counting? >How about |Z_p^n| = p^n, and A = {k such that p | k and k in Z_p^n} >= {k | (k,p) = p and k in Z_p^n} = {qp | q = 1,2, ... ,p^(n-1)}, so >|A| = p^(n-1) and |Z*_p^n| = phi(p^n) = |Z_p^n| - |A| = p^n - p^(n-1) >Well, I guess this is back to the old way of counting. r in (Z_p^n)^* iff (p^n,r) = 1 iff (p,r) = 1 iff not p|r (Z_p^n)^* = (Z_p^n){ pk + p^n.Z | 0 <= pk < p^n } = (Z_p^n){ pk + p^n.Z | 0 <= k < p^(n-1) } ---- === Subject: : Re: Automorphisms of groups === Subject: : Re: Automorphisms of groups, Automorphed These days, whatever they Ôimprove', it's never as good. >different fonts, hide and don't hide quotes, etc., in posts now. >It has problems with sci.math because its auto-formatter doesn't >know how to deal with the ASCII math symbol conventions here.) How modern corporations can turn perfectly working and useful facilities into worthless overpriced pieces of junks, would be the highest of humor if only those corporations weren't off shored onto my home planet. -- the FOAD (Fabulous offhand alien diatribe) Once upon a time computers all spoke Ascii. But in their arrogance acclaiming to know it all, they were cursed to babble in a multitude of confusing formats. They even thought they could talk people, but as they can't, they's gonna make people talk computer. After that, the next quantum leap in productivity will be replacement of consumers with computers. -- Updates are worse than blind dates. Who wants date with Billy Gate who makes debate why him we hate This massage automatically smell checked by MicroSoft. -- Are you soft on MicroSoft? As the [CapitalThorn]rst computer was a [CapitalThorn]ve [CapitalThorn]nger model, historical records are not available. This model was soon superseded by a twice as powerful ten [CapitalThorn]nger model. A supercomputing twenty digit model was designed but soon deemed unmarketable because of a newfangled fad of wearing foot gear. Other computer systems experimented with pebbles. Then a new generation of computers came forth, utilizing a technological innovation, namely bead making. Portable models carried as necklaces soon became popular. However wrangling merchants in passionate greed so often strangled themselves computing potential pro[CapitalThorn]ts that they were soon replaced by a safer table top model, now know to us as the abacas. Mechanical versions of the abacas, using clogs instead of beads, became passing fancy, until with the progression of time, the beads were miniaturized into small electronic charges and even magnetic bubbles. Thus you see the ancient archaic principle of digits, now revered as bits and bytes, established well over 10,000 years ago, remains the established norm. So I ask you, as the basic method of computing has remained the same since the stone age, why would you expect Miraculous MicroSoft Mirages to work any better than a [CapitalThorn]nger? -- Riddle of the day Why couldn't the mathematican fall to sleep counting sheep? ---- === Subject: : Re: Easy number theory problem === Subject: : Re: Easy number theory problem >CRT: (m,n) = 1 and x in Z_m, y in Z_n ==> there is a unique >z in Z_mn such that z = x mod m and z = y mod n. >> As m,n coprime, some a,b with am + bn = 1 >> bn = 1 (mod m); am = 1 (mod n) >> sam + rbn = r (mod m); sam + rbn = s (mod n) >the ideals mZ == (m). >> The actual bijection you want is >> f:Z_mn -> Z_m x Z_n, z + mnZ -> (z + mZ, z + nZ) >> which you need to show is well de[CapitalThorn]ned. >> If x + Z_mn = y + Z_mn, then x = y (mod mn), >> x = y (mod m), x = y (mod n); x + mZ = y + mZ, x + nZ = y + nZ >> (x + mZ, x + nZ) = (y + mZ, y + nZ) >> Thus it's well de[CapitalThorn]ned. >or x + (mn) = y + (mn) ==> x - y in (mn) = mnZ = mZ/nZ . >so x - y in mZ and nZ or (x + mZ, x + nZ) = (y + mZ, y + nZ) >so well de[CapitalThorn]ned. You're so idealistic. >> As m,n coprime, some a,b with am + bn = 1 >> bn = 1 (mod m); am = 1 (mod n) >> sam + rbn = r (mod m); sam + rbn = s (mod n) >> f(sam + rbn) = (r,s) in Z_m x Z_n >> Thus f surjection. >> Rest is direct. f is ring homomorphism >> f(x+y + mnZ) = (x+y + mZ, x+y + nZ) = (x+mZ + y+mZ, x+nZ + y+nZ) >> = (x + mZ, x + nZ) + (y + mZ, y + nZ) = f(x) + f(x) >> f(xy + mnZ) = (xy + mZ, xy + nZ) >> = ((x + mZ)(y + mZ), (x + mZ)(y + nZ)) >> = (x + mZ, x + nZ)(y + mZ, y + nZ) = f(x).f(y) >> If f(z) = (0,0), then z = 0 (mod m), z = 0 (mod n), >> some k with z = km; n|z; n | km; m,n coprime; n | k >> nm | km; nm | z; z = 0 (mod nm) >> Thus ker f = { 0 }; f injection. >Yes, this is the best way. CRT + ker f = 0 (and f a ring homo), so f >1-1 and onto, >of course |Z_mn| = mn = |Z_m x Z_n|, >so 1-1 ==> onto for [CapitalThorn]nite sets. Hm, so proving surjection as above with CRT isn't necessary. Whence CRT directly from Z_mn = Z_m x Z_n. However the proof has one advantage, namely as it's constructive, it offers a direct way of calculating the CRT solution. -- >Now I want to show that f extends to a group isomorphism of Z*_mn >with Z*_m x Z*_n. This should be straightforward it seems, though >I have never done it. i.e. to show >f : Z*_mn --> Z*_m x Z*_n : f(z + mnZ) = (z + mZ, z + nZ) Have you put the saddle on backwards? The restriction of f to Z_mn^* is what you're wanting. if u unit in Z_mn, then some v with uv = 1 (1,1) = f(1) = f(uv) = f(u)f(v), f(u) unit in Z_m x Z_n f(u) = (u_m, u_n); f(v) = (v_m, v_n); u_m = u + mZ (1,1) = (u_m v_m, u_n v_n); u_m, u_n units in Z_m, Z_n f(u) in Z_m^* x Z_n^* We already know f is injection and multiplicative homomorphism. What's left is surjectiveness. If u_m, u_n units in Z_n, Z_m some v_m, v_n in Z_m, Z_n with u_m v_m = 1 = u_n v_n some r,s with f(r) = (u_m, u_n), f(s) = (v_m, v_n) f(rs) = f(r).f(s) = (u_m, u_n)(v_m, v_n) = (u_m v_m, u_n v_n) = (1,1) Thus rs = 1 as f injection, and r is unit. >with (m,n) = 1 and (z,m) = (z,n) = (z,mn) = 1. >I tried to do something like this, but kept having problems. ... >Any comments on this? Am I just confused? ---- === Subject: : Re: Easy number theory problem === Subject: : Re: Easy number theory problem >Proof of the case Z*_p is cyclic for p = 13. >There are 12 = 2^2*3 elements, which are the distinct solns. of >x^(p-1) - 1 = x^12 - 1 = 0. >In general, p - 1 = q = p1^e1 ... pr^er. Let s^e = pi^ei for some i. >Let y = x^s (s = pi for some i). >x^(s^e) - 1 = y^(s^(e-1)) - 1 = 0 x^(pi^ei) = y^(pi^(ei - 1)). That notation is so confusing. >has s^(e-1) distinct solns. Why? >y = x^pi; with o(y) <= s^(e-1), and >s^e - s^(e-1) = s^(e-1)(s - 1) with o(x) > s^(e-1). Huh? What did you say? >Since o(x) | s^e, there are s^(e-1)(s - 1) solns with o(x) = s^e. >Thus there is an element of order pi^ei for each i, and their >product is an element of order > q = p1^e1 ... pr^er, >which generates the whole group which must be cyclic. Where that rabbit go? >Case of p = 13; >For s^e = 2^2 = 4 >y^4 - 1 = 0 has 4 solns ==> 4 elements with orders <= 4 >==> 12 - 4 = 8 elements with orders > 4. It will turn out that >we have Z_12, and the 4 with order <= 4 are (0,3,6,9) in additive >notation. Since 2^6 mod 13 = 64 mod 13 = -1 mod 13, Makes no tangible sense. Do you mean 2^6 = 64 = -1 (mod 13) ? >Z*_13 = <2>, and we could take these 4 elements to Why go on, have you not shown or claimed 2 generates Z_13^* = <2> ? >be the (0,3,6,9)th powers of 2. >If we let s^e = 3^1 = 3, we have x^3 - 1 = 0 ==> 3 elements >with orders <= 3. ( Z_12 = Z_3 x Z_4. ) The product of an element of >order 3 with one of order 4 is one of order 12, which generates the >whole group, so it must be cyclic. For a in Z_p, let d_a = d be the smallest for which a^d = 1. Show o(a^j) = d iff (j,d) = 1. Thus o(a) = d ==> |{ x | o(x) = d}| = phi(d) For all d | p-1, |{ x | o(x) = d}| = 0 or phi(d) sum(d | p-1) phi(d) = p - 1 Thus for all d | p-1, some a with o(a) = d. In paticular, p-1 | p-1, thus some a with o(a) = p-1 This generalizes to show [CapitalThorn]nite [CapitalThorn]elds are cyclic. -- >Consider the group Z*_8 =~ Z_2 x Z_2 (not cyclic). Where does the >argument fail--where does the fact these are not the nonzero >elements of a [CapitalThorn]eld enter. Well, since or if the arguement doesn't fail for Z_8^*, then ... >I guess we can't say that the elements have x^4 - 1 = 0 as a >splitting [CapitalThorn]eld, even though it is true that phi(8) = 4, >and x^4 = 1. (Actually x^2 = 1). >Any comments on this part of it? 7.5.4. Theorem. [Fundamental Theorem of Finite Abelian Groups] Any [CapitalThorn]nite abelian group is isomorphic to a direct product of cyclic groups of prime power order. Any two such decompositions have the same number of factors of each order. 7.5.5. Proposition. Let G be a [CapitalThorn]nite abelian group. Then G is isomorphic to a direct product of cyclic groups such that n_i | n_i-1 for i = 2,3,...k. What do you think that means? 7.5.6. Corollary. Let G be a [CapitalThorn]nite abelian group. If a in G is an element of maximal order in G, then the order of every element of G is a divisor of the order of a. 7.5.7. Lemma. Let p be a prime number, and let k,a,b be integers. (a) If 1 <= k <= p-1, then p is a divisor of the binomial coef[CapitalThorn]cient p!/k!(p-k)! (b) If k >= 1 and a is congruent to b (mod p^k), then a^p is congruent to b^p (mod p^{k+1} ). (c) If k >= 2 and p is an odd prime, then (1 + ap)^(p^(k-2)) = 1 + ap^(k-1) (mod p^k) (d) If p is an odd prime and p is not a divisor of a, then (1 + ap)^(p^(k-1)) = 1 (mod p^k) (1 + ap)^(p^(k-2)) /= 1 (mod p^k) 7.5.8 Theorem. Let p be an odd prime, let k be a positive integer, and let n=p^k. Then Z_n^x is a cyclic group. -- 3.5.6. De[CapitalThorn]nition. Let G be a group. If there exists a positive integer N such that a^N = e for all a in G, then the smallest such positive integer is called the exponent of G. 3.5.8. Proposition. Let G be a [CapitalThorn]nite abelian group. (a) The exponent of G is equal to the order of any element of G of maximal order. (b) The group G is cyclic if and only if its exponent is equal to its order. Zp^* cyclic. Let z be an element with maximal order n; n|p-1 for all x in Zp^*, x^(p-1) = 1 = x^n; n = p-1 for p-1 <= n as 4.1.12 Corollary. A polynomial of degree n with coef[CapitalThorn]cients in the [CapitalThorn]eld F has at most n distinct roots in F. ---- === Subject: : Re: Easy number theory problem Proofs of Fermat's theorem: 1) Calculate (x + 1)^p (mod p) = Sum(m=0,p) C(p,m) x^m (mod p) = x^p + 1 (mod p) since for m = 1, ...,p-1, the p | C(p,m) = binomial coeff. C(p,m) = p!/[m!(p-m)!] = pn/m!, where n = (p-1)!/(p-m)! . (m!,p) = 1 and m! | pn ==> m! | n and p | C(p,m). (I always take mod p to mean mod pZ, i.e., x (mod p) = [x] = x + pZ = equivalence class of x in Z/pZ.) Let f(x) = x^p - x, then f(x + 1) = (x + 1)^p - x - 1 = x^p - x (mod p) so f(x + 1) = f(x) (mod p). f(0) = 0 mod p, so by induction f(x) = x^p - x = 0 (mod p) for all x in Z, and x^(p-1) = 1 (mod p) for x != 0. 2) For x in Z but not in pZ ((x,p) = 1) and 1 <= j < p consider xj = u_j (mod p) Write x = qp + k with 1 <= k < p. Clearly f_j(x) = xj (mod p) = jk mod p = u_j mod p is a bijection f_j : Z*_p --> Z*_p (I don't give the details here, everything is straightforward. Use (x,p) = (j,p) = (k,p) = (u_j,p) = 1). Thus u_1 u_2 ... u_p-1 = (p-1)! . Since u_j = jx for j in Z*_p, this gives x^(p-1) (p-1)! = (p-1)! so x^(p-1) = 1 if (x,p) = 1. 3) In a group G = Z*_p (I will not show that this is a group), we have o(x) | p-1 so x^(p-1) = 1 (mod p) for all x in Z*_p. Van === Subject: : Re: Easy number theory problem === Subject: : Re: Easy number theory problem >Proofs of Fermat's theorem: >> First show (Z_p)^* is a multiplicative group, ie show Z_p is a >> [CapitalThorn]eld. >You like to go back to basics, don't you? >I guess that is good, esp. on sci.math. Yup, it's most good for knowing math. >> First off 1 is an identity. Now for all a not divisible by p, >> some n,m with an + pm = 1; an = 1 (mod p) >> Hence a has a multiplicative inverse. >> Thus as Z_p has p elements, >> (Z_p)^* = (Z_p)0 >> has p-1 elements with o(a) = p-1. >How do you know o(a)? You can say o(a) | p - 1, but >shouldn't you show that (Z_p)^* is cyclic? You don't need to. By Lagrange's Theorem. No, not needed. Let k = (p-1)/o(a) in N a^(p-1) = a^(k.o(a)) = (a^o(a))^k = 1^k = 1 >Let m = p - 1. Then o(x) | m for all x in F* = F0. They are all >solns. of x^m - 1 = 0, which has m distinct solns. (m x^(m-1) not 0 >if x not 0). This seems like a digression or generalization into [CapitalThorn]nite [CapitalThorn]elds. >I like the ring theory approach, eith ideals. I recall >if M = (m) and N = (n) are ideals of Z, (m,n) = 1 becomes >M + N = Z, and M/N = (0) >> Namely >> a^(p-1) = 1 (mod p). >> To use ring theory, show pZ is a maximal ideal of Z, thus by >> theorem Z_p = Z/pZ is a [CapitalThorn]eld (clearly with p elements in it). >Prime ideals are maximal in Z since if pZ is a proper subset of a >maximal ideal M, there is an a in M but not in pZ. But this >means that (a,p) = 1 or pZ + (a) = M = Z, where (a) = aZ is a subset >of M. Thus pZ is maximal. Estranged approach. pZ for prime p is maximal as pZ is an ideal and if n not in pZ, then 1 in ideal , the ideal generated by n an pZ. As n not in pZ, (n,p) = 1; some a,b with an + bp = 1 in . QED. Theorem Every nonzero prime ideal of a principal ideal domain is maximal. pZ is prime ideal for pZ is an ideal and if ab in pZ, then p | ab; p | a or p | b; a in pZ or b in pZ Conversely, a maximal ideal of a communitative ring with identity is prime. -- Four proofs of Fermat's theorem: >1) Calculate (x + 1)^p (mod p) = Sum(m=0,p) C(p,m) x^m (mod p) >= x^p + 1 (mod p) >since for m = 1, ...,p-1, the p | C(p,m) = binomial coeff. >C(p,m) = p!/[m!(p-m)!] = pn/m!, where n = (p-1)!/(p-m)! . >(m!,p) = 1 and m! | pn ==> m! | n and p | C(p,m). >(I always take mod p to mean mod pZ, i.e., x (mod p) = [x] >= x + pZ = equivalence class of x in Z/pZ.) Why not just use as you basically showed above (x + 1)^p = x^p + 1 (mod p) Thus [(x + 1)^p] = [x^p + 1] x = a mod p isn't mathematics, it's nebulous de[CapitalThorn]ned computer notation and nearly ambigous with x = a mod p, the relation. Please use x = [a]. [a] is a coset, a mod p is a number, a bogus number invented by object orientated computer programmers. >Let f(x) = x^p - x, then f(x + 1) = (x + 1)^p - x - 1 = x^p - x (mod >p) so f(x + 1) = f(x) (mod p). f(0) = 0 mod p, so by induction >f(x) = x^p - x = 0 (mod p) for all x in Z, and No, for all x in Z+, x^p - x = 0 (mod p) However descending f(x - 1) = f(x) (mod p), thus for all x in Z, x^p - x = 0 (mod p) >x^(p-1) = 1 (mod p) for x != 0. p^(p-1) = 1 (mod p) ?? Be careful of that division. >2) For x in Z but not in pZ ((x,p) = 1) and >1 <= j < p consider xj = u_j (mod p) >Write x = qp + k with 1 <= k < p. Clearly f_j(x) = xj (mod p) >= jk mod p = u_j mod p is a bijection f_j : Z*_p --> Z*_p None of this is clear including the notation. >(I don't give the details here, everything is straightforward. >Use (x,p) = (j,p) = (k,p) = (u_j,p) = 1). >Thus u_1 u_2 ... u_p-1 = (p-1)! . Since u_j = jx for j in Z*_p, >this gives x^(p-1) (p-1)! = (p-1)! so x^(p-1) = 1 if (x,p) = 1. >3) In a group G = Z*_p (I will not show that this is a group), >we have o(x) | p-1 so x^(p-1) = 1 (mod p) for all x in Z*_p. Finally a simple proof of Fermat's little theorem. g in G iff g unit Z_p if g,h in G: g,h units Z_p; gh unit; gh in G 1 identity in G; for all g in G, some h in G with gh = 1. 4) pZ is maximal ideal, thus Z_p = Z/pZ is a [CapitalThorn]eld. Hence Z_p^* = Z_p0 is a multiplicative group of order p-1. Or if you prefer, pZ is a prime ideal in an integral domain, hence maximal ... ---- === Subject: : Re: Help, need to come up with simple equation Ah just what I needed, thank you Andersen and Raymond. Sorry for not providing all of the information, next time I won't post if I'm half asleep ^^. === Subject: : request for ideas I am planning to write a paper which surveys mathematical results that show that the old axiom->proof->theorem way of doing mathematics does not always yield complete information about mathematics. The prime example of this (which started it all) is Godel's Incompleteness Theorem, but there has been a lot of work in this area since then. For instance, Gregory Chaitin has an incompleteness theorem which shows conclusively that a certain number which he calls Omega, which is really the probability that a computer program halts (de[CapitalThorn]ned in a way that makes sense), is a random number - which implies that there is no [CapitalThorn]nite axiom system that can yield all of the bits of Omega. He concludes from all of his work that sometimes one has to simply perform experiments in mathematics and form conclusions from the experiments without being absolutely certain that the conclusions are correct. It is these types of very original ideas that I am looking for to put in my paper, that there are some problems out there that are so dif[CapitalThorn]cult for us to get a grip on that we might have to approach them like a chemist approaches chemistry, never being 100% sure that his or her theories are always correct. Anyone who knows of results like these or has done work in this area or has original ideas is welcome to respond to me on usenet or if you want, you can email me directly. Craig === Subject: : Re: request for ideas Hi Craig, I've got something that you might be interested in. There is a theory in geology that could do with some mathematical input. Basically all the ancient continents can be joined together to make one continuous land mass, but the only way to do this is to consider the earth was smaller in the past. The concept of an earth that has been expanding in size is such an amazing idea that most people refuse to believe it. But whether you believe it or not there is still the fact that all the ancient continents can only be reconstructed on a smaller diameter earth. This fact has been explained away by saying that some of the ancient ocean ßoors have been consumed within the earth. If you look at my web site on the subject you should get the concept. Look at this page; http://www.dinox.org/english/geoevid.htm and the earth has been expanding in size link from that page for example. http://www.dinox.org/expic/exp-30.htm Anyway, to get back to your question. It's dif[CapitalThorn]cult to prove that the ocean ßoor hasn't been consumed within the earth. Would it be possible to calculate the probability that all the continents would [CapitalThorn]t together on a smaller earth? How likely is this? If we had two identical jigsaw puzzles and one person had [CapitalThorn]tted all the pieces together to make a small picture, but the other had a larger picture with large gaps in his picture (which he said must have been lost) we would surely all know that the small picture was correct. Why is this different from reconstructing the ancient earth as a smaller diameter? Would a mathamathical study of the probablity of this reconstruction occuring by chance help the argument or not? Or will we never be 100% certain who is correct? Hope that gives you a few ideas to be going on with. Stephen Hurrell > I am planning to write a paper which surveys mathematical results that > show that the old axiom->proof->theorem way of doing mathematics > does not always yield complete information about mathematics. The > prime example of this (which started it all) is Godel's Incompleteness > Theorem, but there has been a lot of work in this area since then. > For instance, Gregory Chaitin has an incompleteness theorem which > shows conclusively that a certain number which he calls Omega, which > is really the probability that a computer program halts (de[CapitalThorn]ned in a > way that makes sense), is a random number - which implies that there > is no [CapitalThorn]nite axiom system that can yield all of the bits of Omega. He > concludes from all of his work that sometimes one has to simply > perform experiments in mathematics and form conclusions from the > experiments without being absolutely certain that the conclusions are > correct. > It is these types of very original ideas that I am looking for to put > in my paper, that there are some problems out there that are so > dif[CapitalThorn]cult for us to get a grip on that we might have to approach them > like a chemist approaches chemistry, never being 100% sure that his or > her theories are always correct. > Anyone who knows of results like these or has done work in this area > or has original ideas is welcome to respond to me on usenet or if you > want, you can email me directly. > Craig === Subject: : Re: request for ideas === > === Subject: : Re: request for ideas >Message-id: <4128c98a$0$51857$ed2619ec@ptn-nntp-reader03.plus.netHi Craig, >I've got something that you might be interested in. There is a theory in >geology that could do with some mathematical input. Basically all the >ancient continents can be joined together to make one continuous land mass, >but the only way to do this is to consider the earth was smaller in the >past. The concept of an earth that has been expanding in size is such an >amazing idea that most people refuse to believe it. But whether you believe >it or not there is still the fact that all the ancient continents can only >be reconstructed on a smaller diameter earth. This fact has been explained >away by saying that some of the ancient ocean ßoors have been consumed >within the earth. If you look at my web site on the subject you should get >the concept. Look at this page; >http://www.dinox.org/english/geoevid.htm >and the earth has been expanding in size link from that page for example. >http://www.dinox.org/expic/exp-30.htm So how come the ocean expands while the continents remain the same size? >Anyway, to get back to your question. It's dif[CapitalThorn]cult to prove that the >ocean ßoor hasn't been consumed within the earth. Would it be possible to >calculate the probability that all the continents would [CapitalThorn]t together on a >smaller earth? How likely is this? If we had two identical jigsaw puzzles >and one person had [CapitalThorn]tted all the pieces together to make a small picture, >but the other had a larger picture with large gaps in his picture (which he >said must have been lost) we would surely all know that the small picture >was correct. Why is this different from reconstructing the ancient earth as >a smaller diameter? Would a mathamathical study of the probablity of this >reconstruction occuring by chance help the argument or not? Or will we never >be 100% certain who is correct? >Hope that gives you a few ideas to be going on with. >Stephen Hurrell >> I am planning to write a paper which surveys mathematical results that >> show that the old axiom->proof->theorem way of doing mathematics >> does not always yield complete information about mathematics. The >> prime example of this (which started it all) is Godel's Incompleteness >> Theorem, but there has been a lot of work in this area since then. >> For instance, Gregory Chaitin has an incompleteness theorem which >> shows conclusively that a certain number which he calls Omega, which >> is really the probability that a computer program halts (de[CapitalThorn]ned in a >> way that makes sense), is a random number - which implies that there >> is no [CapitalThorn]nite axiom system that can yield all of the bits of Omega. He >> concludes from all of his work that sometimes one has to simply >> perform experiments in mathematics and form conclusions from the >> experiments without being absolutely certain that the conclusions are >> correct. >> It is these types of very original ideas that I am looking for to put >> in my paper, that there are some problems out there that are so >> dif[CapitalThorn]cult for us to get a grip on that we might have to approach them >> like a chemist approaches chemistry, never being 100% sure that his or >> her theories are always correct. >> Anyone who knows of results like these or has done work in this area >> or has original ideas is welcome to respond to me on usenet or if you >> want, you can email me directly. >> Craig -- Mensanator Ace of Clubs === Subject: : Re: JSH: Weird group > This post is a textbook case of a mind disconnected from what the rest > of us consider reality. It's disturbing. So far my best guess at what is happening is this: James Harris is completely incapable of understanding any written mathematics. You write anything mathematical, and it is pure nonsense to him. So in a typical exchange: You write: You are wrong, because . And that is what I would read, and what most people would read. Harris however reads: You write: You are wrong, because . Of course it would be hard to understand for anyone why, if they post many brilliant bits of mathematics, everyone always responds that they are wrong, and give complete gibberish as the reason. That is what James Harris sees happening, and the only explanation to him is a conspiracy against him. === Subject: : Re: JSH: Weird group >It amazes me how weird sci.math'ers are and it's clearly some weird >social thing where many of you are simply abnormal when it comes to >even basic social function. >Like look at the David Ullrich thing. >[...] >No doubt about it, David Ullrich is clearly a stalker weirdo. >At that point I decided that it'd be a good idea to report his PUBLIC >statements to his university. >And he began using that to play victim. >[...] >By now the ratio of replies of David Ullrich to posts of mine versus >my replies to him is got to be well over 2 to 1, which should be >enough for RATIONAL people to know who the weirdo is here. no doubt. i've never denied being a weirdo. which doesn't change the facts: i. at the present time in in the us it -can- happen that charges of racism get people in trouble, even if the charges are totally bogus. [i was lucky that the relevant authorities at osu took a more rational view of things...] ii. even if [i] were not so, bogus complaints about racism would still be a Bad Thing because they can lessen the impact of legitimate complaints. so far we deduce that it's important to make certain that your complaint to osu about me did not have the desired effect. this raises the question of what you were trying to accomplish, leading to the third fact: iii. you've stated explicitly that your aim in complaining to osu about me was to try to get me to stop replying to your posts. >But maybe the reality of sci.math is that most of you are weirdos like >Ullrich, so his stalking behavior just seems normal to you! >However, most people around the world would [CapitalThorn]nd it extraordinary that >a professor would, after making comments like his talk about racial >slurs, keep after a person who'd felt compelled by his behavior >already to make a complaint! >Stalkers get complained about, and then they try to further victimize >their victims by themselves playing the victim. >On sci.math, David Ullrich found a ready crew of believers to help him >in his game which continues to this day, as he STILL tries to get my >attention. >But to sci.math'ers, he's just behaving normally, eh? what's strange is that you're -still- looking for sympathy, bringing all this up again, in spite of the essentially universal contempt people express for the fact that you complained to my employer about things i said on usenet. why is it that you've never answered my request for contact information on -your- employer, btw? >James Harris ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: Cardinal Exponentiation > For an ordinal a, let a+ and cf(a) denote respectivel the smallest > cardinal greater than a and the con[CapitalThorn]nality of a. Let k and m be > cardinals greater than 1 with at least one of them in[CapitalThorn]nite. Kunen > proves the following theorem of ZFC+GCH. > 1) If k <= m, then k^m = m+. > 2) If cf(k) <= m < k, then k^m = k+. > 3) If m < cf(k), then k^m = k. > Studying the proof, it seems to me that without GCH, one can show > 1) If k <= m, then k^m = 2^m. This is true. > 2) If cf(k) <= m < k, then k^m = 2^k. I don't think you can prove this in ZFC. In particular, I think the case m = aleph_0 and k = aleph_omega is known to be independent of ZFC. If you change the conclusion to k < k^m <= 2^k, then it's correct. > However, if m < cf(k), it seems the best you can say is k <= k^m <= 2^k. Yes, you can't prove that k^m < 2^k, because it's consistent that (aleph_1)^(aleph_0) = 2^(aleph_1). X-mailer: xrn 9.02 === Subject: : Polar integration? Mail-To-News-Contact: abuse@dizum.com It recently occurred to me that the only de[CapitalThorn]nition of integration that I've ever seen relates to rectangular coordinates. I was wondering if there was anything similar for polar coordinates. There's one obvious dif[CapitalThorn]culty with doing this, which is that the graph of a function in polar coordinates is quite likely to have different values of r for theta = t0 +/- 2*PI*n. We can hide this problem by initially limiting ourselves to the restriction of a function to the domain [0,2*PI), and only considering functions that are positive-valued on that domain. Then, after we see what happens under those conditions, we can look into [CapitalThorn]nding elegant ways to eliminate those initial restrictions. Or, we can decide that the whole thing's a waste of time. Normal rectangular integration (what's taught to non-mathematicians in service classes) looks at partitions of an interval, and the limits of lower and upper sums (well, *the* limit, but you don't start out knowing that). It seems to me that something similar could be done in polar coordinates. I'll take just the equivalent of the lower sum to illustrate what I'm getting at. Given a function, f(o), as above, we can de[CapitalThorn]ne an interval (a,b) for o (I'm representing the Greek letter theta with a lower case Roman o), and then partition it. Let's examine the wedge de[CapitalThorn]ned by o_i, o_(i+1). If we represent the minimum value that the functions takes on in this interval as m_i, we can establish an isosceles triangle with an altitude of m_i, and a base of m_i*2*sin( (o_(i+1) - o_i)/2 ). The area of this triangle would then be (m_i)^2 * sin( (o_(i+1) - o_i)/2 ). As we add more points to our partition, (o_(i+1) - o_i) gets very small, and we can replace sin( (o_(i+1) - o_i)/2 ) with (o_(i+1) - o_i)/2, giving us: (m_i)^2 * (o_(i+1) - o_i)/2. It seems intuitive to me that the limit of these lower sums, as the number of points in the partition increases, would be the same as the limit of the upper sums, although I haven't actually proven it. Does such a development of calculus exist? If so, is it not mentioned in the service courses because no practical use has been found for it or because its development would make engineers' heads explode? Or does it have an inherent ability to deal with functions beyond the restrictions that I mentioned above? I'm trying to come up with the analogue of the FTC for this, but I haven't quite [CapitalThorn]gured out what a derivative would be like in polar coordinates. -- Michael F. Stemper #include Have you embraced Windows ... lately? === Subject: : Re: Polar integration? >Does such a development of calculus exist? Yes. See, for example, the textbooks of Thomas and Finney or Leithold. It is also mentioned in passing (equation (7)) at http://mathworld.wolfram.com/Area.html . BTW, AN Niel seems to refer to the integral of a function of two variables (changing from rectangular to polar coordinates in the domain), whereas you and I discuss a function of one variable, r = f(theta) -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: : Re: Polar integration? >It seems intuitive to me that the limit of these lower sums, as the number >of points in the partition increases, would be the same as the limit of >the upper sums, although I haven't actually proven it. >Does such a development of calculus exist? Yes. Look in any standard (3 semester) calculus text. You know, one of those books like Stewart or Thomas that weigh 10 pounds. You are outlining the proof of the standard integral for polar area: A = 1/2 int(alpha..beta) f^2(theta) d theta. --Lynn === Subject: : Re: Polar integration? > It recently occurred to me that the only de[CapitalThorn]nition of integration > that I've ever seen relates to rectangular coordinates. I was wondering > if there was anything similar for polar coordinates. There is a course (in the US, like the third semester of calculus) that deals with this. Basically you compute the Jacobian of the change of coordinates, and insert that in your integral in the appropriate places. > I'm trying to come up with the analogue of the FTC for this, but I > haven't quite [CapitalThorn]gured out what a derivative would be like in polar > coordinates. Maybe you can refer to partial derivatives, gradent, curl, divergence. That same course will have exercises where these are converted to polar coordinates, cylindrical coordinates, and spherical coordinates. === Subject: : Chaos/Fractals: Call for Papers (I apologize if this duplicates a post I tried to make to sci.math on Friday. I'm getting used to this Xnews newsgroup application...) for the Chaos and Graphics Section of the international scienti[CapitalThorn]c journal Computers and Graphics (Elsevier). I edit this section, which appears in each issue of the journal. Topics include the mathematical, scienti[CapitalThorn]c, and artistic application of fractals, chaos, and related. Your papers can be quite short if desired, for example, often a page or two is suf[CapitalThorn]cient to convey an idea and a pretty graphic. (The journal is peer-reviewed, which means that several reviewers will judge whether the paper is suitably written, attractive, relevant, or novel.) I publish color, where appropriate. The goal of my section is to provide visual demonstrations of complicated and beautiful structures which can arise in systems based on simple rules. The section presents papers on the seemingly paradoxical combinations of randomness and structure in systems of mathematical, physical, biological, electrical, chemical, and artistic interest. Topics include: iteration, cellular automata, bifurcation maps, fractals, dynamical systems, patterns of nature created from simple rules, and aesthetic graphics drawn from the universe of mathematics and art. You can [CapitalThorn]nd submission guidelines here: http://sprott.physics.wisc.edu/pickover/guidec.html (The email address used to post this message is nonfunctioning. Please contact me at the address given at http://www.pickover.com .) === Subject: : Re: Chaos/Fractals: Call for Papers > (I apologize if this duplicates a post I tried to make to sci.math > on Friday. I'm getting used to this Xnews newsgroup application...) > for the Chaos and Graphics Section of the international scienti[CapitalThorn]c > journal Computers and Graphics (Elsevier) 1000 pages. For 2005, there are again six issues planned. The price of the journal in 2005 for libraries in the US is $1,852 per year. This [CapitalThorn]gures to over $1.85 per page. If I am mistaken about these prices, I would like to be corrected. Otherwise, I would like to ask the question: given these prices, why should people choose to publish their paper in this journal? Greg Lawler PS: Yes, I know that there is a signi[CapitalThorn]cantly cheaper price for individuals, but most of us rely on libraries for journals, and libraries have limited budgets. === Subject: : Re: Errors in Wiles's proof of Fermat's Last Theorem > [[ This message was both posted and mailed: see > the To, Cc, and Newsgroups headers for details. ]] Our library has Nonlinear Studies available in electronic form. > Oh, how I envy you. > Mathematical Reviews covers this journal. But for the paper > cited above, they simply say: > {There will be no review of this item.} > In fact, for all ten of EEE's papers, MR has only a remark like this. > Your posting has a little sketch of a boy apparently about to lose his > lunch. Appropriate, I think. *** After having all the discussions about EEE and his papers nobody has yet been able to identify any speci[CapitalThorn]c errors in his paper on FLT. He may be cranky but even if a cranky person says 2 + 3 = 5 based on common number system any mathematician having little credibility must accept this. Send a rebuttal to EEE's paper if you prefer. Otherwise, accept his claim and close this chapter once for all. *** === Subject: : Re: Errors in Wiles's proof of Fermat's Last Theorem > [[ This message was both posted and mailed: see > the To, Cc, and Newsgroups headers for details. ]] > Our library has Nonlinear Studies available in electronic form. > Oh, how I envy you. > Mathematical Reviews covers this journal. But for the paper > cited above, they simply say: > {There will be no review of this item.} > In fact, for all ten of EEE's papers, MR has only a remark like this. Your posting has a little sketch of a boy apparently about to lose his > lunch. Appropriate, I think. > *** > After having all the discussions about EEE and his papers nobody has > yet been able to identify any speci[CapitalThorn]c errors in his paper on FLT. He > may be cranky but even if a cranky person says 2 + 3 = 5 based on > common number system any mathematician having little credibility must > accept this. Send a rebuttal to EEE's paper if you prefer. Otherwise, > accept his claim and close this chapter once for all. > *** Unfortunately sci.math'ers don't play by rules. They simply repeat what they wish to believe over and over and over again. Elevating life through science Driving out the phantoms and redundancy By E. E. Escultura, Ph.D. When I posted the question, Is 1 = 0.99? at the newsgroup Sci Math in 1997, I was ridiculed by the mathematical community for engaging in triviality. Everyone else believes, until now, that the answer is obviously yes. And yet no one has been able to prove or explain it. Today, there are long threads of debate on this issue in this and other news groups and it has become even more spirited. Still, it remains unresolved. ... I read through it and agree with his points raised, as for too long people have done things like put sqrt(2) = 1.414... as if they're actually doing something when in reality all calculations in the real world end up being of very large integers like 1414, as you split up a given object into lots of parts. I seriously doubt you'll get any cogent responses from sci.math'ers taught to behave dogmatically on the issues that Escultura raises. They are a cult of true believers. James Harris === Subject: : Re: Errors in Wiles's proof of Fermat's Last Theorem >Driving out the phantoms and redundancy >By E. E. Escultura, Ph.D. >When I posted the question, Is 1 = 0.99? at the newsgroup Sci Math >in 1997, I was ridiculed by the mathematical community for engaging in >triviality. Everyone else believes, until now, that the answer is >obviously yes. And yet no one has been able to prove or explain it. The funny thing is that Mr. Escultura actually Ôproves' it in his in[CapitalThorn]nite number of 9s) is smaller than 1. This assumption leads to a contradiction. The assumption must thus be false. Conclusion: 0.99... = 1. Mr. Escultura just doesn't recognise the Ôreductio ad absurdum'. >Today, there are long threads of debate on this issue in this and >other news groups and it has become even more spirited. Still, it >remains unresolved. Only in Mr. Escultura's mind. >people have done things like put sqrt(2) = 1.414... as if they're >actually doing something when in reality all calculations in the real >world end up being of very large integers like 1414, as you split up a >given object into lots of parts. So you believe that sqrt(2) has no decimal expansion? === Subject: : Re: Errors in Wiles's proof of Fermat's Last Theorem Am 22.08.04 19:03 schrieb Deep K. Deb: > *** > After having all the discussions about EEE and his papers nobody has > yet been able So, if *you* want, everyone has to want? And if not, he's unable? And even you got some hints on discussions times back. You didn't read them, so you even are unable to read, following your own logic. So you even will be unable to read this... Good night. Gottfried Helms === Subject: : Re: Errors in Wiles's proof of Fermat's Last Theorem X-no-archive: yes linux) >> [[ This message was both posted and mailed: see >> the To, Cc, and Newsgroups headers for details. ]] >> Our library has Nonlinear Studies available in electronic form. >> Oh, how I envy you. >> Mathematical Reviews covers this journal. But for the paper >> cited above, they simply say: >> {There will be no review of this item.} >> In fact, for all ten of EEE's papers, MR has only a remark like this. >> Your posting has a little sketch of a boy apparently about to lose his >> lunch. Appropriate, I think. > *** > After having all the discussions about EEE and his papers nobody has > yet been able to identify any speci[CapitalThorn]c errors in his paper on FLT. He > may be cranky but even if a cranky person says 2 + 3 = 5 based on > common number system any mathematician having little credibility must > accept this. Send a rebuttal to EEE's paper if you prefer. Otherwise, > accept his claim and close this chapter once for all. > *** Are you able to read and to understand mathematics ? His argumentation lies on the following fact: FTL's proof would be wrong because of some logical reasons, that is: two axioms about real numbers would be false. Are you realizing the total nonsense of this sentence ? It is not necessary to be specialist in number theory to understand the nonsense of EEE's argumentation. EEE does not say 2+3=5 so I am right but 2+3=5 is false because two axioms about equality are false so the whole number theory is false. pg. === Subject: : Re: Is Meas cartesian closed boundary=------------010700050301050407040700 ------------------------------------------------------------- -------- >Let's be more explicit here. I guess you are asking if, >for any measurable spaces B, C, there is a measurable space (to >be called C^B) so that an appropriate correspondence between measurable >maps A x B -> C and measurable maps A -> C^B is bijective. >Basically, yes. >So: How are we to de[CapitalThorn]ne the measurable space C^B ? >Of course, the best thing would be to [CapitalThorn]nd an explicit candidate for >>the exponential object C^B (that is, some measurable structure on the >>set of measurable functions) >There is something called the Effros Borel structure, I don't >remember exactly what it is, but it may be something like this. >[Perhaps only in the case C = 2 (a two-point space with all four >subsets measurable), so 2^B is the collection of B-measurable >sets.] The Effros structure is a sigma-algebra on the set of closed subsets of a topological space. === Subject: : Re: Easy number theory problem >How do you know o(a)? You can say o(a) | p - 1, but >shouldn't you show that (Z_p)^* is cyclic? > You don't need to. By Lagrange's Theorem. No, not needed. > Let k = (p-1)/o(a) in N > a^(p-1) = a^(k.o(a)) = (a^o(a))^k = 1^k = 1 >Let m = p - 1. Then o(x) | m for all x in F* = F0. They are all >solns. of x^m - 1 = 0, which has m distinct solns. (m x^(m-1) not 0 >if x not 0). Yes. If o(x) | p-1, then x^(p-1) = 1. Same for o(x) | phi(n) in general Z*_n. > This seems like a digression or generalization into [CapitalThorn]nite [CapitalThorn]elds. Yes, I went on a bit--not sure where I was going either. > Estranged approach. pZ for prime p is maximal as pZ is an ideal and if > n not in pZ, then 1 in ideal , the ideal generated by n an pZ. > As n not in pZ, (n,p) = 1; some a,b with an + bp = 1 in . QED. > Theorem > Every nonzero prime ideal of a principal ideal domain is maximal. > pZ is prime ideal for pZ is an ideal and if ab in pZ, then > p | ab; p | a or p | b; a in pZ or b in pZ > Conversely, > a maximal ideal of a communitative ring with identity is prime. > -- Four proofs of Fermat's theorem: >1) Calculate (x + 1)^p (mod p) = Sum(m=0,p) C(p,m) x^m (mod p) >= x^p + 1 (mod p) >since for m = 1, ...,p-1, the p | C(p,m) = binomial coeff. >C(p,m) = p!/[m!(p-m)!] = pn/m!, where n = (p-1)!/(p-m)! . >(m!,p) = 1 and m! | pn ==> m! | n and p | C(p,m). >(I always take mod p to mean mod pZ, i.e., x (mod p) = [x] >= x + pZ = equivalence class of x in Z/pZ.) > Why not just use as you basically showed above > (x + 1)^p = x^p + 1 (mod p) > Thus > [(x + 1)^p] = [x^p + 1] > x = a mod p isn't mathematics, it's nebulous de[CapitalThorn]ned computer > notation and nearly ambigous with x = a mod p, the relation. > Please use x = [a]. [a] is a coset, a mod p is a number, a > bogus number invented by object orientated computer programmers. OK. >Let f(x) = x^p - x, then f(x + 1) = (x + 1)^p - x - 1 = x^p - x (mod >p) so f(x + 1) = f(x) (mod p). f(0) = 0 mod p, so by induction >f(x) = x^p - x = 0 (mod p) for all x in Z, and > No, for all x in Z+, x^p - x = 0 (mod p) > However descending f(x - 1) = f(x) (mod p), > thus for all x in Z, x^p - x = 0 (mod p) >x^(p-1) = 1 (mod p) for x != 0. > p^(p-1) = 1 (mod p) ?? Be careful of that division. Should have said when (x,p) = 1, x^(p-1) = 1 (mod p) >2) For x in Z but not in pZ ((x,p) = 1) and >1 <= j < p consider xj = u_j (mod p) >Write x = qp + k with 1 <= k < p. Clearly f_j(x) = xj (mod p) >= jk mod p = u_j mod p is a bijection f_j : Z*_p --> Z*_p > None of this is clear including the notation. Yes. I got this from someone's notes--I agree its not at all clear. >(I don't give the details here, everything is straightforward. >Use (x,p) = (j,p) = (k,p) = (u_j,p) = 1). >Thus u_1 u_2 ... u_p-1 = (p-1)! . Since u_j = jx for j in Z*_p, >this gives x^(p-1) (p-1)! = (p-1)! so x^(p-1) = 1 if (x,p) = 1. >3) In a group G = Z*_p (I will not show that this is a group), >we have o(x) | p-1 so x^(p-1) = 1 (mod p) for all x in Z*_p. > Finally a simple proof of Fermat's little theorem. > g in G iff g unit Z_p > if g,h in G: g,h units Z_p; gh unit; gh in G > 1 identity in G; for all g in G, some h in G with gh = 1. > 4) pZ is maximal ideal, thus Z_p = Z/pZ is a [CapitalThorn]eld. > Hence Z_p^* = Z_p0 is a multiplicative group of order p-1. > Or if you prefer, pZ is a prime ideal in an integral domain, > hence maximal ... 4) with maximal ideals pZ is easy. I have not seen it before that I recall. Van === Subject: : Re: Automorphisms of groups LOL! But sadly, true. I don't get a warm feeling about MS either. How is it that the marketing people took over and made all the money off the ideas that the geeks had when fooling with computers? I am glad to see that the money is everything philosophy is a hard sell to a segment of computer people, and we still have FSF and Sourceforge, and many others with great free software. I recall when the only Navigator was Netscape--well 1st there was Mosaic and that guy at Cern (was it CERN? bad memory). Anyway, here comes MS with Inet Explorer, tried to wipe out Netscape, and did a pretty good job, though I guess Mozilla is a free continuation. If not for these people providing free stuff, we would be getting charged outrageous prices for everything, browsers, every program--well, I could go on, but that is enough for now. Van === > === Subject: : Re: Automorphisms of groups, Automorphed > These days, whatever they Ôimprove', it's never as good. colors, >different fonts, hide and don't hide quotes, etc., in posts now. >It has problems with sci.math because its auto-formatter doesn't >know how to deal with the ASCII math symbol conventions here.) > How modern corporations can turn perfectly working and useful facilities > into worthless overpriced pieces of junks, would be the highest of humor > if only those corporations weren't off shored onto my home planet. > -- the FOAD (Fabulous offhand alien diatribe) > Once upon a time computers all spoke Ascii. > But in their arrogance acclaiming to know it all, > they were cursed to babble in a multitude of confusing formats. > They even thought they could talk people, but as > they can't, they's gonna make people talk computer. > After that, the next quantum leap in productivity > will be replacement of consumers with computers. > -- > Updates are worse than blind dates. > Who wants date > with Billy Gate > who makes debate > why him we hate > This massage automatically smell checked by MicroSoft. > -- Are you soft on MicroSoft? > As the [CapitalThorn]rst computer was a [CapitalThorn]ve [CapitalThorn]nger model, > historical records are not available. > This model was soon superseded by a twice as powerful > ten [CapitalThorn]nger model. > A supercomputing twenty digit model was designed but > soon deemed unmarketable because of a newfangled > fad of wearing foot gear. > Other computer systems experimented with pebbles. > Then a new generation of computers came forth, utilizing > a technological innovation, namely bead making. > Portable models carried as necklaces soon became popular. > However wrangling merchants in passionate greed so often > strangled themselves computing potential pro[CapitalThorn]ts > that they were soon replaced by a safer table top model, > now know to us as the abacas. > Mechanical versions of the abacas, using clogs instead of beads, became > passing fancy, until with the progression of time, the beads were > miniaturized into small electronic charges and even magnetic bubbles. > Thus you see the ancient archaic principle of digits, now revered as > bits and bytes, established well over 10,000 years ago, remains the > established norm. So I ask you, as the basic method of computing has > remained the same since the stone age, why would you expect > Miraculous MicroSoft Mirages to work any better than a [CapitalThorn]nger? > -- Riddle of the day > Why couldn't the mathematican fall to sleep counting sheep? > ---- === Subject: : Re: abstract algebra..... I guess in the zero ring R = {0}, 0 = 1 since 0 + 0 = 0 ; 0*0 = 0. Van === Subject: : Re: Tricky maths problem >> For real pedantry, try this on: ... >Or this: The question can not be answered because who said we are >spinning/tossing coins being exposed to friction/gravitation, and even >if we assume these, who said that the coin will actually fall on one of >its main sides - moreover, how many sides does the coin have? >I mean, you have to draw the line somewhere. ... Hey, you got your prior, I got mine, everything's cool, man! Lee Rudolph === Subject: : Re: Easy number theory problem I posted yesterday was a mess--I will be more careful (in particular the post where where I slip back and forth between pi^ei and s^e). === > === Subject: : Re: Easy number theory problem >CRT: (m,n) = 1 and x in Z_m, y in Z_n ==> there is a unique >z in Z_mn such that z = x mod m and z = y mod n. >> As m,n coprime, some a,b with am + bn = 1 >> bn = 1 (mod m); am = 1 (mod n) >> sam + rbn = r (mod m); sam + rbn = s (mod n) >the ideals mZ == (m). >> The actual bijection you want is >> f:Z_mn -> Z_m x Z_n, z + mnZ -> (z + mZ, z + nZ) >> which you need to show is well de[CapitalThorn]ned. >> If x + Z_mn = y + Z_mn, then x = y (mod mn), >> x = y (mod m), x = y (mod n); x + mZ = y + mZ, x + nZ = y + nZ >> (x + mZ, x + nZ) = (y + mZ, y + nZ) >> Thus it's well de[CapitalThorn]ned. >or x + (mn) = y + (mn) ==> x - y in (mn) = mnZ = mZ/nZ . >so x - y in mZ and nZ or (x + mZ, x + nZ) = (y + mZ, y + nZ) >so well de[CapitalThorn]ned. > You're so idealistic. >> As m,n coprime, some a,b with am + bn = 1 >> bn = 1 (mod m); am = 1 (mod n) >> sam + rbn = r (mod m); sam + rbn = s (mod n) >> f(sam + rbn) = (r,s) in Z_m x Z_n >> Thus f surjection. >> Rest is direct. f is ring homomorphism >> f(x+y + mnZ) = (x+y + mZ, x+y + nZ) = (x+mZ + y+mZ, x+nZ + y+nZ) >> = (x + mZ, x + nZ) + (y + mZ, y + nZ) = f(x) + f(x) >> f(xy + mnZ) = (xy + mZ, xy + nZ) >> = ((x + mZ)(y + mZ), (x + mZ)(y + nZ)) >> = (x + mZ, x + nZ)(y + mZ, y + nZ) = f(x).f(y) >> If f(z) = (0,0), then z = 0 (mod m), z = 0 (mod n), >> some k with z = km; n|z; n | km; m,n coprime; n | k >> nm | km; nm | z; z = 0 (mod nm) >> Thus ker f = { 0 }; f injection. >Yes, this is the best way. CRT + ker f = 0 (and f a ring homo), so f >1-1 and onto, >of course |Z_mn| = mn = |Z_m x Z_n|, >so 1-1 ==> onto for [CapitalThorn]nite sets. > Hm, so proving surjection as above with CRT isn't necessary. > Whence CRT directly from Z_mn = Z_m x Z_n. However the proof > has one advantage, namely as it's constructive, it offers a direct > way of calculating the CRT solution. > -- >Now I want to show that f extends to a group isomorphism of Z*_mn >with Z*_m x Z*_n. This should be straightforward it seems, though >I have never done it. i.e. to show >f : Z*_mn --> Z*_m x Z*_n : f(z + mnZ) = (z + mZ, z + nZ) > Have you put the saddle on backwards? > The restriction of f to Z_mn^* is what you're wanting. Yes. > if u unit in Z_mn, then some v with uv = 1 > (1,1) = f(1) = f(uv) = f(u)f(v), f(u) unit in Z_m x Z_n > f(u) = (u_m, u_n); f(v) = (v_m, v_n); u_m = u + mZ > (1,1) = (u_m v_m, u_n v_n); u_m, u_n units in Z_m, Z_n > f(u) in Z_m^* x Z_n^* > We already know f is injection and multiplicative homomorphism. > What's left is surjectiveness. > If u_m, u_n units in Z_n, Z_m > some v_m, v_n in Z_m, Z_n with u_m v_m = 1 = u_n v_n > some r,s with f(r) = (u_m, u_n), f(s) = (v_m, v_n) > f(rs) = f(r).f(s) = (u_m, u_n)(v_m, v_n) = (u_m v_m, u_n v_n) = > (1,1) > Thus rs = 1 as f injection, and r is unit. f(r) = (1 + mZ,1 + nZ) with (m,n) = 1, ==> r = 1 + mnZ in Z/mnZ. Ker(f) = (1) in Z_mn. (Should make some decision and clean up notation.) ==> f is into (1-1), and as you showed, its a homomorphism of [CapitalThorn]nite groups, so its onto, and an isomorphism. === Subject: : Re: Easy number theory problem === > === Subject: : Re: Easy number theory problem >Proof of the case Z*_p is cyclic for p = 13. >There are 12 = 2^2*3 elements, which are the distinct solns. of >x^(p-1) - 1 = x^12 - 1 = 0. >In general, p - 1 = q = p1^e1 ... pr^er. Let s^e = pi^ei for some i. Let y = x^s (s = pi for some i). >x^(s^e) - 1 = y^(s^(e-1)) - 1 = 0 > x^(pi^ei) = y^(pi^(ei - 1)). That notation is so confusing. >has s^(e-1) distinct solns. > Why? This post is a mess, sorry. I will try to post something better, esp. for the number of distinct solns. > 7.5.4. Theorem. [Fundamental Theorem of Finite Abelian Groups] > Any [CapitalThorn]nite abelian group is isomorphic to a direct product of cyclic > groups of prime power order. Any two such decompositions have the same > number of factors of each order. > 7.5.5. Proposition. Let G be a [CapitalThorn]nite abelian group. Then G is > isomorphic to a direct product of cyclic groups such that > n_i | n_i-1 for i = 2,3,...k. > What do you think that means? > 7.5.6. Corollary. Let G be a [CapitalThorn]nite abelian group. If a in G is an > element of maximal order in G, then the order of every element of G is > a divisor of the order of a. > 7.5.7. Lemma. Let p be a prime number, and let k,a,b be integers. > (a) If 1 <= k <= p-1, then p is a divisor of the binomial coef[CapitalThorn]cient > p!/k!(p-k)! > (b) If k >= 1 and a is congruent to b (mod p^k), > then a^p is congruent to b^p (mod p^{k+1} ). > (c) If k >= 2 and p is an odd prime, then > (1 + ap)^(p^(k-2)) = 1 + ap^(k-1) (mod p^k) > (d) If p is an odd prime and p is not a divisor of a, then > (1 + ap)^(p^(k-1)) = 1 (mod p^k) > (1 + ap)^(p^(k-2)) /= 1 (mod p^k) > 7.5.8 Theorem. Let p be an odd prime, let k be a positive integer, and > let n=p^k. Then Z_n^x is a cyclic group. > -- > 3.5.6. De[CapitalThorn]nition. Let G be a group. If there exists a positive > integer N such that a^N = e for all a in G, then the smallest such > positive integer is called the exponent of G. > 3.5.8. Proposition. Let G be a [CapitalThorn]nite abelian group. > (a) The exponent of G is equal to the order of any element of G of > maximal order. > (b) The group G is cyclic if and only if its exponent is equal to its > order. > Zp^* cyclic. Let z be an element with maximal order n; n|p-1 > for all x in Zp^*, x^(p-1) = 1 = x^n; n = p-1 for p-1 <= n as > 4.1.12 Corollary. A polynomial of degree n with coef[CapitalThorn]cients > in the [CapitalThorn]eld F has at most n distinct roots in F. > ---- === Subject: : [OT] KASH/KANT: Loading source-[CapitalThorn]les Shame on me, but I can't [CapitalThorn]nd out, how to load a [CapitalThorn]le into the KASH-shell. (KASH is a shell for KANT, an algebra-system of the Technical University of Berlin http://www.math.tu-berlin.de/cgi-bin/kant/calc/calc.py ) Anyone who can help? I didn't [CapitalThorn]nd anything in the manual. mf === Subject: : Intro. to Godel's Theorems Some here might like to check out www.godelbook.net where I've posted the [CapitalThorn]rst 12 chaps of a draft book on Godel's theorems [advanced undergrad/beginning grad student level -- but with some asides others may [CapitalThorn]nd interesting] === Subject: : Re: Easy number theory problem My last post on this was a mess. This (I hope) is a little better. p.55 Chap 11, Primitive Roots Polynomials over Z/pZ (p prime). Prop. 1) A nonzero polynomial f in Z/pZ[x] has at most deg(f) (distinct) elements a in Z/pZ such that f(a) = 0. I won't reproduce the proof of this here. Prop. 2) Let d | p-1 . Then f = x^d - 1 has exactly d solutions. Proof: Let p-1 = de. Then x^(p-1) - 1 = (x^d)^e - 1 = (x^d - 1) g(x) where g(x) = (x^d)^(e-1) + (x^d)^(e-2) + ... + 1, so deg(g) = d(e-1) = p - 1 - d. Prop. 1 ==> g has at most n_g <= p - 1 - d roots, and x^d - 1 has at most n_d <= d roots, we know that x^(p-1) - 1 has p-1 roots in Z/pZ, i.e. the elements of Z*_p == (Z/pZ){0}. Thus n_g + n_d = p-1 which means that n_g = p - 1 - d and n_d = d. Somewhere the fact that Z/pZ is a [CapitalThorn]eld enters, I'm not clear on this right now. W. Elliot had some comments in his post. Lemma. If a and b are in an Abelian group with |a| = r, |b| = s, and (r,s) = 1, then |ab| = rs. There was a proof of this on sci.math recently, so I won't reproduce it now. Theorem. For any prime p, Z*_p is a cyclic group of order p-1. Proof: Let p-1 = q_1^e_1 ... q_r^e_r. Let q^e = (q_i)^(e_i) for some i. Prop 2 ==> x^(q^e) - 1 has exactly q^e roots, and x^[q^(e-1)] - 1 has q^(e-1) roots ==> there is a solution a_i of x^(q^e) - 1 = 0 such that (a_i)^[q^(e-1)] != 1 ==> |a_i| = (q_i)^(e_i) for each factor of (q_i)^(e_i) in p - 1. Let a = a_1 ... a_r. Then |a| = p - 1 by the Lemma, and Z*_p is cyclic. The number of primitive roots of Z*_p is phi(phi(p)) = phi(p-1), since there are phi(p-1) generators of Z_(p-1) =~ Z*_p, = numbers in Z_(p-1) which are prime relative to p-1. Van === Subject: : Re: A strange constant perhaps, Avogadro's Number N_A..... charset=iso-8859-1 Re: A strange question perhaps, but I thought I'd give it a shot..... i.e is there a [CapitalThorn]nite granularity to the movement of energy and matter ? Perhaps I'm not asking the correct question Planck distance. Look it up. No it hasn't and you know that. We are about 13-some orders of magnitude away from that to be able to tell. IMHO, small distances depend on energy level starting at around hbar/m_e*c. There is a natural free space boundary condition. O.K. let me change the question a bit. Could we -in principle- establish that Planck Length is the shortest possible length. Only if you (or anyone) believes that c, hbar, and G is all there is to the Universe. We also have those nagging coupling constants which G is really one of them. What happens if there is a fourth unknown limit and G is a result of hbar, c and the 4th unknown limit? [hanson] atom size then one has to specify what the surface is and where said surface begins/ends, because the smallest distance lays certainly within the fog of the surface thickness. like the electron then Freddy's length, l_F, may apply ::: *** l_F = hbar/(m_e*c) = 3.86E-11 cm ***, which when multiplied with the Finestructure constant [a] , since we describe here an event that take place in EM environs, produces nothing less then the classical radius, r_e, of the electron, m_e, itself. ::: *** r_e = hbar*a /(m_e*c) = 2.82E-13 cm ***, or which when divided the Finestructure constant [a] , we get ::: *** r_H = hbar/(m_e*a*c) = 5.29E-9 cm ***, a length which can be expressed now for the [CapitalThorn]rst time as an experimentally veri[CapitalThorn]able measurement as/of the Lyman series limit wave length of ::: *** l_L= 4*pi*r_H/a = 1/Roo = 9.11E-06 cm *** Now, comparing Freddy's EM unit length of ::: *** l_F = hbar/(m_e*c) = 3.86E-11 cm ***, to unit lengths in the realm where gravitational effects do prevail we get to the Planck Length unit, constructed/proposed in 1899, which says ::: *** l_pl = sqrt (hbar*G/c^3) = 1.62E-33 cm *** But, that is by far not the smallest conceivable unit length we can construct, imagine or conjure up. Consider the Kerr event horizon for the mass of the electron, which gives you a length that you can use as another smallest unit with ::: *** l_e = m_e*G/c^2 = 6.76E-56 cm ***, .... then for good EM measure throw in the Finestructure constant (a), pi and sqrt(3) in, to give it some ßavor and ßair and you get: ::: *** l_a = m_e*G*pi*a*sqrt(3)/c^2 = 2.68-57 cm *** which gives you a new unit length, l_a, that beats the ass of the Planck length l_pl, by some than 6E+23 magnitudes. (see N_A below) A few years back I made a post on this subject of max/min length units and showed that one can drive this game to any desired size up or down. BUT, whether these unit lengths constructs do have any REAL physical signi[CapitalThorn]cance is, at present, in the eye and mind of the beholder. To the OP Powells's issue whether there is a [CapitalThorn]nite granularity MLT, the question is still wide question. We know from daily experience that nature is discrete and ordered (Chaos organizes spontaneously .... otherwise you wouldn't exist...ahahahaha....) and that gross matter consists of atoms one gigantic divisional step downward, atoms being N_A, 6E-23, times smaller then one mole of its ponderable matter. Similarly, heavily bodies can be seen and classi[CapitalThorn]ed the same way, N_A, 6E-23 times larger, 1 mole up the ladder. This N_A, Avogadro's Number, phenomenon is manifest in the above equation of ::: *** l_pl = sqrt (hbar*G/c^3) = 1.62E-33 cm *** as the following, which can be written as ::: *** l_pl = r_H * (N_A*pi*sqrt3)^(-1) *** suggesting that 1 mole amount (N_A) of Planck length units (l_pl) is nothing more than the measurable Bohr radius of the Hydrogen atom. So, a bottle of Hydrogen gas obeying the smooth PV= RT law, shows granularity one N_A step further down in it's luminous emissions and then again another N_A step further down it is manifests again as a granular event, N_A times smaller, at the Plank unit level... ........ahahahaha........AHAHAHAHA......... here are the other N_A connections, for the stuffy morons who believe and think that the N_A-mole thing only describes a speci[CapitalThorn]c amount of mass of C12 carbon.......ahahaha........... ::: *** tau / t_pl = a^(-1) * (N_A*pi*sqrt3) **** 1 mole of Planck time units = 1 atomic time unit ::: *** r_H / l_pl = a^(0) * (N_A*pi*sqrt3) **** 1 mole of Planck length units = 1 H-Bohr radius ::: *** m_pl / m_e = a^(1) * (N_A*pi*sqrt3) **** 1 mole of electron masses = 1 Planck mass ::: *** r_e / l_pl = a ^(2) * (N_A*pi*sqrt3) 1 mole of Plank length units = 1 classical el-radius ......and then naturally the next N_A step further down into ::: *** l_a = m_e*G*pi*a*sqrt(3)/c^2 = 2.68-57 cm ***, where 1 mole of l_a length units = 1 miserly Planck length... ............ahahahahaha.........ahahahahanson [Fredi] Yeah... Some good stuff here, hanson. The concept of length gets very funny at the quantum level. [hanson] Right, just as the concept of length does get funny at high speeds. SR, GR etc. It also show us that this funny behavior is applicable to mass and time too. --- This off the linearity or proportionality behavior may however simply indicate that our observer based story telling aka our current theories are domain limited in their applicability and true capability. --- In this connection it is wise to remember that NO theory is/was ever required nor needed to design & produce any hardware, be it Radar, TV, GPS, or nuclear bombs, etc. Hardware production goes grandfather style, experimenting and building on preexisting proven designs solely per (try/do/check/test)x until you have a saleable item. After the item/instrument is built and ready for Sale (to pay for it and make a buck off it) the Sales-department makes a big song and dance about it with theories galore to sell their newfangled machines. ---- It is then that band-wagon of pathetic physics fanatics and (ex)school teachers do tell their live audiences & in NGs here that theory was essential for the development.........ahahahhahaha.... But let'em sing....all of'em....it's a beautiful choir.........ahahahaha.. [Fredi] We can take the expression for Planck mass and show it this way as a coupling constant; G*mass^2/hbar*c = dimensionless number. Planck mass makes this number equal to one. So what? What this is really showing us is that there might be a very small mass quantum of less than 3 eV/c^2. [hanson] Right, at the =< 3 eV/c^2 mass level you have entered the aetheral/ethereal/aetheric......ahahaha........world of the elusive of neutrinos, photons, gravitons, and others are hotly debated subjects of such objects. [Fredi] A fourth limit to the Universe? G times the mass quantum squared is part of the coupling constant for gravity. [hanson] By which you mean what? Some k_F = G*m_pl^2 construct of yours?..... with m_pl = l_pl*c^2/G wherein m_pl/l_pl = c^2/G is the Kerr converter that links/turns mass expressions into length & visa versa.......therefore ::: *** G = l_pl^2 *c^3/hbar or, and using also e^2= a*hbar we write ::: *** G = a^(-1)*(e/m_pl)^2 or ::: *** G * N_A^2 = [1/3] * [ hbar * c] / [pi* a* m_e]^2 = const = k_F or equivalently: ::: *** G * N_A^2 = [2/(3pi)] * [c^3] * [r_H^2 / h] = const = k_F ::: *** G * N_A^2 = f(tau, etc) = f(Lyman freq, etc) = const = K_F or which of many other such ways ?? ....ahahaha....it's all the same, dude, just looked at and expressed differently from different povs...ahahaha....... There is another much simpler & clearer way to look at the grand show. SELF SIMILARITY: an atom looks similar to a solar system, a solar system looks similar to a galaxy, each one being ~some N_A magnitudes larger then the preceding smaller one. Connect these self-similar domains with the standard coupling constant, N_A, known since the late 1800's, (on which Einstein worked and Perrin was awarded the 1926 Nobel price), like in the ::: *** universal gas constant, R(gas) = k*N_A linking the description of gas behavior in the atomic & gross world, or ::: *** Faraday = e*N_A linking macroscopic currents in a wire to the electron behavior, or ::: *** Hubble, H = (1/2) * ((a^2)/2)^2 * f_L / N_A, linking the cosmic domain with the atomic domain behavior....ahahahaha... this one now, with a bit of rewriting, turns into a rather exquisite form to be loved and embraced by the haters of N_A, for it is also and equally ::: *** Hubble, H = (3/2) * pi *c * l_pl^2 / (2*r_e)^3 linking 3 domains: Planks, QM's and the cosmic realms, where f_L = Lyman series limit frequency, k = Boltzman, e= e-charge, r_e the classical electron radius. See, dude, very many ways to skin the cat....ahahaha....AHAHAHAHA.... It's all just theorizing, = ing story telling about nature...that's all!.... Well, I can't swear about the ing, but it's somewhere in there too... and they knew all this stuff above and its equations presented since around the turn of the 2nd last century, Planck stuff since 1899.... BUT then enter Einstein and the Juden physics ....ahahahaha... with its immense propaganda machine......and we are still stuck in the coul de sac of relativity with no apparent way out......ahahahaha..... Read the hilarious, tormented twists and turns they take in their blurbs in xxx.lanl.gov, arXiv.org/, spr, etc, them trying to escape out of their rel-sac they maneuvered themselves into, by making grand math assumptions, .... only to apply & use at ever turn and corner in their ground-breaking works the simplest of the good, old fashioned NEWTONIAN tools and his de[CapitalThorn]nitions.......ahahahahaha.......which is why their physics comedies made the eminent Professor Carver A. Mead of Caltech assess the situation with: It is my [CapitalThorn]rm belief that the last seven decades of the twentieth century will be characterized in history as the dark ages of physics. ..... or an even more damming view was expressed in the sentiments of F.A Hayek, a Nobel Price winner: In the future, Humanity will see in our Epoch an Era of superstition, essentially associated with the names of Marx, Freud and Einstein ! AHAHAHAHAHA.........ahahahaha.....ahahanson === Subject: : Re: A strange constant perhaps, Avogadro's Number N_A..... charset=iso-8859-1 | Re: A strange question perhaps, but I thought I'd give it a shot..... | i.e is there a [CapitalThorn]nite granularity to the movement of energy | and matter ? Perhaps I'm not asking the correct question | Planck distance. Look it up. | No it hasn't and you know that. We are about 13-some | orders of magnitude away from that to be able to tell. | IMHO, small distances depend on energy | level starting at around hbar/m_e*c. | There is a natural free space boundary condition. | O.K. let me change the question a bit. | Could we -in principle- establish that | Planck Length is the shortest possible length. | Only if you (or anyone) believes that c, hbar, and G is all there is | to the Universe. We also have those nagging coupling constants | which G is really one of them. What happens if there is a fourth | unknown limit and G is a result of hbar, c and the 4th unknown limit? | [hanson] | atom size then one has to specify what the surface is and where | said surface begins/ends, because the smallest distance lays | certainly within the fog of the surface thickness. | like the electron then Freddy's length, l_F, may apply | ::: *** l_F = hbar/(m_e*c) = 3.86E-11 cm ***, which when | multiplied with the Finestructure constant [a] , since we describe here | an event that take place in EM environs, produces nothing less then | the classical radius, r_e, of the electron, m_e, itself. | ::: *** r_e = hbar*a /(m_e*c) = 2.82E-13 cm ***, or which when | divided the Finestructure constant [a] , we get | ::: *** r_H = hbar/(m_e*a*c) = 5.29E-9 cm ***, a length which | can be expressed now for the [CapitalThorn]rst time as an experimentally veri[CapitalThorn]able | measurement as/of the Lyman series limit wave length of | ::: *** l_L= 4*pi*r_H/a = 1/Roo = 9.11E-06 cm *** | | Now, comparing Freddy's EM unit length of | ::: *** l_F = hbar/(m_e*c) = 3.86E-11 cm ***, | to unit lengths in the realm where gravitational effects do prevail we | get to the Planck Length unit, constructed/proposed in 1899, which says | ::: *** l_pl = sqrt (hbar*G/c^3) = 1.62E-33 cm *** | But, that is by far not the smallest conceivable unit length we can | construct, imagine or conjure up. | Consider the Kerr event horizon for the mass of the electron, which | gives you a length that you can use as another smallest unit with | ::: *** l_e = m_e*G/c^2 = 6.76E-56 cm ***, .... then for good | EM measure throw in the Finestructure constant (a), pi and sqrt(3) in, | to give it some ßavor and ßair and you get: | ::: *** l_a = m_e*G*pi*a*sqrt(3)/c^2 = 2.68-57 cm *** | which gives you a new unit length, l_a, that beats the ass of the | Planck length l_pl, by some than 6E+23 magnitudes. (see N_A below) | | A few years back I made a post on this subject of max/min length | units and showed that one can drive this game to any desired size | up or down. BUT, whether these unit lengths constructs do have any | REAL physical signi[CapitalThorn]cance is, at present, in the eye and mind of the | beholder. | | To the OP Powells's issue whether there is a [CapitalThorn]nite granularity | MLT, the question is still wide question. We know from daily | experience that nature is discrete and ordered (Chaos organizes | spontaneously .... otherwise you wouldn't exist...ahahahaha....) | and that gross matter consists of atoms one gigantic divisional | step downward, atoms being N_A, 6E-23, times smaller then one | mole of its ponderable matter. Similarly, heavily bodies can be seen | and classi[CapitalThorn]ed the same way, N_A, 6E-23 times larger, 1 mole up | the ladder. | | This N_A, Avogadro's Number, phenomenon is manifest in the above | equation of | ::: *** l_pl = sqrt (hbar*G/c^3) = 1.62E-33 cm *** | as the following, which can be written as | ::: *** l_pl = r_H * (N_A*pi*sqrt3)^(-1) *** | suggesting that 1 mole amount (N_A) of Planck length units (l_pl) is | nothing more than the measurable Bohr radius of the Hydrogen | atom. So, a bottle of Hydrogen gas obeying the smooth PV= RT law, | shows granularity one N_A step further down in it's luminous emissions | and then again another N_A step further down it is manifests again | as a granular event, N_A times smaller, at the Plank unit level... | | ........ahahahaha........AHAHAHAHA......... here are the other N_A | connections, for the stuffy morons who believe and think that | the N_A-mole thing only describes a speci[CapitalThorn]c amount of mass of | C12 carbon.......ahahaha........... | | ::: *** tau / t_pl = a^(-1) * (N_A*pi*sqrt3) **** | 1 mole of Planck time units = 1 atomic time unit | | ::: *** r_H / l_pl = a^(0) * (N_A*pi*sqrt3) **** | 1 mole of Planck length units = 1 H-Bohr radius | | ::: *** m_pl / m_e = a^(1) * (N_A*pi*sqrt3) **** | 1 mole of electron masses = 1 Planck mass | | ::: *** r_e / l_pl = a ^(2) * (N_A*pi*sqrt3) | 1 mole of Plank length units = 1 classical el-radius | | ......and then naturally the next N_A step further down into | ::: *** l_a = m_e*G*pi*a*sqrt(3)/c^2 = 2.68-57 cm ***, where | 1 mole of l_a length units = 1 miserly Planck length... | | ............ahahahahaha.........ahahahahanson | [Fredi] | Yeah... Some good stuff here, hanson. | The concept of length gets very funny at the quantum level. | [hanson] | Right, just as the concept of length does get funny at high speeds. | SR, GR etc. It also show us that this funny behavior is applicable | to mass and time too. --- This off the linearity or proportionality | behavior may however simply indicate that our observer based | story telling aka our current theories are domain limited in their | applicability and true capability. --- In this connection it is wise to | remember that NO theory is/was ever required nor needed to design | & produce any hardware, be it Radar, TV, GPS, or nuclear bombs, etc. | Hardware production goes grandfather style, experimenting and | building on preexisting proven designs solely per (try/do/check/test)x | until you have a saleable item. After the item/instrument is built and | ready for Sale (to pay for it and make a buck off it) the Sales-department | makes a big song and dance about it with theories galore to sell their | newfangled machines. ---- It is then that band-wagon of pathetic physics | fanatics and (ex)school teachers do tell their live audiences & in NGs here | that theory was essential for the development.........ahahahhahaha.... | But let'em sing....all of'em....it's a beautiful choir.........ahahahaha.. | | [Fredi] | We can take the expression for Planck mass and show | it this way as a coupling constant; | G*mass^2/hbar*c = dimensionless number. | Planck mass makes this number equal to one. So what? | What this is really showing us is that there might be a very | small mass quantum of less than 3 eV/c^2. | [hanson] | Right, at the =< 3 eV/c^2 mass level you have entered the | aetheral/ethereal/aetheric......ahahaha........world of the elusive | of neutrinos, photons, gravitons, and others are hotly debated | subjects of such objects. | [Fredi] | A fourth limit to the Universe? G times the mass quantum | squared is part of the coupling constant for gravity. | [hanson] | By which you mean what? Some k_F = G*m_pl^2 construct of yours? No, no, no. The opposite! G*m_q^2/hbar*c = the true gravitational coupling constant. Where m_q is a mass quantum > 3 eV/c^2. This puts length starting to get funny at atomic scales ~ 10^-8 meters. Uni[CapitalThorn]cation comes from the bottom, not the top. A fourth undiscovered limit to the Universe in addition to hbar, c and G. | with m_pl = l_pl*c^2/G wherein m_pl/l_pl = c^2/G is the Kerr converter | that links/turns mass expressions into length & visa versa.......therefore | ::: *** G = l_pl^2 *c^3/hbar or, and using also e^2= a*hbar we write | ::: *** G = a^(-1)*(e/m_pl)^2 or | ::: *** G * N_A^2 = [1/3] * [ hbar * c] / [pi* a* m_e]^2 = const = k_F | or equivalently: | ::: *** G * N_A^2 = [2/(3pi)] * [c^3] * [r_H^2 / h] = const = k_F | ::: *** G * N_A^2 = f(tau, etc) = f(Lyman freq, etc) = const = K_F | or which of many other such ways ?? ....ahahaha....it's all the same, dude, | just looked at and expressed differently from different povs...ahahaha....... | | There is another much simpler & clearer way to look at the grand show. | SELF SIMILARITY: an atom looks similar to a solar system, a solar system | looks similar to a galaxy, each one being ~some N_A magnitudes larger | then the preceding smaller one. Connect these self-similar domains with | the standard coupling constant, N_A, known since the late 1800's, (on which | Einstein worked and Perrin was awarded the 1926 Nobel price), like in the | ::: *** universal gas constant, R(gas) = k*N_A | linking the description of gas behavior in the atomic & gross world, or | ::: *** Faraday = e*N_A | linking macroscopic currents in a wire to the electron behavior, or | ::: *** Hubble, H = (1/2) * ((a^2)/2)^2 * f_L / N_A, | linking the cosmic domain with the atomic domain behavior....ahahahaha... | this one now, with a bit of rewriting, turns into a rather exquisite form to | be loved and embraced by the haters of N_A, for it is also and equally | ::: *** Hubble, H = (3/2) * pi *c * l_pl^2 / (2*r_e)^3 | linking 3 domains: Planks, QM's and the cosmic realms, where | f_L = Lyman series limit frequency, k = Boltzman, e= e-charge, | r_e the classical electron radius. | | See, dude, very many ways to skin the cat....ahahaha....AHAHAHAHA.... | It's all just theorizing, = ing story telling about nature...that's all!.... | Well, I can't swear about the ing, but it's somewhere in there too... | and they knew all this stuff above and its equations presented since | around the turn of the 2nd last century, Planck stuff since 1899.... | BUT then enter Einstein and the Juden physics ....ahahahaha... | with its immense propaganda machine......and we are still stuck in the | coul de sac of relativity with no apparent way out......ahahahaha..... | Read the hilarious, tormented twists and turns they take in their blurbs | in xxx.lanl.gov, arXiv.org/, spr, etc, them trying to escape out of their | rel-sac they maneuvered themselves into, by making grand math | assumptions, .... only to apply & use at ever turn and corner in their | ground-breaking works the simplest of the good, old fashioned | NEWTONIAN tools and his de[CapitalThorn]nitions.......ahahahahaha.......which is | why their physics comedies made the eminent Professor Carver A. Mead | of Caltech assess the situation with: | It is my [CapitalThorn]rm belief that the last seven decades of the twentieth century | will be characterized in history as the dark ages of physics. ..... or an | even more damming view was expressed in the sentiments of F.A Hayek, | a Nobel Price winner: | In the future, Humanity will see in our Epoch an Era of superstition, | essentially associated with the names of Marx, Freud and Einstein ! being a wave again--more like wavicles. Everything is pointing in that direction due to the discoveries of the last half of last century. FrediFizzx === Subject: : Re: A strange constant perhaps, Avogadro's Number N_A..... ... > | See, dude, very many ways to skin the cat....ahahaha....AHAHAHAHA.... > | It's all just theorizing, = ing story telling about nature...that's > all!.... > | Well, I can't swear about the ing, but it's somewhere in there too... > | and they knew all this stuff above and its equations presented since > | around the turn of the 2nd last century, Planck stuff since 1899.... > | BUT then enter Einstein and the Juden physics ....ahahahaha... > | with its immense propaganda machine......and we are still stuck in the > | coul de sac of relativity with no apparent way out......ahahahaha..... > | Read the hilarious, tormented twists and turns they take in their blurbs > | in xxx.lanl.gov, arXiv.org/, spr, etc, them trying to escape out of their > | rel-sac they maneuvered themselves into, by making grand math > | assumptions, .... only to apply & use at ever turn and corner in their > | ground-breaking works the simplest of the good, old fashioned > | NEWTONIAN tools and his de[CapitalThorn]nitions.......ahahahahaha.......which is > | why their physics comedies made the eminent Professor Carver A. Mead > | of Caltech assess the situation with: > | It is my [CapitalThorn]rm belief that the last seven decades of the twentieth century > | will be characterized in history as the dark ages of physics. ..... or an > | even more damming view was expressed in the sentiments of F.A Hayek, > | a Nobel Price winner: > | In the future, Humanity will see in our Epoch an Era of superstition, > | essentially associated with the names of Marx, Freud and Einstein ! to > being a wave again--more like wavicles. Everything is pointing in that > direction due to the discoveries of the last half of last century. There was a fellow on NPR the other day, that published a paper in which he light... URL:http://www.npr.org/features/feature.php?wfId=3804795 David A. Smith === Subject: : Re: Easy number theory problem > The number of primitive roots of Z*_p is phi(phi(p)) = phi(p-1), > since there are phi(p-1) generators of Z_(p-1) =~ Z*_p, > = numbers in Z_(p-1) which are prime relative to p-1. i.e. prime relative to p-1 when Z_p-1 is written as an additive group, not as a multiplicative cyclic group. e.g. Z*_13 = <2> =~ Z_12 has 2^n, with n = 1,5,7,11 as generators, since (n,p-1) = (n,12) = 1. === Subject: : Re: What am I studying here? > Can anyone give me any pointers to what exactly it is I'm studying here? :) > I am exploring properties of the *[CapitalThorn]nite* set of 2D integer vectors > generated by the matrix multiplication > {a c)(i) modulo (D,D) where D=ad-bc > {b d)(j) > for all i,j in Z, where a,b,c,d are [CapitalThorn]xed integers, and D<>0. > I know the matrix is a linear operator, and I know some basic group theory. > Just some names of things I can look up the web would be enough. A related subject, perhaps not quite what you're looking for, is called the geometry of numbers. It might give you a lead. Ken Pledger. === Subject: : Re: Gauss-Jordan Elimination > .... > algorithm requires row divisions by the elements on diagonal, A( i, i ). My > question: what do I do when A( i, i ) is zero (within a tolerance)? > At the moment I do this: I look for a nonzero (and maximum in absolute > value) element on the lower part of column i, among A( i + 1, i ) ... A( n, > i ) and then I swap rows and the computed inverse is ok. Yes, that's just what the Gauss-Jordan algorithm calls for (although your maximum in absolute value condition doesn't matter. It's usual just to take the next non-zero entry in the column.) > If I can't [CapitalThorn]nd a nonzero element among these, I look on the right of row i, > among A( i, i+1 ) ... A( i, n ). If I [CapitalThorn]nd a nonzero element, I swap > columns.... No. Gauss-Jordan uses row operations only. To see why, look at any text-book explanation. Ken Pledger. === Subject: : Re: Gauss-Jordan Elimination >[...] >> sorry about the inelegant formatting - typing >> one-handed for a few weeks... > ^^^^^^^^^^^^^^^^^^^^^ >So are you going to tell us why? broken shoulder [doesn't seem relevant] >And are you going to post whatever it was the the man said but didn't >appear in sci.bio.evolution? i did - it's appeared in sci.logic on my server... >And yes, can you imagine how many cranks sci.bio.evvolution would attract if >it WASN'T moderated? yes, i don't think i complained about that, just said that i hadn't realized it was moderated. ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === === Subject: : numerically solving simple equation Hi! I need help in writing code to solve this equation : Sum(i=1,n)(1/[N-(i-1)])=n/(N-a) N unknown n,a known It's actually equation from Jelinski-Moranda model. Any pointers or references would be greatly appriciated. === Subject: : Re: numerically solving simple equation > Hi! > I need help in writing code to solve this equation : > Sum(i=1,n)(1/[N-(i-1)])=n/(N-a) > N unknown > n,a known > It's actually equation from Jelinski-Moranda model. > Any pointers or references would be greatly appriciated. It look like you are going to have to solve a polynomial of degree n-1 or n - 2 (depending on a) to solve the equation. Experiments with Maple show that many of these polynomials will be unsolvable by radicals even when a = 1. So you will need to write a routine to [CapitalThorn]nd all the roots of the polynomials. Experiments also indicate that you get n-1 real roots. So there is not a unique solution. Here's an example using Maple: > restart: > n,a:=8,9: > add(1/(N-(i-1)),i=1..n)- n/(N-a); 1 1 1 1 1 1 1 8 1/N + ----- + ----- + ----- + ----- + ----- + ----- + ----- - ----- N - 1 N - 2 N - 3 N - 4 N - 5 N - 6 N - 7 N - 9 > simplify(%); 2 3 5 4 - 4 (-11340 + 49986 N - 69039 N + 44506 N + 2877 N - 15281 N 6 7 - 280 N + 11 N )/(N (N - 1) (N - 2) (N - 3) (N - 4) (N - 5) (N - 6) (N - 7) (N - 9)) > sort(numer(%)); 7 6 5 4 3 2 -44 N + 1120 N - 11508 N + 61124 N - 178024 N + 276156 N - 199944 N + 45360 > fsolve(%,N,complex); 0.3926627584, 1.497454668, 2.577311552, 3.647350277, 4.713260480, 5.778662309, 6.847843410 === Subject: : Re: need help with bell curve > i want to write a program that will approximate the bell curve.. > could someone tell me the best way to do it? or at least a far far > more ef[CapitalThorn]cient way than what i do at this time which is: > i loop 1000 times > each loop i ßip a coin 1000 times and see how many times it lands 0 > (in other words !(rand()%2) ) > that's about it.. i have an array.. and each loop if it gets 0 say.. > 490 times.. then array[490] gets increased by 1... at the end i've got > this array that if u plot it as the ordinate and the indexes of the > array as the absica then you get a very loose very ty very crappy > piece of dog approximation of the bell curve.. it's not even > close.. it's terrible.. i need something more ef[CapitalThorn]cient. Well, I'm not very good at combining maths and English but I think you are talking about the Gauss curve here (after comparing the results from Gauss curve and Bell curve at Google). The function for the Gauss Curve is written as f(x) = e^-(x^2), more information about it can be found at http://www.2dcurves.com/exponential/exponentialg.html. === Subject: : Re: need help with bell curve David, It sounds as if you want a method to compute normally distributed variates. You might want to consult Devroye, Luc. 1986. Non-Uniform Random Variate Generation. New York: Springer-Verlag. He should have some numerically stable methods for generating normal variates. Jeffrey > i want to write a program that will approximate the bell curve.. > could someone tell me the best way to do it? or at least a far far > more ef[CapitalThorn]cient way than what i do at this time which is: > i loop 1000 times > each loop i ßip a coin 1000 times and see how many times it lands 0 > (in other words !(rand()%2) ) > that's about it.. i have an array.. and each loop if it gets 0 say.. > 490 times.. then array[490] gets increased by 1... at the end i've got > this array that if u plot it as the ordinate and the indexes of the > array as the absica then you get a very loose very ty very crappy > piece of dog approximation of the bell curve.. it's not even > close.. it's terrible.. i need something more ef[CapitalThorn]cient. === Subject: : Re: need help with bell curve >i want to write a program that will approximate the bell curve.. > Why would you want to approximate it? Why not just calculate it? > See http://mathworld.wolfram.com/NormalDistribution.html sorry i'm not [CapitalThorn]nding the best word to ask what i want to ask.. yes i know the equation for it.. my problem is.. i don't wish to just graph the well known equation.. i'd like to have a computer simulation.. graph it by.. some kind of repetitive-trials sort of algorithm. ßipping a coin over and over again sucks weener in my opinion.. but maybe it's not so bad and i'm just dealing with the values obtained through that method in an inef[CapitalThorn]cient way. any help apprecitaed. === Subject: : Re: need help with bell curve >i want to write a program that will approximate the bell curve.. >>Why would you want to approximate it? Why not just calculate it? >>See http://mathworld.wolfram.com/NormalDistribution.html > sorry i'm not [CapitalThorn]nding the best word to ask what i want to ask.. > yes i know the equation for it.. my problem is.. i don't wish to just > graph the well known equation.. i'd like to have a computer > simulation.. graph it by.. some kind of repetitive-trials sort of > algorithm. ßipping a coin over and over again sucks weener in my > opinion.. but maybe it's not so bad and i'm just dealing with the > values obtained through that method in an inef[CapitalThorn]cient way. > any help apprecitaed. So, what is it you want to do? You're generating samples of a binomial distribution, right? Is that what you really want? Usually, a person means the so-called normal distribution when he says bell curve. You might think of other distributions (than the binomial one from tossing a coin in[CapitalThorn]nitely many times) to use to approximate the normal distribution; the CLT says that averaging independent random variables from *any* [CapitalThorn]xed distribution with [CapitalThorn]nite variance will work. You might note that it is necessarily inef[CapitalThorn]cient to attempt to generate a truly continuous distribution by taking samples from one with not only a discrete, but a [CapitalThorn]nite, set of values. Further, if you gave some notion of what it is you mean by a good approximation (for instance, what range of values you mean to cover -- after all, the normal distribution has in[CapitalThorn]nite support), what precision of values for the density function, and the method you intend to use to get it (in your case, via the Central Limit Theorem as applied to binomial distributions as the number of trials goes to in[CapitalThorn]nity), then it would at least be possible to give you some form of clue as to what to expect. Dale === Subject: : Re: need help with bell curve > i want to write a program that will approximate the bell curve.. could someone tell me the best way to do it? or at least a far far > more ef[CapitalThorn]cient way than what i do at this time which is: > i loop 1000 times > each loop i ßip a coin 1000 times and see how many times it lands 0 > (in other words !(rand()%2) ) that's about it.. i have an array.. and each loop if it gets 0 say.. > 490 times.. then array[490] gets increased by 1... at the end i've got > this array that if u plot it as the ordinate and the indexes of the > array as the absica then you get a very loose very ty very crappy > piece of dog approximation of the bell curve.. it's not even > close.. it's terrible.. i need something more ef[CapitalThorn]cient. Go to the library, borrow a book on probability and statistics. Read it. > you coudl have just said: i'm an asshole and don't wish to help at > all. Slight mispellings here : What he could have said, but I'll say it for him, was : YOU ARE an asshole and I don't wish to help YOU at all. but even that would be completely a waste of a post. > i know the equation > and as you can see by my algorithm.. i know what the bell curve > represents. i'm just looking for an algorithm to approximate it > better. === Subject: : Re: Lacking Concept Of Numbers >= 3 > Did you hear about this Amazon tribe, the Piraha?: > Their language lacks words for any speci[CapitalThorn]c numbers >= 3. > Even though they supposedly are generally intelligent and understand > such mathematical concepts such as catagory, they cannot easily grasp > the idea of speci[CapitalThorn]c integers much above 3. This makes me wonder how they gained the mathematical maturity necessary to deal with concepts as general as catergories. The idea of categories is relatively simple, but it seems arti[CapitalThorn]cial to just introduce them in a mathematical vacuum. Did these guys learn some Real Analysis or Algebra [CapitalThorn]rst? > And unlike Piraha adults, the Piraha children did not have dif[CapitalThorn]culty > understanding larger numbers. > Makes me wonder how much of what WE know (know) is only a direct > consequence of our language and experience and genetics. > Leroy Quet I think the answer to your wondering is Quite a bit. The issue with the Piraha is not genetic--if it were, the children would have as much trouble with larger numbers as the adults. If you haven't done so already, I suggest you read the Philosphical Investigations by L. Wittgenstein -- in particular, the [CapitalThorn]rst 100 or theses. One of his Cambridge Lectures (featuring a young Alan Turing) also touches on this issue. Ôcid Ôooh === Subject: : Re: Lacking Concept Of Numbers >= 3 Interesting... However, their instincts may be driven by lumped size magnitudes judged by empirical estimates with some way to refer to it. May be a facial expression, more powerful than any quantization when striking a bargain or making a point. In Sanskrit, there is a word to denote we/us two (Vayam,from which the word Ôwe' may have been originally derived), and none such in English. When we say we, referring to our group size, it may be 2 <= size(we) < a very large number.It gets sensed from the number of people present at the referred location. A week ago,I had posted that there was no precise Ôsingle word for enquiring sequence order'in English . Also there no speci[CapitalThorn]c word as an interrogative demonstrative that replies with'this' or Ôthat'. ÔWhich' is too general, although it does not materially affect everyday speech. > Did you hear about this Amazon tribe, the Piraha?: .... > Makes me wonder how much of what WE know (know) is only a direct > consequence of our language and experience and genetics. > Leroy Quet === Subject: : Re: Lacking Concept Of Numbers >= 3 Language without cardinality concepts seems to impede intellectual progress. This sends the implication that the Neanderthals may not have had an adequate language for them to progress to quantitative conceptualizations and even art. This raises the question: Category theory may be a step backward? Since it takes mathematics to a place where counting is forgotten? > Did you hear about this Amazon tribe, the Piraha?: > Their language lacks words for any speci[CapitalThorn]c numbers >= 3. > Even though they supposedly are generally intelligent and understand > such mathematical concepts such as catagory, they cannot easily grasp > the idea of speci[CapitalThorn]c integers much above 3. > And unlike Piraha adults, the Piraha children did not have dif[CapitalThorn]culty > understanding larger numbers. > Makes me wonder how much of what WE know (know) is only a direct > consequence of our language and experience and genetics. > Leroy Quet -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: : Re: Measurable [CapitalThorn]leds of Hilbert spaces >Hi all, >Is there a generalization of measurable [CapitalThorn]elds of (separable) Hilbert >spaces to, let's say, measurable [CapitalThorn]elds of (arbitrary) Banach spaces? >References? (if online so much the better). >G. Rodrigues Most likely the answer is yes. Further details depend on what exactly you mean by a measurable [CapitalThorn]eld of Hilbert spaces. === Subject: : Cum Hoc, Ergo Propter Hoc Looking over my threads I talk a good bit about sci.math'ers who dispute reality--living in their own little fantasy world--and sci.math'ers claim I live in my own little fantasy world, so what's the answer? Well it is a *math* newsgroup and you'd think that math would be enough, but I've noticed repeatedly that certain posters say things that go against rather basic mathematics, and somehow many of you seem to suddenly forget basic mathematics long enough to believe them. But it's not just a sci.math thing, and mathematicians rather bizarrely ignore logic itself in claiming that Andrew Wiles found a proof of Fermat's Last Theorem, and his mistake is so basic there's a name in logic for it: Cum Hoc, Ergo Propter Hoc And just so you know that I'm not just tossing out some technical term which may leave many of you befuddled because you don't know logic, I'll explain carefully, and in detail how Wiles screwed up and why basic logic says it must be so. Now you may know that Wiles was looking to prove that modular forms and elliptic curves are related. Basically there are 4 numbers that you can use to describe a modular describe an elliptic curve, and they noticed the for every set of 4 numbers for a modular form they could [CapitalThorn]nd an elliptic curve with the It's a perfect setup for a logical error, as rather than do what's necessary in such a situation, which is [CapitalThorn]nd out what exactly the relation is--if there is one--between modular forms and elliptic curves, Wiles set out to compare between sets. He set out to compare every elliptic curve against every modular form, which is a logical error called Cum Hoc, Ergo Propter Hoc. Computer scientists can understand the error by considering that what's needed is a superclass which has the 4 numbers, which both modular forms and elliptic curves belong to, but you see, that's not what was done. To nail it down that mathematicians DO make a logical error in considering Wiles's work, consider that often you'll see the claim that Wiles proved that in some sense modular forms and elliptic curves Look up Cum Hoc, Ergo Propter Hoc, and see for yourself. So why would mathematicians ignore a basic logical error? I think it's because they can. If I'm wrong here I'd like a cogent explanation as to why. I've brought this subject up before and noticed a lot of dancing around the actual subject. Posters would make rather vague statements when I could actually go look at Wiles's paper as it's now available on the web and see they were basically saying b.s. and it seems to me that often when challenged mathematicians, and especially sci.math posters, say a lot of b.s. as if no one will check them. What's the satisfaction though? Don't any of you actually want math research that's truly correct versus being some group fubar? Don't ANY of you actually want math proofs without having to deal with smart people who don't like you pointing out over and over again that you're lying losers who when you can't actually prove something can just congratulate yourselves anyway because you're weak lemmings? You people follow your leaders at any and all costs, no matter how wrong they are, how stupid their mistakes, no matter how wrong that makes all of you. You're rather pathetic. What can you possibly believe in? Do any of you care about anything at all? Does truth mean anything to any of you? Don't any of you have souls? James Harris === Subject: : Re: Cum Hoc, Ergo Propter Hoc > Looking over my threads I talk a good bit about sci.math'ers who > dispute reality--living in their own little fantasy world--and > sci.math'ers claim I live in my own little fantasy world, so what's > the answer? They are right and James Harris is wrong at least better than 90% of the time, since JSH posts at least 9 rejections of each correction to his work before [CapitalThorn]nally admitting that the correction is right and he has been wrong all along. === Subject: : multivariate kernel density function and local minima Hi I was wondering whether there is a way to [CapitalThorn]nd all the local minima given a MULTIVARIATE kernel density function or not. If not, is there any way to [CapitalThorn]nd an approximations for all the local minima? For your convience, I cited a brief intro to kernel density estimation here (for univariate only). http://www.xplore-stat.de/tutorials/smoothernode2.html The goal of density estimation is to approximate the probability density function $f(bullet)$ of a random variable $X$. Assume we have $n$ independent observations $x_1,ldots,x_n$ from the random variable $X$. The kernel density estimator $widehat{f}_{h}(x)$ for the estimation of the density value $f(x)$ at point $x$ is de[CapitalThorn]ned as begin{displaymath} widehat{f}_{h}(x)= frac{1}{nh}sum_{i=1}^{n} Kleft(frac{x_i-x}{h}right), end{displaymath} where $K(bullet)$ denoting a so-called kernel function, and $h$ denoting the bandwidth. === Subject: : Re: Analytic functions of two variables >Some pointers, please, on how similar the theory of analytic functions in >two variables is to the single variable case. > Do you refer to complex-analytic functions of two or more complex > variables? > I was thinking about exactly two, hoping to retain some geometric intuition. [snip discussion of domain of convergence issue] > Now consider a function of two variables, the greatest common divisor. It > certainly seems to be intrinsically tied to the prime structure of the > integers, but we should no longer be surprised if it turned out to have a > useful complex extension. Of course we should be able to interpolate a > countable number of isolated points (I suspect, using a Blaschke (sp?) > product), but I'm looking for something natural (or perhaps more > dif[CapitalThorn]cult, a formulation of the nonexistence of a natural extension). [We > shall not speak of a complex version of pr*me c**nting functions, for fear > of awakening those who slumber.] > One clue seems to be gcd(z,z) = z, at least for z = 1,2,3,... After a bit of reßection a better set of clues seems to be: gcd(z,0) = z gcd(z,w) = gcd(z,w-z) periodicity gcd(z,w) = gcd(w,z) symmetry Also, I'm reminded that the Gaussian integers form a Bezout domain, so greatest common divisors are de[CapitalThorn]ned and calculable there by Euclidean algorithm. So we should expect a natural extension to interpolate the GCD on Gaussian integers (in some fashion). -- Chip Hard A T Eastham Math.com === Subject: : Re: Analytic functions of two variables >>Some pointers, please, on how similar the theory of analytic functions in >>two variables is to the single variable case. >Do you refer to complex-analytic functions of two or more complex variables? >>In particular I'm wondering about how singularities might be distributed. >>As a starting point, is it true that in a power series expansion of an >>analytic function of two variables, say centered at the origin for >>simplicity, the radius of convergence is up to the nearest singularity? >What's the radius of convergence, here? Are you assuming that >(as in complex dimension 1) the interior of the domain on which a >power series converges is an open (round) disk? That's false in >dimension > 1; the right generalization is polydisks (maybe >I mean disc and polydisk; I can never get that right), >as a little experimentation with simple power series in two >variables should convince you. actually the interior of the set of convergence need not be a ball or a polydisk. it's a union of polydisks, satisfying a certain log-convexity condition. [i forget the exact condition, it's not too hard to [CapitalThorn]gure out, and i think it's in narasimhan's little book. in particular it -can- be a ball or a polydisk...] >Leaving aside that kind of question, I think that the (fairly >vast) dissimilarities between complex analysis in one or several >variables are (at least morally) all consequences of the simple >fact that complex-analytic varieties of dimension 0 are just >collections of isolated points, whereas a complex-analytic variety >of dimension 1 or more always contains non-trivial open Riemann surfaces; >isolated points don't themselves, support a whole lot of interesting >analysis, so they don't impose *too* much of a restriction on (for >instance) some analytic function of which they are all the singularities, >but of course a non-trivial Riemann surface is chock-a-block with >analytical (and topological) subtleties of its own, so it can make >a big contribution to the subtlety of (for instance) an analytic >function of which *it* belongs to the singularities. (Here, I'm >using singularities in a sense which may not be the sense you're >interested in, given your question about domains of convergence. >But I think my comments probably still apply.) i once heard someone say that some Big Name in several complex variables claimed the difference between n = 2 and n = 3 was larger than the difference between n = 1 and n = 2. [note that i'm certainly not asserting this myself - i suspect that it may be true from the point of view of the problems that BN was concerned with...] >Lee Rudolph ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: Analytic functions of two variables ... >>What's the radius of convergence, here? Are you assuming that >>(as in complex dimension 1) the interior of the domain on which a >>power series converges is an open (round) disk? That's false in >>dimension > 1; the right generalization is polydisks (maybe >>I mean disc and polydisk; I can never get that right), >>as a little experimentation with simple power series in two >>variables should convince you. >actually the interior of the set of convergence need not be a >ball or a polydisk. Well, I'll be darned; I don't think I ever knew that (as opposed to having known and forgotten it). On the other hand, I don't think I've ever actually made use of the false statement, so my hands are still sort of clean. ... >>Leaving aside that kind of question, I think that the (fairly >>vast) dissimilarities between complex analysis in one or several >>variables ... >i once heard someone say that some Big Name in several complex >variables claimed the difference between n = 2 and n = 3 was >larger than the difference between n = 1 and n = 2. [note that >i'm certainly not asserting this myself - i suspect that it >may be true from the point of view of the problems that BN >was concerned with...] Well, I can believe it, for several (vague) reasons. First, in algebraic geometry there appears to be a big leap in dif[CapitalThorn]culty from surfaces (dimension 2 over the ground [CapitalThorn]eld) to 3-folds (for instance, for purposes of resolution of singularities, or of classi[CapitalThorn]cation); so why not in analytic geometry, and all the more so in analysis? Second, speci[CapitalThorn]c to topological aspects of complex analytic geometry, there certainly are various ways where the difference between n = 2 and n = 3 is larger than the difference between n = 1 and n = 2; due to my own bias I have always assumed that the `real reason' for this was the peculiar status of real dimension 4 _vis a vis_ higher and lower real dimensions from the point of view of real differential topology (sometimes expressed in the catchphrase four dimensions gives you enough room to get yourself into trouble, and not enough room to get yourself out), but maybe the `real reason' is analysis, after all. (Example difference: Eliashberg gave very simple necessary and suf[CapitalThorn]cient conditions in real dimensions > 4 for a real 2n-manifold to be the underlying smooth manifold of a Stein manifold of complex dimension n; but in real dimension 2, there are extra necessary conditions. The way I'm used to understanding them, they're knot theory; maybe they really are complex analysis, though.) Lee Rudolph === Subject: : Re: Analytic functions of two variables >... >What's the radius of convergence, here? Are you assuming that >(as in complex dimension 1) the interior of the domain on which a >power series converges is an open (round) disk? That's false in >dimension > 1; the right generalization is polydisks (maybe >I mean disc and polydisk; I can never get that right), >as a little experimentation with simple power series in two >variables should convince you. >>actually the interior of the set of convergence need not be a >>ball or a polydisk. >Well, I'll be darned; I don't think I ever knew that (as >opposed to having known and forgotten it). ok, simple example, lest you forget: it's clear that sum z^n w^n converges precisely where |zw| < 1. what's the real story... ah, i think i got it. take n = 2 and a series centered at the origin. [CapitalThorn]rst, it's clear that if the terms are bounded when |z|=a and |w|=b then the series converges for |z| On the other hand, >I don't think I've ever actually made use of the false statement, >so my hands are still sort of clean. >... >Leaving aside that kind of question, I think that the (fairly >vast) dissimilarities between complex analysis in one or several >variables >... >>i once heard someone say that some Big Name in several complex >>variables claimed the difference between n = 2 and n = 3 was >>larger than the difference between n = 1 and n = 2. [note that >>i'm certainly not asserting this myself - i suspect that it >>may be true from the point of view of the problems that BN >>was concerned with...] >Well, I can believe it, for several (vague) reasons. First, >[vague reasons snipped] i'll just take your word for that stuff... ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: Analytic functions of two variables >>... >>What's the radius of convergence, here? Are you assuming that >>(as in complex dimension 1) the interior of the domain on which a >>power series converges is an open (round) disk? That's false in >>dimension > 1; the right generalization is polydisks (maybe >>I mean disc and polydisk; I can never get that right), >>as a little experimentation with simple power series in two >>variables should convince you. >actually the interior of the set of convergence need not be a >ball or a polydisk. >>Well, I'll be darned; I don't think I ever knew that (as >>opposed to having known and forgotten it). >ok, simple example, lest you forget: it's clear that >sum z^n w^n converges precisely where |zw| < 1. >what's the real story... ah, i think i got it. i got it exactly backwards... >take n = 2 and a series centered at the origin. >[CapitalThorn]rst, it's clear that if the terms are bounded when |z|=a >and |w|=b then the series converges for |z|so the boundary of the set of convergence is the boundary >of the set where the terms are bounded. the result is that >the set where the terms are not bounded is log-convex. actually the set where the terms are bounded is log-convex. >suppose for example that 4^j c[j,k] is unbounded bounded > (in j and >k) and 4^k c[j,k] is unbounded. bounded >then 2^j 2^k c[j,k] is >unbounded, since for each j,k it's at least as large as no larger than [duh] >the larger of 4^j c[j,k] and 4^k c[j,k]. the same argument >shows that in general the set of all (log|z|, log|w|) such >that c[j,k] z^j w^k is unbounded bounded >is convex. >that's the easy half - may think about the Ôconverse' >later... the converse makes much more sense now that i have the result sort of straight. say S is an open set, S is a union of polydisks centered at the origin, and C is the set of (log|z|, log|w|) with (z,w) in S. if C is convex there is a power series that converges in S and in no larger open set. quasi-proof: if S is |zw|<1 as above then C is a half-plane. if C is some other half-plane it's easy to construct the required power series [analogous to the sum z^n w^n above, but with (exponent of z)/(exponent of w) approximating the slope of the boundary of C...] now in general C is an intersection of half-planes; construct the required power series by taking in[CapitalThorn]nitely many terms from each of the series corresponding to those half-planes. >> On the other hand, >>I don't think I've ever actually made use of the false statement, >>so my hands are still sort of clean. >>... >>Leaving aside that kind of question, I think that the (fairly >>vast) dissimilarities between complex analysis in one or several >>variables >>... >i once heard someone say that some Big Name in several complex >variables claimed the difference between n = 2 and n = 3 was >larger than the difference between n = 1 and n = 2. [note that >i'm certainly not asserting this myself - i suspect that it >may be true from the point of view of the problems that BN >was concerned with...] >>Well, I can believe it, for several (vague) reasons. First, >>[vague reasons snipped] >i'll just take your word for that stuff... >************************ >David C. Ullrich >sorry about the inelegant formatting - typing >one-handed for a few weeks... ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: Analytic functions of two variables >Some pointers, please, on how similar the theory of analytic functions in >two variables is to the single variable case. there's really a lot of differences. see a book on Ôseveral complex variables' [for example krantz.] >In particular I'm wondering about how singularities might be distributed. one big difference is there are no isolated singularities. >As a starting point, is it true that in a power series expansion of an >analytic function of two variables, say centered at the origin for >simplicity, the radius of convergence is up to the nearest singularity? no. yes. no. no: the phrase Ôradius of singularity' doesn't usually come up. yes: if f[z,w] is analytic for |(z,w)| < r [where that's the euclidean norm] then f has a power series convergent in that ball. no: the set where a power series converges is no longer a ball. ************************ David C. Ullrich sorry about the inelegant formatting - typing one-handed for a few weeks... === Subject: : Re: JSH: Weird group by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0nvv05888; Hi! I'm James Harris. I don't have a life so every few days I have to post inanities to the sci.math group in order to get attention. Sometimes these have to do with math, today it has to do with how terrible you math people are. Did I tell you I don't have any degrees in math? I'm an amateur. So my motives are pure. Please don't question them. Oh, did I tell you my new discovery? It's great! Astounding. Monumental. I'll be the next Hawkings. I'm amazed that no one in math has thought of this one before. How stupid you people must be. It's all so very clear to me. I am posting it to this newgroup. It's for your approval, not for your careful consideration and reßection. My discovery, which is mine remember, is so perfect, that few of you will be able to understand it. Those that do will see it needs no correction. WHAT? You disagree with my discovery? You say it's been thought of before? You say it is [CapitalThorn]lled with errors and misconceptions? You think it is about equivalent to high school algebra? I can't believe it. Only two possible conclusions exist: 1) Either you are too stupid to understand my theory, or 2) There is a conspiracy among the sci.math people to keep me from my deserved glory! I will not accept criticism from PhDs and others who have spent years studying math. This is my discovery. Not yours. I own it. It's mine. I will name it after me. Clearly you are jealous that a mere amateur is showing you up. You are all anti-social clods. You clearly don't understand everything. I'm leaving...for a day or two. But I'll be back with a new discovery Thursday. And we will do this all over again. === Subject: : Re: JSH: Your fantasy world problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0o1K06084; >One of the things I've noticed over the years is that posters who many >of you seem to admire or look up to for some reason get away with >rather ludicrous lies about math itself, which lead me to realize that >for many of you mathematics is just some kind of social game. >You care more about the social aspect than the actual facts, but I'm >going to demonstrate to you why your ignorance of mathematics as more >than a social game is so misplaced in this subject area. >You see, when you take a false position in mathematics it doesn't >matter how many people side with you or how long you lie about it as >you're just wrong. >Eventually some people come along who care more about the truth than >your lies as, to survive in the REAL world, you can't just use >fantasy. >Like some sci.math poster might believe that if they wish really hard >bullets aren't real, but if someone comes up with a gun and shoots >them, then it doesn't matter how many other sci.math'ers were cheering >them and their particular fantasy on, as the real world doesn't give a >rat's ass what you think. >And whether you realized it or not, mathematics is part of the real >world. >It's not just your own little private fantasy area where your gang can >get together and just believe what it wants. >You don't get to pick and choose what's true mathematically, and it >doesn't matter how much a particular person pisses you off, the MATH >DOES NOT CHANGE!!! >Now I'm not a nice guy. And I think many of you are immature both >intellectually and emotionally with weird [CapitalThorn]xations and a tendency >towards acting out in hostile group behavior. >But what I hate about you is your weird ability to believe false >things that go against even basic mathematics as long as it makes you >feel good and that's what you think your group expects. >To think in the supposedly modern world the [CapitalThorn]eld of mathematics has >somehow been taken over by these odd social creatures who seem to >think it matters if people LIKE you!!! >Gauss would probably just shut himself away somewhere and never talk >to the public again if he were faced with a crop of mathematicians >like you people. >Math is not and never has been a social activity. >If that sounds weird to you, or ludicrous, and you [CapitalThorn]nd yourself >jumping up in frustration and anger that I'd ever say such a thing, >then you're not a mathematician. >You may be called a mathematician. You may call yourself a >mathematician. >But if you think math is a social activity then you're NOT a >mathematician, not really. >Mathematicians never pause to see what the group thinks, or to check >and see if the result in front of them was discovered by this person >or that as what's more important is whether or not it's true. >And truth in mathematics has nothing to do with people. >It has nothing to do with who you know. >It has nothing to do with who likes you. >All of you can really, really, really just not like me, and it doesn't >matter mathematically, but for sci.math posters, as I've heard it time >and time again over the years that I've been posting, liking is more >important than the math. >People here need to like you or they lie about math. >I say, hey, if you don't like someone that much IGNORE THEM but don't >lie about the math! >You people lie about math and cheer each other for doing it! >You have little respect for algebra and act like algebraic properties >and results depend on opinion polls! >You are not mathematicians, not really. >Mathematicians are not social creatures. They don't need opinion >polls. They are not swayed by group dynamics. >They do not check [CapitalThorn]rst to see if they LIKE someone before they'll >believe correct results. >You see, real mathematicians actually like math. >They like its purity. They like its infallibility. They like its >perfection. >You people may call yourselves whatever, but you are not >mathematicians, as if you were you'd never lie about the math. >James Harris James: Try my Creative web site. It will give you a real hangover. Zim Olson http://www.zimmathematics.com I think, therefore....... === Subject: : Re: nitpicking in the complex plane? there isn't such a [CapitalThorn]gment of our lazy imaginations! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0nwD05915; >WHERE INGENUITY COULD NOT PREVAIL. show me a negative anything and >i'll show you a crook.. and that crook would be YOU Interesting comment. I'm not sure what to make of it except to give you two examples. 1. In a few months here in Wisconsin we will undoubtedly have a wind chill of -10oF. So if negative numbers can't exist, what will happen when I step outside to shovel the driveway? 2. My friend just scored a -5 on the new golf course he tried. 18 holes, 5 under par. Do I conclude, since this is a negative number, that he never really plays golf at all? Negatives were introduced because they do exist in physical reality as well as mathematical theory. Ingenuity is not con[CapitalThorn]ned to positive, real numbers. === Subject: : Re: 42 is interesting by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0nwx05901; >While playing with the number 42 on my recent 42nd birthday, >it came to my attention that the average divisor of 42 is >twelve, which exactly equals the Euler Phi function of 42 >(the count of positive integers less than 42 with no common >divisors to 42). >Some smaller numbers (1,3, and 14) share this property, >but I could not [CapitalThorn]nd any larger ones (although I only >searched up to 100,000). >Does anyone know if there are larger numbers with this >property (Average Divisor equals Euler Phi function), >or know of a proof that there are none? >|/|/| || Burnaby South Secondary >|| |orewood@olc.ubc.ca || Beautiful British Columbia >(You should be using a non-proportional font to correctly > read the email address in my signature.) Huh? === Subject: : Re: What am I studying here? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0nw405930; >Can anyone give me any pointers to what exactly it is I'm studying here? :) >I am exploring properties of the *[CapitalThorn]nite* set of 2D integer vectors >generated by the matrix multiplication > {a c)(i) modulo (D,D) where D=ad-bc > {b d)(j) >for all i,j in Z, where a,b,c,d are [CapitalThorn]xed integers, and D<>0. >I know the matrix is a linear operator, and I know some basic group theory. >Just some names of things I can look up the web would be enough. >-- {a,b,c,d} and ad-bc ? I think about homographic functions: (a*x+b)/(c*x+d) where we [CapitalThorn]nd these four parameters. Or product z1*z2 of z1=a+b*I and z2=c+d*I ,I imaginary. Have a try, Alain === Subject: : Re: A plea for help by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0nxA06019; >Well, I feel more and more like I may end up going to my grave >investigating various properties of my dear Floretions- there >is just too much to do for an (English) teacher working as an >after hour amature mathematician. Mabie some of you would like to >help: >Are you, for example, a CAD specialist? >If so, please fold Floret`s cube into a 3-Dim object >as described in my paper and, after entering the exact placement >of the corners, by all means, inform me of it's precice volume >(say, when the radius of the white circle is set to one)- I >don't care how ugly or pretty that number is- I would just love >to have it. (However, what I am in most dire need of is help >from mathematicians, not computer experts...) I posted a photograph of Floret's cube (3D) at the bottom of the page http://www.crowdog.de -> The Floretions Note that the face in contact with the desk is an equilateral triangle. C. Dement >Anyway, here's what I found today: >A beautiful supplement to Chung-shu's division group. >I call them bats (from their geometric appearance on >Floret's star) >choose Options -> Star Bats >choose K x L x M . The result will be diplayed at the bottom. >Now, looking at the result, group (the last word is meant as a verb) >all the elements of a particular coef[CapitalThorn]cient together (for the >time being, do this without regard to the sign of the coef[CapitalThorn]cient). >In the case described, this yields: >'i, Ôjk' -- 0.25 -> (multiplying elements together without regard > to sign) Ôkk' >'kk', 1 -- 0.50 -> Ôkk' >'k, i', k', Ôij', Ôji', Ôkj' -- 0.75 -> Ôkk' >Now, square the result by pressing the button ^2 and >repeat. To me, this is a fasinating result (the >result -> may also be 1). Want to get >'jj' instead of Ôkk' ? Then swap, either L and >M or K and L and do the procedure exactly as above >(actually, I've never tried this- just listening to my gut) >Better still, return to the options menu and choose >Load Floret's Star: A-1, B-1, C-1 >Do the same procedure as above but replace the step >choose K x L x M from the pull down menu with >multiply A by K (by pressing the buttons A and then >K in that order. >If anyone has read my paper and the explanation of Floret`s cube >I gave was good enough to be understood, he/she will note that this >kind of stuff (i.e., there we relied on geometric positions - >instead of analyising like coef[CapitalThorn]cients- to tell us which elements >would multiply to 1) has happened before under a different setting. >Sincerly, >C. Dement === Subject: : Re: division by zero by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0o2K06193; >> Theorem: 1 = 2. >> Proof: It follows from 0 = 0*1 and 0 = 0*2 that 0*1 = 0*2. >> Now divide both sides by 0 to get 1 = 2. >> Dave L. Renfro >Typical invalid syllogism, where two objects/concepts (here 1 and 2) are >equated on the basis of a shared property (giving 0 when multiplied by 0). > All elephants are pink > Socrates is pink > => Socrates is an elephant >-- >Paul Townsend >I put it down there, and when I went back to it, there it was GONE! >Interchange the alphabetic elements to reply Yes, that is right. Dave Renfro's argument would only be correct, if it followed that whenever 0*1= 0*2 then 1= 2. That is, that the shared property is a DEFINING property: which is exactly what saying that division by 0 is de[CapitalThorn]ned would mean! Given the proposition, Dave Renfro was arguing against, his syllogism is valid. === Subject: : Re: 42 is interesting by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0nvw05897; >While playing with the number 42 on my recent 42nd birthday, >it came to my attention that the average divisor of 42 is >twelve, which exactly equals the Euler Phi function of 42 >(the count of positive integers less than 42 with no common >divisors to 42). >Some smaller numbers (1,3, and 14) share this property, >but I could not [CapitalThorn]nd any larger ones (although I only >searched up to 100,000). >Does anyone know if there are larger numbers with this >property (Average Divisor equals Euler Phi function), >or know of a proof that there are none? >|/|/| || Burnaby South Secondary >|| |orewood@olc.ubc.ca || Beautiful British Columbia >(You should be using a non-proportional font to correctly > read the email address in my signature.) Humans have 42 on the brain. Is it the mice's program? === Subject: : Re: 42 is interesting >>While playing with the number 42 on my recent 42nd birthday, >>it came to my attention that the average divisor of 42 is >>twelve, which exactly equals the Euler Phi function of 42 >>(the count of positive integers less than 42 with no common >>divisors to 42). >>Some smaller numbers (1,3, and 14) share this property, >>but I could not [CapitalThorn]nd any larger ones (although I only >>searched up to 100,000). >>Does anyone know if there are larger numbers with this >>property (Average Divisor equals Euler Phi function), >>or know of a proof that there are none? >>|/|/| || Burnaby South Secondary >>|| |orewood@olc.ubc.ca || Beautiful British Columbia >>(You should be using a non-proportional font to correctly >>read the email address in my signature.) > Humans have 42 on the brain. Is it the mice's program? Ah the vogons are coming! -- The life of a repoman is always intense! === Subject: : Re: Question: SPACE of super-complex numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0o0106039; >SNIP >> There are no 3 D division algebras over the reals. You have to go from >> complex number to quaternions. >> All of these manipulations were tried back in the middle of the 19-th >> century. Google on the history of quaternions and octernians. Also >> Google division algebras. Having a quotient is a strong constraint. >Moufang loops (all groups and Octonions) provide that constraint. They >are mxm Cayley multiplication tables with left and right >multiplicative inverses for every element. They multiply and divide >sets of unsigned coef[CapitalThorn]cients A, and have Frobenius conservation, >with Detm[A]Detm[B]=Detm[AB], where Detm is the determinant of the >multiplication table mapped with the coef[CapitalThorn]cients. They become >algebras when another operation (generalized negation) collapses an >mxm table (iff it has r-fold symmetry) to an (m/r)x(m/r) table, >and the m unsigned coef[CapitalThorn]cients to m/r signed coef[CapitalThorn]cients. Their >multiplicative inverses Ai have Detm[A] as divisors; if this >factorises (into conserved sizes) the inverse splits into partial >fractions. Division by zero occurs if any size becomes zero; it can be >avoided by working in a constrained sub-algebra (renormalization). >Real algebras have r=2; the four real division algebras without >divisors of zero (R, C, H, O) conserve the sum of their squared >elements. As they are monosized, they cannot renormalize. They do not >have (non-trivial) real divisors of zero because the sum of squares is >only zero in the {0,0...} case. >Every Group and Octonion de[CapitalThorn]nes a Hoop algebra [1] over the real >(r=2), terplex (r=3), complex (r=4), (etc.) numbers. Most of these are >partial fraction division algebras; Clifford, Davenport, >Pauli-sigma, etc, algebras are Hoops; Wedge (exterior) and Lie >algebras are obtained by constraining particular hoops. >There are many ways to go to multiple dimensional algebras, and some >of them are interesting, despite the mathematicians horror of >division by zero. >Roger Beresford. >[1] http://library .Wolfram.com/infocenter/Mathsource/4894(Mary Shelley). I agree with your last statement. Let t be any real number- then 0*t= 0 and 1*t = t. Pardon my enthusiasm, but wow- what an expression of sheer, raw and unbiased symmetry of the elements 0 and 1 towards, well, absolutely everything! (at least part of this symmetry lies in the fact that the above expressions are themselves justi[CapitalThorn]able). However, if x*y = 0 and x,y and 0 are real numbers, then either x or y is 0. This is an additional symmetry property to the one above- but it also represents a new, third constraint which, in a [CapitalThorn]gurative way, con[CapitalThorn]nes us to speaking of circles instead of ellipses. For algebras with divisors of zero, one can [CapitalThorn]guratively restate the above: If x*y = 0 then either x or y or both x and y must contain some remnants of symmetry. This gives us many more ways of producing symmetry: squares, rectangles, ellipses, etc.- not just circles. Put another way, 0 may be seen as a linear operator 0: reals -> reals, 0(x + y) = 0x + 0y = 0 0(tx) = tOx = 0. The nullspace of this operator is the reals itself. The nullspace of some operator x != 0 is {0}. Accordingly, all such nullspaces are trivial. However, on a real or complex algebra P with zero divisors, choose any z in P. Then z: P -> P, z(x) = z*x is also a linear function whose nullspace may very well be worth investigating. C. Dement http://www.crowdog.de === Subject: : measurable functions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0o1w06123; Let $X$ be a measure space with measure $mu$ and $mu(X)=1$. is it really true that the space of measurable functions $u:Xto mathbb{R}$ is a topological space with basis de[CapitalThorn]ned by the following neighborhoods: $U_{r,sigma}(v)={umid mu(xmid |u(x)-v(x)|1-sigma}$ ? Oleg. === Subject: : Re: Giving up by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0nvE05884; >New comer to this newsgroup. >Good luck >Beadumund Sauerborn >Tel. +1 310 276 2194 >Beadumund@fashionspan.com >Beadumund@soheilroohani.com That was quick. === Subject: : Re: 1+i > i by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0o2D06228; >oh...my god~ >were you born for mathematics ? >or else were mathematics born for you ? >thank you sir. >of course, other doctors, too. Hey, s/he's a mood booster! And if she's a she, we can hire her as sci.math's of[CapitalThorn]cial cheerleader lol === Subject: : Re: cos(x) >= 1 - x^2/2! + x^4/4! - x^6/6! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0o1w06170; I think you can clear up this thing considering the function h(x)=cos(x)-(1 - x^2/2! + x^4/4! - x^6/6!) and looking for its minimums by means of derivatives. >f_0(x) = 1 >f_1(x) = x^2/2! >f_2(x) = x^4/4! >f_3(x) = x^6/6! >f_n(x) = x^(2n) / (2n)! >We know that cos(x) = f_0(x) - f_1(x) + f_2(x) - f_3(x) + ... + >(-1)^n*f_n(x) + ... for all x. >I wonder if it is true that: > cos(x) >= f_0(x) - f_1(x) + .... + (-1)^(2n+1)*f_(2n+1)(x) for all x. >This inequality is true for very small x, because the sequence ( f_0(x), >f_1(x), f_2(x), ... ) is a decreasing sequence when -1This inequality is true for very large x, because then >f_0(x) - f_1(x) + .... + (-1)^(2n+1)*f_(2n+1)(x) >is a negative number far away from 0 while cos(x) is bounded between -1 and >Therefore we can guess that this inequality may be also true for other >values of x. >This inequality comes from a book called Problem-Solving Through Problems. >The author of the book assumes the inequality to be true, for the only >reason that the sequence f_0(x), -f_1(x), f_2(x), -f_3(x), ... is an >alternating sequence. ( That the tayler series for cos(x) is an alternating >series.) But the absolut values of the sequence is not steadily decreasing. >it is not true that f_n(x) >= f_(n+1)(x) for all x and for all nonnegative >integer n. >I failed to prove or disprove the inequality. Can somebody help me on this? === Subject: : Re: When radius point is not accessible. > I work construction as a carpenter and need to lay out interior > partition walls on a curve. More speci[CapitalThorn]cally, this curve is a 25'-0 > radius. The problem is that the focus point or pivot point to swing > the radius is not accessible to me because its position is located > dead center of an existing 10 H-column. > What is the best and most ef[CapitalThorn]cient way to proceed while trying to > maintain the integrity of the 25' radius?.... You've had some fairly technical answers; but why not just buckle your belt in a large loop around the column and attach to it something shorter than the radius? As you swing that around, the belt will slip round the column, giving your circle. It shouldn't be hard for you to adjust the arrangement by trial and error to get the right radius. Ken Pledger. === Subject: : Re: When radius point is not accessible. >I work construction as a carpenter and need to layout interior >partition walls on a curve. More speci[CapitalThorn]cally, this curve is a 25'-0 >radius. The problem is that the focus point or pivot point to swing >the >radius is not accessible to me because it's position is located dead >center of an existing 10 H-column. > What is the best and most ef[CapitalThorn]cient way to proceed while trying to >maintain the integrity of the 25' radius? Mark a quarter of a 25' radius circle (outside, away from obstructions) on a 25' x 25' (or larger) piece of canvas (or similar). Take that canvas indoors and fold up the corner near the H-column and place the canvas so that the center of the circle you have marked is at the center of the column. Rob Johnson take out the trash before replying === Subject: : Re: When radius point is not accessible. > I work construction as a carpenter and need to layout interior > partition walls on a curve. More speci[CapitalThorn]cally, this curve is a 25'-0 > radius. The problem is that the focus point or pivot point to swing > the > radius is not accessible to me because it's position is located dead > center of an existing column. > What is the best and most ef[CapitalThorn]cient way to proceed while trying to > maintain the integrity of the 25' radius? > Chris Grubb If your beam is symmetric it can be inscribed in a circle, and a line attached to that circle would itself produce a circle around the beam's center. A needlepoint hoop of the right size should work well if your beam is small enough. Just put it around the beam and attach a line of 25' - (radius of hoop). === Subject: : Re: Help by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i7N0o3G06320; I have the same question about this problem. If anyone has written you with help on this problem, I would greatly appreciate forwarding some helpful information back to me. Signed, Derek >just cant solve. >can somebody please help me save face (My face) >Problem as follows >A grade three maths test included the rather tough challenge shown. >can you solve it? >If 2 * 6 = 4 > 4 * 1 = 7 > 8 * 3 = 1 >[CapitalThorn]nd the value of > 5 * 5 >thank you in antisipation >Manfred hetech@alphalink.com.au > === Subject: : [JSH] Lunatic fringe .... Re: Your fantasy world problem [Deranged, incoherent, useless nonsense deleted!] > James Harris Mr. Harris, one simple question. Show your prime counting function beat the current record and prove it. All arguments would cease. What is wrong? Why won't you do that? Why won't you admit the truth? Your method is about as an ant crawling through molasses and still you continue to spew! Why is it that you say one thing, but fail to produce results? Could it be that you are a fraud, a liar, a cheat? You bet!