mm-2849 Subject: Re: 3x+1 and a limit of zero question. > I have a question on expressing a relationship of the First evens > after 3x+y and the rest of the values. > What I have done and I don't know if it is right is add up the evens > after 3x+y and add up the values from x/2. So for the sequence beginning with 33: 33 100 50 25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 the evens after 3x+1 would be 100 76 58 88 34 52 40 16 and the values from x/2 would be 50 25 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1 which is all the rest of the numbers excluding the starting value of 33? These lists sum to 464 and 344 respectively. > I divide the first evens after 3x+1 by x/2's 464/344 = 1.348837209 Why is yours off at the 8th decimal place? What is point of so many decimal places if they are all wrong? This makes all your calculations suspect. > and sum the results for each sequence. > I keep count of how many sequences I have added and divide the sum of > the sequences by that count. So you're averaging the averages? > The result drops in value over a distance. For example : x = 33 was > on the screen. > : 33 : > For x == 33 master = 1.64212163955129253345432971400441601872444153 What is master? Is it the average of averages for the numbers less than 33? > the sum is 464.0000000000000000000000 > the eve 344.0000000000000000000000 > the average is 1.3488372564315795898438 > The result for this sequence was 1.3488372564315795898438. > For the first 33 sequences the result was > 1.64212163955129253345432971400441601872444153 Or does it include the result from 33? > And to jump ahead. > : 4967295 : > For x == 4967295 master = > 0.13651073485192624645812031758396187797188759 > the sum is 5370177536.0000000000000000000000 > the eve 3585085184.0000000000000000000000 > the average is 1.4979218244552612304688 > What I am not sure of is, is this a way to show that 3x+1 approaches > zero as a limit? What approaches 0? The master or the average? > That over the distance, as the value of x grows, the average looks to > get closer to zero. Ok, you just said average here, so you don't mean master approaches 0. And this is wrong because 1.49... is larger than 1.34... But if you meant that 0.136... is closer to 0 than 1.64..., then you really should work on making this stuff understandable to others. Right now it is riddled with vague and inconsistant concepts. Do you wonder why no one replies to your queries? > I suppose if the sum of the data related to x/2 was divided by the > even result of 3x+1 a constant would slowly appear. > I just checked ( in a hurry ) and perhaps dividing the sum of all x/2 > by the first even after 3x+1 gives this. > : 4967293 : > For x == 4967293 master = > 979.58880315526323556696297600865364074707031250 > the sum is 68498488.0000000000000000000000 > the eve 50632936.0000000000000000000000 > the average is 0.7391832470893859863281 > ------------- > : 4967294 : > For x == 4967294 master = > 979.58916300312830571783706545829772949218750000 > the sum is 5088031744.0000000000000000000000 > the eve 3396987648.0000000000000000000000 > the average is 0.6676427721977233886719 > ------------- > 4967295 : > For x == 4967295 master = > 979.58892847991523922246415168046951293945312500 > the sum is 5370177536.0000000000000000000000 > the eve 3585085184.0000000000000000000000 > the average is 0.6675915718078613281250 > Any clues? > === Subject: Re: 3x+1 and a limit of zero question. correctly. > I have a question on expressing a relationship of the First evens > after 3x+y and the rest of the values. What I have done and I don't know if it is right is add up the evens > after 3x+y and add up the values from x/2. > So for the sequence beginning with 33: > 33 100 50 25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 > the evens after 3x+1 would be > 100 76 58 88 34 52 40 16 > and the values from x/2 would be > 50 25 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1 Yes that is the idea. I want to study the two sets idea. I have divided the sequence into two data sets. One is the result of a 3x+y and I call it the first even after. The other data are the values that are the result of the x/2 including the resulting odd. > which is all the rest of the numbers excluding the starting value of 33? > These lists sum to 464 and 344 respectively. > I divide the first evens after 3x+1 by x/2's > 464/344 = 1.348837209 > Why is yours off at the 8th decimal place? What is point of so many decimal > places if they are all wrong? This makes all your calculations suspect. I don't know what limits are valid for the settings I have used so perhaps the values are off some. The basic idea I'm aiming at it is getting the algorithm correct. I will work on precision with the proper library. > and sum the results for each sequence. > I keep count of how many sequences I have added and divide the sum of > the sequences by that count. > So you're averaging the averages? Yes. Is that a bogus thing to do? The result drops in value over a distance. For example : x = 33 was > on the screen. > : 33 : > For x == 33 master = 1.64212163955129253345432971400441601872444153 > What is master? Is it the average of averages for the numbers less than 33? Master is my term for average of averages. > the sum is 464.0000000000000000000000 sum of 3x+1 ( first even after ) > the eve 344.0000000000000000000000 Sum of x/2's > the average is 1.3488372564315795898438 sum of result of 3x+1 / sum of x/2 results The result for this sequence was 1.3488372564315795898438. > For the first 33 sequences the result was > 1.64212163955129253345432971400441601872444153 > Or does it include the result from 33? Yes. And to jump ahead. > : 4967295 : > For x == 4967295 master = > 0.13651073485192624645812031758396187797188759 > the sum is 5370177536.0000000000000000000000 > the eve 3585085184.0000000000000000000000 > the average is 1.4979218244552612304688 > What I am not sure of is, is this a way to show that 3x+1 approaches > zero as a limit? > What approaches 0? The master or the average? The average of averages. Stated as master: master average. The 0.1365... > That over the distance, as the value of x grows, the average looks to > get closer to zero. > Ok, you just said average here, so you don't mean master approaches 0. > And this is wrong because 1.49... is larger than 1.34... Over a distance the sum of averages divided by the count of the averages looks to become smaller ( if I have not abused the system ). Is that what approaching a limit of zero means? Referenceing the averages of averages: I am thinking you mean that each term must be less that the one before to qualify as heading toward a limit of zero. I don't think that is what happens. There seems to be some ups and downs. But over a distance it seems to head toward zero. Understand I may not be using the term head toward zero correctly. But that is why I'm exposing myself. > But if you meant that 0.136... is closer to 0 than 1.64..., then you really > should work on making this stuff understandable to others. Right now it is > riddled with vague and inconsistant concepts. Do you wonder why no one replies > to your queries? They don't know. Most people don't want to risk their image so they 1.) hide behind a false persona or 2.) don't get involved. I understand both. I learned of the concept of going toward a limit of zero here in sci.math so I figure it's the place to ask about it. So is my use of averaging the averages correct? ( forgive any errors in the math ) === Subject: Re: 3x+1 and a limit of zero question. > correctly. I have a question on expressing a relationship of the First evens > after 3x+y and the rest of the values. > What I have done and I don't know if it is right is add up the evens > after 3x+y and add up the values from x/2. So for the sequence beginning with 33: 33 100 50 25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 the evens after 3x+1 would be 100 76 58 88 34 52 40 16 and the values from x/2 would be 50 25 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1 Yes that is the idea. I want to study the two sets idea. I have > divided the sequence into two data sets. One is the result of a 3x+y > and I call it the first even after. The other data are the values > that are the result of the x/2 including the resulting odd. Ok, but try to take the time to provide simple examples like I did above so that readers don't have to guess what you mean and have to build a spreadsheet to test various goupings until the sums match what you have below. > which is all the rest of the numbers excluding the starting value of 33? These lists sum to 464 and 344 respectively. > I divide the first evens after 3x+1 by x/2's 464/344 = 1.348837209 Why is yours off at the 8th decimal place? What is point of so many decimal > places if they are all wrong? This makes all your calculations suspect. I don't know what limits are valid for the settings I have used so > perhaps the values are off some. The basic idea I'm aiming at it is > getting the algorithm correct. But how do you know the algorithm is correct? For instance, the number I got above is the default setting in Excel. When I do the same on the Windows calculator I get 1.3488372093023255813953488372093 but note that although this has more digits than the Excel answer, none of the Excel digits are wrong, unlike your calculation. > I will work on precision with the proper library. That may not be the answer. You might be doing something in your program like intermixing single and double precision numbers, in which case a proper math library won't help. It's not so much that we care about the 10th decimal place, it's that the significance of the error can grow as you do repeated operations on the numbers, possibly to the point where the expected convergence doesn't materialize. > and sum the results for each sequence. > I keep count of how many sequences I have added and divide the sum of > the sequences by that count. So you're averaging the averages? Yes. Is that a bogus thing to do? I don't know. For some things it is legit. If you took the average weight of 100 pennies minted in 2002, you'll find that it is higher than the average weight of pennies minted in 1992. Actually, they all started out the same, but average weight is a function of wear which is a function of time. > The result drops in value over a distance. For example : x = 33 was > on the screen. > : 33 : > For x == 33 master = 1.64212163955129253345432971400441601872444153 What is master? Is it the average of averages for the numbers less than 33? Master is my term for average of averages. Ok, the problem was you were saying average when you meant average of averages. That's what I meant about being consistant. Define the terms you are using and use them consistantly. And use the same names and variables in you programs. > the sum is 464.0000000000000000000000 > sum of 3x+1 ( first even after ) > the eve 344.0000000000000000000000 > Sum of x/2's > the average is 1.3488372564315795898438 sum of result of 3x+1 / sum of x/2 results > The result for this sequence was 1.3488372564315795898438. > For the first 33 sequences the result was > 1.64212163955129253345432971400441601872444153 Or does it include the result from 33? Yes. > And to jump ahead. > : 4967295 : > For x == 4967295 master = > 0.13651073485192624645812031758396187797188759 > the sum is 5370177536.0000000000000000000000 > the eve 3585085184.0000000000000000000000 > the average is 1.4979218244552612304688 > What I am not sure of is, is this a way to show that 3x+1 approaches > zero as a limit? What approaches 0? The master or the average? > The average of averages. Stated as master: master average. The > 0.1365... > That over the distance, as the value of x grows, the average looks to > get closer to zero. Ok, you just said average here, so you don't mean master approaches 0. > And this is wrong because 1.49... is larger than 1.34... > Over a distance the sum of averages divided by the count of the > averages looks to become smaller ( if I have not abused the system ). > Is that what approaching a limit of zero means? I'm not a math expert, so I won't risk saying something incorrect. But I don't think becoming smaller necessarily means having 0 as a limit. For example, say you only looked at sequences that start with an odd number and have nothing but evens all the way to 1: 5 16 8 4 2 1 21 64 32 16 8 4 2 1 85 256 128 64 32 16 8 4 2 1 341 1024 512 256 128 64 32 16 8 4 2 1 1365 4096 2048 1024 512 256 128 64 32 16 8 4 2 1 5461 16384 8192 4096 2048 1024 512 256 128 64 32 16 8 4 2 1 1.06666667 1.01587302 1.00392157 1.00097752 1.00024420 1.00006104 and although these are getting smaller, the limit is 1, not 0. > Referenceing the averages of averages: I am thinking you mean that > each term must be less that the one before to qualify as heading > toward a limit of zero. I don't think that is what happens. There > seems to be some ups and downs. But over a distance it seems to head > toward zero. Understand I may not be using the term head toward zero > correctly. But that is why I'm exposing myself. That's what I'm not sure about. Those ups and downs may combine to give you some single number, but I'm not sure what it means. To use the coin analogy again, the average weight of pennies is one bell curve and the average weight of nickels is a different bell curve. Now if you took the average weight of a mixture of pennies and nickels, especially if you didn't know the ratio, what would that tell you? I've been studying binary patterns and find that an overall stopping time doesn't make sense. Stopping times behave like pennies and nickels. The stopping times (S) for m-bit binary numbers: S = m * 13.457 for Mersenne Numbers (2^m - 1) S = m * 8.228 for Fermat Numbers (2^m + 1) S = m * 8.228 for random binary patterns (same as Fermat but for a different reason) I could average those numbers together, but what would it accomplish? There are many more random binary patterns than Mersenne or Fermat so the result would probably be close to m*8.228 but that would obscure the important differences. Why do Mesersenne numbers have so much larger stopping times? I pointed out that Fermat numbers and random numbers are different, but you would not know that just by looking at the averages. But if you meant that 0.136... is closer to 0 than 1.64..., then you really > should work on making this stuff understandable to others. Right now it is > riddled with vague and inconsistant concepts. Do you wonder why no one replies > to your queries? They don't know. Most people don't want to risk their image so they > 1.) hide behind a false persona or If the reason for a false persona was to hide, then they would not be risking their image by replying. Using a psuedonym allows me to reply to anything I feel like without any concern about image. > 2.) don't get involved. I understand both. You missed reason number 3. When I scan through sci.math looking for interesting subjects, I just skip past the stuff that's over my head such as topology questions. I also have a tendency to skip past things that are incomprehensible. > I learned of the concept of going toward a limit of zero here in > sci.math so I figure it's the place to ask about it. But you have to learn how to ask about it in the right way. Many people will simply hit the [Next] button rather than make an effort to decipher your writing. > So is my use of averaging the averages correct? ( forgive any errors > in the math ) > === Subject: Re: Help With a Series... > I am looking for a second simple sequence formula for the following: > 3, 5, 7, 9, 11... > I have found the following: 2n + 1. try the online encyclopedia of integer sequences at: http://www.research.att.com/~njas/sequences/ Make sure to enter in a long enough string of numbers, that is: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 for example. Some interesting ones indeed (may need to throw out some initial values from the sequence). HTH, Flip === Subject: calculate days between two dates - Australian program I'm trying to find a program I used to have on my computer that would calculate the number of days between two dates. I think it was an Australian program. It would do other calculations like add a certain number of days to a given date and give that date. === Subject: Re: calculate days between two dates - Australian program > I'm trying to find a program I used to have on my computer that would > calculate the number of days between two dates. I think it was an > Australian program. It would do other calculations like add a certain > number of days to a given date and give that date. Tucows has a couple of date calculators, one freeware, one commercial. http://www.tucows.com/calculators95_default.html Here's one that does the forward calculation (what date is it X days in the future?) http://www.srosystems.com/sro_wd.htm There's a bunch of other stuff out there. I got the most useful hits with freeware date calculator in Google. - R === Subject: Re: calculate days between two dates - Australian program Yes!! Found it doing that search. http://members.ozemail.com.au/~jaesenj/software/software.html >There's a bunch of other stuff out there. I got the most >useful hits with freeware date calculator in Google. > - R === Subject: Re: calculate days between two dates - Australian program >I'm trying to find a program I used to have on my computer that would >calculate the number of days between two dates. I think it was an >Australian program. It would do other calculations like add a certain >number of days to a given date and give that date. An easy way to that is with your spreadsheet === Subject: Re: Can mathematics unambigiouslly model cause and effect? > (revised) > Hey all, > I have a question, if I have two seperate formulas that model two > phenomena and the two phenomena are related by cause and effect, is it > possible to model the cause and effect relationship between the two > formulas without leaving any question to interpretation as to which is > the cause, and which is the effect? > How about the following: > If, for all x, P(x) causes Q(x), then > (1) there exists an x such that P(x), and > (2) there exists an x such that not Q(x), and > (3) for all x, P(x) implies Q(x). > The converse is not true, however. On second thought, might we define mathematical causality as follows: Suppose P and Q are logical predicates of a single variable. P is said to cause Q iff (1) there exists an (instance) x such that P(x), and (2) there exists an (instance) x such that not Q(x), and (3) for all (instances) x, P(x) implies Q(x). It seems to fit with everyday notions of causality: We can't say that P causes Q if P is never true or if Q is never false. And whenever P is true, Q must be also true. Comments? Dan === Subject: Re: Can mathematics unambigiouslly model cause and effect? Dan Christensen > It seems to fit with everyday notions of causality: We can't say that P > causes Q if P is never true or if Q is never false. And whenever P is true, > Q must be also true. Comments? Dan, this works well enough for me, but unfortunately you are using the language of logic here and not the language of mathematics. Personally I think logic is exactly what the solution requires, but I've been told that only mathematics will suffice. Here is the situation: A. Classical Mechanics has rules for the objects it describes We have discovered these rules by observing something, finding the pattern, and deriving the formula. For example, x = 1/2 at^2 B. Quantum Mechanics has rules for the objects it describes C. I suggest that the rules of Quantum Mechanics produces the result of Classical Mechanics. The rules of CM that we have discovered are contained in the results of the rules that we have discovered for QM. D. I have been told that to justify my suggestion I must present an equation that shows this correlation. I feel this is impossible. Because the rules of CM are found in the results of QM and not in the rules of QM, finding an equality between the two sets of rules is meaningless. To justify why I feel it is impossible, we consider rule 16 of cellular automata. If we have the following cells (1 = black, 0 = white) 100000000 Rule 16 says that the next row can be found by making black every cell that was previouslly to the right of a black, and all others white. The result is something like: 100000000 010000000 001000000 000100000 000010000 When exectued, we have observed that it produces a result. We also have recognized a pattern in this result that can be described as a line described by the formula: y = -x. The question is, can I write an equation that describes all of the following: 1. Rule 16 2. y = -x 3. That y = -x is a result of rule 16 As far as I know this is impossible. If I am wrong, given how simple the formula and program are, it should be simple to prove me wrong by providing an equation. Because this simple request has been impossible to fulfil, and because the program represents itself (1) and produces its result (2) we see that any patterns we observe in the result are an effect of the program (3), it is my conclusion that no equation is necessary as the program itself is capable of satisfying all 3 of the conditions. E. My conclusion is that I do not need to present an equation but a program to model the relation I wish to communicate My conclusion could be easily falsified by providing an equation (or set of) that satisfy the three conditions above. Is it possible? Mike Helland === Subject: Re: Can mathematics unambigiouslly model cause and effect? the language of logic here and not the language of mathematics. No, Logic is part of Mathematicsa. The real problem is that you failed to define what you mean by cause. Absent such a definition, people like Dan have to guess at your meaning. >C. I suggest that the rules of Quantum Mechanics produces the result >of Classical Mechanics. >D. I have been told that to justify my suggestion I must present an >equation that shows this correlation. That doesn't sound like a philosophical discussion of cause and effect, it sounds like you need to show that the solution to one set of equations approximates the solution to another. It also sounds like a homework assignment. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be 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: Re: Can mathematics unambigiouslly model cause and effect? > On second thought, might we define mathematical causality as follows: > Suppose P and Q are logical predicates of a single variable. > P is said to cause Q iff > (1) there exists an (instance) x such that P(x), and > (2) there exists an (instance) x such that not Q(x), and > (3) for all (instances) x, P(x) implies Q(x). > It seems to fit with everyday notions of causality: We can't say that P > causes Q if P is never true or if Q is never false. And whenever P is true, > Q must be also true. Comments? > Dan Suppose someone claimed that thunder caused lightning. How would you use your definition to support or refute it? -- 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: Can mathematics unambigiouslly model cause and effect? My proposed definition clearly doesn't work. It doesn't, for example, take into account delayed effects, which account for most effects in the real world. With delays, the cause and effect need never be present at the same instant. Sorry about that. Dan > On second thought, might we define mathematical causality as follows: > Suppose P and Q are logical predicates of a single variable. > P is said to cause Q iff > (1) there exists an (instance) x such that P(x), and > (2) there exists an (instance) x such that not Q(x), and > (3) for all (instances) x, P(x) implies Q(x). > It seems to fit with everyday notions of causality: We can't say that P > causes Q if P is never true or if Q is never false. And whenever P is true, > Q must be also true. Comments? > Dan > Suppose someone claimed that thunder caused lightning. How would you use your > definition to support or refute it? > -- > 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: good introduction to representation theory resources I'm starting an independent study of representation theory and I was wondering if anyone could suggest some good references (textbooks, online lecture notes) to have a look at. B === Subject: Re: Questions on Infinite Sets > I stand by my earlier assertion that a path-connected space cannot > be partitioned into countably many, but at least 2, disjoint nonempty > closed sets. I can't find a published proof of this at the moment, > but I have found > http://www.math.nus.edu.sg/~urops/Projects/connected.PDF > which contains proofs that the following classes of spaces cannot be so > partitioned: > connected compact Hausdorff spaces > connected locally connected locally compact Hausdorff spaces > connected locally connected complete metric spaces > R^2 is, of course, in two of these classes. The property for path- > connected spaces follows from that for [0,1]. I haven't checked > the proofs in the above paper, but I have a separate proof for [0,1]. You're right. I withdraw my claim. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: question on hypothesis test-- when the distribution unknown or not normal The question is: There are two populations whose distributions are unknown or not normal. What I want to do is to test the difference of the expectation and variance. The sample size may be a few hundred, e.g. 200. Is there any effective way to do that? I know for t-test, the assumption is the data is normal and the variance of two population is the same. But what if the assumption is violated? === Subject: Inverse of Lagrange enterpolation I have system that takes X as input and gives me Y as output (Y=F(X)). I have interpolated (lagrange) three points to find its transfer function. But I need to calculate X from Y values. I think I need the inverse function of lagrange interpolation polinomial. I don't know if it is inversible but I know each seperate x value corresponds to a seperate y value. Please help === Subject: Re: Inverse of Lagrange enterpolation > I have system that takes X as input and gives me Y as output > (Y=F(X)). I have interpolated (lagrange) three points to > find its transfer function. But I need to calculate X from Y > values. I think I need the inverse function of lagrange > interpolation polinomial. I don't know if it is inversible > but I know each seperate x value corresponds to a seperate y > value. Assuming it's one-to-one over the range of interest (I think that's what your last sentence says), I'd say it's easiest to interpolate in the inverse direction from the start. That is, calculate X = G(Y) as a polynomial. Obviously it's not the same polynomial but it should be just as reasonable an approximation as your first polynomial. Another approach that will more accurately follow your first polynomial is to invert by table lookup: Calculate Y for a bunch of interpolated X's. Then when you want a particular value, find the corresponding place in the table. You can use the X = G(Y) just in that place in the table, say by interpolating among 3 neighboring points. Yet another approach is to use successive approximations with a binary search. Keep evaluating F(X) at different places, narrowing the search interval, until you've zeroed in on the value of interest. If time is a major concern, I'd go with table lookup and even precalculate the interpolating coefficients. If simplicity, I'd go with the binary search or the X = G(Y) approach. Hope some of that was clear. - R === Subject: Re: solving a class diophantine equations >Consider the Diophantine equations of the form: >y = P(x)/Q(x) >where y and x are integers and P(x) and Q(x) are polinomials with integer >coefficients. There is a general and efficient algorithm to solve this kind >of equations? Use the division algorithm in k(x) (where k is the field of rational numbers (I wish you hadn't used Q already! (Oh no, nested parentheses! I'm starting to write like Lee Rudolph!))). You can then clear denominators to find an equation among integer polynomials of the form N P(x) = Q(x) D(x) + R(x) for some integer N, where R has lower degree than Q. Then if both x and y=P(x)/Q(x) are integers, so must R(x)/Q(x) be. But |R(x)/Q(x)| < 1 for all sufficiently large x, which will give a contradiction unless x is a root of R (or R is identically zero). You then have only the finitely many small x to check manually. dave === Subject: Re: yet another lebesgue measure problem >Let E subset R be a lebesgue measurable subset such that if x, y in >E, x != y then 1/2 (x+y) is _not_ in E. Show that E has measure zero. Hint: Suppose (a,b) is an interval the great majority of whose measure is in E. Let F = (a,b) intersect E. Take any x in F, and consider {(x+y)/2: y in F}. 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: yet another lebesgue measure problem > Let E subset R be a lebesgue measurable subset such that if x, y in > E, x != y then 1/2 (x+y) is _not_ in E. Show that E has measure zero. > Any idea? Let I be an interval and let F = E intersect I. If x is in F, then Fx = (x/2) + (1/2)*(F {x}) is disjoint from F. On the other hand, Fx must lie in I, because I is convex. Note that m(Fx) = (1/2)m(F). This tells you that E can never take up more than 2/3 of an interval, which leads to m(E) = 0 fairly quickly. === Subject: |R^2 vs |E^2 vs |C It's a shame that I don't seem to understand the root of things. But again, I'm just a first year student, I'm allowed to look silly. Before these days are over, I better ask questions. All of these things are in general considered as sets of all (x, y) [x, y (obviously) real]. I have trouble distinguishing them. The definitions are quite distinct, I can see that. Please, kind soul, take time to let me understand the *idea* behind the distinction as the *results* which make each of them worthy of study. If you have a quick search, you'll see I have asked the question before (under the name of some 'Pirated Dreams', perhaps), but was largely ignored. This is not fair, I am not James Harris. Let me tell you what I *do* know. |R^2 has much less structure than the other two. Correct me if I am wrong - |R^2 has only a linear space structure. There is no notion of distance (therefore, orthogonality), only linear combination (over |R). Clearly shows my question could ignore this one. Now, both |E^2 and |C has a notion of distance, but that of |C, is defined in a funny way (but in the end, it has the same *form* as in |E^2, and that might be where I am confused, if they were different, I would know straightaway that |E^2 and |C are different). You say, Can't you see? There is no multiplication defined of |E^2! I see that. But what does that have to do with topological differences between |C and |E^2? Here it goes. d(a, b) = sqrt[(a_x - b_x)^2 + (a_y - b_y)^2] in |E^2, so that we have a down to earth situation here. d(a, b) = |a - b| in |C. I have seen many books that have the |E^2 distance definition as the definition of |a - b|. This makes them (from a topological point of view) exactly the same. But there is an alternative, define |z| by sqrt(z^* z), a definition *via* multiplication. Not to forget it also involves conjugation, so that, in a way, the x-axis has a preference, unlike |E^2. Does that have any consequence later on? I mean, the only thing I can *see* different in |C is that the real axis is treated in sort of a special way here. But why? And what are the topological implications of this? Tell me, kind soul, why can we integrate 1/z? Isn't there a counterpart of that in |E^2? Or, am I asking the wrong question? Are these guys topologically the same, but that weird *extended* complex plane comes into play here? Best wishes (the more explanatory and exemplified the replies are, the better!) === Subject: Re: |R^2 vs |E^2 vs |C at 09:23 PM, uchchwhash@hotmail.com ( Tashdid ul Alam) said: >It's a shame that I don't seem to understand the root of things. But >again, I'm just a first year student, I'm allowed to look silly. >Before these days are over, I better ask questions. You're not going to like the answer. Different authors use different notations. Normally a textbook will define the author's notation, either at the beginning or as he goes along. If you're reading several books on the same subject, you need to keep straight which notation applies to which. >Now, both |E^2 and |C has a notion of distance, but that of |C, is >defined in a funny way (but in the end, it has the same *form* as in >|E^2, and that might be where I am confused, if they were different, >I would know straightaway that |E^2 and |C are different). You say, >Can't you see? There is no multiplication defined of |E^2! I see >that. But what does that have to do with topological differences >between |C and |E^2? Assuming that by |E^2 and |C you mean the Euclidean Plane and the Complex Field with the usual topologies, then there are *NO* topological differences. >But there is an alternative, define |z| by sqrt(z^* z), a definition >*via* multiplication. Not to forget it also involves conjugation, so >that, in a way, the x-axis has a preference, unlike |E^2. Does that >have any consequence later on? Not topological consequences. >But why? Because it makes Analysis simpler. >And what are the topological implications of this? None. >why can we integrate 1/z? Why not, as long as the origin is not on the path you're integrating over? >Or, am I asking the wrong question? Are these guys topologically the >same, but that weird *extended* complex plane comes into play here? What weird extended complex plane? The one point compactification? You can do the same thing with Rn. It's just more useful in the case of C. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be 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: Re: |R^2 vs |E^2 vs |C > It's a shame that I don't seem to understand the root of things. > But again, I'm just a first year student, I'm allowed to look silly. > Before these days are over, I better ask questions. > All of these things are in general considered as sets of all (x, y) > [x, y (obviously) real]. I have trouble distinguishing them. The > definitions are quite distinct, I can see that. Please, kind soul, > take time to let me understand the *idea* behind the distinction as > the *results* which make each of them worthy of study. If you have a > quick search, you'll see I have asked the question before (under the > name of some 'Pirated Dreams', perhaps), but was largely ignored. This > is not fair, I am not James Harris. > Let me tell you what I *do* know. |R^2 has much less structure than > the other two. Correct me if I am wrong - |R^2 has only a linear space > structure. There is no notion of distance (therefore, orthogonality), > only linear combination (over |R). Clearly shows my question could > ignore this one. It depends on how you are speaking of R^2. If you're just viewing it as RxR with no operations, as just a set, then no, you don't have many of the things like length and the inner product. Occasionally people will talk about R^2 endowed with the Euclidean norm, adding to R^2 a bit of structure. You can add addition to it and even define your own multiplication, as long as you make it explicit. If you want to view R^2 as a topological space, you're going to need to either hand pick open and closed sets or create a norm to do it for you. To reiterate - the operations you give to a space are largely a function of the way in which you are viewing that space, and there are several possibilities - as a simple set, topologically, algebraically, etc. > Now, both |E^2 and |C has a notion of distance, but that of |C, is > defined in a funny way (but in the end, it has the same *form* as in > |E^2, and that might be where I am confused, if they were different, I > would know straightaway that |E^2 and |C are different). You say, > Can't you see? There is no multiplication defined of |E^2! I see > that. But what does that have to do with topological differences > between |C and |E^2? It _all_ depends on what operations you're going to attach to the space. I can endow, for instance, RxR with an operation . so that (RxR, .) will be isomorphic to C. Isomorphism is a really important concept that you should quickly grasp, because you'll come across things in math all the time that are isomorphic, but defined differently. If you're speaking strictly topologically, I should note, I don't think multiplication on C is relevant in the definition of C as a topological space. It might be helpful to list the definition of a topology on a space and to note that the multiplication does not appear. To state it again, when you're viewing C as a topological space, the multiplication on C isn't necessary, as is the norm (which can be used to define the topology). In that way, since the norms are equivalent, the spaces are topologically equivalent. Algebraically, however, there's a large difference. What's the difference between R and (R, +)? You can't do anything with two elements in R. In (R, +), however, you can add them, find inverses, etc. This gives them a structure beyond a topology. > Here it goes. > d(a, b) = sqrt[(a_x - b_x)^2 + (a_y - b_y)^2] in |E^2, so that we have > a down to earth situation here. > d(a, b) = |a - b| in |C. I have seen many books that have the |E^2 > distance definition as the definition of |a - b|. This makes them > (from a topological point of view) exactly the same. > But there is an alternative, define |z| by sqrt(z^* z), a definition > *via* multiplication. Not to forget it also involves conjugation, so > that, in a way, the x-axis has a preference, unlike |E^2. Does that > have any consequence later on? I mean, the only thing I can *see* > different in |C is that the real axis is treated in sort of a special > way here. Let z = a + bi. Then sqrt(z^ * z) = sqrt((a - bi)(a + bi)) = sqrt (a^2 + b^2). This is exactly the same thing, when you actually go through the calculation; it's just stated more concisely. > But why? And what are the topological implications of this? Tell me, > kind soul, why can we integrate 1/z? Isn't there a counterpart of that > in |E^2? Why can we integrate 1/z? Why can't we? 1/z is defined for all z != 0, so unless you're passing through the origin, there's nothing that should immediately imply that it would be impossible. A course in complex analysis would be helpful at this point, for I think the answer to what you're trying to ask involves the idea of a residue and a few associated theorems. > Or, am I asking the wrong question? Are these guys topologically the > same, but that weird *extended* complex plane comes into play here? This is what makes me believe you might be considering 1/z at the origin. By the extended complex plane, do you mean C union {infinity}? As with most infinities, it cannot be used naively, so I'd save this question until you've dealt with some path integrals over C, then closed path integrals, then closed path integrals containing bad points. > Best wishes (the more explanatory and exemplified the replies are, the > better!) > === Subject: Re: |R^2 vs |E^2 vs |C > It depends on how you are speaking of R^2. If you're just viewing it > as RxR with no operations, as just a set, then no, you don't have many > of the things like length and the inner product. Occasionally people > will talk about R^2 endowed with the Euclidean norm, adding to R^2 a > bit of structure. You can add addition to it and even define your own > multiplication, as long as you make it explicit. If you want to view > R^2 as a topological space, you're going to need to either hand pick > open and closed sets or create a norm to do it for you. > To reiterate - the operations you give to a space are largely a > function of the way in which you are viewing that space, and there are > several possibilities - as a simple set, topologically, algebraically, > etc. Your answers, lifeform, did not add to my confusion (such a relief!), but did not reduce it either (alas!). Specifically, I am talking about (actually I thought it was a common practice): |R^2 defined to be |R times |R, with + and multiplication by elements of |R defined. (treated as a vector space over |R, basically) |E^2 defined to be |R^2 with the standard metric (Euclidean space, I hate it when they say euclidean) |C defined to be |R^2 along with multiplication, conjugation defined and (x, 0) identified as x in |R (people tell me that things are a bit fishy when I don't have a vector space structure over |R here, so field |C has a few more [maybe infinite number of] field automorphisms. I largely ignored them). Isn't it the standard practice? (Many of the books I have gone through seem to follow the convention). > It _all_ depends on what operations you're going to attach to the > space. I can endow, for instance, RxR with an operation . so that > (RxR, .) will be isomorphic to C. Isomorphism is a really important > concept that you should quickly grasp, because you'll come across > things in math all the time that are isomorphic, but defined > differently. I already denoted the set by |C. What is there to be confused about if I don't treat |R^2 and |C differently? I don't see the distinction between the field |C and the set |R^2 endowed with complex multiplication to be relevant to my question. > If you're speaking strictly topologically, I should note, I don't > think multiplication on C is relevant in the definition of C as a > topological space. It might be helpful to list the definition of a > topology on a space and to note that the multiplication does not > appear. To state it again, when you're viewing C as a topological > space, the multiplication on C isn't necessary, as is the norm (which > can be used to define the topology). In that way, since the norms are > equivalent, the spaces are topologically equivalent. That's what I *thought* (happy to know I have got reasons to be confused). Then justify this comment of the wise men The Euclidean plane and the complex plane, speaking topologically, are entirely different > Algebraically, however, there's a large difference. What's the > difference between R and (R, +)? You can't do anything with two > elements in R. In (R, +), however, you can add them, find inverses, > etc. This gives them a structure beyond a topology. You mean, inverse with respect to addition? > Let z = a + bi. Then sqrt(z^ * z) = sqrt((a - bi)(a + bi)) = sqrt (a^2 > + b^2). This is exactly the same thing, when you actually go through > the calculation; it's just stated more concisely. Isn't this *exactly* the same as what I said? The multiplication disappears in the end, but the point is, you *can* define the norm via multiplication. I suspected that there might be a difference resulting from that fact. > Why can we integrate 1/z? Why can't we? 1/z is defined for all z != 0, > so unless you're passing through the origin, there's nothing that > should immediately imply that it would be impossible. A course in > complex analysis would be helpful at this point, for I think the > answer to what you're trying to ask involves the idea of a residue and > a few associated theorems. Precisely. I want to know if the idea of poles relates to any property that |E^2 did not have. > This is what makes me believe you might be considering 1/z at the > origin. By the extended complex plane, do you mean C union {infinity}? > As with most infinities, it cannot be used naively, so I'd save this > question until you've dealt with some path integrals over C, then > closed path integrals, then closed path integrals containing bad > points. Yes I do mean C* (is it the standard notation, again?). I have dealt with some path integrals over |C, then closed path integrals, then closed path integrals containing bad points. But the thing is, I missed a crucial point, I missed why I shouldn't have all these fascinating things in the framework of Euclidean plane. I *can* write down 1/z as (x e_x - y e_y)/(x^2 + y^2), can't I? Is it because f(z) dz has *complex* multiplication between f(z) and dz? Whereas in the Euclidean regime, I have f(r) cdot dr? === Subject: Re: |R^2 vs |E^2 vs |C > That's what I *thought* (happy to know I have got reasons to be > confused). Then justify this comment of the wise men > The Euclidean plane and the complex plane, speaking topologically, > are entirely different Can you identify who said that, and in what context? It would help to figure out what they really meant. Clearly there's no topological difference between the complex plane and the Euclidean plane. They have the same topology, ie they have the same open sets. === Subject: Re: Mathematics conventions |Oh bummer. I saw the subject line and I was thinking of mathematics |conventions as in science fiction conventions (big social get-togethers |of professionals and fans). | |Would be fun. :) Keith Ramsay === Subject: Re: Proof: Problem with algebraic integers > Why do you take so much trouble to expose such a reasoner as > Mr. Smith? I answer as a deceased friend of mine used to answer > on like occasions - A man's capacity is no measure of his power > to do mischief. Mr. Smith has untiring energy, which does > something; self-evident honesty of conviction, which does more; > and a long purse, which does most of all. He has made at least > ten publications, full of figures few readers can critize. A great > many people are staggered to this extend, that they imagine there > must be the indefinite something in the mysterious all this. > They are brought to the point of suspicion that the mathematicians > ought not to treat all this with such undisguised contempt, > at least. > -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan > Arturo Magidin > magidin@math.berkeley.edu As you say, my good knight! There ought to be laws to protect the body of acquired knowledge. Take one of our good pupils, for example: modest and diligent, from his earliest grammar classes he's kept a little notebook full of phrases. After hanging on the lips of his teachers for twenty years, he's managed to build up an intellectual stock-in-trade; doesn't it belong to him as if it were a house or money? --P. Claudel, _Le Soulier du Satin_ (cited in Pierre Bourdieu, _Language and Symbolic Power_, p. 43) === Subject: Re: Math rules REVISED > I feel like I'd better agree it's correct lest he say something > like fu David Ullrich - that would be too much to bear. > Giggle. Nothing makes a pig happier than a roll in the wallow, as this giggling porker knows! http://www.shop4egifts.com/target.asp?item=/jpg/25112.jpg&width=314&height=4 00# === Subject: A Trigonometric Challenge? For every natural number N, X(N)=cos(N), where N is in units of radians. Find a function Y(L) such that for every natural number L, Y(L) For every natural number N, X(N)=cos(N), where N is in units of radians. > Find a function Y(L) such that for every natural number L, Y(L) every Y(L) uniquely equals some X(N). Also, every X(N) uniquely equals some > Y(L). > R > http://www.rlgerl.com If I understood your question correctly, the answer is that it is trivially impossible to find such a function Y. The reason is that the X(N) form a dense set in the interval (-1,1). (This is an immediate consequence of the fact that Pi is irrational). Let's say that you have decided Y(1)=X(k) and Y(2)=X(t) for some k and t such that X(k)>For every natural number N, X(N)=cos(N), where N is in units of radians. >>Find a function Y(L) such that for every natural number L, Y(L)>every Y(L) uniquely equals some X(N). Also, every X(N) uniquely equals some >>Y(L). >>R >>http://www.rlgerl.com > If I understood your question correctly, the answer is that it is trivially > impossible to find such a function Y. The reason is that the X(N) form a dense > set in the interval (-1,1). (This is an immediate consequence of the fact that > Pi is irrational). Let's say that you have decided Y(1)=X(k) and Y(2)=X(t) for > some k and t such that X(k) face the problem that there exists an n such that Y(1)=X(k) so X(n) cannot be any of the Y(j):s. A contradiction. > Please rephrase the problem, or accept that no solution is possible. > Jyrki I think more generally, it exists, but it is not possible to explicitly write it out. It sounds like finding a particular well-ordering of a countable subset of [-1,1], but there is a difference between asserting that Y exists and writing down what Y is. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: A Trigonometric Challenge? > I think more generally, it exists, but it is not possible to explicitly > write it out. It sounds like finding a particular well-ordering of a > countable subset of [-1,1], but there is a difference between asserting > that Y exists and writing down what Y is. No. The way I read the OP seems to indicate that we cannot choose any well-ordering of this countable set, but are stuck with the natural < instead (and as you know that is not a well-ordering of a dense set). So as {cos(n)| n in N} is dense it isn't well-ordered WITH RESPECT TO <, but N is, so you cannot match the two. So Y doesn't exist. Changing to the set {cos(n+1-Pi) | n in N} doesn't change this > -- > Will Twentyman > email: wtwentyman at copper dot net Jyrki === Subject: Re: A Trigonometric Challenge? >>For every natural number N, X(N)=cos(N), where N is in units of radians. >>Find a function Y(L) such that for every natural number L, Y(L)>every Y(L) uniquely equals some X(N). Also, every X(N) uniquely equals some >>Y(L). >>R >>http://www.rlgerl.com write it out. It sounds like finding a particular well-ordering of a > countable subset of [-1,1], but there is a difference between asserting > that Y exists and writing down what Y is. Maybe I don't understand the question, but heres my 2 cents worth. Let S = the set of cos(N) for all natural N Set Y(1) = min[S] Set Y(2) = min[S - {Y(1)}] Set Y(3) = min[S - {Y(1), Y(2)}] Set Y(4) = min[S - {Y(1), Y(2), Y(3)}] ... The question is: do minimum values exist for ALL of these Y(L)? If S is nowhere dense - yes. Otherwise - no. === Subject: Re: A Trigonometric Challenge? >>For every natural number N, X(N)=cos(N), where N is in units of radians. >>Find a function Y(L) such that for every natural number L, Y(L) and >>every Y(L) uniquely equals some X(N). Also, every X(N) uniquely equals > some >>Y(L). >>R >>http://www.rlgerl.com > I think more generally, it exists, but it is not possible to explicitly >>write it out. It sounds like finding a particular well-ordering of a >>countable subset of [-1,1], but there is a difference between asserting >>that Y exists and writing down what Y is. > Maybe I don't understand the question, but heres my 2 cents worth. > Let S = the set of cos(N) for all natural N > Set Y(1) = min[S] > Set Y(2) = min[S - {Y(1)}] > Set Y(3) = min[S - {Y(1), Y(2)}] > Set Y(4) = min[S - {Y(1), Y(2), Y(3)}] > ... > The question is: do minimum values exist for ALL of these Y(L)? If S is > nowhere dense - yes. Otherwise - no. Ahh, good point. Y(1) cannot actually be defined. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: A Trigonometric Challenge? > Maybe I don't understand the question, but heres my 2 cents worth. > Let S = the set of cos(N) for all natural N > Set Y(1) = min[S] > Set Y(2) = min[S - {Y(1)}] > Set Y(3) = min[S - {Y(1), Y(2)}] > Set Y(4) = min[S - {Y(1), Y(2), Y(3)}] > ... > The question is: do minimum values exist for ALL of these Y(L)? If S is > nowhere dense - yes. Otherwise - no. > Ahh, good point. Y(1) cannot actually be defined. Good points from all posters. But I don't believe Y(L) exists if we replace X(N) in the original problem with X(N)=cos(N-1+PI). R http://www.rlgerl.com === Subject: Re: A Trigonometric Challenge? >For every natural number N, X(N)=cos(N), where N is in units of radians. >Find a function Y(L) such that for every natural number L, Y(L)every Y(L) uniquely equals some X(N). Also, every X(N) uniquely equals some >Y(L). It seems to me (though I offer no formal proof) that -1 is a limit point of cos(N), where I use N to denote the *set* of natural numbers. Thus, it is impossible for you to define Y(0). -- === Subject: Re: A Trigonometric Challenge? >>For every natural number N, X(N)=cos(N), where N is in units of radians. >>Find a function Y(L) such that for every natural number L, Y(L)>every Y(L) uniquely equals some X(N). Also, every X(N) uniquely equals some >>Y(L). >It seems to me (though I offer no formal proof) that -1 is a limit point >of cos(N), where I use N to denote the *set* of natural numbers. This is a well-known true fact; in fact cos(N) is dense in [-1,1]. The easiest proof is probably to show that exp(iN) is dense in the circle T. In fact the set of all exp(int), n = 0, 1, ... is dense in the circle whenever t/pi is irrational. This follows from the fact that int f = lim_N (1/N) sum_1^N f(exp(itn)) for any f continuous on T. To prove _that_, note that it's true by a direct calculation if f(t) = exp(imt) for some integer m; hence it's true when f is a trigonometric polynomial, and hence for all continuous f (because the trigonometric polynomials are uniformly dense in C(T).) >Thus, >it is impossible for you to define Y(0). ************************ David C. Ullrich === Subject: Re: Boy, it's crowded in here today Mensanator >Larry Hammick > Did you follow the link in the sig? http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm Okay, I just did. Quite the beard. I'm sorry I stuck in the punchline without waiting for you to finish the joke, as it were ;) LH === Subject: Re: Boy, it's crowded in here today === >Subject: Re: Boy, it's crowded in here today >Message-id: >Larry Hammick >> Did you follow the link in the sig? >http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm >Okay, I just did. >Quite the beard. Yeah, it ensures that I'm the one who gets randomly selected to be searched at the airline gate. The guy with the three foot long orange beard, shaved head with skullcap, wearing white monk's robes and a large gold crucifix on a chain around his neck waltzes right through. Apparently _he_ doesn't fit the profile of a religious fanatic. >I'm sorry I stuck in the punchline >without waiting for you to finish the joke, as it were ;) -- Mensanator 2 of Clubs http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm === Subject: Re: Boy, it's crowded in here today Mensanator http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm >Quite the beard. > Yeah, it ensures that I'm the one who gets randomly selected to be searched > at the airline gate. The guy with the three foot long orange beard, shaved head > with skullcap, wearing white monk's robes and a large gold crucifix on a chain > around his neck waltzes right through. Apparently _he_ doesn't fit the profile > of a religious fanatic. :) It's totally OT but I'll tell a little story. Many people want to attract attention. They want to be recognized by strangers wherever they go. I, on the other hand, have an extraordinary talent for inconspicuousness. Nobody suspects me of _anything_. During the APEC summit in Vancouver here, a few years ago, I wanted to get a taste of the big security buildup around the expensive waterfront hotel where most of the muckamucks were staying. I heard there were snipers on rooves, and even frogmen in the water. Getting close, I saw that they were setting up a cordon with metal-detecting gates. I got to the entrance of the hotel, and was approached by a young female employee. I said I had been jogging and needed to use a restroom. With no hesitation, she pointed through the lobby and gave directions. I walked through the lobby to the restroom as directed. The guy at the next urinal looked familiar. He was the Prime Minister of Japan. LH === Subject: Question about Significant Digits I keep finding discrepancies between my texts, my logic and my memory on how to determine the number of signifigant digits for a solution. I know that, when adding or subtracting, we round off our sum or difference at the same number of decimal places as the least precise measurement, and when multiplying or dividing we round off at the least number of signifigant digits, but how do we handle multiple operations? For example, in this problem: 320.55 - (6104.5/2.3) if we account for significant digits at each step, we get the following: = 320.55 - (2654.130435) which rounds off to two s.f = 320.55 - 2700 = -2379.45 which is only precise to the hundreds place = -2400 But if we do the math entirely, disregarding sf until the end, we get: 320.55 - (6104.5/2.3) = -2333.630435 which rounds off to 2 sf, at -2300. This is a small amount of difference (only 4%) but often, calculations can be off much more, especially when exponents are involved. On top of that, it seems ludicrous that all my data was precise to at least one decimal place, but my answer can only be expressed in hundreds. Can anyone explain the proper way to deal with sf in multi-step calculations? --riverman === Subject: Re: Question about Significant Digits > I keep finding discrepancies between my texts, my logic and my memory on how > to determine the number of signifigant digits for a solution. I know that, > when adding or subtracting, we round off our sum or difference at the same > number of decimal places as the least precise measurement, and when > multiplying or dividing we round off at the least number of signifigant > digits, but how do we handle multiple operations? For example, in this > problem: > 320.55 - (6104.5/2.3) > if we account for significant digits at each step, we get the following: > = 320.55 - (2654.130435) which rounds off to two s.f > = 320.55 - 2700 > = -2379.45 which is only precise to the hundreds place > = -2400 > But if we do the math entirely, disregarding sf until the end, we get: > 320.55 - (6104.5/2.3) > = -2333.630435 > which rounds off to 2 sf, at -2300. > This is a small amount of difference (only 4%) but often, calculations can > be off much more, especially when exponents are involved. On top of that, it > seems ludicrous that all my data was precise to at least one decimal place, > but my answer can only be expressed in hundreds. Can anyone explain the > proper way to deal with sf in multi-step calculations? > --riverman There's an assumed error of +/- 50. The errors will *generally* balance out with some being +, some -. As I recall from when I did s.f., your work is correct. There are times when these errors *can* accumulate, which becomes an issue with certain floating point computations in programs. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Question about Significant Digits riverman > I keep finding discrepancies between my texts, my logic and my memory on how > to determine the number of signifigant digits for a solution. I know that, > when adding or subtracting, we round off our sum or difference at the same > number of decimal places as the least precise measurement, and when > multiplying or dividing we round off at the least number of signifigant > digits, but how do we handle multiple operations? For example, in this > problem: > 320.55 - (6104.5/2.3) > if we account for significant digits at each step, we get the following: > = 320.55 - (2654.130435) which rounds off to two s.f > = 320.55 - 2700 Error here. Should be 320.55 - 2654.1 (significant figures _after_ the decimal). The rule of thumb for sig. digits in a product derives from the equation (A +- a)(B +- b) = AB +- (aB+Ab) + ab together with the assumption that ab is negligible relative to AB. A somewhat similar argument can be applied to other differentiable functions, such as A^B. LH === Subject: Re: Question about Significant Digits > riverman >> I keep finding discrepancies between my texts, my logic and my memory on > how >> to determine the number of signifigant digits for a solution. I know that, >> when adding or subtracting, we round off our sum or difference at the same >> number of decimal places as the least precise measurement, and when >> multiplying or dividing we round off at the least number of signifigant >> digits, but how do we handle multiple operations? For example, in this >> problem: >> 320.55 - (6104.5/2.3) >> if we account for significant digits at each step, we get the following: >> = 320.55 - (2654.130435) which rounds off to two s.f >> = 320.55 - 2700 > Error here. Should be 320.55 - 2654.1 (significant figures _after_ the > decimal). Not sure what you're talking about there. That 2654.130435 is only good to 2 digits. The _relative_ error in a quotient goes as the worst _relative_ error in the dividend and the divisor. It is NOT the case that the _absolute_ error in a quotient goes as the worst _absolute_ error in the dividend and the divisor. The two reliable digits in the quotient happen to be in the thousands and hundreds places. That 2.3 could have been anywhere between 2.25 and 2.35 for a range of plausible quotients between 2713 and 2597 (roughly 2654 +/- 60). Relative error translates to number of reliable digits in a result. Absolute error translates to position of the rightmost reliable digit. If you're multiplying or dividing, you pay attention to significant figures -- relative error -- total digits. If you're adding or subtracting you pay attention to decimal places -- absolute error -- digits to the right of the point. Note: If it were me, I'd be carrying some suspect digits through the intermediate calculations just so that I wouldn't be introducing rounding error into the computation in addition to the measurement error implicit in the inputs. I might round to 2700 in a rough run through to determine sig figs. But I'd probably retain 2654.13 in my final calculation and round off to two digits only at the end. > The rule of thumb for sig. digits in a product derives from the equation > (A +- a)(B +- b) = AB +- (aB+Ab) + ab ~= AB +- (AB*a/A+AB*b/B) ~= AB +- max(AB*a/A , AB*b/B) = AB +- AB * max(a/A, b/B) The relative error in a product (or quotient as it turns out) is roughly equal to the worst relative error in the inputs. Give or take a factor of 2. > together with the assumption that ab is negligible relative to AB. A > somewhat similar argument can be applied to other differentiable functions, > such as A^B. Or such as A/B. Try it. John Briggs === Subject: Re: Question about Significant Digits > riverman > I keep finding discrepancies between my texts, my logic and my memory on > how > to determine the number of signifigant digits for a solution. I know that, > when adding or subtracting, we round off our sum or difference at the same > number of decimal places as the least precise measurement, and when > multiplying or dividing we round off at the least number of signifigant > digits, but how do we handle multiple operations? For example, in this > problem: > 320.55 - (6104.5/2.3) > if we account for significant digits at each step, we get the following: > = 320.55 - (2654.130435) which rounds off to two s.f > = 320.55 - 2700 > Error here. Should be 320.55 - 2654.1 (significant figures _after_ the > decimal). > The rule of thumb for sig. digits in a product derives from the equation > (A +- a)(B +- b) = AB +- (aB+Ab) + ab > together with the assumption that ab is negligible relative to AB. A > somewhat similar argument can be applied to other differentiable functions, > such as A^B. > LH Incorrect. The intermediate result 2654.130435 should indeed be rounded to 2700. The error is definitely in the 100's place digit. Consider: d(x/y) = (1/y)dx - (x/y^2)dy, where dx,dy are taken to be the errors in x and y. Taking absolute values: |d(x/y)| <= |(1/y)dx| + |(x/y^2)dy|. Entering x=6104.5, y=2.3, dx=0.1, dy=0.1 gives an error of about 115, plus or minus. Similarly, maximizing the quotient over the error range gives about 2775.818, and minimizing the quotient over the error range gives 2543.5 Either way, the error is in the 100's place digit. The digits after that are not significant. As for the O.P.'s question, I would suggest the following: 1) Round as you go, according to the rules, to propagate the error. Or, 2) Calculate the answer without rounding, then compute the error separately using partial derivatives. A good treatment is given in Experiments in Physical Chemistry by Shoemaker, Garland, and Nibler in the section Treatment of Experimental Data. Hope this helps. === Subject: Re: Question about Significant Digits > riverman > I keep finding discrepancies between my texts, my logic and my memory on > how > to determine the number of signifigant digits for a solution. I know that, > when adding or subtracting, we round off our sum or difference at the same > number of decimal places as the least precise measurement, and when > multiplying or dividing we round off at the least number of signifigant > digits, but how do we handle multiple operations? For example, in this > problem: > 320.55 - (6104.5/2.3) > if we account for significant digits at each step, we get the following: > = 320.55 - (2654.130435) which rounds off to two s.f > = 320.55 - 2700 > Error here. Should be 320.55 - 2654.1 (significant figures _after_ the > decimal). > The rule of thumb for sig. digits in a product derives from the equation > (A +- a)(B +- b) = AB +- (aB+Ab) + ab > together with the assumption that ab is negligible relative to AB. A > somewhat similar argument can be applied to other differentiable functions, > such as A^B. texts tell me. Is there a 'more advanced' way to deal with them that we learn after HS? For reference, this is what the texts say: I quote from Chemistry, the Central Science, Prentice Hall, 7th ed (1997): In multiplication and division, the result must be reported with the same number of significant figures as the measurement with the fewest significant figures. And as an example, it gives: For example, the area of a rectangle whose edge lengths are 6.221 cm and 5.2 cm should be reported as 32cm^2. Area = (6.221 cm)(5.2 cm) = 32.3492 cm^2 ------> rounded off to 32 cm^2 And I quote from Chemistry, Addison Wesley, 2nd ed (1990): In calculations involving multiplication and division, the answer must contain no more signifigant figures than the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the number of significant figures. And as an example, it gives: 2.4536 m / 8.4 m = 0.291976 m = 0.29 m [8.4 has two significant figures] It seems clear enough, but there's a real logical glip here. If we consider a compound operation as a series of one-step operations, then the rules for sf at each step ought to apply. OTOH, that means we are introducing limitation at each step (I don't want to call them 'errors', since technically the extra sf are errors) but those limitations compound until the final answer is *really* limited.... --riverman === Subject: Re: Question about Significant Digits riverman > if we account for significant digits at each step, we get the following: = 320.55 - (2654.130435) which rounds off to two s.f > = 320.55 - 2700 > Error here. Should be 320.55 - 2654.1 (significant figures _after_ the > decimal). > The rule of thumb for sig. digits in a product derives from the equation > (A +- a)(B +- b) = AB +- (aB+Ab) + ab > together with the assumption that ab is negligible relative to AB. > texts tell me. Is there a 'more advanced' way to deal with them that we > learn after HS? For reference, this is what the texts say: Yes, you're right. I was a bit hasty, making an assumption about the origin of the data. It's not just figures after the decimal point, but sig. figures in all, as the equation about AaBb suggests. I'll have a look at the other responses, and get back if needed. LH === Subject: Re: Question about Significant Digits >>riverman >I keep finding discrepancies between my texts, my logic and my memory on >how >to determine the number of signifigant digits for a solution. I know that, >when adding or subtracting, we round off our sum or difference at the same >number of decimal places as the least precise measurement, and when >multiplying or dividing we round off at the least number of signifigant >digits, but how do we handle multiple operations? For example, in this >problem: >320.55 - (6104.5/2.3) >if we account for significant digits at each step, we get the following: >= 320.55 - (2654.130435) which rounds off to two s.f >= 320.55 - 2700 >Error here. Should be 320.55 - 2654.1 (significant figures _after_ the >>decimal). >>The rule of thumb for sig. digits in a product derives from the equation >>(A +- a)(B +- b) = AB +- (aB+Ab) + ab >>together with the assumption that ab is negligible relative to AB. A >>somewhat similar argument can be applied to other differentiable >functions, >>such as A^B. >texts tell me. Is there a 'more advanced' way to deal with them that we >learn after HS? For reference, this is what the texts say: >I quote from Chemistry, the Central Science, Prentice Hall, 7th ed (1997): >In multiplication and division, the result must be reported with the same >number of significant figures as the measurement with the fewest significant >figures. >And as an example, it gives: >For example, the area of a rectangle whose edge lengths are 6.221 cm and >5.2 cm should be reported as 32cm^2. >Area = (6.221 cm)(5.2 cm) = 32.3492 cm^2 ------> rounded off to 32 cm^2 >And I quote from Chemistry, Addison Wesley, 2nd ed (1990): >In calculations involving multiplication and division, the answer must >contain no more signifigant figures than the measurement with the least >number of significant figures. The position of the decimal point has nothing >to do with the number of significant figures. >And as an example, it gives: >2.4536 m / 8.4 m = 0.291976 m = 0.29 m [8.4 has two significant figures] >It seems clear enough, but there's a real logical glip here. Yup. > If we consider a compound operation as a series of one-step operations, then the rules for sf at each step ought to apply. OTOH, that means we are introducing >limitation at each step (I don't want to call them 'errors', since technically the extra sf are errors) but those limitations compound until the final answer is *really* limited.... And? There's a reason s.f. are not discussed in math or cs contexts. s.f. is a convenient quick and dirty way to deal with error analysis. Quick and dirty means that it's every now and again erroneous. If the risk of error is too great to you, then you have to analyze the errors and then give your error estimates at the end of the calculations. Jon Miller === Subject: =?ISO-8859-1?Q?Conditions_d'intersection_entre_une_sph=E8re_et_une_droite?= bonjour, Le probl.8fme est le suivant : Soient 3 points O, A, B dans l'espace et la sph.8fre S de centre B de rayon R. Les coordonn.8ees cart.8esiennes respectives de ces point dans un rep.8fre orthonorm.8e de centre O sont : - O(0,0,0) - A(Xa,Ya,Za) - B(Xb,Yb,Zb) Quelles sont les conditions n.8ec.8essaires et suffisantes sur Xa,Ya,Za,Xb,Yb,Zb pour que la droite (OA) coupe la sph.8fre S? === Subject: Re: =?ISO-8859-1?Q?Conditions_d'intersection_entre_une_sph=E8re_et_une_droite?= Soit H le pied de la perpendiculaire tir.8ee de point B .88 droite OA, et soient a, b, h vecteurs de position de points A, B, H respectivement. Puisque h = ta pour quelque nombre r.8eel t, et OA et BH sont perpendiculaires, on a 0 = (a, h - b) = (a, ta - b) = t(a, a) - (a, b), i.e., t = (a, b)/(a, a). ((a, b) est le produit scalaire de a et b.) Alors, la distance d entre B et OA est donn.8ee par d^2 = |h - b|^2 = |ta - b|^2 = t^2(a, a) - 2t(a, b) + (b, b) = [(a, a)(b, b) - (a, b)^2]/(a, a). Donc OA coupe S si et seulement si d^2 <= R^2. (On a l'.8egalit.8e quand OA touche S.) Tad === Subject: Re: =?ISO-8859-1?Q?Conditions_d'intersection_entre_une_sph=E8re_et_une_droite?= > bonjour, > Le probl.8fme est le suivant : > Soient 3 points O, A, B dans l'espace et la sph.8fre S de centre B de rayon R. > Les coordonn.8ees cart.8esiennes respectives de ces point > dans un rep.8fre orthonorm.8e de centre O sont : > - O(0,0,0) > - A(Xa,Ya,Za) > - B(Xb,Yb,Zb) > Quelles sont les conditions n.8ec.8essaires et suffisantes sur > Xa,Ya,Za,Xb,Yb,Zb pour que la droite (OA) coupe la sph.8fre S? Points on the line OA have coordinates P(s) = (s*Xa, s*Ya, s*Za) where 0 <= s <= 1. OA crosses the sphere S if and only if there exists s in [0,1] such that |P(s)-B|^2 <= R^2. OA coupe la sphere <=> existe s en [0,1] avec |P(s)-B|^2 <= R^2. |P(s)-B|^2 = (s*Xa - Xb)^2 + (s*Ya - Yb)^2 + (s*Za - Zb)^2 <= R^2 s^2 * |A|^2 - s * [2 Xa Xb + 2 Ya Yb + 2 Za Zb] + (|B|^2-R^2) <= 0 OA cuts S if and only if there exists a solution to: (s - r1)(s - r2) <= 0 and s in [0,1] where r1 = [2 + sqrt( ^2 - |A|^2(|B|^2 - R^2))] / |A|^2 r2 = [2 - sqrt( ^2 - |A|^2(|B|^2 - R^2))] / |A|^2 where = dot product of A and B, |A| = magnitude of A, |B| = magnitude of B. - R === Subject: Re: Conditions d'intersection entre une sph.8fre et une droite > bonjour, > Le probl.8fme est le suivant : > Soient 3 points O, A, B dans l'espace et la sph.8fre S de centre B de rayon > R. > Les coordonn.8ees cart.8esiennes respectives de ces point > dans un rep.8fre orthonorm.8e de centre O sont : > - O(0,0,0) > - A(Xa,Ya,Za) > - B(Xb,Yb,Zb) > Quelles sont les conditions n.8ec.8essaires et suffisantes sur > Xa,Ya,Za,Xb,Yb,Zb pour que la droite (OA) coupe la sph.8fre S? Que abs(A.B/|A|) < R, o.9d A.B = (Xa*Xb + Ya*Yb + Za*Zb) et |A| = sqrt(Xa^2 + Ya^2 + Za^2) -- Jim Heckman === Subject: Re: Conditions d'intersection entre une sph.8fre et une droite in message : > bonjour, > Le probl.8fme est le suivant : > Soient 3 points O, A, B dans l'espace et la sph.8fre S de centre B de > rayon > R. > Les coordonn.8ees cart.8esiennes respectives de ces point > dans un rep.8fre orthonorm.8e de centre O sont : > - O(0,0,0) > - A(Xa,Ya,Za) > - B(Xb,Yb,Zb) > Quelles sont les conditions n.8ec.8essaires et suffisantes sur > Xa,Ya,Za,Xb,Yb,Zb pour que la droite (OA) coupe la sph.8fre S? > Que abs(A.B/|A|) < R, o.9d > A.B = (Xa*Xb + Ya*Yb + Za*Zb) et > |A| = sqrt(Xa^2 + Ya^2 + Za^2) Gloups, j'ai m.8elang.8e sin et cos. D'autres ont donn.8e la bonne r.8eponse, qui est.be: Que sqrt((A.A)(B.B) - (A.B)^2)/|A| < R. -- Jim Heckman === Subject: A Simple Exercise from Rudin's Functional Analysys I have been reading Rudin's Functional Analysys lately and I've come up against a problem I cannot solve by myself. It is a simplified version of original Exercise I.22. Let f:[0,1]->R be a Lipschitz function, let L_f denote number inf{L: |f(x)-f(y)|I have been reading Rudin's Functional Analysys lately and I've come up >against a problem I cannot solve by myself. It is a simplified version of >original Exercise I.22. >Let f:[0,1]->R be a Lipschitz function, let L_f denote number >inf{L: |f(x)-f(y)|and let X be the space of all functions f on [0,1] for which L_fand f(0)=0. >Show that (X, L_f) is a Banach space. >Naturally the completeness part makes problem the most. Well to start you should note this: Say ||f|| is the norm of f in C([0,1]), ie the supremum of |f|. Show that ||f|| <= L_f. (Hence if f_n is Cauchy in X it is Cauchy in C([0,1]), hence converges uniformly to some f. Now you still have to show that it converges in X, but knowing what it converges _to_ may help...) >I'd be grateful for some hints about that. >olej ************************ David C. Ullrich === Subject: Trivia ladder problem The following problem came to me during a sleepless night. Consider a ladder standing vertically against a wall into a corner such that one side of the ladder is touching the orthogonal (white) wall. This side of the ladder is soaked with (red) paint. Now, someone begin continuously dragging the ground touching end of the ladder such that it is continously touching both walls and the ground. The dragging ends when the ladder is horisontally laying on the ground. Due to the red paint, the orthogonal wall has been painted with a concave triangular like shape. Problem: Find a formula for the border of this shape. I in fact struggled with it that night and the following day _with_ paper and pencil. But the next night the solution was there... It is, I think, a suprisingly simple formula. -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: Trivia ladder problem > The following problem came to me during a sleepless night. Consider a > ladder standing vertically against a wall into a corner such that one > side of the ladder is touching the orthogonal (white) wall. This side > of the ladder is soaked with (red) paint. Now, someone begin > continuously dragging the ground touching end of the ladder such that > it is continously touching both walls and the ground. The dragging > ends when the ladder is horisontally laying on the ground. Due to the > red paint, the orthogonal wall has been painted with a concave > triangular like shape. Problem: Find a formula for the border of this > shape. > I in fact struggled with it that night and the following day _with_ > paper and pencil. But the next night the solution was there... It > is, I think, a suprisingly simple formula. What you are looking for is the envelope of the lines created by the ladder. In other words the envelope for the family of curves f(x,y,c) = y + sqrt(R^2-c^2) * (x-c) /c, where R is the length of the ladder. You can imagine the ladder as the section of the line between the positive x- and y-axses, where c determines how far the ladder has been pulled. You can solve the equation of the envelope by eliminating c between 1) and 2) below: 1) f(x,y,c) = 0 2) first partial derivative of f(x,y,c) w.r.t c = 0 Hope this helps, -Sunny === Subject: Re: Trivia ladder problem * sunnyelloco@hotmail.com > What you are looking for is the envelope of the lines created by the > ladder. > In other words the envelope for the family of curves f(x,y,c) = y + > sqrt(R^2-c^2) * (x-c) /c, where R is the length of the ladder. You can > imagine the ladder as the section of the line between the positive x- > and y-axses, where c determines how far the ladder has been pulled. > You can solve the equation of the envelope by eliminating c between 1) > and 2) below: > 1) f(x,y,c) = 0 > 2) first partial derivative of f(x,y,c) w.r.t c = 0 > Hope this helps, Actually, I _have_ solved the problem. I just wantet to share it with others. You know, share and enjoy. :) -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: Trivia ladder problem > The following problem came to me during a sleepless night. Consider a > ladder standing vertically against a wall into a corner such that one > side of the ladder is touching the orthogonal (white) wall. This side > of the ladder is soaked with (red) paint. Now, someone begin > continuously dragging the ground touching end of the ladder such that > it is continously touching both walls and the ground. The dragging > ends when the ladder is horisontally laying on the ground. Due to the > red paint, the orthogonal wall has been painted with a concave > triangular like shape. Problem: Find a formula for the border of this > shape. > I in fact struggled with it that night and the following day _with_ > paper and pencil. But the next night the solution was there... It > is, I think, a suprisingly simple formula. Yes, indeed. The astroid. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Trivia ladder problem * The Last Danish Pastry > Yes, indeed. The astroid. Eh? Really? Can you point to me some source of the asteroid formula or problem? -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: Trivia ladder problem Yes, indeed. The astroid. > Eh? Really? Can you point to me some source of the asteroid > formula or problem? http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html Enjoy, ZVK(Slavek) > -- > Jon Haugsand > Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no > http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: Trivia ladder problem > * The Last Danish Pastry > Yes, indeed. The astroid. > Eh? Really? Can you point to me some source of the asteroid > formula or problem? Not asteroid but astroid... :) http://mathworld.wolfram.com/Astroid.html -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Trivia ladder problem * The Last Danish Pastry > Not asteroid but astroid... :) > http://mathworld.wolfram.com/Astroid.html Oh... (Why can't I ever invent a problem which is not inventet before. Maybe the harris track is more rewarding?) Anyway, I couldn't find a formula, that is _my_ formula. -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: Trivia ladder problem > * The Last Danish Pastry > Not asteroid but astroid... :) > http://mathworld.wolfram.com/Astroid.html > Oh... (Why can't I ever invent a problem which is not inventet > before. Maybe the harris track is more rewarding?) Unlikely. > Anyway, I couldn't find a formula, that is _my_ formula. I had assumed that you meant the memorable x^(2/3)+y^(2/3) = a^(2/3) which is certainly on that page. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Trivia ladder problem * The Last Danish Pastry > I had assumed that you meant the memorable > x^(2/3)+y^(2/3) = a^(2/3) > which is certainly on that page. Yes, of course. Silly me. I just had y as a function of x. But (* back to the drawing board *) -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: A formal solution to Hilbert's 1st and 6th problems. You can find it here: http://www.geocities.com/complementarytheory/CATpage.html Yours, Doron Shadmi === Subject: Re: A formal solution to Hilbert's 1st and 6th problems. > You can find it here: http://www.geocities.com/complementarytheory/CATpage.html > Yours, > Doron Shadmi I don't see how the set theory you are describing on your page is used to model the real numbers. You state that it talks about discreteness and continuity, but I'm not clear on how your definition corresponds to the usual definitions. You also have some symbols which are not defined, such as |a| where a is an element of a set. -- Will Twentyman email: wtwentyman at copper dot net Supersedes: === Subject: Invariant Galilean Transformations (FAQ) On All Laws Summary: All laws/equations are Galilean invariant when expressed in the generalized cartesian coordinates demanded by basic analytic geometry, vector algebra, and measurement theory. Originator: faqserv@penguin-lust.MIT.EDU Disclaimer: approval for *.answers is based on form, not content. Opponents of the content should first actually find out what it is, then think, then request/submit-to arbitration by the appropriate neutral mathematics authorities. Flaming the hard- working, selfless, *.answers moderators evidences ignorance and despicable netiquette. Archive-Name: physics-faq/criticism/galilean-invariance Version: 0.04.03 Posting-frequency: 15 days Invariant Galilean Transformations (FAQ) On All Laws (c) Eleaticus/Oren C. Webster Thnktank@concentric.net An obvious typo or two corrected. The Brittanica section revised to less 'pussy-footing' and to more directly anticipate the elementary measurement theory and basic analytic geometry that is applied to the transformation concept. ------------------------------ === Subject: 1. Purpose The purpose of this document is to provide the student of Physics, especially Relativity and Electromagnetism, the most basic princ- iples and logic with which to evaluate the historic justification of Relativity Theory as a necessary alternative to the classical physics of Newton and Galileo. We will prove that all laws are invariant under the Galilean transformation, rather than some being non-invariant, after we show you what that means. We shall also show that another primal requirement that SR exist is nonsense: Michelson-Morley and Kennedy-Thorndike do indeed fit Galilean (c+v) physics. ------------------------------ === Subject: 2. Table of Contents 1. Foreword and Intent 2. Table of Contents 3. The Principle of Relativity 4. The Encyclopedia Brittanica Incompetency. 5. Transformations on Generalized Coordinate Laws 6. The data scale degradation absurdity. 7. The Crackpots' Version of the Transforms. 8. What does sci.math have to say about x0'=x0-vt? 9. But Doesn't x.c'=x.c? 10. But Isn't (x'-x.c')=(x-x.c) Actually Two Transformations? 11. But Doesn't (x'-x.c+vt) Prove The Transformation Time Dependent? 12. But Isn't (x'-x.c')=(x-x.c) a Tautology? 13. But Isn't (x'-x.c')=(x-x.c) Almost the Definition of a Linear Transform? 14. But The Transform Won't Work On Time Dependent Equations? 15. But The Transform Won't Work On Wave Equations? 16. But Maxwell's Equations Aren't Galilean Invariant? 17. First and Second Derivative differential equations. ------------------------------ === Subject: 3. The Principle of Relativity and Transformation If a law is different over there than it is here, it is not one law, but at least two, and leaves us in doubt about any third location. This is the Principle of Relativity: a natural law must be the same relative to any location at which a given event may be perceived or measured, and whether or not the observer is moving. The idea of location translates to a coordinate system, largely because any object in motion could be considered as having a coordinate system origin moving with it. If you perceive me moving relative to you - who have your own coordinate system - will your measurements of my position and velocity fit the same laws my own, different measurements fit? If a law has the same form in both cases it is called covariant. If it is identical in form, var- ables, and output values, it is called invariant. What we're asking is that if the x-coordinate, x, on one coordinate axis works in an equation, does the coordinate, x', on some other, parallel axis work? Speaking in terms of the axis on which x is the coordinate, x' is the 'transformed' coordinate. The situation is complicated because we're talking about coordinates - locations - but in most mean- ingful laws/equations, it is lengths/distances (and time intervals) the equations are about, and x coord- inates that represent good, ratio scale measures of distances are only interval scale measures on the x' axis. [See Table of Contents for discussion of scales.] So, if we have an x-coordinate in one system, then we can call the x' value that corresponds to the same point/location the transform of x. In particular, the Principle of Relativity is embodied in the form of the Galilean transformation, which relates the original x, y, z, t to x', y', z', t' by the transform equations x'=x-vt, y'=y, z'=z, t'=t in the simplified case where attention is focused only on transforming the x-axis, and not y and z. In the case of Special Relativity, the x' transform is the same except that x' is then divided by sqrt(1-(v/c)^2), and t'=(t-xv/cc)/sqrt(1-(v/c)^2). In either case, v is the relative velocity of the coordinate systems; if there is already a v in the equations being trans- formed use u or some other variable name. ------------------------------ === Subject: 4. The Encyclopedia Brittanica Incompetency. One example of the traditional fallacious idea that an equation is not invariant under the galilean transformation comes from the Encyclopedia Brittanica: Before Einstein's special theory of relativity was published in 1905, it was usually assumed that the time coordinates measured in all inertial frames were identical and equal to an 'absolute time'. Thus, t = t'. (97) The position coordinates x and x' were then assumed to be related by x' = x - vt. (98) The two formulas (97) and (98) are called a Galilean transformation. The laws of nonrelativ- istic mechanics take the same form in all frames related by Galilean transformations. This is the restricted, or Galilean, principle of relativity. The position of a light wave front speeding from the origin at time zero should satisfy x^2 - (ct)^2 = 0 (99) in the frame (t,x) and (x')^2 - (ct')^2 = 0 (100) in the frame (t',x'). Formula (100) does not transform into formula (99) using the transform- ations (97) and (98), however. ................................................. Besides the trivially correct statement of what the Galilean 'transform' equations are, there is exactly one thing they got right. I. Eq-100 is indeed the correct basis for discussing the question of invariance, given that eq-99 is the correct 'stationary' (observer S) equation. [Let observer M be the 'moving'system observer.] In particular, eq-100 is of exactly the same form [the square of argument one minus the square of argument two equals zero (argument three).] II. It is nonsense to say eq-99 should be derivable from eq-100; for one thing, the transforms are TO x' and t' from x and t, not the other way around, and the idea that either observer's equation should contain within itself the terms to simplify or rearrange to get to the other is ridiculous. As the transform equations say, the relationship of t', x' to t, x is based on the relative velocity between the two systems, but neither the original (eq-99) equation nor the M observer equation is about a relationship between coordinate systems or observers. One might as well expect the two equations to contain banana export/import data; there is no relevancy. The 'transform' equations are the relationships between x' and x, t' and t and have nothing to do with what one equation or the other ought to 'say'. The equations' content is the rate at which light emitted along the x-axes moves. III. Most remarkable, the True Believer SR crackpots who most despise the consequences of measurement theory (demonstrable fact) contained in this document are those who want to argue against our saying the Britt- anica got eq-100 right; They insist that the correct equation is derived directly from x'=x-vt and t'=t. Solve for x=x'+vt and replace t with t', then substitute the result in eq-99: (x'+vt')^2 - (ct')^2 = 0. Besides the fact that this results in an equation with arguments exactly equal to eq-99, they will insist the transform is not invariant. IV. A major justification they have for their idea of the correct M system equation on which to base the the discussion of invariance, is that the variables are M system variables, never mind the fact that the arguments are S system values. That argument of theirs is arrant nonsense. The velocity v that S sees for the M system relative to herself is the negative of what the M system sees for the S system relative to himself. In other words, x'+vt' is a mixed frame expression and it is x'+(-v)t' that would be strictly M frame notation, and that equation is far off base. [Work it out for yourself, but make sure you try out an S frame negative v so as not to mislead yourself.] V. In I. we said: given that eq-99 is the correct 'stationary' equation. Let's look at it closely: x^2 - (ct)^2 = 0 (99) This whole matter is supposed to be about coordinate transforms. Is that what t is, just a coordinate? No. It isn't, in general. Suppose you and I are both modelling the same light event and you are using EST and I'm using PST. 'Just a time coordinate' is just a clock reading amd your t clock reading says the light has been moving three hours longer than my clock reading says. Well, that's what the idea that t is a coordinate means. Eq-99 works if and only if t is a time interval, and in particular the elapsed time since the light was emitted. Thus, that equation works only if we understand just what t is, an elapsed time, with emissioon at t=0. However, we don't have to 'understand' anything if we use a more intelligent and insightful form of the equation: (x)^2 - [ c(t-t.e) ]^2 = 0, where t.e is anyone's clock reading at the time of light emission, and t is any subsequent time on the same clock. Similarly, x is not just a coordinate, but a distance since emission. (x-x.e)^2 - [ c(t-t.e) ]^2 = 0 (99a) VI. In the spirit of 'there is exactly one thing they got right', the correct M system version of eq-99a is eq-100a: (x'-x.e')^2 - [ c(t'-t.e') ]^2 = 0 (100a) Every observer in the universe can derive their eq-100a from eq-99a and vice versa, not to mention to and from every other observer's eq-99a. Now, THAT's invariance. [You do realize that every eq-100a reduces to eq-99a, when you back substitute from the transforms, right? t.e'=t.e, x.e'=x.e-vt.] ------------------------------ === Subject: 5. Transformations on Generalized Coordinate Laws The traditional Gallilean transform is correct: t' = t x' = x - vt. But remember this: a transform of x doesn't effect just some values of x, but all of them, whether they are in the formula or not. This is important if you want to do things right. The crackpot position is strongly against this sci.math verified position, and the apparently standard coordinate pseudo-transformation they suggest is perhaps the result. {See Table of Contents.] Let's use a simple equation: x^2 + y^2 = r^2, which is the formula for a circle with radius r, centered at a location where x=0. But what if the circle center isn't at x=0? Well, we'd want to use the form analytic geometry, vector algebra, and elementary measurement theory tells us to use, a form where we make explicit just where the circle center is, even if it is at x=x0=0: (x-x0)^2 + (y-y0)^2 = r^2. The circle center coordinate, x0, is an x-axis coordinate, just like all the x-values of points on the circle. So, in proper generalized cartesian coordinate forms of laws/equations we want to transform every occurence of x and x0 - by whatever name we call it: x.c, x_e, whatever. So, what is the transformed version of (x-x0)? Why, (x'-x0'); both x and x0 are x-coordinates, and every So, what is the value of (x'-x0') in terms of the original x data? is also true for x0'=x0-vt: (x'-x0')=[ (x-vt)-(x0-vt) ]=(x-x0). In other words, when we use the generalized coordinate form specified by analytic geometry, we find that the value of (x'-x0') does not depend on either time or velocity in any way, shape, form, or fashion. Similarly for (y-y0). We can treat time the same way if necessary: (t-t0). The above is a proof that any equation in x,y,z,t is invariant under the galilean transforms. Just use the generalized coordinate form, with (x-x0)/etc, in the transformation process, not the incompetently selected privileged form, with just x/etc. [The form is privileged because it assumes the circle center, point of emission, whatever, is at the origin of the axes instead at some less convenient point. After transform the coordinate(s) of the circle center/origin are also changed but the privileged form doesn't make this explicit and screws up the calculations, which should be based on (x'-x0') but are calculated as (x'-0).] The value of (x'-x0') is the same as (x-x0). That makes sense. Draw a circle on a piece of paper, maybe to the right side of the paper. On a transparent sheet, draw x and y coordinate axes, plus x to the right, plus y at the top. Place this axis sheet so the y-axis is at the left side of the circle sheet. Now answer two questions after noting the x-coordinate of the circle center and then moving the axis sheet to the right: (a) did the circle change in any way because you moved the axis sheet (ie because you transformed the coordin- nate axis)? (b) did the coordinate of the circle center change? The circle didn't change [although SR will say it did]; that means that (x'-x0') does indeed equal (x-x0). The coordinate of the circle center did change, and it changed at the same rate (-vt) as did every point on the circle. That means that x0'<>x0, and the fact the circle center didn't change wrt the circle, means that the relationship of x0' with x0 is the same as that of any x' on the circle with the corresponding x: x'=x-vt; x0'=x0-vt. This is to prepare you for the True Believer crackpots that say 'constant' coordinates can't be transformed; some even say they aren't coordinates. These crackpots include some that brag about how they were childhood geniuses, btw. QED: The galilean transformation for any law on generalized Cartesian coordinates is invariant under the Galilean transform. The use of the privileged form explains HOW the transformed equation can be messed up, the next Subject explains what the screwed up effect of the transform is, and how use of the generalized form corrects the screwup. ------------------------------ === Subject: 6. The data scale degradation absurdity. The SR transforms and the Galilean transforms both convert good, ratio scale data to inferior interval scale data. The effect is corrected, allowed for, when the transforms are conducted on the generalized coordinate forms specified by analytic geometry and vector algebra. Both sets of transforms are 'translations' - lateral movements of an axis, increasing over time in these cases - but with the SR transform also involving a rescaling. It is the translation term, -vt in the x transform to x', and -xv/cc in the t transform to t', that degrades the ratio scale data to interval scale data. In general, rescaling does not effect scale quality in the size-of-units sense we have here. SR likes to consider its transforms just rotations, however - in spite of the fact Einstein correctly said they were 'translations' (movements) - and in the case of 'good' rotations, ratio scale data quality is indeed preserved, but SR violates the conditions of good ro- tations; they are not rigid rotations and they don't appropriately rescale all the axes that must be rescaled to preserve compatibility. The proof is in the pudding, and the pudding is the combination of simple tests of the transformations. We can tell if the transformed data are ratio scale or interval. Ratio scale data are like absolute Kelvin. A measure- ment of zero means there is zero quantity of the stuff being measured. Ratio scale data support add- ition, subtraction, multiplication, and division. The test of a ratio scale is that if one measure looks like twice as much as another, the stuff being measured is actually twice as much. With absolute Kelvin, 100 degrees really is twice the heat as 50 degrees. 200 degrees really is twice as much as 100. Interval scale data are like relative Celsius, which is why your science teacher wouldn't let you use it in gas law problems. There is only one mathematical operation interval scales support, and that has to be between two measures on the same scale: subtraction. 100 degrees relative (household) Celsius is not twice as much as 50; we have to convert the data to absolute Kelvin to tell us what the real ratio of temperatures is. However, whether we use absolute Kelvin or relative Celsius, the difference in the two temperature readings is the same: 50 degrees. Thus, if we know the real quantities of the 'stuff' being measured, we can tell if two measures are on a ratio scale by seeing if the ratio of the two measures is the same as the ratio of the known quant- ities. If a scale passes the ratio test, the interval scale test is automatically a pass. If the scale fails the ratio test, the interval scale test becomes the next in line. It isn't just the bare differences on an interval scale that provides the test, however. Differences in two interval scale measures are ratio scale, so it is ratios of two differences that tell the tale. Let's do some testing, and remember as we do that our concern is for whether or not the data are messed up, not with 'reasons', excuses, or avoidance. ------------------------------------------------------ Are we going to take a transformed length (difference) and see whether that length fits ratio or interval scale definitions? Of course, not. Interval scale data are ratio after one measure is subtracted from another. That is the major reason the SR transforms can be used in science. Let there be three rods, A, B, C, of length 10, 20, 40, respectively. These lengths are on a known ratio scale, our original x-axis, with one end of each rod at the origin, where x=0, and the other end at the coordinate that tells us the correct lengths. Note that these x-values are ratio scale only because one end of each rod is at x=0. That may remind you of the correct way to use a ruler or yard/meter-stick: put the zero end at one end of the thing you are measuring. Put the 1.00 mark there instead of the zero, and you have interval scale measures. Let A,B,C, be 10, 20, 40. Let a,b,c be x' at v=.5, t=10. x'=x-vt. A B C a b c ---------------- -------------------- 10 20 40 5 15 35 ---------------- -------------------- B/A = 2 b/a = 3 C/A = 4 c/a = 7 C/B = 2 c/b = 2.333 Obviously, the transformed values are no longer ratio scale. The effect is less on the greater values. C-A = 10 b-a = 10 C-A = 30 c-a = 30 C-B = 20 c-b = 20 Obviously, the transformed values are now interval scale. This will hold true for any value of time or velocity. (C-A)/(B-A) = 3 (c-a)/(b-a) = 3 (C-B)/(B-A) = 2 (c-b)/(b-a) = 2 Obviously, the ratios of the differences are ratio scale, being identical to the ratios of the corresponding original - ratio scale - differences. The main difference between these results and the SR results is that the differences do not correspond so neatly to the original, ratio scale, differences. This is due only to the rescaling by 1/sqrt(1-(v/c)^2). The ratios of the differences on the transformed values do correspond neatly and exactly to the ratio scale results. Using the generalized coordinate form, such as (x-x0), the transform produces an interval scale x' and an interval scale x0'. That gives us a ratio scale (x'-x0'), just like - and equal to - (x-x0). ------------------------------ === Subject: 7. The Crackpots' Version of the Transforms. It has become apparent - whether misleading or not - that the crackpot responses to the obvious derive from a common source, whether it be bandwagoning or their SR instructors. Below, in the sci.math subject, we see that all sci.math respondents agree with the basic controversial position of this faq: every coordinate is transformed, whether a supposed constant or not. Think about it, the generalized coordinate of a circle center, x0, applies to infinities upon infinities of circle locations (given y and z, too); it is a constant only for a given circle, and even then only on a given coordinate axis. And even variables are often held 'constant' during either integration or differentiation. The utility of a variable is that you can discuss all possible particular values without having to single out just one. That utility does not make particular - singled out - values on the variable's axis not values of the variable just because they have become named values. In any case, all that is preamble to the incompetent idea they have proposed for a transform of coordinates. It is based on the idea that the circle center, point of emission, whatever, has coordinates that cannot be transformed. Let there be an equation, say (x)^2 - (ict)^2 = 0. What is the transformed version of that equation? Answer: (x')^2 - (ict')^2 = 0. That's the one thing the Brittanica got right. Note that the leading crackpot just criticized this faq for presuming to correct the Britt- anica, but it then and before poses the incompetent pseudo- transform we analyze here in this section. x to x' and t to t' are obviously coordinate transforms; the x and t coordinates have been replaced by the coord- inates in the primed system. A tranform of an equation from one coordinate system to another is NOT a substitution of the/a definition of x for itself; that is not a coordinate transformation. The most that can said for such a substitution is that it is a change of variable. But the crackpots are calling this a coordinate trans- form of the original equation: (x'+vt)^2 - (ict')^2 = 0. It is not a coordinate transform, of course, except accidentally. (x'+vt) is not the primed system coordinate, it is another form/expression of x. They get that substitution by solving x'=x-vt for x; x=x'+vt. So, by incompetent misnomer, they accomplish what they have been railing against all along. It has been the generalized coordinate form in question all this time: (x-x0)^2 - (ict)^2 = 0. Here they substitute for x instead of transforming to the primed frame: (x'+vt-x0)^2 - (ict')^2. ----- ^ | ^ | It is still x ^ but see what they have accomplished by their mis/malfeasance: [x'+vt-x0]=[x'+(vt-x0)]=[x'-(x0-vt)]. =[x'-x0'] The crackpots have been bragging about how you don't have to transform the circle center's coordinate to transform the circle center's coordinate. Bragging that what they were doing was not what they said they were doing. This does give us insight as to some of the crackpot variations on their x0'<>x0-vt theme, which in all the variations will be discussed in later sections.. They are used to seeing the mixed coordinate form, (x'+vt-x0) without realizing what it respresented, so - accompanied with a lack of understanding of the term 'dependent' - they are used to seeing just the one vt term, and not the one hidden in the defi- nition of x' and are used to imagining it makes the whole expression time dependent and thus not invariant. About which, let x=10, let, x0=20, v=10, and t variously 10 and 23: (x-x0)=-10. Using their (x'+vt-x0): For t=10, we have (x'+vt-x0) = [ (10-10*10) + (10*10) - (20) ] = -90 + 100 - 20 = -10 = (x-x0) For t=23, we have (x'+vt-x0) = [ (10-10*23) + (10*23) - (20) ] = -220 + 230 - 20 = -10 = (x-x0) The result depends in no way on the value of time; we showed the obvious for a couple of instances of t just so you can see that the crackpots not only do not understand the obvious logic of the algebra { (x'-x0')=[ (-vt)-(x0-vt) ]=(x-x0) } - which shows that the transform has no possible time term effect - but they don't understand even a simple arithmetic demonstration of the facts. Oh. Their (x'+vt-x0) or (x'+vt'-x0) reduces the same way since t'=t: (x-vt+vt-x0)=(x-x0). Their process, which says (x'+vt') is the transform of x, says that (x'+vt') is the moving system location of x, but it can't be because x is moving further in the negative direction from the moving viewpoint. That formula will only work out with v<0 which is indeed the velocity the primed system sees the other moving at. However, that formula cannot be derived from x'=x-vt, the formula for transformation of the coordinates from the unprimed to the primed, ------------------------------ === Subject: 8. What does sci.math have to say about x0'=x0-vt? The crackpots' positions/arguments were put to sci.math in such a way that at least two or three who posted re- sponses thought it was your faq-er who was on the idiot's side of the questions. Their responses: ---------------------------------------------------------- I. x0' = x0. In other words: x0' <> x0-vt, or constant values on the x-axis are not subject to the transform. No. x0' = x0 - vt. Well, if you want, you could define constant values on the x-axis, but in the context of the question that is not relevant. The relevant fact is that if the unprimed observer holds an object at point x0, then the primed observer assigns to that object a coordinate x0' which is numerically related to x0 by x0'= x0 -vt. What does this mean? The line x=x0 will give x'=x-v*t=x0-vt', so if x0' is to give the coordinate in the (x',t',)-system, it will be given by x0'=x0-v*t': ie., it is not given by a constant. Thus, being at rest (constant x-coordinate) is a coordinate-dependent concept. Sounds very false. We can say that the representation of the point X0 is the number x0 in the unprimed system, and x0' in the primed system. Clearly x0 and x0' are different, if vt is not zero. However one may say that (though it sounds/is stupid) the point X0 itself is the same throughout the transformation. However that expression sounds meaningless, since a transform (ok, maybe we should call it a change of basis) is only a function that takes the point's representation in one system into the same point's representation in another system. It is preferrable to use three notations: X0 for the point itself and x0 and x0' for the points' representations in some coordinate systems. ------------------------------ === Subject: 9. But Doesn't x.c'=x.c? That idea is one of the most idiotic to come up, and it does so frequently. And in a number of guises. The idea being that x.c' <> x.c-vt, with x.c being what we have called x0 above; the notation makes no difference. Some crackpots have managed to maintain that position even after graphs have illustrated that such an idea means that after a while a circle center represented by x.c' could be outside the circle. The leading crackpot just make that explicit, as far as one can tell from his befuddled post in response to a line about active transforms, which are actually moving body situations, not coordinate transformations: -------------------------------------------------------------------- e>An active transform is not a coordinate transform, ... Right, it is a transform of the center (in the opposite direction) done to effect the change of coordinates without a coordinate transform. ... E: Transform of the center? Center of a circle? He really is saying a circle center moves in the opposite direction of the circle! Right? -------------------------------------------------------------------- If r=10 and x.c was at x.c=0, then the points on the circle (10,0), (-10,0), (0,10) and (0,-10) could at some time become (-10,0), (-30,0), (-20,10), and (-20,-10), but with x.c'=x.c, the circle center would be at (0,0) still! The circle is here but its center is way, way over there! Indeed, although a change of coordinate systems is not movement of any object described in the coordinates, the x.c'=x.c crackpottery is tantamount to the circle staying put but the center moving away. Or vice versa. ------------------------------ === Subject: 10. But Isn't (x'-x.c')=(x-x.c) Actually Two Transformations? One crackpot puts the (x'-x.c')=(x-vt - x.c+vt) relationship like this: (x-vt+vt - x.c). See, he says, that is transforming x (with x-vt - x.c) and then reversing the transform (x-vt+vt - x.c). That's just another crackpot form of the idiocy that x.c' <> x.c-vt. You'll have noticed the implication is that there is no transform vt term relating to x.c. ------------------------------ === Subject: 11. But Doesn't (x'-x.c+vt) Prove The Transformation Time Dependent? That particular crackpottery is perhaps more corrupt than moronic, since it includes deliberately hiding a vt term from view, and pretending it isn't there. [However, we have seen above that it is a familiar incompetency, and not likely an original.] Look, the crackpots say, there is a time term in the transformed (x' - x.c+vt). The transform isn't invariant! It's time dependent! Just put x' in its original axis form, also, which reveals the other time term, the one they hide: (x'-x.c+vt) = (x-vt - x.c+vt) = (x-x.c). So, at any and all times, the transform reduces to the original expression, with no time term on which to be dependent.