mm-1279 === Subject: Re: Algebra and Analysis : Thoughts? > I was wondering : I feel that I am very strong in Algebra, but not as much > in Analysis, and I would say Topology is in the middle (I havent studied > Differential Geometry yet, that is next semester). Is it me, or is thinking > in an Analysis type way different than in an Algebraic type way? I just > feel that they are very different. I feel that Algebra is much more visual > than Analysis. For me I can just see Linear Algebra, representation theory > (not too advanced, but advanced enough), introductory Field theory, etc etc. > It is much harder for me to see Analysis. Does anyone have a recommendation > for me as to this problem? I feel I should just pick up Rudins Analysis > textbook and read through it and do all of the problems, or maybe a > different textbook, and maybe that would help me see Analysis better. > Moshe You might find it interesting to read Paul Halmoss _Automathology_, which is his autobiography of his professional career. If I remember correctly, he states that people draw to algebra are more language oriented and people draw to analysis are more visually oriented, which seems to be the reverse of your opinion. Of course, you are still very early in your career. By the way, I am not a professional mathematician, so I may not know what I am talking about at all. -Mike === Subject: Re: Shannon defeats Cantor = single infinity type He hasnt really provided a number because you havent really > provided a list. Its trivial to provide the number once you have > the list. Its not trivial to provide the list. It might even be > impossible. Thats irrelevant to showing that, _if you could > provide the list_, it would be incomplete. Fresh post to get me just the algorithm I will take apart..... > wipe! Providing the list is trivial. UTM(Z). Get any UTM apply to > each integer in sequence, assume they all halt nicely. Different > UTMs might get a different list but any will do. Use the one in > the book emporers new mind. calculations so far after conversion to decimal 650897098709879878 > 549789870098709879803978 > 39087709898709879875987879 > 4987897987987987987987987987987 > 978987079809874908980798749874579845 The algorithm please? Apparently, I do not understand UTMs sufficiently. I see this discussion of whether or not a particular UTM(n) halts and after how long. I havent seen the importance of this. Ive just been considering your UTM to be a mapping from the natural numbers to the reals. If the output halts after so many digits (either by definite halting or by unproductive crunching without numbers) then I would think thats the same as producing infinite trailing zeros. It could be there are subtleties Im not aware of. However, for the purposes of you providing me with a list, there has to be an understanding of some kind like I mention, or else you have not yet provided me with a list of reals. With that in mind, I understand the output above as: r_1 = 0.6508970987098798780000... r_2 = 0.5497898700987098798039780000... r_3 = 0.390877098987098798759878790000... r_4 = 0.49878979879879879879879879879870000... r_5 = 0.9789870798098749089807987498745798450000... Using the function neq(x) I defined yesterday, x | 0 1 2 3 4 5 6 7 8 9 neq(x) | 1 2 3 4 5 4 5 6 7 8 and defining d[n] = neq(r_n[n]), I get d = 0.55167... Note that d is not any of the first five reals in your list. As you provide more of UTM(Z), I will be able to provide more digits of d. d will not be any of the other reals you put in your list. I say this not by assumption but because d[n] != r_n[n] and d has no 0s or 9s as digits, and so d != r_n. all n. Jim Burns === Subject: Re: Shannon defeats Cantor = single infinity type [...] > However, its easy to see that it could be that there is > no closest. If the list is contains all the rationals, > (something we agree is possible), the rationals are dense > in the reals. Then there would be rationals arbitrarily > close to the diagonal number, whatever that is. I expect youll find that wierd. I know I do. Finding > something wierd is not the same as disproving it. Not really. > any given diagonal_transform_number - proven number_on_list = 0 > therefore its the same number. Its not the same number. for any e >0, there can be examples of number_on_the_list where |diagonal_transform_number - number_on_list| < e. (this assumes the list is dense in R. This is not the same as |diagonal_transform_number - number_on_list| = 0. Jim Burns === Subject: Re: Shannon defeats Cantor = single infinity type --------------------------------- <^> <()> <^> ----------------------------------- [...] > However, its easy to see that it could be that there is > no closest. If the list is contains all the rationals, > (something we agree is possible), the rationals are dense > in the reals. Then there would be rationals arbitrarily > close to the diagonal number, whatever that is. > > I expect youll find that wierd. I know I do. Finding > something wierd is not the same as disproving it. Not really. > any given diagonal_transform_number - proven number_on_list = 0 > therefore its the same number. > Its not the same number. for any e >0, there can be > examples of number_on_the_list where > |diagonal_transform_number - number_on_list| < e. > (this assumes the list is dense in R. > This is not the same as > |diagonal_transform_number - number_on_list| = 0. Ill try a new approach Well call the new number : (x,y) = ((x,y) + 1) mod 10 Imagine this is the list I provide : (?,1) = 0 (1,?) = 1 otherwise (x,y) = (x-1,y-1) + 1 This results in : 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 3 3 3 3 3 3 1 2 3 4 4 4 4 4 1 2 3 4 5 5 5 5 That would make the diagonal : 1 2 3 4 5 6 ... Is this number on the list? Answer carefully Herc === Subject: Re: Shannon defeats Cantor = single infinity type > Ill try a new approach Well call the new number : > (x,y) = ((x,y) + 1) mod 10 Imagine this is the list I provide : > (?,1) = 0 > (1,?) = 1 otherwise > (x,y) = (x-1,y-1) + 1 This results in : > 0 0 0 0 0 0 0 0 > 1 1 1 1 1 1 1 1 > 1 2 2 2 2 2 2 2 > 1 2 3 3 3 3 3 3 > 1 2 3 4 4 4 4 4 > 1 2 3 4 5 5 5 5 That would make the diagonal : > 1 2 3 4 5 6 ... Is this number on the list? Answer carefully No. If you choose epsilon > 10^(-n), all rows after the nth will be within epsilon of the diagonalized number, but none will be equal to it. (So, it is the limit of the sequence defined by the list.) Or, to see this another way: Every element of the actual list can be written as m/(9*10^n) where m and n are non-negative integers. The new number generated from the diagonal is 1234567890/9999999999 = 137174210 / 1111111111 in lowest terms. Let me try to explain the problem with obtaining an epsilon before presenting an equal list element which is merely within epsilon. Consider the following list: n_1 = 1/2 n_2 = 1/4 n_3 = 3/4 n_4 = 1/8 n_5 = 3/8 n_6 = 5/8 n_7 = 7/8 ... In general, if 2^k <= m < 2^(k+1), n_m = (1+2*(m-2^k)) / 2^(k+1) Do you think this list contains all real numbers between 0 and 1? Do you think a list entry can be found arbitrarily close to any real number in that interval? -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Problem thats sent me for a loop y = x + ln(x) solve for x in terms of y Ive struggled with this for about a week off and on (Im almost positive at one point I knew how to solve this!). Im about 1 week away from going up to the local university and bugging a student or prof out of frustration. Anyone have advice on how to solve that? Oh yeah and really the goal is y = x + A*ln(x)/ln(D) solve for x in terms of y where D and A are constants. I was hopping if anyone could solve/help me solve the original I could get the 2nd one easy enough. any direction would be appreciated. === Subject: Re: Problem thats sent me for a loop > y = x + ln(x) > solve for x in terms of y > Ive struggled with this for about a week off and on (Im almost > positive at one point I knew how to solve this!). Im about 1 week > away from going up to the local university and bugging a student or > prof out of frustration. Anyone have advice on how to solve that? Oh > yeah and really the goal is > y = x + A*ln(x)/ln(D) > solve for x in terms of y > where D and A are constants. I was hopping if anyone could solve/help > me solve the original I could get the 2nd one easy enough. > any direction would be appreciated. This type of equation is call transcendental. As you noticed, they can sometime be hard to solve. There might be some that are not even solvable. Steven === Subject: Re: Problem thats sent me for a loop >y = x + ln(x) >solve for x in terms of y >Ive struggled with this for about a week off and on (Im almost >positive at one point I knew how to solve this!). Im about 1 week >away from going up to the local university and bugging a student or >prof out of frustration. Anyone have advice on how to solve that? Oh >yeah and really the goal is >y = x + A*ln(x)/ln(D) >solve for x in terms of y >where D and A are constants. I was hopping if anyone could solve/help >me solve the original I could get the 2nd one easy enough. Exponentiate both sides of the equation: y x e = x e Then take a look at http://mathworld.wolfram.com/LambertW-Function.html and perhaps http://www.whim.org/nebula/math/lambertw.html for a method to compute LambertW. To handle y = x + a ln(x), note that y/a - ln(a) = (x/a) + ln(x/a) is the same thing, and solve for x/a as you would solve y = x + ln(x) for x. Rob Johnson take out the trash before replying === Subject: Re: lots of balls = 0 balls > If it is not in at the end, and being not in is true of every ball, > which (allegedly large number of) balls are in the bucket at the end? Only the finitely labeled balls are absent. The infinite number of > balls remaining all have infinite labels. You got a problem with that ? Lew Mammel, Jr. Since initially all balls have finite labels and no new labels are > introduced, where do those infinite labels come from? Well, suppose the operation is simply to replace the nth ball with a new one, so there is always one ball. Youre bound to claim that if you do this an infinite number of times, there is no ball left, since every ball is removed by the next operation. Im thinking that if you DO do it an infinite number of times, it makes as much sense as anything to say that you have a ball left from an ordinally non-finite operation. Well, I know all the integers are finite, even though there are an infinite number of them, but somethings got to give! ... but then suppose you have two balls, and simply replace one with the other an infinite number of times. Is infinity odd or even ? Here you dont have the argument that each ball is removed and never put back. ... or suppose you have a slate and for each operation you erase the number written on it, and write its successor. If you do this an infinite number of times, is the slate blank because every number is erased, even though each of the infinite operations leaves a number on the slate ? Actually, I liked Rob Johnsons remarks on the Ten balls in ... thread about the need to define a topology before taking a limit, otherwise the idea of carrying out an infinite number of operations remains intuitive. Lew Mammel, Jr. === Subject: Re: lots of balls = 0 balls >> If it is not in at the end, and being not in is true of every ball, >> which (allegedly large number of) balls are in the bucket at the end? >> Only the finitely labeled balls are absent. The infinite number of >> balls remaining all have infinite labels. >> You got a problem with that ? >> Lew Mammel, Jr. >> Since initially all balls have finite labels and no new labels are >> introduced, where do those infinite labels come from? > Well, suppose the operation is simply to replace the nth ball with > a new one, so there is always one ball. Youre bound to claim that > if you do this an infinite number of times, there is no ball left, > since every ball is removed by the next operation. You got a problem with that? > Im thinking that if you DO do it an infinite number of times, it > makes as much sense as anything to say that you have a ball left from > an ordinally non-finite operation. Well, I know all the integers are > finite, even though there are an infinite number of them, but somethings > got to give! It would make sense, *if* the problem were stated is such a way that ordinally non-finite operations were included. But they werent. For an example of a related problem that *does* include transfinite operations, consider the Transfinite Subway, as described by Scott Huddleston in sci.math on 2 Aug. 1991: ------------------------------------------------------------- --------------- There is a subway line from the airport to the Hilbert hotel which operates as follows: there is a station at each ordinal number, and every station is assigned a unique ordinal. The subway stops at each station, in order. At each station people disembark and board, in order, as follows: i). if any passengers are on the subway, exactly 1 disembarks, then ii). aleph_0 passengers board the subway. Station 0 is at the airport, and the Hilbert hotel is at station w_1 (the first uncountable cardinal). The subway starts its journey empty. Aleph_0 passengers board the subway to the Hilbert hotel at the airport (station 0), and off it goes. The puzzle: when the subway pulls up to the Hilbert hotel at station w_1, how many passengers are on it? Is it 0, aleph_1, some determinate value in between, or indeterminate? ------------------------------------------------------------- ------------- You can find the answer at Google groups. > ... but then suppose you have two balls, and simply replace one with the > other an infinite number of times. Is infinity odd or even ? Here you > dont have the argument that each ball is removed and never put back. Correct. And therefore the problem is not well posed. But your objection does not apply to the original. And it doesnt apply to the transfinite subway, if we stipulate that no passenger boards more than once. > ... or suppose you have a slate and for each operation you erase the > number written on it, and write its successor. If you do this an infinite > number of times, is the slate blank because every number is erased, even > though each of the infinite operations leaves a number on the slate ? If it isnt blank, then what number is on it? > Actually, I liked Rob Johnsons remarks on the Ten balls in ... thread > about the need to define a topology before taking a limit, otherwise the > idea of carrying out an infinite number of operations remains intuitive. But are all topologies equally valid? What about Newtons first law? -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: lots of balls = 0 balls > > If it is not in at the end, and being not in is true of > every ball, which (allegedly large number of) balls are in > the bucket at the end? > > Only the finitely labeled balls are absent. The infinite > number of balls remaining all have infinite labels. > > You got a problem with that ? > > Lew Mammel, Jr. Since initially all balls have finite labels and no new labels > are introduced, where do those infinite labels come from? Well, suppose the operation is simply to replace the nth ball > with a new one, so there is always one ball. Youre bound to > claim that if you do this an infinite number of times, there is > no ball left, since every ball is removed by the next operation. Since the set of all balls used in your operation is in 1 to 1 correspondence with the ( 1 origin) natural numbers, and any non-empty set of natural numbers must have a first (or smallest) member, if you claim that there ARE balls left, you should be able to name the first one. Im thinking that if you DO do it an infinite number of times, it > makes as much sense as anything to say that you have a ball left > from an ordinally non-finite operation. Well, I know all the > integers are finite, even though there are an infinite number of > them, but somethings got to give! ... but then suppose you have two balls, and simply replace one > with the other an infinite number of times. Is infinity odd or > even ? Here you dont have the argument that each ball is removed > and never put back. The result cannot be said to converge to either parity any more that a divergent sequence may be required to converge to some particular finite value. ... or suppose you have a slate and for each operation you erase > the number written on it, and write its successor. If you do this > an infinite number of times, is the slate blank because every > number is erased, even though each of the infinite operations > leaves a number on the slate ? If the slate is not blank, what number is written upon it? Actually, I liked Rob Johnsons remarks on the Ten balls in ... > thread about the need to define a topology before taking a limit, > otherwise the idea of carrying out an infinite number of > operations remains intuitive. Lew Mammel, Jr. === Subject: Does the word Ôvacuously mean something special in maths? I am working through an MIT opencourseware math course. (yum!!) However the lecturere has started to use the word Ôvacuously in his notes. In the dictionary the word vacuous is said to mean: vacent, unintelligent. However I dont see how this fits, I was wondering whether this is a word that is common in Math circles, or is it just the lecturer likes it? Cassandra. ********************** Brief summary of his notes below ********************** The problem is a postage stamp (3c and 5c stamps) problem to learn how to do .....the base case, P(0) wont be interesting because P9n) is vacuously true for all n < 8. ...... 1. n+1<8: So P(n+1) holds vacuously 2. n+1 = 8: P(n+1) holds because we produces 8c postage using one 3c and one 5c stamp..... === Subject: Re: Does the word Ôvacuously mean something special in maths? > I am working through an MIT opencourseware math course. (yum!!) Good for you. This use of vacuously corresponds to the first definition, vacant. For example, > .....the base case, P(0) wont be interesting because P(n) is vacuously true > for all n < 8. This means, for n<8 there are no cases, or no non-trivial cases, to be considered, so the hypothesis P is true. Without seeing the complete paragraph its hard to tell whether the use is strictly correct, or whether the lecturer has just used vacuously to mean obviously. === Subject: Re: Does the word Ôvacuously mean something special in maths? Probably the topic of another thread, but what does non-trivial refer to? > I am working through an MIT opencourseware math course. (yum!!) > Good for you. > This use of vacuously corresponds to the first definition, vacant. > For example, > .....the base case, P(0) wont be interesting because P(n) is vacuously true > for all n < 8. > This means, for n<8 there are no cases, or no non-trivial cases, to be > considered, so the hypothesis P is true. Without seeing the complete > paragraph its hard to tell whether the use is strictly correct, or > whether the lecturer has just used vacuously to mean obviously. === Subject: Re: Does the word Ôvacuously mean something special in maths? >I am working through an MIT opencourseware math course. (yum!!) >>Good for you. >>This use of vacuously corresponds to the first definition, vacant. >>For example, >.....the base case, P(0) wont be interesting because P(n) is vacuously true > >for all n < 8. >>This means, for n<8 there are no cases, or no non-trivial cases, to be >>considered, so the hypothesis P is true. Without seeing the complete >>paragraph its hard to tell whether the use is strictly correct, or >>whether the lecturer has just used vacuously to mean obviously. > > Probably the topic of another thread, > but what does non-trivial refer to? Here, no non-trivial cases means empty or ...well ... vacuous. trivial in math has many tendencious meanings (like obviously), but in this context it is really the dual of vacuous a vacuous argument is p->q and p = F (there are no instances satisfying the antecedent) a trivial argument is p->q and q = T (all instances of the consequent are true (so the antecedent is irrelevant)). It is trivial (in the nonmathematical sense) because no deduction steps are necessary to support the consequent, it is already true. Mitch === Subject: Re: Does the word Ôvacuously mean something special in maths? > I am working through an MIT opencourseware math course. (yum!!) However the lecturere has started to use the word Ôvacuously in his notes. > In the dictionary the word vacuous is said to mean: vacent, unintelligent. However I dont see how this fits, I was wondering whether this is a word > that is common in Math circles, or is it just the lecturer likes it? Cassandra. > ********************** > Brief summary of his notes below > ********************** > The problem is a postage stamp (3c and 5c stamps) problem to learn how to do > .....the base case, P(0) wont be interesting because P9n) is vacuously true > for all n < 8. > ...... > 1. n+1<8: So P(n+1) holds vacuously > 2. n+1 = 8: P(n+1) holds because we produces 8c postage using one 3c and one > 5c stamp..... An if-then statement is vacuously true whenever the if clause is false. An all xs are ys statement is vacuously true whenever there are no xs. === Subject: Re: Does the word Ôvacuously mean something special in maths? > I am working through an MIT opencourseware math course. (yum!!) > However the lecturere has started to use the word Ôvacuously in his notes. > In the dictionary the word vacuous is said to mean: vacent, unintelligent. > However I dont see how this fits, I was wondering whether this is a word > that is common in Math circles, or is it just the lecturer likes it? > Cassandra. > ********************** > Brief summary of his notes below > ********************** > The problem is a postage stamp (3c and 5c stamps) problem to learn how to do > .....the base case, P(0) wont be interesting because P9n) is vacuously true > for all n < 8. > ...... > 1. n+1<8: So P(n+1) holds vacuously > 2. n+1 = 8: P(n+1) holds because we produces 8c postage using one 3c and one > 5c stamp..... A statement is said to be vacuously true if there is no counter examples to the statement and no examples to support the statement either. For example, if A is a non empty set then the empty set is a subset of A since you cant find an element in the empty set which is not in A. That empty set is a subset of A is said to be vacuously true In your example above P(n) is vacuously true for n<8 since n must be 8 or above. I hope this clears things up for you. Steve === Subject: Re: Does the word Ôvacuously mean something special in maths? I like the racoon example it made it very clear... When the term Ôvacuously is used, does it always mean that the set is empty? > I am working through an MIT opencourseware math course. (yum!!) However the lecturere has started to use the word Ôvacuously in his > notes. > In the dictionary the word vacuous is said to mean: vacent, unintelligent. However I dont see how this fits, I was wondering whether this is a word > that is common in Math circles, or is it just the lecturer likes it? Cassandra. > ********************** > Brief summary of his notes below > ********************** > The problem is a postage stamp (3c and 5c stamps) problem to learn how to > do > .....the base case, P(0) wont be interesting because P9n) is vacuously > true > for all n < 8. > ...... > 1. n+1<8: So P(n+1) holds vacuously > 2. n+1 = 8: P(n+1) holds because we produces 8c postage using one 3c and > one > 5c stamp..... > A statement is said to be vacuously true if there is no counter examples > to the statement and no examples to support the statement either. For > example, if A is a non empty set then the empty set is a subset of A since > you cant find an element in the empty set which is not in A. That empty set > is a subset of A is said to be vacuously true > In your example above P(n) is vacuously true for n<8 since n must be 8 or > above. > I hope this clears things up for you. > Steve === Subject: Re: Does the word Ôvacuously mean something special in maths? > I like the racoon example it made it very clear... Glad you liked it. :-) > When the term Ôvacuously is used, does it always mean that the set is > empty? Yes. Theres nothing in a vacuum; similarly, there are no elements in the empty set. David === Subject: Re: Does the word Ôvacuously mean something special in maths? Last question Is the term used alot in math circles? I mean I was surprised by the way the lecturer used it, as if I should have known what it meant. > I like the racoon example it made it very clear... > Glad you liked it. :-) > When the term Ôvacuously is used, does it always mean that the set is > empty? > Yes. Theres nothing in a vacuum; similarly, there are no elements in the > empty set. > David === Subject: Re: Does the word Ôvacuously mean something special in maths? > Last question > Is the term used alot in math circles? Entry: vacuous Function: adjective Definition: empty Synonyms: airheaded, bare, birdbrain, blank, clear, dorky, drained, dull, dumb, dumb bunny, dumbbell, dumdum, emptied, foolish, half-baked, inane, lamebrain, menus, nerdy, nobody home, nothing upstairs, nutty, shallow, silly, stark, stupid, superficial, uncomprehending, unfilled, unintelligent, unreasoning, vacant, void Maybe mathematicians should vary their language... The case n=1 is vacuously true. The case n=2 is airheadedly true. The case n=3 is barely true. The case n=4 is foolishly true. The case n=5 is shallowly true. The case n=6 is superficially true. The case n=7 is dorkily true. === Subject: Re: Does the word Ôvacuously mean something special in maths? > Last question > Is the term used alot in math circles? Yes. In fact, offhand, I cant think of instances when Ive heard it used outside of math/logic. David I mean I was surprised by the way > the lecturer used it, as if I should have known what it meant. > I like the racoon example it made it very clear... Glad you liked it. :-) > When the term Ôvacuously is used, does it always mean that the set > is empty? Yes. Theres nothing in a vacuum; similarly, there are no elements in > the empty set. David === Subject: Re: Does the word Ôvacuously mean something special in maths? > Yes. In fact, offhand, I cant think of instances when Ive heard it used > outside of math/logic. David Used Sherlock on it. It turns out there is also a medical meaning. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Does the word Ôvacuously mean something special in maths? > Last question > Is the term used alot in math circles? > Yes. In fact, offhand, I cant think of instances when Ive heard it used > outside of math/logic. Let me also recommend -- to give you a broader picture, so to speak -- that you do a Google search for vacuously true. I only looked at the first two links it gave me, but they were both worthwhile. David === Subject: Re: Does the word Ôvacuously mean something special in maths? just read http://www.sciencedaily.com/encyclopedia/Vacuously_true great site, well for me anyway, seems to explain it all simply and clearly. > Last question > Is the term used alot in math circles? Yes. In fact, offhand, I cant think of instances when Ive heard it used > outside of math/logic. > Let me also recommend -- to give you a broader picture, so to speak -- that > you do a Google search for vacuously true. I only looked at the first two > links it gave me, but they were both worthwhile. > David === Subject: Re: Does the word Ôvacuously mean something special in maths? > I am working through an MIT opencourseware math course. (yum!!) > However the lecturere has started to use the word Ôvacuously in his > notes. In the dictionary the word vacuous is said to mean: vacent, > unintelligent. > However I dont see how this fits, I was wondering whether this is a word > that is common in Math circles, or is it just the lecturer likes it? > Cassandra. > ********************** > Brief summary of his notes below > ********************** > The problem is a postage stamp (3c and 5c stamps) problem to learn how to > .....the base case, P(0) wont be interesting because P9n) is vacuously > true for all n < 8. > ...... > 1. n+1<8: So P(n+1) holds vacuously > 2. n+1 = 8: P(n+1) holds because we produces 8c postage using one 3c and > one 5c stamp..... Since you havent told us what proposition P is, I cant address your specific problem of the postage. But vacuously is often used in mathematics and logic. For example, the statement All the racoons that have ever bitten me have been rabid. is true, but (mercifully!) only vacuously so, since Ive never been bitten by a racoon. In other words: The set of racoons that have bitten me is empty. We can truly say that all members of that set have property X. After all, no member of that set can be exhibited which lacks property X. But it is just vacuously true since the set is empty. David === Subject: Re: Applications of Eigenvectors!? said: >It get the manipulations involved, but cant imagine the applications Classical Mechanics Economics Quantum Mechnics Sadistics and many more. -- 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 === Subject: Re: Applications of Eigenvectors!? > said: > >>It get the manipulations involved, but cant imagine the applications >You seem to forget that the Canadians kicked our ass in two conßicts.... | | >Really? Which two? | | One of them was vaguely contemporary with the War of 1812, as I | recall, but I dont remember *another* one. | | However, I doubt very much that, given the *current* ratio of military | might between our two countries, that Canadians are optimistic about | being able to defend themselves against a determined attack from the | south. Rather, they have faith in the good, peaceful, and democratic | intentions of that nation. | | Perhaps, unfortunately, unlike some of its brothers to the south. | | But just today in the National Post, I found a *reason* for the United | States to annex Canada. | | It seems that the sensible ordinary people in both countries, who want | the same things, are being forced to put up with governments of | opposite kinds which are in many ways different from what they would | want because of bloc voting groups. In the States, they have the Bible | Belt. In Canada, we have Quebec. | yeah, real bummer..that thing called Ôdemocracy , eh! === Subject: Re: Lenny Susskinds important physics ideas My Commentary on Lennys lecture wow... Wow is right. Why Onearth Would you post the entire pile of crap post again? === Subject: Re: Vedic Mathematics --- Myth and Reality > No, that is wrong. The Indian philosophical thought - Sanatana > dharma, or the way of life beyond the scope of time - is completely > different from the modern and dominant Jewish thinking [...] > This frame of mind, of course, serves to lend additional credence > to the otherwise unbelievable notion that the Swastika actually > originated in India. > The sign of the swastika relates to good health and well being, from > the Indian perspective. This only a brahmin perspective. This might be true for brahmins who > constitue > less than 5% of Indian population. We, Dalits (constitute more than > 20% of Indian popualtion) have no relation to swastika. I do not know > its relation > to Indian Muslims, Indian Christians, Indian Sikhs, Indian Buddhists > etc. > Interestingly brahmins are fire worshippers. Fire is unavidable for > their rituals. In contrast, Dalits do not give importance to fire like > Muslims and Chrstians, and Dalits do not have fire as essential thing > for their religious and spritual rituals and duties. Please note, Mr Arindam Banerjee. You write about your religion, do > not give > distorted picture of India to others. Brahmins/hindus have no right to talk about Dalits. Brahmins/hindus > are not representatives > of Dalits. Nor is any anonymous coward. When unable to answer, fault finding is expected from a racist coward. > [Timofeev]: What can you tell about changes in the measuring techniques of quantities of planetary masses between 1980 and 1990? > [Timpfeev]: What can you add in this context for Mercury, Venus, Earth-Moon, Mars, and asteroid Icarus & so on Those planetary masses were determined by earlier ßy-bys. The orbits of these bodies are determined by a combination of planetary radar ranging and optical observations. -|Tom|- Tom Van Flandern - Washington, DC - see our web site on replacement astronomy research at http://metaresearch.org === Subject: Re: Operator ambiguity > One of those hard lessons to learn in mathematics is operator > ambiguity. > For instance the square root and cuberoot operators are ambiguous, and > theres nothing you can do about it. > Given (1)^{1/3} there are *three* solutions, and not one, which is the > operator ambiguity that gave me fits for a while back last year when I > posted and posted trying to find some trick around it. > The square root operator has ambiguity in that it gives *two* > solutions, even if you only want one. > I managed to get myself in trouble yet again today trying yet again to > escape operator ambiguity by making an earlier post trying to go with > the sign convention of taking the positive solution of the square > root operator. > That doesnt work. > It bothers me that I keep fighting operator ambiguity and trying to > find ways around it, as if some part of me just cant accept that if > you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which > refuse to go away, no matter how hard you wish. James, youre using the wrong terminology here, which no doubt generates replies that seem like James-baiting to you. You should be saying that sqrt(x), or x^{1/3}, have multiple roots, not multiple solutions. Keith > James Harris === Subject: Secret Santa probability question Last month my wife and five of her girlfriends randomly and secretly selected names in a Secret Santa gift exchange. On Sunday they exchanged their gifts and discovered that whenever lady A had chosen lady B, lady B had also chosen lady A. In other words, there were three pairs of ladies who had chosen each others names. What is the probability of this happening? (Note: The ladies were not allowed to pick their own name. If they did, they threw it back into the hat.) Looking forward to a lively thread! --Rich === Subject: Re: Secret Santa probability question > Last month my wife and five of her girlfriends randomly and secretly > selected names in a Secret Santa gift exchange. On Sunday they > exchanged their gifts and discovered that whenever lady A had chosen > lady B, lady B had also chosen lady A. In other words, there were > three pairs of ladies who had chosen each others names. What is the > probability of this happening? (Note: The ladies were not allowed to > pick their own name. If they did, they threw it back into the hat.) So what would they have done if the choices went like this: A: B B: C C: D D: E E: A F: F Lady F can put her name back in the hat as often as she likes - there are no other ladies left to choose from! The probability depends, it seems to me, on how you handle rejections. If you just declare the whole round invalid whenever any one lady picks her own name, the other answers in the thread are correct. If each lady puts her own name back in as soon as she gets it though, before the end of her turn, and letting the previous ladies keep their names, then the answer may well be different. Your answer of 1/15 would be correct from among the cases where lady F didnt get left with her own name, and I think failing to consider that case is where your error lies. === Subject: Re: Secret Santa probability question scenario. I did, in fact, take into consideration that the last person to pick may have only her name left to pick. In my simulation, that lady switches with another lady, but the lady with whom she switches is picked at random, also. This, of course, would not be how it would be done in real life as everyone would now know that the lady picked to switch has Lady 6s name and the lady picked to switch knows what name Lady 6 has. So in reality, they would probably all throw all names back in the hat and repick. Since I dont follow that pattern (starting completely over), but the lady that Lady 6 switches with is picked AT RANDOM, does this affect the results of my program? Because my program runs simulations rather than zero in on the solution mathematically, the results seem to hover around the percentage this group seems to be agreeing upon, although possibly just slightly lower (high 4 to mid 5 percentage rate). Don >> Last month my wife and five of her girlfriends randomly and secretly >> selected names in a Secret Santa gift exchange. On Sunday they >> exchanged their gifts and discovered that whenever lady A had chosen >> lady B, lady B had also chosen lady A. In other words, there were >> three pairs of ladies who had chosen each others names. What is the >> probability of this happening? (Note: The ladies were not allowed to >> pick their own name. If they did, they threw it back into the hat.) >So what would they have done if the choices went like this: >A: B >B: C >C: D >D: E >E: A >F: F >Lady F can put her name back in the hat as often as she likes - there are no >other ladies left to choose from! >The probability depends, it seems to me, on how you handle rejections. If you >just declare the whole round invalid whenever any one lady picks her own name, >the other answers in the thread are correct. If each lady puts her own name >back in as soon as she gets it though, before the end of her turn, and letting >the previous ladies keep their names, then the answer may well be different. >Your answer of 1/15 would be correct from among the cases where lady F didnt >get left with her own name, and I think failing to consider that case is where >your error lies. === Subject: Re: Secret Santa probability question >Last month my wife and five of her girlfriends randomly and secretly >selected names in a Secret Santa gift exchange. On Sunday they >exchanged their gifts and discovered that whenever lady A had chosen >lady B, lady B had also chosen lady A. In other words, there were >three pairs of ladies who had chosen each others names. What is the >probability of this happening? (Note: The ladies were not allowed to >pick their own name. If they did, they threw it back into the hat.) The number of ways that the gift exchange can occur is the number of derangements of 6 items, or 265. See http://mathworld.wolfram.com/Derangement.html The number of ways for each person to give to the person from whom they received is 5*3*1 = 15. That is, person A can choose any of 5 others; the next available person can choose any of 3 others; the next available person is left with only 1. Thus, the probability is 15/265 = 3/53 or 5.66%. Rob Johnson take out the trash before replying === Subject: Re: Secret Santa probability question > My reasoning led me to different and evidently wrong answer. Call > the ladies A, B, C, D, E and F. A can choose any of the others, but > the probability that her partner, say B, picked her is 1/5. > ... > What am I doing wrong? You are assuming that the five possible choices for B are equally likely. But you can easily check that if B chooses A, the remaining ladies can form 9 fixpoint-free permutations, but if B chooses C, there are 14 possibilities (if my count is correct). Esme === Subject: Re: Secret Santa probability question My reasoning led me to different and evidently wrong answer. Call the ladies A, B, C, D, E and F. A can choose any of the others, but the probability that her partner, say B, picked her is 1/5. C can choose any of D, E or F, but the probability that her partner, say D, picked her is 1/3. Then E and F must choose each other. Therefore the probability that they pick each other in pairs is (1/5)(1/3) = 1/15. What am I doing wrong? I really want to know! === Subject: Re: Secret Santa probability question > Last month my wife and five of her girlfriends randomly and secretly > selected names in a Secret Santa gift exchange. On Sunday they > exchanged their gifts and discovered that whenever lady A had chosen > lady B, lady B had also chosen lady A. In other words, there were > three pairs of ladies who had chosen each others names. What is the > probability of this happening? (Note: The ladies were not allowed to > pick their own name. If they did, they threw it back into the hat.) > Looking forward to a lively thread! This can be done by hand, but a computer is handier :-) All permutations: 720 Allowed: 265 (by inclusion/exclusion, sum (-1)^i/i!) Good: Cycles of type (12)(34)(56) (5*3*1=15, also generalizable) Result: 3/53 (~1/e/(2n!!)) -- Hauke Reddmann <:-EX8 Private email:fc3a501@math.uni-hamburg.de For our chemistry workgroup,remove math from the address === Subject: Re: Paul Erdos Has A Kevin-Bacon Number! === >Subject: Re: Paul Erdos Has A Kevin-Bacon Number! >Message-id: On lighter side... >>By the way, interestingly,.. >>Paul Erdos Has A Kevin-Bacon Number... >>and, of course, vice-versa (but in the filmic, not mathematical, >>sense). >>Any guesses as to what the shared number is? >>Solution here: >>http://www.cs.virginia.edu/cgi-bin/oracle/movielinks? firstname=Paul+Erdo s&secondname=Kevin+Bacon&using=1&game=0 >>;) >> >The site says >> Kevin Bacon has a Paul Erdos number of... >But that doesnt seem quite right to me: The link is by co-appearance >in films, not by a co-authorships. Thats why its called a Paul Erdos number and not an Erdos number. Any actor can be the reference point for these link calculations. Depending on how you reference the question, you get either Kevin Bacon has a Paul Erdos number of 4 or Paul Erdos has a Kevin Bacon number of 4 which are the same number, but if you look at the aggregate stats, it makes a difference who the reference is. See The Center of the Hollywood Universe on the Oracle of Bacon home page. Kevin Bacon is closer to the center of the Hollywood universe than Paul Erdos is. Since the last time I checked, the center has shifted from Christopher Lee to an actor who died in 2002. Even though he is dead, he moves closer to the center as the people he is linked to appear in new movies. >Leroy phrased it correctly: Erdos >has a Bacon number. >Who knew Erdos ever appeared in film?... >-- >Stephen J. Herschkorn herschko@rutcor.rutgers.edu -- Mensanator Ace of Clubs === Subject: Re: JSH: Equation has no memory > -1 IS a solution for sqrt(1). Hint: sqrt(x) is a *function*. To be a function, it must produce a unique output for any possible input. If you disagree, go tell SUN that their sqrt() function in Java is broken and see how far you get. Its interesting that is a point of debate, but not surprising for the > sci.math newsgroup! It is only a point of debate because you wish it to be. Have you ever noticed that you frequently stand alone on one side of the debate? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Rationality test 2, math In sci.logic, James Harris factorization > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > where b_3(x) = a_3(x) - 3 and the as are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. > Now consider the factorization shown again, but with the 49 multiplied > through: > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = > 14706125 x^3 - 900375 x^2 - 17640 x + 1078 > and since a_1(0)= a_2(0) = b_3(0) = 0, its not surprising that the > values thus shown to be constant in the factors on the left side i.e. > 7, 7 and 22 are in fact factors of whats constant on the right side > i.e. 1078. > Now if I divide both sides by 49, I end up with a change where now I > have constant factors 1, 1, and 22 on the left which are still factors > of 22 on the right. Yes, one ends up with the rather interesting expression ( (5/7) * a_1(x) + 1 ) * ( (5/7) * a_2(x) + 1 ) * (5 * b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 The question now might be: are (5/7) * a_1(x) and (5/7) * a_2(x) algebraic integers, and for what x? Since we know (a - a_1(x) ) * (a - a_2(x)) * (a - a_3(x)) = a^3 + 3*(-1 + 49*x)*a^2 - 49*(2401*x^3 - 147*x^2 + 3*x) = N(a,x) by your claim above (which I verified correct in a prior post), we can make certain deductions regarding the following somewhat contrived expression (c - (5/7) * a_1(x) ) * (c - (5/7) * a_2(x)) * (c - (5/7) * a_3(x)) by computing a = 7/5*c, or N(7/5*c, x)*(5/7)^3. On the left side, one gets (7/5*c - a_1(x) ) * (7/5*c - a_2(x)) * (7/5*c - a_3(x)) * (5/7)^3 = (c - 5/7 * a_1(x)) *(c - 5/7 * a_2(x)) *(c - 5/7 * a_3(x)) as required. On the right side, this expression yields N(7/5*c, x)*(5/7)^3 = c^3 - (15/7 - 105*x)*c^2 - (42875*x^3 - 2625*x^2 + 375/7*x) Therefore, c_1(x) = (5/7) * a_1(x) and c_2(x) = (5/7) * a_2(x) cannot be algebraic integers for all x. Some x, maybe (e.g., x=0), but not all x. Therefore also, a_1(x) and a_2(x) are not automatically divisible by 7, either. For x=0 one gets the subequation c^3 - 15/7*c^2. Obviously not all the roots for this equation are algebraic integers, but the equation is not irreducible, either; two of the roots are 0. (The third is c = 15/7. Not horribly surprising when one realizes that a_3(0) = 3.) > Does that fact tell you that *two* of the factors on the left, the > ones that have 7 as a constant factor were each divided by 7, or does > it tell you *nothing* at all? It tells me that youre being slightly careless regarding factors of something. After all, 1 = 7 * 1/7 is a perfectly valid factorization (over Q, of course, as opposed to J or N). > Test is of the ability to understand constant factors as independent > of variables, with a check of ability to convince large groups. Note > sci.logic is included to emphasize logical thinking and see if it > matters to readers. > That is, the test is of self-doubt. > James Harris My methodology is fairly sound, I would think. (If anyone can find a ßaw in it, feel free to point it out, of course.) -- #191, ewill3@earthlink.net Its still legal to go .sigless. === Subject: Re: Rationality test 2, math > Given, where x is in the ring of algebraic integers, Ive shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now consider the factorization shown again, but with the 49 multiplied > through: (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 and since a_1(0)= a_2(0) = b_3(0) = 0, its not surprising that the > values thus shown to be constant in the factors on the left side i.e. > 7, 7 and 22 are in fact factors of whats constant on the right side > i.e. 1078. > Correct. > Now if I divide both sides by 49, I end up with a change where now I > have constant factors 1, 1, and 22 on the left which are still factors > of 22 on the right. > *Only if* you divide 49 into 7 * 7 * 1 and divide the first factor by 7, the second factor by 7, and the third factor by 1. But there are infinitely many other ways to factor 49 into three parts. *Any* such factorization will give you the right overall constant term. The factorization will depend on the value of x. After all, why shouldnt it? Clearly the values of a_1, a_2, and b_3 themselves are dependent on x. The factorization, for a given value of x, is not required to respect the divisibility of the constant terms of the factors. The important thing is the entire number in the linear factor, not just the constant term. What *must* be true here is that, for each value of x, 49 factors into three parts, f1(x), f2(x), and f3(x), each dependent on x and each an algebraic integer, in such a way that (5*a_1(x) + 7)/f1(x), (5*a_2(x) + 7)/f2(x), and (5*b_3(x) + 22)/f3(x) are all algebraic integers. This might be easier to think about if you restrict x to a specific value, say x = 1. Then (5*a_1(1) + 7)/f1(1), (5*a_2(1) + 7)/f2(1), and (5*b_3(1) + 22)/f3(1) are all algebraic integers. This is now just a statement about three *numbers*, not three *functions*. The nice thing about this is, it should take your concerns about the constant terms out of the picture for the time being. Of course it is true that the number 22 has no nonunit factors in common with 7. That might cause you to think that f3(1) must be a unit. But the key thing here is not 22; the key thing is the entire *number*, (5*b_3(1) + 22). Even of *both parts* of this were coprime to 7, it does not follow that the whole thing is coprime 7. After all, 5 and 9 are both coprime to 7, but 5 + 9 is not. Thus it is quite possible that (5*b_3(1) + 22) is divisible by a non-unit factor f3(1) of 49, even though 22 is not. The thing here is, the constant terms are *not important* in themselves. The important thing is that for each individual value of x, the arithmetic works out. It will work out differently for each value of x. You may find this implausible. You may say, how does the arithmetic know how the factorization will be done? It comes from the fact that the as are roots of a polynomial whose coefficients are functions of x: the as depend on x. Therefore also their divisibility by factors of 7 *also* depends on x. When x = 0, you get 7, 7, 1. When x = 1, you get something quite different. Implausible though it may seem, I think you can see by now that you do not have a proof that this *cannot* happen. There is a tiny little chance that it might, eh? You cannot quite close the door. There *is* a proof that if f1(1) = 7, then you get a contradiction. You have seen it many times. The logical principle that you have to follow here was enunciated by Sherlock Holmes: when you have rigorously eliminated most explanations for a given observation, the truth must be among whatever is left. That is the situation here. There is only one explanation left, and it is not that f1(1) = 7, f2(1) = 7, and f3(1) = 1. That one has been eliminated. It is that f1(1), f2(1), and f3(1) are all nonunit divisors of 49. > Does that fact tell you that *two* of the factors on the left, the > ones that have 7 as a constant factor were each divided by 7, or does > it tell you *nothing* at all? > It tells you nothing. The factorization has to be computed for each x individually. There is an algorithm (due to Dedekind), but there is no nice formula. For *most values* of x, 49 is factored into three non- unit divisors. The values of x for which that is NOT true correspond to reducibility of a certain polynomial. One such value happens to be x = 0. The irreducible cases are much more common than the reducible ones. > Test is of the ability to understand constant factors as independent > of variables, with a check of ability to convince large groups. Note > sci.logic is included to emphasize logical thinking and see if it > matters to readers. That is, the test is of self-doubt. > Mine, or yours? Nora B. James Harris === Subject: Re: Rationality test 2, math > Given, where x is in the ring of algebraic integers, Ive shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now consider the factorization shown again, but with the 49 multiplied > through: (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 and since a_1(0)= a_2(0) = b_3(0) = 0, its not surprising that the > values thus shown to be constant in the factors on the left side i.e. > 7, 7 and 22 are in fact factors of whats constant on the right side > i.e. 1078. Now if I divide both sides by 49, I end up with a change where now I > have constant factors 1, 1, and 22 on the left which are still factors > of 22 on the right. > This is one way to divide the LHS by 49. It is by no means the only way. > Does that fact tell you that *two* of the factors on the left, the > ones that have 7 as a constant factor were each divided by 7, or does > it tell you *nothing* at all? > Making a reasonable assumption about your use of the term constant factor it tells me that of the infinite number of ways to divide by 49 you chose to divide the first and second terms by 7. This is not *nothing* at all, but neither is it of great significance -William Hughes === Subject: Interesting Square Root Identity The other day I accidentally discovered that Sqrt[6]-Sqrt[2] = Sqrt[8-4Sqrt[3]] (1) It turns out that if a^2-b is a perfect square, then there are integers c and d such that Sqrt[a-Sqrt[b]]=(Sqrt[c]-Sqrt[d])/k, where k is either 1 or 2. For example: if a = 5 and b = 21, we have a^2-b=4, a perfect square. And Sqrt[5-Sqrt[21]] = (Sqrt[14] - Sqrt[6])/2 (2) This leads to some very nonintuitive simplifications, and would make interesting competition problems (e.g. Putnam). For example, heres a problem: Find a polynomial with integer coefficients, one of whose roots is the number Sqrt[7]-Sqrt[4-2Sqrt[3]]-2Sqrt[10-2Sqrt[21]] Answer: P(x)=x-1 Proof: Obviously we need to show that Sqrt[7]-Sqrt[4-2Sqrt[3]]-2Sqrt[10-2Sqrt[21]]=1, or equivalently Sqrt[7]-1=Sqrt[4-2Sqrt[3]]+2Sqrt[10-2Sqrt[21]] Add equations (1) and (2) above. We get Sqrt[14] - Sqrt[2] = 2Sqrt[2 - Sqrt[3]] + 2Sqrt[5-Sqrt[21] Divide both sides by Sqrt[2]. QED. Does anyone else find this interesting? === Subject: Re: Interesting Square Root Identity > The other day I accidentally discovered that > Sqrt[6]-Sqrt[2] = Sqrt[8-4Sqrt[3]] (1) > It turns out that if a^2-b is a perfect square, then there are > integers c and d such that > Sqrt[a-Sqrt[b]]=(Sqrt[c]-Sqrt[d])/k, where k is either 1 or 2. > For example: > if a = 5 and b = 21, we have a^2-b=4, a perfect square. > And Sqrt[5-Sqrt[21]] = (Sqrt[14] - Sqrt[6])/2 (2) > This leads to some very nonintuitive simplifications, and would make > interesting competition problems (e.g. Putnam). For example, heres a > problem: Find a polynomial with integer coefficients, one of whose > roots is the number > Sqrt[7]-Sqrt[4-2Sqrt[3]]-2Sqrt[10-2Sqrt[21]] > Answer: P(x)=x-1 You meant Sqrt[7] - Sqrt[4 - 2Sqrt[3]] - Sqrt[10 - 2Sqrt[21]] = Sqrt[7] - Sqrt[4 - Sqrt[12]] - Sqrt[10 - Sqrt[84]] = Sqrt[7] - (Sqrt[3] - 1) - (Sqrt[7] - Sqrt[3]) = 1 -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: New Scandal in Maths? For those of you who have been following the Peter Lynds controversy, > you may be interested on knowing that a new scandal has happened > recently in the field of Maths. > You can check here (and leave your opinion in the forum if you like): Looks like the same old silliness to me. Well, at least there are formulas in this paper!!! I am not sure it is a very similar case, after all... === Subject: Re: Periodic sequences--is this obvious? Kevin Searle > I have been told the following fact by someone who didnt know > where they knew it from: > Every periodic sequence can be expressed as the sum of a > constant sequence and an irreducible sequence; > where an irreducible sequence is one such that if you take > successive differences iteratively, you eventually end up with > the sequence you started with; > a constant sequence is one whose terms are all equal; > and the sum of two sequences is the sequence whose nth term > is the sum of the nth terms of the two summand sequences. > In other words, and stated even less rigorously than the > above: every periodic sequence is offset by some constant > amount from an irreducible sequence. Somethings wrong with the statement, or maybe I misunderstand it. If a periodic sequence is irreducible, then the sum of the terms in any period must be zero (because that is true of any derived sequence). So, if this periodic sequence: 2 3 -5 2 3 -5 ... (period=3) is the sum of a constant and an irreducible, then the constant is zero. But heres what happens when we take the first few derivatives: 2 3 -5 ... 7 1 -8 ... 15 -6 -9 ... etc. Hereafter, all the terms are multiples of 3, and therefore we never get back the original sequence. LH === Subject: (Z/p^nZ) If p is a prime number, it is well know that (Z/pZ)* (the group of unities of Z/pZ))is a cyclic group. But its also true that (Z/pZ)* is also a cyclic group. Maibe its possible to prove directly tnat X^k=1 has at most k solutions in Z/pZ to copy the usual proof but that seems difficult. Any hefp ? === Subject: Re: Probability question >> I wonder if someone can help me with the following probability >> problem, and show me the method by which it can be solved. If there >> is an event with three possible outcomes, and each outcome has an >> equal probability of occurring, what are the odds that over 80 trials, >> one of the three possible outcomes only results on two occasions >> (while the other two possible outcomes are successful on the remaining >> 78 trials). Also, is there a significance test that can be applied to >> this type of probability problem, such that it is possible to declare >> that the results are statistically significant? >> Anon > >As far as I know, given the way the question is asked, you can do no >more than calculating the probability (I know Ill be corrected if Im >wrong): P(one outcome occurs twice in 80) = bin(80,2)*(1/3)^2*(2/3)^78 = >6.46e-12 This is the probability that a particular outcome occurs > twice; there are three such outcomes, so multiplying this > by 3 gives a not quite correct answer. The reason it is not quite correct is that two oucomes > could each occur twice. The probability of this is for > a particular ordering of the outcomes is > [80!/2!2!76!]*(1/3)^80, and 3 times this needs to be > subtracted. This is MUCH smaller. There are standard significance tests, such as the > chi-squared test, and exact tests. -- > This address is for information only. I do not claim that these views > are those of the Statistics Department or of Purdue University. > Herman Rubin, Department of Statistics, Purdue University > hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 Yup, I see. I think theres even more. In my opinion, the question as stated is ambigious. It could be read as: 1) I pick one of the three outcomes before asking questions (say I choose outcome S1 from possible {S1, S2, S3}), and then I ask: whats the probability that only S1 occurs exactly twice out of 80? 2) I do not pick an outcome before asking questions. Whats the probability that either S1, S2 or S3 occurs exactly twice out of 80? Maybe a native speaker has another opinion on this? These two answers differ a factor three if I correctly understood. I interpreted the question as option 1, but my calculation was not correct (although numerically very close, but that doesnt count in mathematics). Jeroen === Subject: Re: Probability question >> I wonder if someone can help me with the following probability >> problem, and show me the method by which it can be solved. If there >> is an event with three possible outcomes, and each outcome has an >> equal probability of occurring, what are the odds that over 80 trials, >> one of the three possible outcomes only results on two occasions >> (while the other two possible outcomes are successful on the remaining >> 78 trials). Also, is there a significance test that can be applied to >> this type of probability problem, such that it is possible to declare >> that the results are statistically significant? > > >> Anon > >As far as I know, given the way the question is asked, you can do no >more than calculating the probability (I know Ill be corrected if Im >wrong): > > P(one outcome occurs twice in 80) = bin(80,2)*(1/3)^2*(2/3)^78 = >6.46e-12 This is the probability that a particular outcome occurs > twice; there are three such outcomes, so multiplying this > by 3 gives a not quite correct answer. The reason it is not quite correct is that two oucomes > could each occur twice. The probability of this is for > a particular ordering of the outcomes is > [80!/2!2!76!]*(1/3)^80, and 3 times this needs to be > subtracted. This is MUCH smaller. There are standard significance tests, such as the > chi-squared test, and exact tests. -- > This address is for information only. I do not claim that these views > are those of the Statistics Department or of Purdue University. > Herman Rubin, Department of Statistics, Purdue University > hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 > Yup, I see. I think theres even more. In my opinion, the question as > stated is ambigious. It could be read as: 1) I pick one of the three outcomes before asking questions (say I > choose outcome S1 from possible {S1, S2, S3}), and then I ask: whats > the probability that only S1 occurs exactly twice out of 80? > 2) I do not pick an outcome before asking questions. Whats the > probability that either S1, S2 or S3 occurs exactly twice out of 80? Maybe a native speaker has another opinion on this? These two answers differ a factor three if I correctly understood. I > interpreted the question as option 1, but my calculation was not correct > (although numerically very close, but that doesnt count in > mathematics). Jeroen mathematics is virtually zero, so I have some queries about your responses, and I also seek to clarify what it is I am asking exactly. I performed a simple experiment, in which I divided up a geographical area into three equal zones, and then plotted the distribution of a particular (x) phenomenon. x is supposedly a random phenomenon, so it should theoretically have an equal probability of falling into any one of the three zones. Over the course of a year there were 80 occurrences of x, but only two occurrences of x occurred in one zone, with the other 78 events occurring in the other two zones. Obviously, I dont believe this to be a random phenomenon, so I want to state the probability of such a distribution occurring by chance (i.e. as being one in a million or whatever). So Ive treated the events as though they were trials with three possible outcomes. In the previous year, there were 96 occurrences of x in the same region, with all 96 instances occurring in the same two zones. So I presume that the probability of this is 0.666^96, or one in 18 069 400 000 000 000 000 000 - Im not sure. Given my lack of knowledge of probability theory, I really need things explained to me in laymen terms. For example, I am not even sure how to understand the notation given in the equations above, and what the precise answer to the problem is. Finally, I know what a chi-square test is, but what is an exact test? Is this something like Fishers exact test? === Subject: Re: Probability question > My understanding of probability mathematics is virtually zero, Then you should probably not be making any claims (such as not random) at all. === Subject: (Z/p^nZ)* If p is a prime number, it is well know that (Z/pZ)* (the group of unities of Z/pZ))is a cyclic group. But its also true that (Z/p^nZ)* is also a cyclic group when p<>2. Maibe its possible to prove directly tnat X^k=1 has at most k solutions in Z/p^nZ to copy the usual proof but that seems difficult. Any hefp ? === Subject: Re: (Z/p^nZ)* > If p is a prime number, it is well know that (Z/pZ)* (the group of > unities of Z/pZ))is a cyclic group. But its also true that (Z/p^nZ)* is also a cyclic group when p<>2. > word. Yes. The buzzword is primitive root. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: (Z/p^nZ)* For example http://pma106.math.cuhk.edu.hk/Mat3080/MAT3080.htm Robin Chapman a .8ecrit: >>If p is a prime number, it is well know that (Z/pZ)* (the group of >>unities of Z/pZ))is a cyclic group. >>But its also true that (Z/p^nZ)* is also a cyclic group when p<>2. >>word. > Yes. The buzzword is primitive root. > === Subject: books or websites for inequalities in transencedental functions in C by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBE4VGH13676; Please suggest me good books or websites for inequalities in transencedental functions in C (complex plane). thanx msk === Subject: Re: Ordered odd cf. of n = [2:3,5,7,9,11,13...] by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEHTqM02337; >> >> >Interesting how the center column progress by 12 and the two columns >on either side progress by 3. >> >> >> Im unable really to read Perrons book on continued fractions >> because I dont know any German, but Perron appears to make >> a study of the numbers whose continued fraction is sort of close >> to periodic in this way: [a1,...,an, b1,...,bn, b1+c1, b2+c2, >> b3+c3,..., bn+cn, b1+2c1, b1+2c2,..., bn+2cn, ...], with each >> element of the cycle increasing in an arithmetic progression. >> >> Gosper has explained how to compute with continued fraction >> numbers, and you can calculate the expansion of multiples >> of e and so on easily enough, and youll find a bunch of >> related numbers which have the same type of continued fraction. >> See for example >> >> http://www .inwap.com/pdp10/hbaker/hakmem/cf.html#item101b >> Keith Ramsay >> >If you allow negative coefficients, you seem to get a pretty general scheme >for > 1/k > e for k= ... -3,-2,-1,{0},1,2,3,... >------------------------------------------------------------ --------------- --------------- > - 1 3 5 7 9 11 >------------------------------------------------------------ --------------- --------------- >cf(e^(1/-2)): [1,-3, 1, 1, -7, 1, 1,-11, 1, 1,-15, 1, 1,-19, 1, 1,-23, 1 ] >cf(e^(1/-1)): [1,-2, 1, 1, -4, 1, 1, -6, 1, 1, -8, 1, 1,-10, 1, 1,-12, 1 ] >cf(e^(1/ 0)): [1,-1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1 ] divergent (oscillates on 0 and 1) >cf(e^(1/ 1)): [1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1 ] >cf(e^(1/ 2)): [1, 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, 21, 1 ] >cf(e^(1/ 3)): [1, 2, 1, 1, 8, 1, 1, 14, 1, 1, 20, 1, 1, 26, 1, 1, 32, 1 ] >cf(e^(1/ 4)): [1, 3, 1, 1, 11, 1, 1, 19, 1, 1, 27, 1, 1, 35, 1, 1, 43, 1 ] >cf(e^(1/ 5)): [1, 4, 1, 1, 14, 1, 1, 24, 1, 1, 34, 1, 1, 44, 1, 1, 54, 1 ] >cf(e^(1/ 6)): [1, 5, 1, 1, 17, 1, 1, 29, 1, 1, 41, 1, 1, 53, 1, 1, 65, 1 ] >------------------------------------------------------------ --------------- --------------- > + 1 3 5 7 9 11 >------------------------------------------------------------ --------------- --------------- >for > 2/k > e for k= ... -3,-2,-1,{0},1,2,3,... >------------------------------------------------------------ --------------- -------------------------------------------- >delta: - 1 12 5 7 36 11 13 60 17 19 84 23 25 >------------------------------------------------------------ --------------- -------------------------------------------- >cf(e^(2/-1)); [1,-1, -6, -3, 1, 1, -4, -18, -6, 1, 1, -7, -30, -9, 1, 1, -10, -42, -12, 1, 1,-23 ] >cf(e^(2/1)); [1, 0, 6, 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12 ] >cf(e^(2/3)); [1, 1, 18, 7, 1, 1, 10, 54, 16, 1, 1, 19, 90, 25, 1, 1, 28, 126, 34, 1, 1, 37 ] >cf(e^(2/5)); [1, 2, 30, 12, 1, 1, 17, 90, 27, 1, 1, 32, 150, 42, 1, 1, 47, 210, 57, 1, 1, 62 ] >cf(e^(2/7)); [1, 3, 42, 17, 1, 1, 24, 126, 38, 1, 1, 45, 210, 59, 1, 1, 66, 294, 80, 1, 1, 87 ] >cf(e^(2/9)); [1, 4, 54, 22, 1, 1, 31, 162, 49, 1, 1, 58, 270, 76, 1, 1, 85, 378, 103, 1, 1, 112 ] >cf(e^(2/11)); [1, 5, 66, 27, 1, 1, 38, 198, 60, 1, 1, 71, 330, 93, 1, 1, 104, 462, 126, 1, 1, 95 ] >------------------------------------------------------------ --------------- -------------------------------------------- >delta: + 1 12 5 7 36 11 13 60 17 19 84 23 25 >------------------------------------------------------------ --------------- -------------------------------------------- >Also, allowing fractions for coefficients, the primary expansion of e = e^(1/1) = e^(2/2) can be inserted in the >previous table: >------------------------------------------------------------ --------------- -------------------------------------------- >... >cf(e^(2/1)): [1, 0, 6, 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1 ... >cf(e^(2/2)): [1, 0.5, 12, 4.5, 1, 1, 6.5, 36, 10.5, 1, 1, 12.5, 60, 16.5, 1, 1, 18.5, 84, 22.5, 1 ... >cf(e^(2/3)): [1, 1, 18, 7, 1, 1, 10, 54, 16, 1, 1, 19, 90, 25, 1, 1, 28, 126, 34, 1 ... >... >------------------------------------------------------------ --------------- -------------------------------------------- >delta + 0.5 6 2.5 3.5 18 5.5 6.5 60 8.5 9.5 42 11.5 >------------------------------------------------------------ --------------- -------------------------------------------- >Perhaps this allowing of negative and/or fractional coefficients enables also to find >more simple regularities for e^k with abs(k)>2 >With the golden-ratio (phi) you can have also some regularities: >Golden-Ratio >x = phi ~ 1.61803398874989484820... >------------------------------------------------------------- >?? cf(x^-2) >?? cf(x^-1); [ -1, -1, ?????? >------------------------------------------------------------ >cf(x^1); [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ... >cf(x^3); [ 4, 4, 4, 4, 4, 4, 4, 4, 4 ... = a1*3-a_(-1) >cf(x^5); [ 11, 11, 11, 11, 11, 11, 11, 11, 11 ... = a3*3-a1 >cf(x^7); [ 29, 29, 29, 29, 29, 29, 29, 29, 29 ... = a5*3-a3 >cf(x^9); [ 76, 76, 76, 76, 76, 76, 76, 76, 76 ... = a7*3-a5 >cf(x^11); [199, 199, 199, 199, 199, 199, 199, 199 ... >cf(x^0): [ 1, 0, 0, 0, 0, 0, 0, 0, 0, >cf(x^2); [ 2, 1, 1, 1, 1, 1, 1, 1, 1 ... = a0 + a1 >cf(x^4); [ 6, 1, 5, 1, 5, 1, 5, 1, 5 ... = a2 + a3 >cf(x^6); [ 17, 1, 16, 1, 16, 1, 16, 1, 16 ... = a4 + a5 >cf(x^8); [ 46, 1, 45, 1, 45, 1, 45, 1, 45 ... = a6 + a7 >------------------------------------------------------------ ----- >Maybe one can formulate a regular proof and systematize this to a greater extend. >Gottfried Helms Interesting web site and also what you spun from it! I will add this little tidbit which is trivial but relevant to the golden mean and all the related means and there cfs. Where n = 1,2,3,4,5,6,7,..n Then one simple closed form does it all for these irrationals. n=1 Golden mean = (sqrt(((n/2)^2)+1))+ n/2 n=2 Silver mean = n=3 3rd metal mean = n=4 4th metal mean = n=5 5th metal mean = n=6 6th metal ...... etc. Also they are the simplest order of all the cfs ---->oo cf of phi =[1:1,1,1,1,1,1,1,1,1,1,1,1,1...] cf of silver =[2:2,2,2,2,2,2,2,2,2,2,2,2,2...] cf of 3rd mean =[3:3,3,3,3,3,3,3,3,3,3,3,3,3...] cf of 4th mean =[4:4,4,4,4,4,4,4,4,4,4,4,4,4...] cf of 5th mean =[5:5,5,5,5,5,5,5,5,5,5,5,5,5...] cf of nth mean =[n:n,..etc. These converge faster and faster for each succeeding mean. For each term in its cf phi has the slowest convergence of all. I guess that is why they call phi the most irrational of the irrationals. Dan === Subject: Re: Was: Convergence on a space with no topology by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEDudc20217; === >Subject: Was: Convergence on a space with no topology >>Every sequence of subsets of N convergent to N, isnt >>uniformly convergent to N unless the sequence is eventually N. >>Every sequence of subsets of R convergent to R, isnt uniformly >>convergent to N unless the sequence is eventually dense. >R is the reals. That N likely a typo, read R instead. If you have a proof of this just lying around, could you please send it? I havent sat down to proof this yet, but I have a feeling it might take me a while... >>You claim A_n ->* A iff >>A_n subset A forall n and for all p > 0, p in R exists q in N >>for all a in A for all n > q exists b in A_n: d(a,b) < p >>For all n, let A_n = D be a dense subset of R. >>Let A be any set with D subset A subset R. >>Then A_n ->* A for all such As. >>Your uniform convergence, and hence your convergence doesnt have unique >>limits. Moreover, a sequence can even converge to different sets with >>different cardinalities. Im curious: what does converge to different sets with different cardinalities mean? (In no way am I implying here that you dont know what it means). >All of this stuff has topology in it, contary to your title topic. Just a beginning quote from another posting: I have, in no way, defined an open set or neighborhood - much less a distance function- how can I have reinvented the Hausdorff metric? For a space to have a topology, open sets on that space must somehow be defined. How are they defined here- this is not a rhetorical question, i.e. if you know how- please show me. Btw., it is not as if I lie awake at night wondering Is it really Ôconvergence on a space with *no* topology concerning P(X) or does the fact that we used a topology on X make a difference... I.e., semantics are not a primary concern for me. What I want to know is: How do the notions of convergence introduced here relate to the Hausdorff metric and to the various notions of set theoretic convergence (both you and I have) introduced below. O.k., just between you and me: if I could retitle this posting, Id call it Four types of convergence on a space P(X): How are they the same, how are they different? Four refering to Hausdorff convergence, set theoretic convergence, the pointwise convergence and the uniform convergence discussed here. I would also like to see what it means to say P(X) complete using the Cauchy-sequences defined here. C. Dement >For limits without topology, for A_n, A subset S, theres: >limsup A_n = /{ { / Aj | j > k } | k in N } >liminf A_n = /{ { / Aj | j > k } | k in N } >liminf A_n subset limsup A_n; / / intersect union >A_n -> A when A = limsup A_n = liminf A_n > lim A_n = liminf A_n = limsup A_n >descending, ascending A_n > ==> lim A_n = /{ A_n | n in N }, /{ A_n | n in N } >Whatever math I dream up is already old hat. > -- Williams Metatheorem >---- === Subject: Re: JSH: Consider Dik Winter by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEHTqU02329; >> > One of the more persistent posters in replying to me has been Dik >> > Winter, a person who also has several webpages up about some of my old >> > work, which I notice when I do Google searches. >> >> Yes. Are those pages in error? If so, what is the error on those >> pages? >> >I didnt say they were in error as I havent checked them. >However, your need to put up webpages talking about old failed >arguments of mine isnt lost on me. >> > However, he not only seems to lack a throrough mathematical >> > understanding, he often screws up on basics, and Ive yet to see any >> > other poster correct him. >> >> Well, you apparently go for personnal abuse this time (but not the first >> time). >> >I want to emphasize your *lack* of expertise to counteract your >current position of having expertise. >That is, my take on the situation is that with all your webpages >talking about old *failed* arguments of mine, and your tendency to >post a lot in my threads, youve gained a certain positive reputation >which I now find an interest in trying to take away. >> > In my experience, posters on sci.math are labeled crank NOT for being >> > wrong, but for upsetting other posters! >> > >> > Now then Ill give a simple example to show one of Winters mistakes >> > in reasoning to show you exactly what I mean. >> >> Lets see. >> >Good, so now that the preamble is over, its time to get to the >mathematics. >> > For a while I posted about the factorization >> > >> > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = >> > >> > 49(300125 x^3 - 18375 x^2 - 360 x + 22) >> > >> > where b_3(x) = a_3(x) - 3 and the as are roots of >> > >> > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). >> > >> > Notice you have three factors of the polynomial: >> > >> > (5 a_1(x) + 7), (5 a_2(x) + 7), and (5 b_3(x) + 22) >> > >> > where if x=0, a_1(0) = a_2(0) = b_3(0) = 0. >> > >> > Dik Winter adamantly argued that 49 divided those factors as a >> > function of x, so that you had something like >> > >> > w_1(x) w_2(x) w_3(x) = 49 >> > >> > where the ws are functions that vary as x varies, so the >> > factorization, after dividing both sides by 49 is >> > >> > ((5 a_1(x)+ 7)/w_1(x))((5 a_2(x) + 7)/w_2(x))((5 b_3(x) + 22)/w_3(x)) >> > = >> > >> > 300125 x^3 - 18375 x^2 - 360 x + 22 >> > >> > where he was always rather vague about the ws, and in fact never gave >> > them. >> >> You lie. I have given explicit expressions for the ws, not once but >> author dik winter, and terms w3 and gcd. >I went on that search and Google gave 15 results. Looking over those >I guess that you are talking about > w1(x) = gcd(5 a1(x) + 7, 49) > w2(x) = gcd(5 a2(x) + 7, 49) > w3(x) = gcd(5 b3(x) + 22, 49) > k(x) = w1(x).w2(x).w3(x)/49. > These three are easily shown to be algebraic integers for all x. > We factor as: > [k(x).(5 a1(x)+7)/w1(x)] * (5 a2(x)+7)/w2(x) * (5 >b3(x)+22)/w3(x) = > 300125 x^3 - 18375 x^2 - 360(x) + 22. >ws with those statements Dik Winter? >I want you to be VERY clear on that point. >Are you here claiming that you have given an explicit definition for >the ws with your gcd statements? >> >> > Thats a classic crank strategy--being vague about the details--but >> > here it turns out that Winter is easily foiled by considering constant >> > terms. >> > You see, you now have the factors >> > ((5 a_1(x)+ 7)/w_1(x)), >> > ((5 a_2(x) + 7)/w_2(x)), and >> > ((5 b_3(x) + 22)/w_3(x)) >> > which *look* terribly complicated, and Winter probably feels that hes >> > safe with his assertion with expressions that are hard to resolve. >> > >> > However, now let me consider the possibility that the ws are >> > algebraic integer functions and that the factors shown are algebraic >> > integer functions for which Ill use fs. >> >> I have *defined* them as algebraic integer functions. Which you would >> know if you had read and understood the definitions. And similarly I >> have *shown* that your fs are algebraic integer functions, which you >> would have known if you had read and understood the definitions. >> >Well thats interesting given some of your *other* statements Dik >Winter. >> > Then I have >> > f_1(x) = ((5 a_1(x)+ 7)/w_1(x)), >> > f_2(x) = ((5 a_2(x) + 7)/w_2(x)), and >> > f_3(x) = ((5 b_3(x) + 22)/w_3(x)) >> > and letting x=0, I have >> > ((5 a_1(0)+ 7)/w_1(0))((5 a_2(0) + 7)/w_2(0))((5 b_3(0) + 22)/w_3(0)) >> > = >> > 300125 0^3 - 18375 0^2 - 360 (0) + 22 >> > so >> > f_1(0) = 7/w_1(0) = 1, >> > f_2(0) = 7/w_2(0) = 1, and >> > f_3(0) = 22/w_3(0)= 22. >> > and it follows that w_1(0) = 7, w_2(0) = 7, and w_3(0) = 1. >> >> Yup, entirely correct. >> >> > Now using gs to again separate out constant terms as the constant >> > terms were separated out in my original factorization, I have >> > f_1(x) = g_1(x) + 1, f_2(x) = g_2(x) + 1, and f_3(x) = g_3(x) + 22. >> > So now I have >> > (g_1(x) + 1)(g_2(x) + 1)(g_3(x) + 22) = >> > 300125 x^3 - 18375 x^2 - 360 x + 22 >> > where the gs all equal 0, when x=0, which compares nicely with the >> > original. >> > >> > Now then you might suppose that there exist gs that will in general >> > be algebraic integers, but in fact, they do not exist. >> >> Now you have lost me. When you claim that the gs do not exist you >> are actually claiming that the fs do not exist as algebraic integer >> functions. >> >Yup. >> > I would find it intriguing if anyone is willing to dispute that fact. >> >> What fact? That the gs cannot be algebraic integer functions? >> Pray show it using the definition of the ws I have given, which you >> now deny do exist. >So, youre now claiming that there exists gs such that >(g_1(x) + 1)(g_2(x) + 1)(g_3(x) + 22) = > 300125 x^3 - 18375 x^2 - 360 x + 22 >where the gs are algebraic integer functions? >I want you to be VERY clear on that Dik Winter. >Are you now claiming that gs exist with that factorization where the >gs are algebraic integer functions? Yep: g_1(x) = 0, g_2(x) = 0, and g_3(x) = 300125 x^3 - 18375 x^2 - 360 x. B.P. >James Harris === Subject: Re: A noncommunative associative. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEDueA20230; >> Def. >> Let X be a set. Call *: X times X -> X an >> associative if (a*b)*c = a*(b*c) for all a,b,c in X. >Correction: use the know term Ôsemigroup. Ive also heard the expression >associative system, but only once. >> Let Z be the whole numbers. Then normal multiplication >> and ab = a+b-ab (ab normal multiplication) are associatives. >Advise: dont use anything but simple ascii for best readibilty by all >readers in the newsgroup. The symbol that you chose for the operator >appears as an ulgy graphic symbol. Use something plain text, like % and >write a%b = a+b-ab or a$b, a@b, a++b, a+b, etc. >> Earlier, I had asked if the set of all associatives of >> a given set had a particular name (still not sure): >> Let As(Z) be such a set on Z. >Theres no such set in ZF. >> Every x in N (naturals) has a unique representation as >> x =(x_1)(x_2)...(x_n) where >> x_1, x_2, ...,x_n are (positive) primes and >> x_1 <= x_2 <= ... <= x_n >> (normal multiplication) >> If x in Z and x < 0, then x also has a (unique positive) prime >> representation as >> x =-(x_1)(x_2)...(x_n) where >> x_1, x_2, ...,x_n are prime and x_1 <= x_2 <= ... <= x_n >> I would like to show that * in As(Z) where * is defined as: >Just say: ÔId like to show (Z,*) with * as defined below is a semigroup >> For x,y > 1 >> x*y = [(-1)^j]xy = [(-1)^j](x_1)(x_2)...(x_n)(y_1)(y_2)...(y_m) >> = [(-1)^j](z_1)(z_2)...(z_{n+m}) >> = [(-1)^j]z >> when (x_1)(x_2)...(x_n) >> (y_1)(y_2)...(y_m) >> (z_1)(z_2)...(z_{n+m}) are the (unique) prime representations >> of x, y and z (=xy) respectively and j is the >> minimum number of shifts necessary to order >> (x_1)(x_2)...(x_n)(y_1)(y_2)...(y_m) >> into (z_1)(z_2)...(z_{n+m}). >> If x or y = 0, then define x*y = 0. >> Define 1*x = x = x*1 and (-1)x = -x = x(-1) >> If x < 1 or y < 1, then do the multiplication exactly as if >> both were positive, but substitute j with (j+k) were >> k = 1 <-> (x < 0 and y > 0) or (x > 0 and y < 0) >> k = 2 <-> x,y < 0. >Clumbersome and with no intuitive motivation. Im not interested in >proving or even considering if * is associative. Too bad. A motivation could be the parity property discussed below... granted, Im no expert on what all the particularly motivating properties of Z are. Note that it is similar to a long division algorithm in the sense that it is clumbersome to write down but *very* easy to use once youve worked a few examples. Furthermore, I dont understand how I am to limit the process to the positive integers. However * is defined, we have to have *:X times X -> X. Since negative numbers are involved ex. 3*2 = -2*3 = -(2)(3) = -6, we would not have *:N times N -> N C. Dement For a semigroup, youd >save lots of details limiting yourself to positive integers N. As to >uniqueness of your definition, I suspose so, however that needs to be >proven. Again a task of no personal interest. So before you establish >(Z,*) is a semigroup, I advise first show (N,*) is a semigroup. Then >extend it to (Z,*) if you must. So whats so astonding about your >proposed semigroup? Noncommutative semigroups are a dime a dozen, even >countable ones. So heres a pesky problem for that new fangled *: show >its distributive over ordinary +. The reverse operator appears suddenly >as a digression. Is there any reason for introducing it? The parity of a >number may have some uses. Given the parity x,y, whats the parity of xy, >x*y, x+y? >> Notes: >> Whether the shift is to the left or the right does not >> play a role, i.e. j is the same number in either case. >> We cannot replace For x,y > 1 above with >> For x,y > 0; i.e., the rules for 1 must be >> defined differently for the law of association >> to hold (which they were). >> Otherwise, we would have, for example: >> (3*1)*1 = (-3)*1 = 3 != -3 = 3*1 = 3*(1*1) >> In the first equation above, 3 was shifted once to >> the right... -> j = 1 >> This does not happen when 1 is replaced by >> any other whole number, for example 2: >> (3*2)*2 = (-6)*2 = (-(2)(3))*2 = (2)(2)(3) = 12 >> and >> 3*(2*2) = 3*4 = [(-1)^2](2)(2)(3) = 12 >> In the second equation, j = 2 since each >> 2 was shifted once to the left to order >> (3)(2)(2) into (2)(2)(3) so that there >> were two total shifts. Alternatively, >> 3 was shifted twice to the right. >> Other examples: >> (3*10)*14 = (3*((2)(5))*14 = (-(2)(3)(5))*14 = (-(2)(3)(5))*((2)(7)) >> = -(2)(2)(3)(5)(7) = -420 >> Reason: in the 2nd equation, one shift was necessary; >> 2 was shifted once to the left. >> in the 4th equation, two shifts were necessary; >> 2 was shifted twice to the left. >> 3*(10*14) = 3*((2)(5)*(2)(7)) = 3*(-(2)(2)(5)(7) >> = -(2)(2)(3)(5)(7) = -420 >> Reason: in the 2nd equation, 2 was shifted once to the left. >> alternatively, 5 was shifted once to the right. >> in the 3rd equation, 3 was shifted twice to the right. >> We can now assign each x in N a parity according to: >> x in P^{+} <-> (x*x)/(xx) = 1 >> x in P^{-} <-> (x*x)/(xx) = -1 >> Ex. >> 10 in P^{-}, 70 in P^{-}, but 560 in P^{+}. >> Let R be the reverse operator, i.e. >> if x = (x_1)(x_2)...(x_n)= (x_1)*(x_2)*...*(x_n) is the prime >> representation of x, then Rx := (x_n)*...*(x_2)*(x_1) >> A few properties and conjectures (in the order of decreasing >> obviousness, as I see it now): >> i) RRx = x >> ii) x*(Rx) in P^{+} >> iii) x in P^{+}, y in P^{-} -> x*y in P^{-} >> x in P^{-}, y in P^{+} -> x*y in P^{-} >> x in P^{-}, y in P^{-} -> x*y in P^{+} >> x in P^{+}, y in P^{+} -> x*y in P^{+} >> iv) x*y = y*x <-> x and y of same parity >> v) x*(y+z) = x*y + x*z <-> x*y and x*z of same parity >> C. Dement === Subject: Computer Algebra Systems (CAS) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBF1Cb201627; Hi Everybody, === Subject: Re: Computer Algebra Systems (CAS) Hi there > Hi Everybody, > I would like to know what properties define a computer algebra system such I hear of being used in certain Texas Instrument calculators Id say that the requirement would simply be to be able to manipulate algebraic expressions. (TI 89? or TI 83?)and certain computing programs such as Maple and Mathematica. Yep, I use a TI89, HP49G+ and Mathematica. What are the advantages of using a computing device that uses CAS? It does allow you to explore more rapidly than doing everything by hand would permit, although it doesnt remove the need to understand what you are doing, nor _how_ to do it by hand. My boss seems unable to grasp this simple fact: Using a CAS does not make him the equal of a practised mathematician! Oh, CAS are also very useful in recording your ideas and work...they often have a WYSISWG (ÔPretty Print) display so they can be used very effectively in producing documents with lots of maths in them. Also, I have done simple programming on a TI 83. How much different would it be to do programming on Maple or Mathematica? It looks different but the ideas are the same (sort of!), so you shouldnt have much difficulty. Using some CAS can be quite difficult as they often have Ôa steep learning curve but it is usually well worth the effort, particularly when you see how much experimentation you can do in a short space of time. Are there other resources where I can find out more info on these Just have a bit of a Google, also look at sites like wolfram.com and mathworks.com (amongst others), not forgetting the GPL/Open-Source side of things! Enjoy! CAS are very useful but do remember that you still need to be able to understand what it is youre trying to get the CAS to do, so youll need a reasonable understanding of the maths! ttfn JasonG === Subject: Re: Simple Problem in Information Theory by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEEX7t23183; Sorry, H(X|Y) is the conditional entropy of random variable X given random variable Y. It is defined as minus Sum[over y] { p(Y=y) * Sum[over x] (p(X=x|Y=y) * log p(X=x|Y=y)) } ...which looks a lot more comprehensible in proper notation. H(X) is the entropy of random variable X, defined as minus Sum[over x] { p(X=x) * log p(X=x) } For continuous distributions specified by a probability density function, replace the sums with integrals. NB: * logarithms are by convention taken to base 2 * zero log zero is taken to be equal to zero === Subject: Re: Length of a function by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEHTqK02333; >> Hey, havent been here in a while... >> >> Anyway, Thursday and Friday I was working on a problem, basically, >> given a fuciton and a continuous interval of that function, I wanted >> to find the length of its curve. I eventually came up with this: >> The length of the curve f(x) on the continuous curve [a,b] is the >> integral taken from a to b of the squareroot of the sum of the >> derivitive of the function, squared, plus one. >> >> >> Bobby Simione >That is a standard result in calculus. If you discovered it >on your own, rather than reading it in a Calculus text, >that is quite good, and shows that you have a much better >than average grasp of the material. >Kepp trying to discover things new to you. It does not matter >if you are not the first. >Martin Cohen On the other hand, only an amateur starts each problem from the beginning. Learn how to research a bit Good luck phil O / ------o o === Subject: Re: squaring a circle by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEFUWa27216; >> Sub: Unsolvabve Geometrical Problems >> Squaring a given circle is unsolvable . There are volumes of pages on the subject which I have read - from Archimedes to SrinivasaRamanujan. >> Well! I have found out a solution. Could not believe it ? My logic is very simple. >> Pi is an irrational number. So is square root of 2. There is a simple geometrical construction to solve for square root of 2. Similarly I have developed a simple geometrical construction to solve for pi. >> My question is Where to send it? Is there any Maths Forum where I can present my papers and answer questions of the experts? >> >Theres already a geomtrical construciton of pi, take a compass and >construct a circle with a diameter of 1. This is the best I, have heard. Hammering on the nail! Panagiotis Stefanides http://www.stefanides.gr/quadcirc.htm http://www.stefanides.gr/quad.htm http;//www.stefanides.gr/theo_circle.htm http://www.stefanides.gr/piquad.htm === Subject: Re: squaring a circle by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEGA2w29880; === >>Subject: squaring a circle >> Sub: Unsolvabve Geometrical Problems >> Squaring a given circle is unsolvable . >And its unsolvabilty has been proven. One has to digest this alleged proof. Have You? I tried to follow those guys who made Marathons to succeed this goal as they could not reach the Sour Grapes. What some of them devised light-heartedly ,the invention and use(God knows how-The King is Naked...)of the HALF PART of EULERS Equation: e^(ipi)=-1 but for JUSTICE to prevail since EULERS Equation states that: e^(iPi)=cos(pi)+i[sin(pi)] then e^(ipi)=-1 + i(0) , thus the other solution is e^(ipi)=0. So take Your pick! This is what this group of guys did not take into consideration! Panagiotis Stefanides http://www.stefanides.gr/theo_circle.htm http://www.stefanides.gr http://www.stefanides.gr_why_logarithm >>There are volumes of pages on the >>subject which I have read - from Archimedes to SrinivasaRamanujan. >And none of them proved it. So what? >> Well! I have found out a solution. >No, you havent. >>Could not believe it ? >Belief is irrelevant. Its been proven impossible. >>My logic is very simple. >And wrong. >>Pi is an irrational number. >Its also transcendental. >>So is square root of 2. >But the square root of 2 is _not_ transcendental. Look up the difference. >>There is a simple >>geometrical construction to solve for square root of 2. >Nobody said that was impossible. >>Similarly I have >>developed a simple geometrical construction to solve for pi. >If you constructed it, what you have is _not_ pi, because pi cannot be >constructed. >> My question is Where to send it? >The circular file. >>Is there any Maths Forum where I can >>present my papers and answer questions of the experts? >Sure, right here on sci.math. But instead of crowing about your proof, try >asking for >help in understanding why its wrong. >> May I request you to kindly guide me in this regard. >Sorry, I wouldnt be able to find your fallacy even though I know it exists. >-- >Mensanator >Ace of Clubs === Subject: Re: squaring a circle === >Subject: Re: squaring a circle === >Subject: squaring a circle > Sub: Unsolvabve Geometrical Problems > Squaring a given circle is unsolvable . >>And its unsolvabilty has been proven. >One has to digest this alleged proof. >Have You? You can point out where the ßaw is in this proof? >I tried to follow those guys who made Marathons >to succeed this goal as they could not reach the Sour Grapes. ? >What some of them devised light-heartedly ,the invention >and use(God knows how-The King is Naked...)of the HALF PART >of EULERS Equation: > e^(ipi)=-1 >but for JUSTICE to prevail since EULERS Equation >states that: e^(iPi)=cos(pi)+i[sin(pi)] >then e^(ipi)=-1 + i(0) , thus the >other solution is e^(ipi)=0. >So take Your pick! >This is what this group of guys did not take into consideration! Rubbish. >Panagiotis Stefanides >http://www.stefanides.gr/theo_circle.htm >http://www.stefanides.gr >http://www.stefanides.gr_why_logarithm So where have you had this published? >There are volumes of pages on the >subject which I have read - from Archimedes to SrinivasaRamanujan. >>And none of them proved it. So what? > Well! I have found out a solution. >>No, you havent. >Could not believe it ? >>Belief is irrelevant. Its been proven impossible. >My logic is very simple. >>And wrong. >Pi is an irrational number. >>Its also transcendental. >So is square root of 2. >>But the square root of 2 is _not_ transcendental. Look up the difference. >There is a simple >geometrical construction to solve for square root of 2. >>Nobody said that was impossible. >Similarly I have >developed a simple geometrical construction to solve for pi. >>If you constructed it, what you have is _not_ pi, because pi cannot be >>constructed. > My question is Where to send it? >>The circular file. >Is there any Maths Forum where I can >present my papers and answer questions of the experts? >>Sure, right here on sci.math. But instead of crowing about your proof, try >>asking for >>help in understanding why its wrong. > May I request you to kindly guide me in this regard. >>Sorry, I wouldnt be able to find your fallacy even though I know it exists. >>-- >>Mensanator >>Ace of Clubs -- Mensanator Ace of Clubs === Subject: Analytical integral by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBF1Ccd01667; Hi I am looking for this integral integral { exp( sinx) dx from zero to x) thnaks === Subject: Re: Analytical integral > Hi I am looking for this integral integral { exp( sinx) dx from zero to x) thnaks > Presumably not elementary. Heres one: integral exp(sin(x)) dx from -pi/2 to pi/2 is pi*I_0(1), where I_0 is the Bessel function. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Hilbert space related ... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBF1Cck01659; H a Hilbert space, K a closed linear subspace. Let x0 be any pt in H. w.t.s. Min{||x0-y|| : y in K} = Max{|| : y in K^perp, ||y||=1} Im getting stuck but should be pretty easy :( know from given info that ||x0 -y|| >= ||x0|| for all y in K Also that there exists a y in K, z in K^perp s.t. x0 = y + z so ||x0 - y|| = ||z|| > = ||x0|| ... I think Im going wrong way with this. === Subject: Re: vertices of a cube by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBF1Cbh01632; > A cube has 8 vertices (corners). === Subject: Re: Ten Balls In - One Ball Out - Repeat - How Many Remain? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEFjt327934; >There are various versions of this. I give three possible solutions at the >end. Please comment. >Infinitely many balls, each numbered (#1,#2,#3, etc.) are to be placed into >a bucket, ten at a time, by the scheme given below. Immediately after each >group of ten are placed in the bucket, one is removed and discarded. The >process is as described below. >11am: Balls #1 - #10 placed into the bucket. Ball #1 is removed >and discarded. >11:30am: Balls #11-#20 placed into the bucket. Ball #2 removed and >discarded. >11:45am: Balls #21-#30 placed into the bucket. Ball #3 removed and >discarded. >11:52.5am: Balls #31-#40 placed into the bucket. Ball #4 is removed and >discarded. >Etc. >The process continues by halving the remaining time until 12 noon. Then ten >are placed in and one is removed and discarded by the above scheme. The >remaining time is halved again, etc. There is a ßurry of activity just >prior to 12 noon. The process does not continue at or beyond 12 noon. >Question: How many balls remain in the bucket at 12 noon? >There are three common, though not necessarily correct, replies. >1) In that the net gain is +9 balls per event, and there are infinitely >many events, there are infinitely remaining balls in the bucket. >2) None remain. Any given ball, say ball #k, is removed and discarded at a >specific time prior to 12 noon. >3) The question is meaningless as the process can not be extended to or >beyond 12 noon. >Comments? am suprised that none in this thread, at least, have ascribed any significance to the series, as time t ---> T: 1 + 1/2 + 1/4 + 1/8 + 1/16 ... = 2 for that is surely the finite progression of the infinite series of time steps corresponding to each iteration, Add 10, subtract 1, and it converges quite quickly. Because the net gain (9) is constant, the natural number describing an iteration is of equal size (if the process is infinite) to its corresponding number of increase, and the infinite process is mapped as a unit square, thus: Let n = sequence of net gain, 0,9,18,27 ... N. Let i = sequence of iterations 1,2,3,4 ... I. Represent n on a vertical line intersecting at right angle 0 (or 1), the horizontal line i. For the diagonal, we derive sqrt2, noting that the bound of the diagonal is a finite 2 corresponding to the sum of the infinite series, which in the unit square by the Pythagorean Theorem is 2^1/2 (as easily seen graphically when one draws a diagonal line between the ith and nth integers, and then draws a complete square between the two). Therefore, because N = I = 1 in the context of an infinitely expanding unit square,it is apparent that the two dimensional mapping of the phenomenon will not give up information concerning how many? Though the sum of the given infinite series of time divisions is finite, the expanding unit square is finite but unbounded. IOW, we cant say how many balls exist in a moment because we cant say what a moment is. In our normal treatment of time, moments have a natural number correspondence, but not in this problem. In this problem, the iteration is itself the measure -- 9i exactly corresponds to n -- but at what time? At 11:59.x for sufficiently large x, we have lost count of our moments because there is no integer correspondence between I and T. The questions how many and what time are then meaningless, for being independent. As in the Heisenberg Uncertainty Principle, one can answer one or the other question, but not both simultaneously. Tom === Subject: Re: Are irrational numbers green? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBEDudJ20221; >>Besides being a mathematician, I am also a musician. As such, I am aware >>that some people experience synesthesia. The composer Scriabin was >>famous for this, experiencing certain color sensations when he heard >>certain pitches. >Your final comment below made me think again about Scriabin. I may well be >mistaken, but I seem to think now that he also may have associated various >smells with pitches and colors. >>I would not be greatly surprised if there were some >>people who experience certain color sensations when contemplating >>certain numbers. Does anyone know of any instances of such? >> Well, I dont see colors every time I see a number. But >> personally, the following list (expect for the number 1) >> has been completely obvious to me ever since I was about 14. >> In fact, even though I havent thought about it directly for >> years, I would say the list probably seems more obvious >> to me now than ever. >Fascinating! >> 1 dont know, white? >> 2 red >> 3 yellow >> 4 green >> 5 blue >> 6 a yellowish orange >because 6 = 2*3 perhaps? >> 7 brown >> 8 black >> 9 yellow >because 9 = 3*3? >Are numbers which you recognize as being divisible by 3 always at least >tinged with yellow? >David I wondered exactly that when I was typing it in. Unfortunately, I dont have enough information to go on to answer propertly. Im definitely not like Rainman or something, i.e. I dont see 27 and immediately think Ah, 3*3*3- yellow!. Numbers greater than 10 are too complicated for me to see colors. For example 18: is it black because of the 8 -the second digit of 18 or orange because of the factors 2 and 3? -which makes the criteria for seeing a color less and less transparent... but maybe there are some Rainmen and -women, Ramanjuans, out there without that proplem. >> By the way, there was a story >> about synesthesia on the BBC a few years back- >> I believe it centered around people who saw colors >> whenever the smelled certain smells. >> C. Dement === Subject: Re: <<<< by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBF6swG24529; >I have been searching for many weeks for the following information. I >have only found a few answers. It is a project for my Enriched Algebra >II class and is due next week. I need to find these dates. I cant >seem to find dates anywhere all I find are the events but no dates. So >1. Hugh C. Willaims and Harvey Dunbar determine that the number formed >by writing 1031 ones in a row is prime >2. Descartes creates analytic geometry >3. Leibniz invents calculus >4. Riemann creates elliptical geometry >5. Recorde introduces the equal sign >6. Pythagoreans discover irrational numbers >7. Four-Color map problem solved >8. Whitehead and Russel write Principia Mathematica >9. John von Neumann develops his minimax theory >10. Wallace introduces the symbol for infinity >11. Archimedes determines formulas for the area and volume of a sphere >12. Lambert proves pi is irrational >13. Cantor creates transfinate numbers >14. Al-Khowarizimi uses zero >16. Fermat leaves last theorem >17. Tartaglia solves cubic equations >19. Argand graphs imaginary numbers >20. Eratosthenes determines the circumference of the earth >21. Adleman and Rumley develop a new and improved test for prime numbers >23. Newton invents calculus >24. Euler shows that e^pi+i = 0 >25. ENIAC, the first electronic computer is invented >26. Harriout introduces the inequality signs >27. Goldbach states famous conjecture >28. Napier invents logarithms >29. G.9adel publishes incompleteness theorems >30. Emmy Noethers Abstract construction of ideal theory in the domain >of Algebraic number fields becomes a cornerstone of modern abstract >Algebra >31. Pascal and Fermat discuss theory of probability in their >correspondence >32. Kovalevski is the first woman to earn a doctorate in mathematics >33. Oughtred invents the slide rule >34. M.9abius strip is discovered >35. Appollonius studies conic sections >36. Rudolff introduces the radical sign >37. Stevin introduces decimals >38. Lovelace describes how to program Babbages Analytical Engine >39. Gauss determines the convergence of infinite series > go to ask jeeves and type them in i found al of them at www.chuckiii.com/Reports === Subject: Re: Prime numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFG7LK29636; El 11 de oct 1997 de Unam escribi.97: >
 > Joe escribi.97
en el
Yampolskiy escribi.97: > hace cualquier persona saben que un
algoritmo eficaz para encontrar n.9cmeros primeros > yo ha
hecho dos
algoritmos: > primer, aplicaciones el tamiz de Erathosthenes
(es deletre.97 correctamente? en > el espa.96ol es Criva de
Erat.97stenes), y lo utilizo conjuntamente con dividirse >
solamente por
n.9cmeros impares. > el otro algoritmo (que sucede ser m.87s
lento),
tambi.8en utiliza concepto principal de > el primer, pero en
vez de
dividirse por n.9cmeros impares, se divide por ya >
encontrado prepara >
yo es seguro que hay algoritmos mejores y m.87s eficientes,
pero .8estos
trabajo > fino... por lo menos para los n.9cmeros no enormes >
Fernando
Gonz.87lez del Cueto. > Matem.87ticas Aplicadas > Instituto
Tecnol.97gico
Aut.97nomo de M.8exico > Ciudad de M.8exico >
=== Subject: Basic ring theory by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFG7LZ29642; I am currently reading An introduction to Ring Theory by PM Cohn (Springer) in which appears the following question: Show that in any ring R, the sum of two ideals A and B, defined as A+B = {a+b|a in A, b in B} is again an ideal, as is the product AB = {sigma over i of a_i.b_i|a_i in A, b_i in B}. If further A + B = R, show that AB = A intersection B. I am puzzled by the solution outlined for the last part. It easy to show that AB is a subset of A intersection B. To show that A intersection B is a subset of AB, I follow the hint given in the text and get the following: Since A + B = R, we can find a in A and B in B such that a + b =1 and so if c is in A intersection B then c = (a+b)c(a+b) = aca + bca + acb + bcb. Now we consider each term in this sum. aca: a is in A; ca is in B since c is in B and B is an ideal similarly for acb and bcb But how do we show that the term bca belongs to AB ? I can see that, since bc is in A, bca is in AA ; and also that since ca is in B, bca is also in BB. But how do we get it into AB? Any suggestions? Clio === Subject: Re: Basic ring theory > I am currently reading An introduction to Ring Theory by PM Cohn (Springer) > in which appears the following question: > Show that in any ring R, the sum of two ideals A and B, defined as A+B = > {a+b|a in A, b in B} is again an ideal, as is the product AB = {sigma over i > of a_i.b_i|a_i in A, b_i in B}. If further A + B = R, show that AB = A > intersection B. This question has been posted here (and answered) before; do a Google search for a thread called called exercice on ideals. It turns out that the statement that you were trying to prove is false. Jose Carlos Santos === Subject: Re: Request for an approval for mathematical research by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFG7MQ29675; >I was advised to write you this letter hoping you would consider my >research in the field of algebra with emphasis on permutation groups >and remarks on the divisors of counting numbers. Although I have >produced papers on these issues, I have not yet brought to the >attention of the concerned entity who has the knoledge about the >subject matter and authorization to certify the validity of the idea. >In the light of the above,I am graaateful for your best consideration >and accomplish the task of getting approval and publication of the >topic. >Hoping an answer from you, I remain >Cordiallly yours, >Samson Lulu Approval denied. Call the next case. phil === Subject: Maximal Ideals by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFEVJ122580; I request help with completing the below proof. If R is the ring of all real valued continuous functions on the closed interval and M is a maximal ideal of R, prove that there exists some p in [0,1] such that M={ all f(x) in R/ f(p)=0} I can prove it assuming that there is atleast one function in M that has only a finite number of zeros. But how might I prove this? RK. === Subject: Re: Maximal Ideals I request help with completing the below proof. If R is the ring of all real valued continuous functions on the closed > interval and M is a maximal ideal of R, prove that there exists some p in > [0,1] such that M={ all f(x) in R/ f(p)=0} I can prove it assuming that there is atleast one function in M that > has only a finite number of zeros. But how might I prove this? RK. > Non-constant linear functions can have at most one zero. === Subject: Re: Maximal Ideals >I request help with completing the below proof. >If R is the ring of all real valued continuous functions on the closed >interval and M is a maximal ideal of R, prove that there exists some p in [0,1] such that M={ all f(x) in R/ f(p)=0} Say Z(M) = {p : f(p) = 0 for all f in M}. You need to show that Z(M) contains exactly one point. If Z(M) contains more than one point its easy to see that M is not maximal. So you only need to show that Z(M) is nonempty. Suppose Z(M) is empty: For every p in [0,1] there exists f = f_p such that f_p(p) <> 0. Note that f_p is actually non-zero on some neighborhood of p. So by compactness there are finitely many g_1, ... g_n in M such that for every p in [0,1] there exists j with g_j(p) <> 0. Now use the fact that M is an ideal: Let g = _______, and g is an element of M with no zero. So g is invertible and hence M is not a proper ideal. > I can prove it assuming that there is atleast one function in M that >has only a finite number of zeros. But how might I prove this? > RK. ************************ David C. Ullrich === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFIJrS07749; >Dave, you chose not to respond to the main point of my >argument (which you cut), which is that one can make >definitive statements about *models* which are well-defined >mathematical objects, but that the choice of model is a >nonmathematical endeavor which is open to opinion and >discussion. When a system is realizable, one can presumably >test the accuracy of a model. When the system is not >realizable, there is not necessarily a correct answer. >I have yet to hear a response to this position. I agree that there is not necessarily a correct answer. I believe that sane, reasonable people understand that the amount of balls in the bucket never decreases and therefore will never be zero, even if we can make unprovable and also disputable statements to the contrary. Note that PA (The label guys) cannot prove anything about the number of balls that were put in the bucket or removed from the bucket. Noon forces us beyond the natural numbers. I choose to stick with the notion that the bucket gets very full. === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls >>Dave, you chose not to respond to the main point of my >>argument (which you cut), which is that one can make >>definitive statements about *models* which are well-defined >>mathematical objects, but that the choice of model is a >>nonmathematical endeavor which is open to opinion and >>discussion. When a system is realizable, one can presumably >>test the accuracy of a model. When the system is not >>realizable, there is not necessarily a correct answer. >>I have yet to hear a response to this position. Already answered elsewhere. > I agree that there is not necessarily a correct answer. I > believe that sane, reasonable people understand that the > amount of balls in the bucket never decreases and therefore > will never be zero, even if we can make unprovable and > also disputable statements to the contrary. Note that PA > (The label guys) cannot prove anything about the number > of balls that were put in the bucket or removed from the > bucket. Noon forces us beyond the natural numbers. I > choose to stick with the notion that the bucket gets very > full. The bucket does indeed get very full. And then it gets empty. The number of balls put in the bucket is exactly aleph_0. The number of balls removed from the bucket is exactly aleph_0. This is not sufficient information to answer the question that was posed, but fortunately, additional information was given, which some people prefer to ignore. Why is it that all the people who claim there is not a unique answer always have to alter the problem in some way to prove their point? Why has not a single person proposed a model that conforms to the original problem statement (and certain natural assumptions such as Newtons first law) such that a different answer is indicated? -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls The bucket does indeed get very full. And then it gets empty. When exactly? If every operation increases the number of balls in the bucket, the number of balls never decreases. Lew Mammel, Jr. === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls > The bucket does indeed get very full. And then it gets empty. When exactly? If every operation increases the number of > balls in the bucket, the number of balls never decreases. Exactly at noon. At any time prior to noon, no matter how small, there exists an N such that that time takes place between 1/2^N sec before noon and 1/2^(N+1) sec before noon. At that point, the Nth iteration, there are still balls there, and an infinite number yet to be put in and taken out. Therefore, only at noon does it get empty. (Im a little surprised that this is even a question to be asked.) Jonathan Hoyle Gene Codes Corporation === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls >> >> The bucket does indeed get very full. And then it gets empty. > When exactly? If every operation increases the number of > balls in the bucket, the number of balls never decreases. For each n let a_n be the time when ball n is placed in the bucket. Let b_n be the time when ball n is removed from the bucket. Letting t=0 represent noon, we have b_n = -2^(1-n) and a_n looks something like -2^(1-ceiling(n/10)), but the important thing is that -1 <= a_n <= b_n < 0 for each n. Now let chi_n : R -> {0,1} be the characteristic function of ball n. That is, chi_n(t) = 1, if a_n <= t < b_n = 0, otherwise. Thus chi_n(t) is 1 if ball n is in the bucket at time t, and 0 otherwise. Here I have made chi_n right-continuous, but using a different convention wont make any difference in the long run. Notice that according to the definition, chi_n(0) = 0 for each n. But we can say more. For each t we can say the number of balls in the bucket at time t is f(t) = sum_n{n=1}^oo chi_n(t) and therefore it is easily seen that f has the following characteristics: (1) f(t) = 9*n for a_n <= t < a_(n+1). (2) f(t) = 0 for t < -1 and for t >= 0. (3) f is nondecreasing on (-oo,0). (4) lim_{t->0-} = +oo. (left-hand limit) Thus f = sum chi_n has an infinite discontinuity at t = 0, despite the fact that each of the chi_n is continuous there. No balls are moved at noon, but the bucket nevertheless becomes empty. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls the original problem statement. Yes, it certainly seems that there can be no balls remaining in the bucket at noon according to that statement. And yet... (See below.) >> The bucket does indeed get very full. And then it gets empty. Might it be significant that your last sentence was stated passively? (See below.) > When exactly? If every operation increases the number of > balls in the bucket, the number of balls never decreases. > For each n let a_n be the time when ball n is placed in the bucket. Let > b_n be the time when ball n is removed from the bucket. Letting t=0 > represent noon, we have b_n = -2^(1-n) and a_n looks something like > -2^(1-ceiling(n/10)), but the important thing is that -1 <= a_n <= b_n < > 0 for each n. > Now let chi_n : R -> {0,1} be the characteristic function of ball n. > That is, > chi_n(t) = 1, if a_n <= t < b_n > = 0, otherwise. > Thus chi_n(t) is 1 if ball n is in the bucket at time t, and 0 otherwise. > Here I have made chi_n right-continuous, but using a different convention > wont make any difference in the long run. > Notice that according to the definition, chi_n(0) = 0 for each n. But we > can say more. For each t we can say the number of balls in the bucket at > time t is > f(t) = sum_n{n=1}^oo chi_n(t) > and therefore it is easily seen that f has the following characteristics: > (1) f(t) = 9*n for a_n <= t < a_(n+1). > (2) f(t) = 0 for t < -1 and for t >= 0. > (3) f is nondecreasing on (-oo,0). > (4) lim_{t->0-} = +oo. (left-hand limit) > Thus f = sum chi_n has an infinite discontinuity at t = 0, despite the > fact that each of the chi_n is continuous there. No balls are moved at > noon, but the bucket nevertheless becomes empty. Passive again. Why? Is it perhaps because you do not wish to identify the mechanism by which the bucket is suddenly emptied? (Yes, I know, we emptied the bucket, taking the balls out -- not suddenly, but one by one -- at times prior to noon. That seems clear from the original problem statement.) And yet, saying No balls are moved at noon, but the bucket nevertheless becomes empty. makes it sound as if aleph_0 balls somehow just vanish -- -- right at noon, without any specifiable causative agent. (But of course my previous parenthetical comment presumably explains what you and I both already knew to have happened.) I really do think that there is something which seems paradoxical here. (No, certainly theres no antinomy.) And I would think that, for someone who considers Newtons First Law to be pertinent here, the sense of paradox might even be heightened. The number of balls in the bucket was increasing without bound as noon approached, and then, suddenly, at noon, there were none! So then (rhetorically, to indicate the seeming paradox): Exactly _when_ did we remove all those balls? We cant have removed them at any time before noon. And we removed no balls at noon or afterwards. And yet we had to have been removed them. So when did we do it? Maybe this doesnt seem at all paradoxical to you, Dave. But it has the ßavor of paradox to me. David Cantrell === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls >Exactly _when_ did we remove all those balls? We didnt remove all of them. We removed each of them (and, at least on some accounts, we know--of each of them--when it was removed.) Actually, *I* had nothing to do with it, dont look at me, my hands are clean, nothing up my sleeves. But I do suggest that part of your feeling of paradox is caused by the connotative differences in common English between each and all, which evaporate when both are rendered into mathematics as universal quantifiers. Lee Rudolph === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls The bucket does indeed get very full. And then it gets empty. > When exactly? If every operation increases the number of > balls in the bucket, the number of balls never decreases. [Im sure I will regret getting involved in this discussion. But somehow, finally, I cant resist...] I must agree with Lews point. But first let me insert what Dave S. said in full: The bucket does indeed get very full. And then it gets empty. The number of balls put in the bucket is exactly aleph_0. The number of balls removed from the bucket is exactly aleph_0. This is not sufficient information to answer the question that was posed, but fortunately, additional information was given, which some people prefer to ignore. Why is it that all the people who claim there is not a unique answer always have to alter the problem in some way to prove their point? Why has not a single person proposed a model that conforms to the original problem statement (and certain natural assumptions such as Newtons first law) such that a different answer is indicated? This thread and its progenitors have mentioned _many_ somewhat different problems. So Dave, when you refer to the question that was posed and the original problem statement, Im sorry, but Im not at all certain what that was. Could you please (re)state (at least your version) of the question that was posed originally? Not knowing yet what you consider to be the original problem, the following may well be premature. But suppose that the transactions were to occur as, say, in the statement of Problem 1 given by Stephen H. at the beginning of this thread: Problem 1. At times 12 - 2^(1-n) (in hours) for n = 1, 2 ,..., you add ten identical balls to the basket, then remove one of balls in the basket. How many balls are in the basket at noon (= 12)? And now lets go back to Lews point. Note that the only additions/removals of balls occur _before_ noon. According to the statement of Problem 1, nothing happens -- no additions or removals of balls -- _at_ noon (or thereafter). Were all in agreement, I presume, that the number of balls in the basket increases without bound as noon approaches. So _if_ there were to be no balls in the basket at (or, perhaps some would rather say, immediately following) noon, lots of balls would have had to have been removed _at some time_. Precisely what is that time? It is certainly not any time prior to noon; there were lots of balls in the basket at any time shortly before noon. And it cannot be at noon or thereafter since nothing is supposed to happen then. Confusedly yours, David Cantrell === Subject: Re: Attn: Poker Joker Re: lots of balls = 0 balls >> The bucket does indeed get very full. And then it gets empty. >> When exactly? If every operation increases the number of >> balls in the bucket, the number of balls never decreases. > [Im sure I will regret getting involved in this discussion. But somehow, > finally, I cant resist...] > I must agree with Lews point. But first let me insert what Dave S. said > in full: > The bucket does indeed get very full. And then it gets empty. The > number of balls put in the bucket is exactly aleph_0. The number of > balls removed from the bucket is exactly aleph_0. This is not sufficient > information to answer the question that was posed, but fortunately, > additional information was given, which some people prefer to ignore. > Why is it that all the people who claim there is not a unique answer > always have to alter the problem in some way to prove their point? Why > has not a single person proposed a model that conforms to the original > problem statement (and certain natural assumptions such as Newtons first > law) such that a different answer is indicated? > This thread and its progenitors have mentioned _many_ somewhat different > problems. So Dave, when you refer to the question that was posed and > the original problem statement, Im sorry, but Im not at all certain > what that was. Could you please (re)state (at least your version) of the > question that was posed originally? Sorry. This is the original problem statement I was referring to: >>There are various versions of this. I give three possible solutions at the >>end. Please comment. >>Infinitely many balls, each numbered (#1,#2,#3, etc.) are to be placed into >>a bucket, ten at a time, by the scheme given below. Immediately after each >>group of ten are placed in the bucket, one is removed and discarded. The >>process is as described below. >>11am: Balls #1 - #10 placed into the bucket. Ball #1 is removed >>and discarded. >>11:30am: Balls #11-#20 placed into the bucket. Ball #2 removed and >>discarded. >>11:45am: Balls #21-#30 placed into the bucket. Ball #3 removed and >>discarded. >>11:52.5am: Balls #31-#40 placed into the bucket. Ball #4 is removed and >>discarded. >>Etc. >>The process continues by halving the remaining time until 12 noon. Then ten >>are placed in and one is removed and discarded by the above scheme. The >>remaining time is halved again, etc. There is a ßurry of activity just >>prior to 12 noon. The process does not continue at or beyond 12 noon. >>Question: How many balls remain in the bucket at 12 noon? > Not knowing yet what you consider to be the original problem, the following > may well be premature. But suppose that the transactions were to occur as, > say, in the statement of Problem 1 given by Stephen H. at the beginning of > this thread: > Problem 1. At times 12 - 2^(1-n) (in hours) for n = 1, 2 ,..., you > add ten identical balls to the basket, then remove one of balls in the > basket. How many balls are in the basket at noon (= 12)? This is a variant of the original problem. In this version there is not enough information to determine an answer. It could be 0, aleph_0, or anything in between. If the ball to be removed is randomly selected at each stage, then with probability 1, the bucket is empty at noon. > And now lets go back to Lews point. Note that the only additions/removals > of balls occur _before_ noon. According to the statement of Problem 1, > nothing happens -- no additions or removals of balls -- _at_ noon (or > thereafter). See my other post regarding the characteristic functions chi_n. Note that each chi_n is continuous at 0, but f = sum chi_n turns out to have an infinite discontinuity there. Your question amounts to asking which of the chi_n is discontinuous, to account for the discontinuity in f. The answer is that none of them is discontinuous at t=0. > Were all in agreement, I presume, that the number of balls in the basket > increases without bound as noon approaches. So _if_ there were to be no > balls in the basket at (or, perhaps some would rather say, immediately > following) noon, lots of balls would have had to have been removed _at > some time_. Precisely what is that time? It is certainly not any time prior > to noon; there were lots of balls in the basket at any time shortly before > noon. And it cannot be at noon or thereafter since nothing is supposed to > happen then. Precisely for what time is f = sum chi_n decreasing? The answer is that the function decreases at 0, despite the fact that each chi_n is continuous and nondecreasing (constant, in fact) on an interval containing 0. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: proof: THIS TIME FOR REAL by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBF6swG24533; >> Yes. That logic, which I did not pay attention to when first >> suggested by W. Dale Hall (simply because of equivalency of a >> statement), is certainly more elegant. >The more I think about this parenthetical remark, the less I find myself >understanding it. >Just think: you have something youre trying to prove, and you present a >sequence of unnecessarily complicated and convoluted arguments dedicated >to making that proof happen, all apparently because youre not really >satisfied with the argument as it stands. Along comes an alternate tack, >suggesting another way to read the thing youre trying to prove. There is no offense in what I am going to say. I am not stupid enough (well sort of used to be...) to accept what is the most elegant, simple, cute. Recently, I have been capable of throwing away any idea that is wrong, in spite of any effort consumed. Actually, why did you not put your simplest argument when this problem was first proposed? I am not saying that you would be as bad at distinguishing what is worthwhile with what is trivial than me (Believe me, I am terribly bad at that... But at least I am always trying to say something correct). It was the truth that I put the thread on correction of my argument before you put your simpler argument. I got the same argument (no reason to use ln ) I do not know if there exists a theorem in game theory that states that being extremely, offensively, argumentative, critical could give the player a benefit. Well, actually there is no award in this game, or even not a game. My intuition (again, do not believe :) but listen to me ) says that being aggresive in a game is not a benefitial strategy overall. I am a little bit interested in why you did not put your simple proof in the first place. === Subject: Re: Please help.... Area between curve problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFH60Z01861; >Here is the problem: >What is the area between the curve x=sqrt(16 - y^2) and the y-axis? >I realize this is a semi-circle, Ive solved for y (+ and -), yet I >dont see the area between the curve and the axis this is looking for. > Is there a solution to this problem? Maybe Im missing something here, but the area you seek is just half the area of a circle whose radius is 4, no? You dont need integral calculus for that phil === Subject: Re: lots of balls = 0 balls by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFEVKV22609; >> What you are saying is that processes involving an infinite number >> of steps in a finite interval of time creates problems of >> understanding. But Zeno knew that millennia ago. >> >> Cool. Zeno didnt understand a millennia ago. Real Cool. >> I dont recall mentioning a finite time interval. >What do you mean a millennia? One can have a millennium, but one >cannot have less then two millennia. But Zeno understood better than >you seem to. Havent you ever learned that the punctuation is to be included inside the quotes. Also, the string of words between the last two periods is not a sentence. Zeno understood better than you seem to. >If the time interval for the infinite process is not finite, the >infinite process never ends, Incorrect. Infinite processes may have no beginning and a definite end. Besides, I only stated that I didnt mention a finite time interval. Zeno understood better than you seem to. > and there is no problem about what >happens when such a process is only carried out for a finite number >of steps. >It seems as if the poker just poked himself in the eye on this one. OOPS! Is your eye okay? Zeno understood better than you seem to. === Subject: Re: lots of balls = 0 balls >> What you are saying is that processes involving an infinite number >> of steps in a finite interval of time creates problems of >> understanding. But Zeno knew that millennia ago. Cool. Zeno didnt understand a millennia ago. Real Cool. >> I dont recall mentioning a finite time interval. What do you mean a millennia? One can have a millennium, but one >cannot have less then two millennia. But Zeno understood better than >you seem to. > Havent you ever learned that the punctuation is to be included > inside the quotes. Id be willing to bet that Virgil learned that convention long ago, as did many of us mathematicians. But it is illogical. For example, if he had typed a millenia?, it would have made it seem, logically, as if you had used a millenia at the end of a question, which of course was not the case. Due to the illogical nature of the convention, some mathematicians have consciously decided, in their own writing, to use a logical method instead. Thus, for example, in the last sentence of the previous paragraph, I purposely did not type a millenia?,, but rather a millenia? followed by a comma. David === Subject: Re: lots of balls = 0 balls > Due to the illogical nature of the convention, some mathematicians have > consciously decided, in their own writing, to use a logical method instead. > Thus, for example, in the last sentence of the previous paragraph, I > purposely did not type a millenia?,, but rather a millenia? followed by > a comma. I am one who consciously does this as well, even if it is not syntacticly correct. As language evolves, many of the new rules of grammar come from its frequent usage. So, I take the hit of being incorrect for now with the hope that I help inspire a change. Not likely one I will see soon, I understand, but I have not lost any sleep over it yet. :-) === Subject: Ack! More Difficult Problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFIJrl07757; >Heres the problem: - >Given three random variables A, B and C, and given >H(A|B) = 0 >(i.e. A is just a deterministic function of B) >prove >H(B|C) >= H(A|C) Damn, its not true for general continuous distributions... This is clear when A and B are related by a bijective function which does not preserve entropy, and C is statistically independent of A (and therefore B). Fortunately I have some additional facts which rule out such a case and may be sufficient for a proof. The problem revisited: Given three random variables A, B and C, and given H(A|B) = 0 0 < H(B|A) < H(B) H(A) < H(B) prove H(B|C) > H(A|C) Notation: H(X) is the Shannon entropy of X H(X|Y) is the conditional Shannon entropy of X given Y Simon McGregor sm66blue.sussexpink.ac.uk Remove colours from the preceding line and replace first . with @ === Subject: Re: lots of balls = 0 balls by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFG7MU29655; >But you need to prove that LBR and LBI instantiated after >infinitely many steps are completed. One can easily define >a unicorn, but it is more difficult to instantiate one. LBR area begins with a ball. The LBR can only be changed by replacing the ball with another ball. Its not difficult (except for you) to see that there always will be a ball in the LBR area. Of coarse you have some sort of monster in your head that seems to eat the entire LBR area. >And in order to provide such instantiation, you will also have to >provide a last (largest) natural number. So what is your version of >the last natural number? No I dont. You are the one making the claim about a last natural number. I think you are claiming the LBR contains a last natural number. Prove that it must. >> Heres another argument for the label guys: Without useing labels, we can say: (1) If the balls are removed in FIFO order, then the bucket is >> empty at noon. (2) If the balls are removed in LIFO order, then there are infinitely >> many balls left at noon. >> >> Yeah right, LIFO. Whats the first one removed? The truth is, this >> is so far from the OP, Im going to disregard LIFO order. >FIFO, for the ignorant, means first in, first out, in which case >each ball will be removed in a finite number of steps. >LIFO, for the ignorant, means last in, first out, in which case 9 >of 10 balls put in will never be taken out. Now that we know what FIFO and LIFO mean to you, well assume the same meaning. >> The balls numbered greater than LBR and up to and including LBI >> have labels. >But have you shown (and can you show) that either LBI or LBR exist >after an infinite number of steps? The LBR area is a place. It neither comes nor goes. It is always there. It has a ball in it. There is one operation: Replace the ball with another ball. The only operation that is possible does not change the fact that the LBR exists with a ball in it. === Subject: Re: lots of balls = 0 balls But you need to prove that LBR and LBI instantiated after >infinitely many steps are completed. One can easily define >a unicorn, but it is more difficult to instantiate one. LBR area begins with a ball. The LBR can only be changed > by replacing the ball with another ball. Its not difficult > (except for you) to see that there always will be a ball > in the LBR area. Of coarse you have some sort of monster > in your head that seems to eat the entire LBR area. If there is a ball in either, the first ball in either should correspond to some natural number. As erudite as you claim to be, you should have no trouble in giving the precise value of such netural numbers. But perhaps you colaimed erudition is only a facade. > >And in order to provide such instantiation, you will also have to >provide a last (largest) natural number. So what is your version of >the last natural number? No I dont. You are the one making the claim about a last > natural number. I think you are claiming the LBR contains > a last natural number. Prove that it must. It is your claim that the process ends, not mine. > >> Heres another argument for the label guys: Without useing labels, we can say: (1) If the balls are removed in FIFO order, then the bucket is >> empty at noon. (2) If the balls are removed in LIFO order, then there are infinitely >> many balls left at noon. >> >> Yeah right, LIFO. Whats the first one removed? The truth is, this >> is so far from the OP, Im going to disregard LIFO order. FIFO, for the ignorant, means first in, first out, in which case >each ball will be removed in a finite number of steps. LIFO, for the ignorant, means last in, first out, in which case 9 >of 10 balls put in will never be taken out. Now that we know what FIFO and LIFO mean to you, well assume the > same meaning. > >> The balls numbered greater than LBR and up to and including LBI >> have labels. But have you shown (and can you show) that either LBI or LBR exist >after an infinite number of steps? The LBR area is a place. It neither comes nor goes. It > is always there. It has a ball in it. There is one > operation: Replace the ball with another ball. The > only operation that is possible does not change the fact > that the LBR exists with a ball in it. You claim that after the process of replacing one ball with another has stopped (after infinitely many iterations in finite time) there is still a ball in LBR. All I want you to do is to state which ball it is. If you cant, perhaps you should rethink your position. === Subject: Re: Probabilty Question by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBG2RBk12739; there are 4 red marbles,5 white marbles and 6 blue marbles in a bag. Once a marble is selected, it is not replaced. Find the probablity of each outcome? 1. a red marble and then a white one 2. 2 blue in a row ? === Subject: Re: Probabilty Question adam a .8ecrit : > there are 4 red marbles,5 white marbles and 6 blue marbles in a bag. Once a marble is selected, it is not replaced. Find the probablity of each outcome? 1. a red marble and then a white one > 2. 2 blue in a row ? > 1. 4/15 * 5/14 2. 6/15 * 5/14 === Subject: cubic spline interpolation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBG2RCl12777; I was wondering if anyone could help me out with cubic splines. I have written a program in Visual Basic that displays chromatographic data in intensity verus time. For the sake of simplicity, first assume that the chromatogram is a very simple Gaussian curve. I call a subroutine in Visual Basic to interpolate values between each data point (all are evenly spaced). If the user selects cubic splines, the program calculates a second derivative using Savitzky-Golay smoothing. The calculated spline-based interpolated data using the second derivative from the Savitzky-Golay smooth correlates exactly with the original function. The problem is not with a Gaussian however, but with a square wave. Now assume a square wave. When the second derivative is calculated for a square wave function, the second derivative function is zero for a portion of time, rises from zero, falls below zero, rises back to zero and remains there for a portion of time, falls below zero, rises back above zero, and falls back to and remains at zero again. Using this second derivative data creates defects in the original data. This gives the original square wave a dog ear like appearance on the top left and right edges of the square wave after cubic spline interpolation is done. To combat this, I set the second derivative equal to zero for the entire chromatogram and this dog ear phenomenon has gone away. There is now perfect correlation between the original data and the interpolated data. My question is... will there be a problem if I Steve === Subject: Re: best calculus book by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFJmIY14625; >> >> I bought the best calculus book ever written...couldnt >> understand a word of it...after 12 years of extensive sudy I >> went back to it and found it was indeed the best one!! >which book is it? courants? So, whats the name of the book?!! Inquiring minds want to know! I hear that the late Dick Feynman (from Cal Tech) was a pretty smart cookie. Does anyone know what calculus book he thought was the best? Hmm....how about Einstein, does anyone know what calculus book he used as a reference? Why is it that the best calculus books seem to have been written in the pre-modern math days? Is there anyone from the modern math generation who has authored high quality calculus books? Also, it seems that the Nobel prize winners in science still seem to be from the pre-WWII era. I dont think any have been born post 1970 at all so far. === Subject: Re: Maze: of 1 through 25 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBFJmIR14616; >I made an error in my solution-reply I just posted. >I fix the error below. My prolog program found two solutions, here is the screen: File Edit Buffers Info Debug Switch Help +--------------------------------| M A I N +---------------------------------+ |Reconsulting ... D:ARITYARISRCLEROYLEROY.ARI [ buffer 1 ]. | | | |yes | |?- main. | | | |[3,4,5,10,9,8,7,12,7,6,1,0] | |[4,5,10,15,14,9,8,7,12,7,6,1,0] | |yes | |?- | | | | | | | | | | | | | | | | | | | | | | | | | +------------------------------------------------------------ --------------- ---+ Arity/Prolog Intepreter v6.00.39 Copyright (C) 1989-1991 Arity Corporation And this is my program: %%%%% main:-line1([0],Path),nl,write(Path),fail. main. line1(L,Res):-line2([1|L],Res). line2([C|T],Res):- adjacent(C,N), N mod 3=:=0, line3([N,C|T],Res). line3([C|T],Res):- adjacent(C,N), prime(N), line4([N,C|T],Res). line4([C|T],Res):- adjacent(C,N), prime(P), N is C+P, line5([N,C|T],Res). line5([C|T],Res):- first, (retract((first)); assert((first)),!,fail), divisor(C,D), L is D//2, case([L=:=1 -> line1([C|T],Res), L=:=2 -> line2([C|T],Res), L=:=3 -> line3([C|T],Res), L=:=4 -> line4([C|T],Res), L=:=5 -> line5([C|T],Res), L=:=6 -> line6([C|T],Res), L=:=7 -> line7([C|T],Res), L=:=8 -> line8([C|T],Res), L=:=9 -> line9([C|T],Res), L=:=10 -> line10([C|T],Res), L=:=11 -> line11([C|T],Res), L=:=12 -> line12([C|T],Res), L=:=13 -> line13([C|T],Res), L=:=14 -> line14([C|T],Res), L=:=15 -> line15([C|T],Res), L=:=16 -> line16([C|T],Res), L=:=17 -> line17([C|T],Res), L=:=18 -> line18([C|T],Res)]); not(first), line6([C|T],Res). line6([C|T],Res):- adjacent(C,N), divisor(C,D), N is C+D, line7([N,C|T],Res). line7([C|T],Res):- (N is C+1; N is C+2), adjacent(C,N), line8([N,C|T],Res). line8([C|T],Res):- dadjacent(C,N), not(member(N,T)), line9([N,C|T],Res). line9([C|T],Res):- phi(C,F), N is F+C-1, adjacent(C,N), line10([N,C|T],Res). line10([C|T],Res):- N is C+1, line11([N,C|T],Res). line11([C|T],Res):- largest(C,P), N is C-P, adjacent(C,N), line12([N,C|T],Res). line12([C|T],Res):- N is integer(sqrt(C))+10, adjacent(C,N), line13([N,C|T],Res). line13([C|T],Res):- adjacent(C,N), gcd(C,N)=:=1, line14([N,C|T],Res). line14([C|T],Res):- adjacent(C,N), P is C-N, prime(P), line15([N,C|T],Res). line15([C|T],Res):- adjacent(C,N), not(prime(N)), not(member(N,T)), (line16([N,C|T],Res); line15([N,C|T],Res)). line16([C|T],Res):- N is (C+1)//2, adjacent(C,N), line17([N,C|T],Res). line17([C|T],Res):- adjacent(C,N), not(member(N,T)), line18([N,C|T],Res). line18([C|T],Res):- sum([C|T],S), ifthenelse(S mod 3=:=0, (left(C,N),Res=[N,C|T]), Res=[C|T]). %%%%% first. %%%%% prime(2). prime2(4,2). primepower(8,2,3). prime(3). prime2(9,3). primepower(16,2,4). prime(5). prime2(25,5). prime(7). prime(11). prime(13). prime(17). prime(19). prime(23). %%%% primeprime(X,P1,P2):- prime(P1), X mod P1=:=0, P2 is X//P1, P1P1,!,Y is P; Y is P1). %%%% left(X,Y):-X mod 5 ==1, Y is X-1. right(X,Y):-X mod 5==0, Y is X+1. up(X,Y):-X>5, Y is X-5. down(X,Y):-X<21, Y is X+5. %%%% adjacent(X,Y):-left(X,Y);right(X,Y);up(X,Y);down(X,Y). %%%% dadjacent(X,Y):- left(X,L),(up(L,Y);down(L,Y)); right(X,R),(up(R,Y);down(R,Y)). %%%% divisor(1,1):-!. divisor(X,Y):-prime(X),!,Y is X+1. divisor(X,Y):-prime2(X,P),!,Y is X+P+1. divisor(X,Y):-primeprime(X,P1,P2),!,Y is X+P1+P2+1. divisor(X,Y):-primepower(X,P,N), Y is (P^(N+1)-1)//(P-1). divisor(X,Y):-prime(P), X mod P=:=0, P1 is X//P, P1 mod P==0,!, divisor(P,D),divisor(P1,D1), Y is D*D1. %%%% gcd(A,A,A):-!. gcd(A,0,A):-!. %???????????????? gcd(A,B,Res):- A > B,A1 is A mod B,gcd(B,A1,Res). gcd(A,B,Res):- B > A,gcd(B,A,Res). %%% sum([],0):-!. sum([N],N):-!. sum([H|T],S):-sum(T,S1),S is H+S1. %%%%----%%%% member(H, [H|_]). member(H, [_|T]) :- member(H, T). %%%% Maybe, Ive made some mistakes in translating your sophisticated rules into prolog clauses. Aleksey === Subject: Re: Solve Benedict Webb Rubin Equation of state by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBGBdTI17009; >Can any body assist me in solving the BWR equation for all its roots >analytically. >John sorry guy, i also face this problem. you found the solution already? === Subject: Re: cauchy-schwarz inequality, normed fields and vector spaces by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBGBdTq17020; >Let k be a field. Is k(T) a normed field? >I was wondering whether the Cauchy-Schwarz inequality holds over >normed k-vector spaces for k other than R or C. If the field k carries the norm |.| you can try to define a norm on the field of rational functions k(T) through |p(T)|:=max (|a| : a runs through the coefficients of p) (*) where p is a polynomial. (The extension to k(T) is then obvious.) However the map thus defined is not multiplicative in general. But if the original norm is ultra-metric, that is it satisfies the strong triangular inequality |x+y|<=max(|x|,|y|) (**), then (*) defines a norm on k(T), that satisfies (**) too. H === Subject: Re: Topologies implied by limits of sets by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBGBdUr17051; Sorry, Re: convergence... should have been Was: convergence... In any case, heres the listing see www.mathforum.org/discuss/sci.math/t/548648 good day, C. Dement === Subject: Re: Topologies implied by limits of sets >>All of this stuff has topology in it, contary to your title topic. >For a space to have a topology, open sets on that space must >somehow be defined. How are they defined here- this is not >a rhetorical question, i.e. if you know how- please show me. Heres my approach to topologizing P(S) For Aj in P(S), j in N define liminf Aj = /{ /{ Aj | j >= k } | k in N } limsup Aj = /{ /{ Aj | j >= k } | k in N } When liminf Aj = limsup Aj define lim Aj = liminf Aj = limsup Aj For A subset P(S), let cl A = { a in P(S) | some a1,a2,... in A with lim aj = a } Thus we have cl nulset = nulset; A subset cl A A subset B ==> cl A subset cl B Does cl cl A = cl A ? That is the crutial question. If it does, then a set is closed when cl A = A and open when P(S)A is closed. The four properties above for cl, assure the so defined open sets are a topology for P(S). The topology for P(S) would be T1 as cl {a} = {a} forall a in P(S) and the set of cofinite subsets of P(N) would be dense. -- Convergence on a space with no topology >Every sequence of subsets of N convergent to N, isnt >uniformly convergent to N unless the sequence is eventually N. >Every sequence of subsets of R convergent to R, isnt uniformly >convergent to N unless the sequence is eventually dense. >>R is the reals. That N likely a typo, read R instead. >If you have a proof of this just lying around >>You claim A_n ->* A iff >>A_n subset A forall n and for all p > 0, p in R exists q in N >>for all a in A for all n > q exists b in A_n: d(a,b) < p Look close at your definition. For A = R, it assures A_n is dense. Theorem: D subset R is dense when for all open U, U/D nonnul. Then look where you basically say b in B(a,p) >>For all n, let A_n = D be a dense subset of R. >>Let A be any set with D subset A subset R. >>Then A_n ->* A for all such As. >>Your uniform convergence, and hence your convergence doesnt have >>unique limits. Moreover, a sequence can even converge to different >>sets with different cardinalities. >Im curious: what does converge to different sets with >different cardinalities mean? (In no way am I implying >here that you dont know what it means). Not only does your convergence not have unique limits, its doesnt even assure cardinality. A convergence of a sequence can be to two different sets with different cardinality. Thats a lot of divergence to swallow for a vague notion of convergence. -- A noncommunative semigroup. >> Def. >> Let X be a set. Call *: X times X -> X an >> semigroup if (a*b)*c = a*(b*c) for all a,b,c in X. >> Every x in N (naturals) has a unique representation as >> x =(x_1)(x_2)...(x_n) where >> x_1, x_2, ...,x_n are (positive) primes and >> x_1 <= x_2 <= ... <= x_n >> (normal multiplication) >> If x in Z and x < 0, then x also has a (unique positive) prime >> representation as >> x =-(x_1)(x_2)...(x_n) where >> x_1, x_2, ...,x_n are prime and x_1 <= x_2 <= ... <= x_n >> I would like to show that (Z,*) with * as defined below is a semigroup >> For x,y > 1 >> x*y = [(-1)^j]xy = [(-1)^j](x_1)(x_2)...(x_n)(y_1)(y_2)...(y_m) >> = [(-1)^j](z_1)(z_2)...(z_{n+m}) >> = [(-1)^j]z >> when (x_1)(x_2)...(x_n) >> (y_1)(y_2)...(y_m) >> (z_1)(z_2)...(z_{n+m}) are the (unique) prime representations >> of x, y and z (=xy) respectively and j is the >> minimum number of shifts necessary to order >> (x_1)(x_2)...(x_n)(y_1)(y_2)...(y_m) >> into (z_1)(z_2)...(z_{n+m}). >> If x or y = 0, then define x*y = 0. >> Define 1*x = x = x*1 and (-1)x = -x = x(-1) >> If x < 1 or y < 1, then do the multiplication exactly as if >> both were positive, but substitute j with (j+k) were >> k = 1 <-> (x < 0 and y > 0) or (x > 0 and y < 0) >> k = 2 <-> x,y < 0. >Furthermore, I dont understand how I am to limit the process >to the positive integers. However * is defined, we have to >have *:X times X -> X. Since negative numbers are involved >ex. 3*2 = -2*3 = -(2)(3) = -6, we would not have >*:N times N -> N Indeed, my oversight. Permutations isnt my cup of tea. The parity definition you gave seemed on key. ---- === Subject: Re: Integer Puzzle by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBGBdUm17061; >
> My advisor, Dr. John Conway, has come up with what he
calls the
> ultimate naming scheme for numbers, which includes numbers
like one
> millillion (10^3003 in American English, 10^6000 in
British). I dont
> remember many details; I might be able to get them if
anyone is
> interested.
> --
> Will Schneeberger
>>He probably includes the numbers millionilion (10^3000003
in American
English,
>>10^6000000 in British English) and billionillion
(10^3000000003 and
>>10^6000000000000, respectively)?
>>Rene
>No, actually 10^3000003 / 10^6000000 is known as one
billillion,
>10^3000000003 / 10^6000000000 is known as one trillillion,
>10^3000000000003 / 10^6000000000000 is known as one
quadrillillion,
>etc. As I remember, -llion cannot appear in the middle of a
word in
>this scheme, but Im not sure about that.
>--
>Will Schneeberger
>william@math.Princeton.EDU
>http://www.math.princeton.edu/~william
>
=== Subject: Re: Integer Puzzle >
> My advisor, Dr. John Conway, has come up with what he
calls the
> ultimate naming scheme for numbers, which includes numbers
like one
> millillion (10^3003 in American English, 10^6000 in
British). I
dont
> remember many details; I might be able to get them if
anyone is
> interested.
> --
> Will Schneeberger
He probably includes the numbers millionilion (10^3000003 in
American
English,
>>10^6000000 in British English) and billionillion
(10^3000000003 and
>>10^6000000000000, respectively)?
Have these numbers any practical use?
--
Paul V. S. Townsend
Interchange the alphabetic elements to reply
===
Subject: Re: proof: THIS TIME FOR REAL
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id hBG2RBa12743;
>I am not stupid enough (well sort of used to be...) to accept
>what is the most elegant, simple, cute. Recently, I have been
>capable of throwing away any idea that is wrong, in spite of
>any effort consumed.
wrong. I was trying to write
I am not stupid enough (...) to not accept ...
Just for clarification so that you will be able to understand
what I said (not The more I read, the more I could not
understand.).
===
Subject: What goes up ...
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id hBGGPMO05387;
What goes up comes down
What goes in comes out
What goes around comes around
Holiday cheer
phil
O
/
----o o
===
Subject: problem solving
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id hBGGPMA05399;
you are the manager of a movie theatre 488 tickets are sold.
$8.oo dollars
for adult, and $6.00 dollars for seniors. tha takein was
$3,3336 dolars.
How many tickets were sold to seniors? And how many were sold
to adults?
===
Subject: Re: problem solving
cisco a .8ecrit :
> you are the manager of a movie theatre 488 tickets are
sold. $8.oo
dollars for adult, and $6.00 dollars for seniors. tha takein
was $3,3336
dolars.
> How many tickets were sold to seniors? And how many were
sold to adults?
>
There is a mistake in what you say, this is not possible that
takein was
$3,3336 dollars because there is only 488 tickets and
33336/488 = 68.3
This is certainly 3336 unstead of 33336
a the number of adult
b the number of seniors
we have: 8a+6b=3336 and a+b=488
4a+3b=1668 and a=488-b
4(488-b)+3b=1668
1952-4b+3b=1668
b=284
a=488-b=204
===
Subject: Re: Equivalence relation with infinite many 
infinite
classes
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id hBGIoxO17152;
>L is a langage containing only one symbol of binary relation
R.
>T is the theory of equivalence relation with infinite many
infinite
classes
>expressed in L.
>M and N are two models of T and M is a sub-structure of N.
It has to be
>proved that N is an elementary extension of M, with 3 steps:
or rather three cases.
>-1. Every equivalence class of N is represented in M.
>-2. All the equivalence classes of N have the same cardinal.
>-3. General case.
>Any help appreciated !
I suspect that this is homework, and the first two cases are
supposed to
indicate a solution. However, here is a solution to the
general case, which
probably wont get you many marks if it is homework.
Show that, given
a) any M a model of T
b) S a substructure of M
c) b, an element from M
d) an |S|^+-saturated model N and
e) an L-embedding f of S into N
we can find an extension of f which is an L-embedding of S cup
a into N.
Thus T has qe, and is model complete so all embeddings are
elementary.
===
Subject: Difficult Analysis Problem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id hBGKVV125504;
may I propose the following problem to you:
Let (f_n) be a sequence of real continuous functions and
x real number such that in every neighbourhood of x,there is
number
in which (f_n) converges,that is
for every d > 0,there is y in (x-d,x+d),such that lim f_n(y)
exist.
Does it follow that lim f_n(x) exist?
In other words :
If lim f_n(x_k) exist for k=1,2,... and lim x_k = x
can we then conclude that lim f_(x) exist?
I dont believe such a conclusion is valid
(if it is please give a proof),but I cant find 
any
counterexample.
I appreciate any suggestion on this.
Martin
===
Subject: Re: Difficult Analysis Problem
>may I propose the following problem to you:
>Let (f_n) be a sequence of real continuous functions and
>x real number such that in every neighbourhood of x,there is
number
>in which (f_n) converges,that is
>for every d > 0,there is y in (x-d,x+d),such that lim f_n(y)
exist.
>Does it follow that lim f_n(x) exist?
No.
>In other words :
>If lim f_n(x_k) exist for k=1,2,... and lim x_k = x
>can we then conclude that lim f_(x) exist?
No.
>I dont believe such a conclusion is valid
>(if it is please give a proof),but I cant find 
any
counterexample.
>I appreciate any suggestion on this.
Hint: Let x = 0. Look for a counterexample f_n with the
property that f_n(t) = 0 for all t with |t| > 1/n.
(Dont try to write down a formula for f_n; instead
give a description as a piecewise-linear function
or some such...)
>Martin
>
************************
David C. Ullrich
===
Subject: Re: Difficult Analysis Problem
> may I propose the following problem to you:
Let (f_n) be a sequence of real continuous functions and
> x real number such that in every neighbourhood of x,there
is number
> in which (f_n) converges,that is
for every d > 0,there is y in (x-d,x+d),such that lim f_n(y)
exist.
Does it follow that lim f_n(x) exist?
In other words :
If lim f_n(x_k) exist for k=1,2,... and lim x_k = x
> can we then conclude that lim f_(x) exist?
I dont believe such a conclusion is valid
> (if it is please give a proof),but I cant 
find any
counterexample.
> I appreciate any suggestion on this.
> Martin
Your conclusion is valid. Heres a proof:
Take any z in R. We will show that lim f_n(z) exists. Let d =
1 if x
equals z and let d = 2 |x-z| otherwise. In either case, z is
in (x-d,
x+d), and d > 0. So lim f_n(z) exists. z was arbitrary, so
lim f_n
exists.
~ Chris
===
Subject: Re: TRUTH, RELIGION, MATH AND EINSTEIN (Was: Re:
Suzy . . .)
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id hBH2Pbg19328;
Yadda yadda yadda. - Me, listening to your conversation.
===
Subject: Combinatorics? Re: Einsteins (??) 5x5 problem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id hBH5jro00340;
This ones for anyone as puzzled about this as I was when I
looked at this
problem- it may provide an insight into combinatorics.
Archimedes is attributed to have written a paper listing
puzzles which shed
light on different ways in which a problem can be solved.
>I could get the solutions German, Dane and Norwegian that
satisfied
>the criteria of this problem.
>Making the usual assumptions about 
Ôfirst and Ôgreen left 
of
white,
>green ends up being an important key. There are only two
places that
>green can meet given conditions- Position 1 and position 4.
>The Swede is ruled out as he owns a dog. IMHO, the Brit
cannot be the
>owner of the fish. I will be more than glad to see a solution
that
>proves otherwise.
>One question bothers me though. Why would Eistein spend time
>formulating this problem?
>
=== Subject: Re: calculus - tan limit question by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBH5jrZ00346; >Find the following limit: >lim tanx - 1 >x->pi/4 ------------- > x - pi/4 >i am stuck on the above problem and despite staring at it and >re-staring at it i am unable to find the answer. any help would be >greatly appreciated. thank you for your time. >scott elisaon Have you tried LHospitals Rule? phil O / ----o o === Subject: Re: calculus - tan limit question > >> >> Find the following limit: >> lim tanx - 1 >> x->pi/4 ------------- >> x - pi/4 >> i am stuck on the above problem and despite staring at it and >> re-staring at it i am unable to find the answer. any help would be >> greatly appreciated. thank you for your time. >> scott elisaon >> Dare one suggest LHopitals rule? >> Heresy! >> How could you use LHopital to calculate that > without already knowing the result? >> -- > Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html > Needless to say, I had the last laugh. > Alan Partridge, _Bouncing Back_ (14 times) >> >> Take the derivative of top and bottom to get sec^2 x/1. Then evaluate >> at pi/4 to get 2. >> >> How do you evaluate the derivative of tan x at x = pi/4 without >> using the stated limit? Use tan(x) = sin(x)/cos(x) with the quotient rule to get 1/cos(x)^2, > then substitute x = pi/4 ? So you know that tan pi/4 = whatever. Now what does that *mean*? (Can you recall the *definition* of derivative?) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: calculus - tan limit question >> >> Find the following limit: >> lim tanx - 1 >> x->pi/4 ------------- >> x - pi/4 >> i am stuck on the above problem and despite staring at it and >> re-staring at it i am unable to find the answer. any help would be >> greatly appreciated. thank you for your time. >> scott elisaon >> Dare one suggest LHopitals rule? >> Heresy! >> How could you use LHopital to calculate that > without already knowing the result? >> -- > Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html > Needless to say, I had the last laugh. > Alan Partridge, _Bouncing Back_ (14 times) >> >> Take the derivative of top and bottom to get sec^2 x/1. Then evaluate >> at pi/4 to get 2. >> >> How do you evaluate the derivative of tan x at x = pi/4 without >> using the stated limit? Use tan(x) = sin(x)/cos(x) with the quotient rule to get 1/cos(x)^2, > then substitute x = pi/4 ? So you know that tan pi/4 = whatever. > Now what does that *mean*? > (Can you recall the *definition* of derivative?) It means that you can apply lHopitals rule without knowing in advance what lim_{x -> pi/4} (tan(x) - 1)/(x - pi/4) is. === Subject: Re: calculus - tan limit question >>Find the following limit: >>lim tanx - 1 >>x->pi/4 ------------- >> x - pi/4 >>i am stuck on the above problem and despite staring at it and >>re-staring at it i am unable to find the answer. any help would be >>greatly appreciated. thank you for your time. >>scott elisaon Have you tried LHospitals Rule? You cant apply LH to this without already knowing the limit! -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: Difficult Analysis Problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBH8kI413049; >may I propose the following problem to you: >Let (f_n) be a sequence of real continuous functions and >x real number such that in every neighbourhood of x,there is number >in which (f_n) converges,that is >for every d > 0,there is y in (x-d,x+d),such that lim f_n(y) exist. >Does it follow that lim f_n(x) exist? >In other words : >If lim f_n(x_k) exist for k=1,2,... and lim x_k = x >can we then conclude that lim f_(x) exist? >I dont believe such a conclusion is valid >(if it is please give a proof),but I cant find any counterexample. >I appreciate any suggestion on this. >Martin Hi Martin, take x = 0 and define the f_n as hat functions: f_n(y) = 0 , if |y|<1/n f_n(y) = n*(ny+1) , if -1/n<=y<=0 f_n(y) = -n*(ny-1), if 0<=y<=1/n. Then f_n(y) converges to 0 for y<>0, but lim (n->oo) f_n(0) = oo. (If you interpret convergence towards +oo as convergence, let the direction of the hat (up or down) alternate dependent on odd or even n.) Best wishes Torsten. === Subject: Re: Difficult Analysis Problem > f_n(y) = 0 , if |y|<1/n > f_n(y) = n*(ny+1) , if -1/n<=y<=0 > f_n(y) = -n*(ny-1), if 0<=y<=1/n. Your definition needs some work. I assume that you inteded to write |y| > 1/n for your first case. === Subject: Re: Difficult Analysis Problem >Hi Martin, >take x = 0 and define the f_n as hat functions: >f_n(y) = 0 , if |y|<1/n > For |y|>1/n naturally >f_n(y) = n*(ny+1) , if -1/n<=y<=0 >f_n(y) = -n*(ny-1), if 0<=y<=1/n. >Then f_n(y) converges to 0 for y<>0, but lim (n->oo) f_n(0) = oo. >(If you interpret convergence towards +oo as convergence, >let the direction of the hat (up or down) alternate dependent >on odd or even n.) > Even wilder: let (q_n:n in N) enumerate the rationals and for each n let f_n be the piecewise linear function that is zero outside the interval [-1/n,1/n] and q_n at 0. Now the sequence of f_n(0) is all over the place. KP === Subject: source code for pseudoinverse in C by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBHDtnW02728; Hi I am looking for a source code for pseudo inverse written in C. Can anyone help me with this? thanking you in advance Kumari >Im developing a statistical application based on the solution of the >least squares problem. >Im writing the source code in C language, and my compiler is Digital >Mars Compiler version 8.35n . >The environment of the final version of the application is still not >sure, so my aim is to develop some portable code: the OS will be Win32 >or Dos (still not sure which version), with the largest memory model >available, and the architecture will be Intel based, probably a >Pentium 2. >The application is more or less complete, now, with the core >statistical routines working in two alternative way: >1) the first version uses a naive definition of the pseudoinverse >matrix : >(A*A)^-1 * A >which has obvious problems of precision and robustness to >close-to-singular matrix. Anyway it works fine in many cases and its >pretty easy to use, letting me easily write all the code by myself and >compile it in the environment well choose for the final version. >2)the second version uses the Matlab C Library function pinv(..), >which is based on the singular value decomposition of the matrix, >which is precise enough even in case of rank deficiency, but is much >slower since the computation steps are more complex. The main problem >of this second way is the fact that Matlab C routines comes in >precompiled libraries, hence I am able to use them only under some >combination of the environment variable (i.e. compiler, memory >model, OS,..) and this fact precludes to some of the environment we >are considering (DOS, for example). >First Question: do other version of the Matlab libraries exist, apart >from those included in Matlab (R12)? >Ive considered the idea of writing my own procedure of singular value >decomposition and started studying text like Numerical Recipes or >Golubs Matrix Computation, but it seems to me their approach is too >complex for me, since my typical data situation does not require to >stress too much on the side of time and memory resources consume. >Ive been told on this group that Matlab core routines (SVD included) >are based on a set of Fortran routines called Lapack, and that these >routines have been traslated in C, under the name of CLapack. >Ive found this stuff at netlib.org, including the dgelss.c source >code and some .tar file like Atlas3.4.2 , Atlas330_winnt_p4.., >Clapack3-windows or CLAw32 , but, honestly , I dont have a clue of >what to do with them. >The first line of the dgells.c routine are >#include blaswrap.h >/* -- translated by f2c (version 19990503). > You must link the resulting object file with the libraries: > -lf2c -lm (in that order) >*/ >#include f2c.h >which seem to suggest that I need at least two header files and two >library file. >Would that be correct to just put these headers and libraries in my >work directory, add the link command to my make file, call the >CLapack routine in my code routine and expect it to work? >and if this is correct where do I found these file? netlib? >Otherwise if this is not enough, what are the correct steps? >If you are so kind to help me, please keep in mind that this is my >first serious C software, that ive been compiling the code so far by >commnd prompt and im not confident with such concepts like projects, >dependencies... >Moreover now Im not concerned with the efficiency of the algorithm, >so even using some existing library instead of building my own with >Atlas is definitely enough (especially since I would not have a clue >about how to do it). >badly-expressed i apologize, feel free to point out what Im >mistaking. >Filippo Venturi >Universit.88 di Bologna === Subject: Re: 1+1=3 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBHDtnK02732; >I remember many years at school, a math teacher proving this. >Has anyone got a math example of how this is possible? No, no one has a math example of how 1+ 1= 3 because it isnt true. There are a number of different ways of writing an INVALID proof mostly involving dividing by 0. === Subject: Re: 1+1=3 > >>I remember many years at school, a math teacher proving this. >>Has anyone got a math example of how this is possible? > No, no one has a math example of how 1+ 1= 3 because it isnt true. There are a number of different ways of writing an INVALID proof mostly involving dividing by 0. > Heres a field with 5 elements, where 1 + 1 = 3. + | 0 1 2 3 4 ---+----------- 0 | 0 1 2 3 4 1 | 1 3 0 4 2 2 | 2 0 4 1 3 3 | 3 4 1 2 0 4 | 4 2 3 0 1 * | 0 1 2 3 4 ---+----------- 0 | 0 0 0 0 0 1 | 0 1 2 3 4 2 | 0 2 1 4 3 3 | 0 3 4 2 1 4 | 0 4 3 1 2 === Subject: Re: Basic ring theory (query reposted) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBHFACP08891; I am reposting in order to move my query to the top of the queue and in the hope of getting a reply. Clio >I am currently reading An introduction to Ring Theory by PM Cohn >(Springer) in which appears the following question: >Show that in any ring R, the sum of two ideals A and B, defined as > A+B = {a+b|a in A, b in B} is again an ideal, as is > the product AB = {sigma over i of a_i.b_i|a_i in A, b_i in B}. >If further A + B = R, show that AB = A intersection B. >I am puzzled by the solution outlined for the last part. >It easy to show that AB is a subset of A intersection B. >To show that A intersection B is a subset of AB, >I follow the hint given in the text and get the following: >Since A + B = R, we can find a in A and B in B such that a + b =1 >and so if c is in A intersection B then c = (a+b)c(a+b) >= aca + bca + acb + bcb. >Now we consider each term in this sum. >aca: a is in A; ca is in B since c is in B and B is an ideal. >Similarly for acb and bcb >But how do we show that the term bca belongs to AB ? >I can see that, since bc is in A, bca is in AA ; >and also that since ca is in B, bca is also in BB. >But how do we get it into AB? Any suggestions? >Clio === Subject: Re: Reposting (was: Basic ring theory (query reposted)) > I am reposting in order to move my query to the top of the queue > and in the hope of getting a reply. This does not make much sense. Where your post appears within a thread depends on the particular newsreader someone uses. Marc === Subject: Re: JSH: Operator ambiguity by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBHI3uc22980; Are the solutions to (1)^(1/3) and x^(1/3) the complex 3rd roots of unity? Kathy >> One of those hard lessons to learn in mathematics is operator >> ambiguity. >> >> For instance the square root and cuberoot operators are ambiguous, and >> theres nothing you can do about it. >> >> Given (1)^{1/3} there are *three* solutions, and not one, which is the >> operator ambiguity that gave me fits for a while back last year when I >> posted and posted trying to find some trick around it. >> >> The square root operator has ambiguity in that it gives *two* >> solutions, even if you only want one. >> >> I managed to get myself in trouble yet again today trying yet again to >> escape operator ambiguity by making an earlier post trying to go with >> the sign convention of taking the positive solution of the square >> root operator. >> >> That doesnt work. >> >> It bothers me that I keep fighting operator ambiguity and trying to >> find ways around it, as if some part of me just cant accept that if >> you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which >> refuse to go away, no matter how hard you wish. >> >> >> James Harris >You say that sqrt(x) has two solutions, but a function doesnt have >solutions, only an equation. sqrt(x) is a function, and as such, has >only values. >The equation x^2 = 5 has two solutions, namely sqrt(5) and -sqrt(5), but >thats just a result of solving the equation. Its not inherent in the >functions which make up the solution. >For example, the equation 2x = 2sqrt(5) has exactly one solution, namely >sqrt(5). >In the same way, while x^3 = 1 has three solutions, 1^y for any y is >exactly 1. === Subject: Re: Maximal Ideals by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBHJTAY29230; Im rewriting the question in a more detailed way hoping for some help. >I request help with completing the below proof. >If R is the ring of all real valued continuous functions on the closed interval, [0,1] and M is a maximal ideal of R, prove that there exists some p in [0,1] such that M={ all f(x) in R/ f(p)=0} > I can prove it assuming that there is atleast one function in M that >has only a finite number of zeros. But how might I prove that there is atleast one function with a finite number of zeros? { If such a function did exist, then number the zeros of that function as z1,z2,....zn. If M were not of the specified form, there exist functions f1,f2,....fn such that fi(zi) is non zero. Then h := f1^2 + f2^2 + ......+ fn^2 + f^2 belongs to the ideal and is non-zero everywhere on [0,1]. So 1/h and hence 1 are in M implying M=R, not a maximal ideal. } > RK. === Subject: Large Exponents by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBHMXv911378; Maybe there is a simple solution to this that I have over looked. I have an equation that I am trying to solve for x 5^1000000 - 2^2321928 = 3^x Initially I tried using log() to help solve this but that didnt seem to get me anywhere. x = log( 5^1000000 - 2^2321928 ) / log( 3 ) Most systems will not be able to evaluate the contents of the first log but I know x itself should not be very large comparably. Is there a way to do this that will yield an accurate result? === Subject: Re: Large Exponents In sci.math, Drake over looked. I have an equation that I am trying to solve for x > 5^1000000 - 2^2321928 = 3^x > Initially I tried using log() to help solve this but that didnt > seem to get me anywhere. > x = log( 5^1000000 - 2^2321928 ) / log( 3 ) x = log( exp(1000000 * log(5)) - exp(2321928 * log(2)) ) / log( 3 ) In this form the expression is almost trivial, and yields x = 1464971.013661600309756169721 in GP/Pari. Admittedly, I would hope for a more exact solution, but considering the size of the numbers it may be the best we can do. The power of 2 is fairly simple to code on a binary computer, yielding a number with 72561 32-bit unsigned ints. In fact, its probably not even worth bothering; one can subtract in place. The power of 5, on the other hand, took some work. Its not difficult to multiply two std::vector in C using unsigned long long, just tricky -- but its clear that I may need to ask Santa for a faster computer this Christmas... :-) > Most systems will not be able to evaluate the contents of the > first log but I know x itself should not be very large comparably. > Is there a way to do this that will yield an accurate result? -- #191, ewill3@earthlink.net Its still legal to go .sigless. === Subject: Re: Large Exponents > Maybe there is a simple solution to this that I have over looked. I have an equation that I am trying to solve for x 5^1000000 - 2^2321928 = 3^x Initially I tried using log() to help solve this but that didnt seem to get me anywhere. x = log( 5^1000000 - 2^2321928 ) / log( 3 ) Most systems will not be able to evaluate the contents of the first log but I know x itself should not be very large comparably. Is there a way to do this that will yield an accurate result? Mathematica doesnt complain and gives 1464971.0136616003097561697216090621 as a rough estimate. === Subject: Re: Large Exponents Maybe there is a simple solution to this that I have over looked. I have an equation that I am trying to solve for x 5^1000000 - 2^2321928 = 3^x Initially I tried using log() to help solve this but that didnt seem to get me anywhere. x = log( 5^1000000 - 2^2321928 ) / log( 3 ) Most systems will not be able to evaluate the contents of the first log but I know x itself should not be very large comparably. Is there a way to do this that will yield an accurate result? 3^x = 5^1000000 - 2^2321928 3^x = exp(1000000*log(5)) - exp(2321928*log(2)) 3^x = exp(2321928*log(2)) * [exp(1000000*log(5) - 2321928*log(2)) - 1] 3^x = exp(2321928*log(2)) * [exp(0.06577090756036) - 1] 3^x = exp(2321928*log(2)) * 0.06798202253119 x = [2321928*log(2) + log(0.06798202253119)] / log(3) x = 1.464971013661599e+006 Jeroen === Subject: Re: Large Exponents Drake a .8ecrit : > Maybe there is a simple solution to this that I have over looked. I have an equation that I am trying to solve for x 5^1000000 - 2^2321928 = 3^x Initially I tried using log() to help solve this but that didnt seem to get me anywhere. x = log( 5^1000000 - 2^2321928 ) / log( 3 ) Most systems will not be able to evaluate the contents of the first log but I know x itself should not be very large comparably. Is there a way to do this that will yield an accurate result? > x is supposed to be integer ? === Subject: Re: integer -> prime mapping by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBI5xv912747; >Does anyone know if there is an efficient one-to-one mapping function >from a subset of integers (say Z_n) to primes? The closest and simplest which will contain primes but will also contain all prime products =>7^2. It automatically omits all evens and 0 (mod 3)s and integers ending in 5. Just start by adding to prime 7 --- [+4,+2,+4,+2,+4,+6,+2,+6] and repeat this sequence summation --->oo. This is the prime footprint. The problem is the product (composites) of previous primes (prime factors) =>7 will also be in this footprint. No primes will fall outside of this summation except 2,3 and 5. Now your sieve operation only has to look at about 26.67% of all the integers up to x to sieve out the primes >7. :-) Dan === Subject: Re: Computers as a tool in foundations research |Goodness. I realize that this is cross-posted to sci.logic, so |maybe Im about to break a taboo local to a group I dont know, |but: who is it who believes that philosophy is a scientific |discipline? When an objectivism newsgroup (devoted to the philosophy of Ayn Rand) was to be created, one poster argued vigorously that it should be named sci.objectivism. ;-) More likely you would find people who feel that the foundations of mathematics belongs to the humanities. In one university library I had trouble finding the Journal of Symbolic Logic until I discovered it was with the humanities journals. Keith Ramsay === Subject: Re: Computers as a tool in foundations research by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBGGPMi05391; >I suspect that the foundations of mathematics is perhaps the only remaining >major scientific discipline in which computers do not play a fundamental >role in research. I am posting this in part to find out if I am mistaken. >The impression I get is that mathematicians working in this area believe >that intuition about large cardinals is the most powerful way to extend >foundations and computer models are not relevant. You seem to assume that basic research in mathematics mainly involves set theory. This is of course not true. Algebra is a counter example. In algebra computers are used in research. >No matter how pretentious the cardinals a formal system claims to deal with, >it is still a computer program for enumerating theorems and computer >simulations are, at least in theory, relevant to understanding its >structure. I have long suspected that directly attacking the combinatorial >content of formal systems will ultimately lead to far more powerful ways of >extending mathematics than large cardinal axioms. This is true precisely >because the combinatorial content is something you can do computer >experiments on to test your intuition. The result of course will be systems >far more complex and less transparent than existing mathematics. That is the >inevitable price to be paid for more powerful systems. A comment on this: science in general and mathematics in particular is done for at least two reasons: to gain insight, and to develop tools for manipulation/problem solving. I agree with your statement in the context of the second reason. But in the context of the first reason any approach that is not transparent is useless. We humans want to understand things. This means that we try to create a possibly complex representation of the topic under consideration within our brain. Simulations may help here - but the real work is not done through simulation. An example: the theory of finite groups is still existing, although nowadays with a quick computer you can easily calculate all finite groups of order lower than a given number n. The reason is that creating long lists of groups does not give insight into how groups are structured. The Sylow theorems for example give such an insight. And all that is finite mathematics - no large cardinals! Of course you can built sophisticated computer algebra systems that are capable of automatically proving theorems. You can automatically send them to a math journal and get them published automatically. But whats the use of all that? If no human being ever reads the stuff, enjoys its elegance, or discovers new connections or possibilities of application? >Large cardinal axioms extend the combinatorial power of mathematics in an >indirect way so that one does not have to directly deal with the immense >combinatorial complexity implicit in those axioms. But only through directly >dealing with that complexity can we know what directions for extending >foundations make sense and which do not. Mathematical intuition like >scientific intuition easily turns into philsophical speculations without the >guiding hand of experiments. In which way could a computer simulation ever help to decide, whether a certain theory makes sense or not? In mathematics there is no reality to compare with. Mathematicians create structures and study them, be it because they like the structure or because they are motivated by facts from biology, physics or whatever.The latter is not better than the first as a lot of examples show: one of the modern famous examples is non-commutative algebraic geometry. It was >invented< by >pure mathematicians< but nowadays it is an important tool in physics. H === Subject: Re: Complex Numbers - Best Method? By computing z_1 + z_2 + z_3, or otherwise, determine the > set for which the statement z_1^3 = z_2^3 = z_3^3 is false. a) z_1 = -sqrt3 + i, z_2 = sqrt3 + i, z_3 = -2i > b) z_1 = 4, z_2 = -2 +sqrt12i, z_3 = -2 -sqrt12i > c) z_1 = sqrt27 + 3i, z_2 = -sqrt27 - 3i, z_3 = 6i > d) z_1 = -sqrt12i, z_2 = -3 +sqrt3i, z_3 = 3 +sqrt3i What is the best method of solving this problem? There must be > a better method than laboriously cubing and comparing them all...? Each set has distinct z_i, so if the statement is true of a set then its elements comprise all three roots of z^3 - z_1^3 = 0, so they must sum to zero (= -coef of z^2). So, contrapositively, if their sum is nonzero then the statement is false for the set, which happens for one and only one of the sets specified above. -Bill Dubuque === Subject: Re: Functions with compact support > Hello everybody, does anybody know about properties concerning the zeros of the > derivatives of smmoth symmetric functions f(t)=f(1-t) with compact > support? I was wondering if he zeros of the derivatives of a function > of the form f(t)=exp(-1/(t*(1-t))), t in (0,1) > Thomas The derivative is going to be a rational function in t times an > exponential and therefore cannot have more than finitely many. This is right, nevertheless if one goes up to infinite many derivatives (the function is smooth on (0,1)) one formally obtains an infinite set of rational functions. This might imply a dense set of zeros in (0,1). But Im not really sure about that. === Subject: Re: Functions with compact support > The derivative is going to be a rational function in t times an > exponential and therefore cannot have more than finitely many. I believe hes looking at D1 U D2 U ..., where Dn is the zero set of the nth derivative. === Subject: Largest ball in a convex polyhedron Hi Im trying to find the largest ball I can fit into a convex polyhedron, represented as the intersection of a number of halfspaces. I guess that the center of mass of the polyhedron might be used as the coordinates of the center of this ball, and that the radius is then simply found by minimizing distance to the halfplanes defining the polyhedron. However, as this is merely a guess at a method, I would be most grateful if someone could state whether it is correct, and perhaps provide a pointer to some theorem. Alternatively, if someone knows a simpler method I would love to hear about it. Sren Nielsen === Subject: Re: Largest ball in a convex polyhedron Sren Holbech > Im trying to find the largest ball I can fit into a convex polyhedron, > represented as the intersection of a number of halfspaces. I guess that > the center of mass of the polyhedron might be used as the coordinates of > the center of this ball, and that the radius is then simply found by > minimizing distance to the halfplanes defining the polyhedron. That cannot be the whole answer. If we start with a polyhedron and its maximal sphere, we can snip away a small amount of the polyhedron, at one of its vertices, by adding one more halfspace; the center of mass of the polyhedron will change, but the sphere will stay the same. The centre of the sphere will be equidistant from at least four of the planes. I suppose one could calculate that distance for each four-element set of planes, and then just take the biggest. LH === Subject: Re: Largest ball in a convex polyhedron 3QLpj-NoP*NzsIC,boYU]bQ]Hy<#4ga3$21: > The centre of the sphere will be equidistant from at least four of the > planes. I suppose one could calculate that distance for each four-element > set of planes, and then just take the biggest. This is a linear program: find (x,y,z,d) such that, for each plane P, the distance from (x,y,z) to the plane is at most d (two linear constraints), maximizing d (linear objective function). Since it is a linear program with a constant number of variables, it can be solved in time linear in the number of planes (your suggestion would instead take O(n^4). -- David Eppstein http://www.ics.uci.edu/~eppstein/ Univ. of California, Irvine, School of Information & Computer Science === Subject: Re: Largest ball in a convex polyhedron >Hi >Im trying to find the largest ball I can fit into a convex polyhedron, >represented as the intersection of a number of halfspaces. I guess that >the center of mass of the polyhedron might be used as the coordinates of >the center of this ball, and that the radius is then simply found by >minimizing distance to the halfplanes defining the polyhedron. >However, as this is merely a guess at a method, I would be most grateful >if someone could state whether it is correct, and perhaps provide a >pointer to some theorem. Your method is certainly not correct. (For example imagine a polyhedron that looks more or less like a very long thin cone; the largest ball is going to touch the base, while the center of mass is far from the base.) >Alternatively, if someone knows a simpler method I would love to hear >about it. I doubt that there exists a simple method to give the exact answer. >Sren Nielsen ************************ David C. Ullrich === Subject: The noncompact Jordan separation theorem Every properly embedded in R^n connected manifold of dimension n-1 separates R^n into two open connected sets. (properly embedded = embedded as a closed subset) If the manifold is compact, we obtain the Jordan separation theorem. The general case may be handled by Alexander duality passing to the one-point compactification . On the other hand, for an elementary exposition it is not a good idea to refer to Alexander duality. My question is: Does anybody know some elementary exposition of this topic, without too much algebraic topology? Maybe there is some obvious modification of the index theory for noncompact hypersurfaces... Simeon === Subject: Aliens stole the 0 Oh no ! We cant use math anymore on earth since aliens abducted the 0. Everytime someone mentions the 0 , a photon torpedo from outer space will hit his head ! Ouch ! Ouch ! OUUUCHHH ! === Subject: Re: Aliens stole the 0 In sci.math, Adam Ben Nalois 0. > Everytime someone mentions the 0 , a photon torpedo from outer space > will hit his head ! > Ouch ! Ouch ! OUUUCHHH ! Oooooookaaaaaaaaay..... -- #191, ewill3@earthlink.net -- insert random masochism here Its still legal to go .sigless. === Subject: Re: Topologies implied by limits of sets === Subject: Re: Topologies implied by limits of sets >Let X be a set. Consider the following three topologies on P(X). >U1: where set-theoretic limits coincide with topological limits. >U2: that induced by all functions f in [0,1]^P(X) such that >lim f(Aj) = f(lim Aj) for all sequences (nets?) A such that lim Aj >exists. >U3: that coinduced by all functions f in P(X)^I such that >lim f(Aj) = f(lim Aj) for all set-convergent nets A indexed by I, >where I is one of N U {infty}, [0,1], or R. >The question on the table: Is U1 the same as any of the others? Let X be {a,b}. The U1 topology for P(X) is discrete. For the same X, what are the U2 and U3 topologies for P(X)? Set convergent sequences of P({a,b}) are the eventually constant sequences which is consist with P({a,b}) having discrete topology. What are the limit preserving maps of [0,1]^P(X) and P(X)^I ? Why I? Why not a long line with cardinality P(X) ? -- set limit topology of P(S) For A subset P(S), let cl A = { a in P(S) | some a1,a2,... in A with lim aj = a } cl nulset = nulset; A subset cl A A subset B ==> cl A subset cl B Does cl cl A = cl A ? If not, then take closure A = lim cl^j A = ... cl cl A cl {a} = {a}, ie P(S) is T1. Thus if S is finite, P(S) is discrete. ---- === Subject: Re: Topologies implied by limits of sets by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBGBdUb17042; >Let X be an arbitrary set, and let I be one of R[0,1], R, or N U >{infty} under the usual topology. Let F = {f in P(X)^I: f(j) -> >f(i) as j -> i for all i in I}. Is there a simple way to describe >the topology on P(X) (or a base or subbase for this topology) >coinduced by F? >I suspect that the answer is more interesting when X is infinite, and >that the answer probably varies with the cardinality of X (at least >countable vs. uncountable). >[Since there was confusion about limits of sets in a recent thread, I >repeat the intended definition here: For example, >> Let I indicate the intersection operator. Given a sequence S of >> sets, by definition >> lim inf S = U{I{S_m: m >= n}: n in N}, and >> lim sup S = I{U{S_m: m >= n}: n in N}. >> Iff the limits inferior and superior are equal, this set is by >> definition the limit of S. >> One can generalize this definition to nets of sets. >P(X) denotes the power set of X.] >-- >Stephen J. Herschkorn herschko@rutcor.rutgers.edu see also: Re: Convergence on a space with no topology for a somewhat similar discussion. C. Dement === Subject: Re: JSH: Operator ambiguity > : For instance the square root and cuberoot operators are ambiguous, and > : theres nothing you can do about it. Pray tell, whats your definition of operator? The sqrt(c) *function* is defined to be the positive real number whose > square is c. There is no *solution* here because this is not an > *equation*. There is no ambiguity. The *equation* x^2-c=0 has *two solutions*, one is sqrt(c) and the other > is -sqrt(c). How do *you* define operator? James, sometimes people denote z^(1/2) as the set of solutions of w^2=z. Where the principal value of z^(1/2) is denoted as sqrt(z). If z is positive (1 for example), the principal value of 1^(1/2) denoted as sqrt(1) equals 1. The other value of 1^(1/2) equals -1. Wilbert === Subject: Re: JSH: Operator ambiguity > > > > [1] Indeed its the right lane. > > Except where the left lane is the right lane. > > But suppose you dont drive in the left lane: > then right is left, right? I think not. I suppose that when you left right, you now are left, right? And when you left left, you are now right, right? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Operator ambiguity > > > [1] Indeed its the right lane. > > Except where the left lane is the right lane. > > But suppose you dont drive in the left lane: > > then right is left, right? I think not. I suppose that when you left right, you now are left, right? > And when you left left, you are now right, right? But if you have to choose a lane from the two, and you choose left, then right is left, while if you choose right, left is left. If you choose wrong, it is ambiguous. Gib === Subject: Re: JSH: Operator ambiguity > It bothers me that I keep fighting operator ambiguity and trying to > find ways around it, as if some part of me just cant accept that if > you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which > refuse to go away, no matter how hard you wish. |x| = sqrt(x^2) === Subject: The Linus sequence algorithm? I dont get it. The sequence composed of 1s and 2s obtained by starting with the number 1, and picking subsequent elements to avoid repeating the longest possible substring. http://mathworld.wolfram.com/LinusSequence.html http://www.research.att.com/cgi-bin/access.cgi/as/njas/ sequences/eisA.cgi?An um=A006345 I try to evaluate it but I cant seem to replicate the sequence they give. For example, I dont see how one could ever put 3 in a row. Is there ever a choice? i.e. is it deterministic, could one reach a postion where one could put either a 1 or 2? What is a systematic algorithm for producing the sequence? Must one do exhaustive search everytime? Is there a recurrence? Mitch. === Subject: Re: The Linus sequence algorithm? > I dont get it. The sequence composed of 1s and 2s obtained by starting with the number > 1, and picking subsequent elements to avoid repeating the longest > possible substring. http://mathworld.wolfram.com/LinusSequence.html > http://www.research.att.com/cgi-bin/access.cgi/as/njas/ sequences/eisA.cgi?Anu m=A006345 I try to evaluate it but I cant seem to replicate the sequence they > give. For example, I dont see how one could ever put 3 in a row. It looks like that because you follow a simple rule and cant plan ahead, sometimes youre forced to do something which will hurt you later. > Is there ever a choice? i.e. is it deterministic, could one reach > a postion where one could put either a 1 or 2? I couldnt get off the ground at all, due to this choice! After 2 terms, you have <1,2>, and there are 3 substrings, <1>, <2>, and <1,2>. Appending a 2, forming <2>, <2,2>, and <1,2,2>, avoids repeating <1> and <1,2>. Appending a 1, forming <1>, <2,1>, and <1,2,1>, avoids repeating <2> and <1,2>. I cant see why the former satisfies avoid repeating the longest possible substring better than the latter. > What is a systematic algorithm for producing the sequence? Must one do > exhaustive search everytime? Is there a recurrence? I think I need a clearer definition before looking into an algorithm. Phil -- Unpatched IE vulnerability: window.open search injection Description: cross-domain scripting, cookie/data/identity theft, command execution Reference: http://safecenter.net/liudieyu/WsFakeSrc/WsFakeSrc-Content.HTM Exploit: http://safecenter.net/liudieyu/WsFakeSrc/WsFakeSrc-MyPage.htm === Subject: Re: The Linus sequence algorithm? |> What is a systematic algorithm for producing the sequence? Must one do |> exhaustive search everytime? Is there a recurrence? | |I think I need a clearer definition before looking into an algorithm. Yes, I think the description both on Math World and in the entry in Sloans collection of sequences could be better stated. It helps a bit that they give terms of a second sequence, the Sally sequence, giving the length of the subsequence whose repetition is being avoided. Start with 1. Given the terms a1,...,an, for each of a_{n+1}=1,2, find the longest substring s with the property that the string a1,...,a_{n+1} ends with ss. (If nothing longer works, the empty string is the longest.) Choose the next term to be whichever gives you the shorter such substring. The first term is 1. The second is 2, because 12 has s= while 11 has s=1. The third is 1, because 121 has s= while 122 has s=2. The fourth is 1, because 1211 has s=1 while 1212 has s=12. The fifth is 2, because 12112 has s= while 12111 has s=1. The sixth is 2, because 121122 has s=2 while 121121 has s=121. The corresponding terms of the Sally sequence are 0,1,1,2,1,3. terms of both sequences given in Sloan. Keith Ramsay === Subject: Re: The Linus sequence algorithm? > the entry in Sloans collection of > sequences could be better stated. The wonderful site http://www.research.att.com/~njas/sequences/ is Maintained by N. J. A. Sloane as you can see there. Lets give him a big hand for his fine work, which surely takes him many hours a day. At least let us spell his nam correctly :-) Rainer Rosenthal r.rosenthal@web.de === Subject: Re: Peano arithmetic and multiplication >>1. 0 in N (zero is a natural number) >>2. x in N -> s(x) in N (the successor of any # is also a #) >> these first two just construct the naturals >>3. 0 != s(x) (0 is the smallest #) >>4. s(x) = s(y) -> x = y (successor is a one-to-one function) >>5. [P(0) and forall x, P(x) -> P(s(x)) ] -> forall x, P(x) > |OK, after some reading (just the questionable web search), it seems that > | what I stated above (rules 1 through 5) are definitely what is > |traditionally called the Peano Axioms, but that by themselves they are > |not enough to express arithmetic. Beware vague phrases like express arithmetic. Yes, I am asking just these questions specifically to clear up the vagueness of such phrases for myself (I suppose I am using them in a parotting fashion, hoping that they fit the right syntax and usage that others who know better would expect). > Its really important in this setting to distinguish between formal systems > and other axiom systems. I have been thinking about this statement for a while? An axiom system is not formal? There are some axioms systems that are formal and some that are not? What is formal? What is a formal system that is -not- axiomatic (involves the manipulation of axioms with inference rules)? > The above axiom 5 refers to arbitrary predicates P > on the domain. No rules for deducing consequences of such an axiom are > presented. Thats because (while there are popular sets of rules for such > deductions) theres no canonical such set of rules. Its not a formal > system. OK, so its not a formal system until you add logical operators and rules of inference. By canonical I suppose you mean universally accepted. Cant we take a popular vote of say first order logic (FOL)? (I realize I may be confounding things here. I do know that validity of FOL statements is recursively enumerable but not recursive. I just cant seem to strighten out where FOL lies with respect to Peano Arithmetic.) So what would be a generally accepted logic/rules of inference to use with PA? So would you consider these Peano axioms along with a logic a formal system? > It *is* enough to express arithmetic in a certain sense.... The rest of your post was very helpful and illuminating, but also raised more questions which I will save for a later date. Mitch Harris === Subject: Reimann Geometry I am a student of Mathematics and Astrophysics and am investigating the non - Eucldian Geometries as a part of my BSc Project. Please spare some time to tell me more about Reimann Geometry and about its general signifiacnce. === Subject: Re: Reimann Geometry > I am a student of Mathematics and Astrophysics and am investigating > the non - Eucldian Geometries as a part of my BSc Project. > Please spare some time to tell me more about Reimann Geometry and > about its general signifiacnce. Google on Riemannian geometry. === Subject: Comparative sigma-fields In general topology, one sometimes compares different topologies on the same set. In particular, I am thinking of the various topologies on function spaces, but there are several toplogies (mainly for examples, it seems) even on the reals (e.g., the Sorgenfrey line). In what I have seen of analysis (not much beyond first-year graduate study), it seems that one compares measures on the same measurable space (e.g., in the context of absolute continuity), but I have rarely seen topics where one compares different sigma-fields on the same set. The only example I can think of is conditional expectation in probability, which is defined with respect to a sub-sigma-field of the underlying probability space. Are there any interesting and/or important examples where one compares implication of different sigma-fields on the same set? I suspect one of the issues here is that with unrelated sigma-fields, one cannot compare measures. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Comparative sigma-fields > In general topology, one sometimes compares different topologies on the > same set. In particular, I am thinking of the various topologies on > function spaces, but there are several toplogies (mainly for examples, > it seems) even on the reals (e.g., the Sorgenfrey line). In what I have seen of analysis (not much beyond first-year graduate > study), it seems that one compares measures on the same measurable space > (e.g., in the context of absolute continuity), but I have rarely seen > topics where one compares different sigma-fields on the same set. The > only example I can think of is conditional expectation in probability, > which is defined with respect to a sub-sigma-field of the underlying > probability space. Are there any interesting and/or important examples where one compares > implication of different sigma-fields on the same set? I suspect one of > the issues here is that with unrelated sigma-fields, one cannot compare > measures. Heres one (Blackwells theorem): Let (Omega, F) be a countably generated measurable space with the property that for each F-measurable map f: Omega -> R, and each B in F, the image f(B) is an analytic subset of R. Let G and H be sub sigma fields of F, H assumed to be countably generated. Then G is contained in H if and only if each G-atom is a union of H-atoms. [The G-atom (for example) containing x (a point of Omega) is the intersection of all sets in G that contain x.] Corollary: A random variable X (aka, an F-measurable function from Omega to R) is H-measurable if and only if X is constant on each H-atom. -- A. === Subject: Re: Prime Numbers expansion canals > does it? For the clear horizontal channel starting at 6, I got 6 21 44 75 114 161 216 279 350 429 516 611 714 825 944 1071 1206 > 1349 1500 1659 1826 2001 2184 which alternates between even and odd numbers and can be generated > by 4x^2 + 11x + 6 or (x+2)(4x+3) Is there a way to tell from this that all the results will be > composite? Is there a way of telling that (x+2)(4x+3) will always be composite? If only there was some way of factoring it... If you want to know what the lines on Ulams Spiral correspond to, then why not just run Dario Alperns java implementation? Phil -- Unpatched IE vulnerability: Web Archive buffer overßow Description: Possible automated code execution. Reference: http://msgs.securepoint.com/cgi-bin/get/bugtraq0303/107.html === Subject: Re: Prime Numbers expansion canals > does it? For the clear horizontal channel starting at 6, I got 6 21 44 75 114 161 216 279 350 429 516 611 714 825 944 1071 1206 > 1349 1500 1659 1826 2001 2184 which alternates between even and odd numbers and can be generated > by 4x^2 + 11x + 6 or (x+2)(4x+3) Is there a way to tell from this that all the results will be > composite? Is there a way of telling that (x+2)(4x+3) will always be composite? Im not a mathematician, so I ask dumb questions. Its how I learn. It factors to two terms, neither of which is 1, so the result will always be composite. But the polynomial 4n^2 + 12n + 7 which generates the diagonal 7 23 47 79 119 167 223 287 doesnt factor, and thus contains primes. But not factoring is neccessary but not sufficient to get primes as shown below: fn = 4*n**2 + 12*n + 7 n: 0 f(n): 7 prime n: 1 f(n): 23 prime n: 2 f(n): 47 prime n: 3 f(n): 79 prime n: 4 f(n): 119 composite n: 5 f(n): 167 prime n: 6 f(n): 223 prime n: 7 f(n): 287 composite n: 8 f(n): 359 prime n: 9 f(n): 439 prime n: 10 f(n): 527 composite n: 11 f(n): 623 composite n: 12 f(n): 727 prime n: 13 f(n): 839 prime n: 14 f(n): 959 composite n: 15 f(n): 1087 prime n: 16 f(n): 1223 prime n: 17 f(n): 1367 prime n: 18 f(n): 1519 composite n: 19 f(n): 1679 composite Have I got that right? > If only there was some way of factoring it... > If you want to know what the lines on Ulams Spiral correspond to, > then why not just run Dario Alperns java implementation? > Phil === Subject: Re: Prime Numbers expansion canals Well, about the html code: here it works fine with both Mozilla 1.x and IE , but Ill try to fix this netscape issue. about the numbers: I forgot to mention that all the constructions/canals resist to an even-compression of the spiral. It was first thing I tried. Also Ive shown the numbers values to quite few people, and they couldnt tell at glance what rule there was behind. Unfortunately Ive lost somewhere the document with the values, Im searching for it so I can add downloadable file of the values. eheh, I apologize for the typo, but it was late and Im not > English :). I looked around and I found hes mainly > recognized officially as mathematician, so I assumed that > statement for good. Let me know if you think its ok:) a few minor items to fix: Researchs, simmetry, regural, > matematical, develope, expecially. Another webpage problem -- The html code for your pictures is > broken. When I use Netscape 4.8 to view the page, all the > .gifs appear at the beginning, before any text, rather than > inline near their captions. Perhaps replace (eg) >



Re the content -- although I dont know precisely what you refer > to in logarithmic expansion of the canals where prime numbers > never fall, I think most empty channels are easily explained > by noting that all the numbers in them are even. Perhaps you > refer to some deeper property, but for major diagonals, > continuing evenness is forced after an even beginning because > the jth complete turn of the spiral adds 8j+4 cells, ie, always > adds an even number of cells. > -jiw === Subject: Re: Prime Numbers expansion canals by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBI5xxQ12825; >> >> eheh, I apologize for the typo, but it was late and Im not >> English :). I looked around and I found hes mainly >> recognized officially as mathematician, so I assumed that >> statement for good. Let me know if you think its ok:) >> >> a few minor items to fix: Researchs, simmetry, regural, >> matematical, develope, expecially. >> >> Another webpage problem -- The html code for your pictures is >> broken. When I use Netscape 4.8 to view the page, all the >> .gifs appear at the beginning, before any text, rather than >> inline near their captions. Perhaps replace (eg) >>
src=spiral_clean.gif
by >>

>> Re the content -- although I dont know precisely what you refer >> to in logarithmic expansion of the canals where prime numbers >> never fall, I think most empty channels are easily explained >> by noting that all the numbers in them are even. Perhaps you >> refer to some deeper property, but for major diagonals, >> continuing evenness is forced after an even beginning because >> the jth complete turn of the spiral adds 8j+4 cells, ie, always >> adds an even number of cells. >> -jiw >But that doesnt apply to the horizontal and vertical channels >does it? For the clear horizontal channel starting at 6, I got >6 21 44 75 114 161 216 279 350 429 516 611 714 825 944 1071 1206 >1349 1500 1659 1826 2001 2184 >which alternates between even and odd numbers and can be generated >by >4x^2 + 11x + 6 or (x+2)(4x+3) >Is there a way to tell from this that all the results will be >composite? Yes, But first read my explanation below! Strange, but the triangle numbers come into play here! The triangular numbers k =n(n+1)/2 which are the top left lower right diagonal in the pyramid below. Where this sequence begins -- 1,3,6,10.. (1) (2) 3 4 5 6 (7) 8 9 10 11 12 13 14 15 16 17 18 19 20 21 (22) 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 (56) 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 Etc.. Where the beginning of each diagonal sequence (top left lower right) that are parenthisized (n) is a prime canal sequence. Where as an exception these prime numbers 2,3,5 and 7 fall into these sequences. The sequences where the first integer marked (n) will never generate a prime other than the first 4 primes. Basically these sequences start with the sum of a sum +1. sum of 0 = 0 = sum of 0 +1 = (1)3,6,10,15.. sum of 1 = 1 = sum of 1 +1 = (2)2,5,9,14,20.. sum of 2 = 3 = sum of 3 +1 = (7)12,18,25,33.. sum of 3 = 6 = sum of 6 +1 = (22)30,39,49,60.. sum of 4 = 10 = sum of 10 +1 = (56)68,81,95,110.. Etc. Also for large enough triangle number it is just k +[-1,-3,-6,-10,-15, -21,-28,-36.. limit > k-1 (previous k) that will choose an integer to its limit from each of these non-prime sequences. Ok! Your sequence that is relevent to the triangle numbers and why they never will be prime! 6 21 44 75 114 161 216 279 350 429 516 611 714 825 944 1071 1206 1349 1500 1659 1826 2001 2184 sum 6 = 21 sum 9 = 45 -1 =44 sum 12 = 78 -3 =75 sum 15 = 120 -6 =114 sum 18 = 171 -10 =161 sum 21 = 231 -15 =216 sum 24 = 300 -21 =279 sum 27 = 378 -28 =350 Ect. Note: The negated triangle numbers --- 1,3,6,10,15,21,28.. Your sequence above --->oo can never be prime. Dan === Subject: Fractal shows annoying white space? 4 years ago, my machine was too slow to run certain Maple code, which graphs a fractal of the iterates of the function f(z)=c^z. The fractal code and the image sit on my webpage (the second fractal, with the 3 buds on the bottom and the kidney shaped yellow region on its center): The image was sent to me by Robert Israel, who did the Maple code optimization. Now that I need to replicate the fractal, either code on my web page, produces some annoying white space inside the buds, instead of the solid deep blue buds that Roberts image shows. Since those pixels are probably unknown areas and Robert doesnt answer emails, does anyone have a clue on how to modify his (or my) code so that the white noise is eliminated and the buds and their thin subbuds show up nicely in deep blue, like in the image on my webpage? I fiddled with the bailout value and the iteration no, but no go. I figure I have to add a second clause to the if statement after the loop, but I really have no clue how to color the white space. -- Ioannis http://users.forthnet.gr/ath/jgal/ === Subject: Re: Fractal shows annoying white space? > 4 years ago, my machine was too slow to run certain Maple code, which > graphs a fractal of the iterates of the function f(z)=c^z. The fractal > code and the image sit on my webpage (the second fractal, with the 3 > buds on the bottom and the kidney shaped yellow region on its center): The image was sent to me by Robert Israel, who did the Maple code > optimization. Now that I need to replicate the fractal, either code on my web page, > produces some annoying white space inside the buds, instead of the > solid deep blue buds that Roberts image shows. Since those pixels are probably unknown areas and Robert doesnt > answer emails, does anyone have a clue on how to modify his (or my) > code so that the white noise is eliminated and the buds and their thin > subbuds show up nicely in deep blue, like in the image on my webpage? I fiddled with the bailout value and the iteration no, but no go. I > figure I have to add a second clause to the if statement after the > loop, but I really have no clue how to color the white space. -- > Ioannis http://users.forthnet.gr/ath/jgal/ I tried Roberts code as given on the web page, and got the picture; With -1..1, your white spaces are violet for me, the rest looks about the same. Then I did the -4..4 version, and it looks like yours except more violet and less blue. But no white space. I used Maple 9. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: decomposition of C^infty map with constant rank The following theorem is intuitively true; but does anybody know a proof? Let U be open in R^n, f:U->R^m a C^infty map so that the derivative Tf has constant rank on U. Then forall x in U there is an open neighborhood V subset U such that: f can be decomposed into a diffeomorphism d_2:R^m->R^m, a linear projection R^n->R^m and a diffeomorphism d_1:V->V. So the following holds on V f = d_2 p d_1 where is composition of maps. I think that this proposition would allow to immediately deduce theorems like the regular value theorem or the immersion theorem easily, but I didnt check this in detail yet. TIA, Tobias -- Phyics is much too hard for physicists. reverse my forename for mail! === Subject: Re: decomposition of C^infty map with constant rank >The following theorem is intuitively true; but does anybody know a proof? >Let U be open in R^n, f:U->R^m a C^infty map so that the derivative Tf has >constant rank on U. >Then forall x in U there is an open neighborhood V subset U such that: >f can be decomposed into a diffeomorphism d_2:R^m->R^m, a linear projection >R^n->R^m and a diffeomorphism d_1:V->V. So the following holds on V >f = d_2 p d_1 >where is composition of maps. >I think that this proposition would allow to immediately deduce theorems >like the regular value theorem or the immersion theorem easily, but I >didnt check this in detail yet. >TIA, >Tobias This is Lemma 3 (Chapter 1, p. 5) of Differential Geometry and Topology, Jacob T. Schwartz, 1968, Gordon and Breach. A proof is given there. John === Subject: maths spreadsheet modelling I often use spreadsheets for prototyping methods before programming them up directly (or even never leaving their prototyped stages!) I find where they really get useful is for small scale Markov chain, dynamic programming, simulation and general optimization problems. Obviously by nature such models will never be computationally very efficient but this is often offset by the rapidity with which they can developed. It often strikes that commonly used spreadsheets (that is in 99% of cases, Excel!) have a whole load of functionality that is irrelvant for this kind of application (e.g. lots of formatting, web stuff) whereas by tightening up certain other aspects they could be considerably more potent. For example: - better handling for array formulas (these are what really makes use of spreadsheets in this way possible, in my opinion) - more intelligent execution plans - perhaps even partial complilation of certain sets of cells and to a lesser extent - support for linear algebra (Excel seems to allows multuplication, inversion and determinant calculation only for matrices of dimension less than 70 or so) My question is, has anyone ever written a spreadsheet application with this type of use in mind? I gather use of spreadsheets is popular with cellular automata researchers, are there any specialized tools out there? Tom === Subject: Re: maths spreadsheet modelling .. >- better handling for array formulas (these are what really makes >use of spreadsheets in this way possible, in my opinion) Details! Excel does a pretty good job with them already, but much of its array semantics are undocumented except in newsgroup threads, and then only empirically. >- more intelligent execution plans Meaning user-specified recalculation order? Customizable circular recalculation? >- perhaps even partial complilation of certain sets of cells Partial *calculation*? >and to a lesser extent >- support for linear algebra (Excel seems to allows multuplication, >inversion and determinant calculation only for matrices of >dimension less than 70 or so) You mean support for larger arrays. See Laurent Longres MOREFUNC.XLL add-in, available at http://longre.free.fr/english/. It contains extended matrix functions as well as other useful functions. >My question is, has anyone ever written a spreadsheet application with >this type of use in mind? I gather use of spreadsheets is popular with >cellular automata researchers, are there any specialized tools out >there? Having some experience with APL and now R in addition to spreadsheets, its not so obvious to me that spreadsheets are ideal at rapid algorithm prototyping. The only thing they do better is present results more completely. Its much easier to design a spreadsheet to see all intermediate steps in calculations. That, however, is a counterargument to your claim that the formatting features get in the way. Anyway, from a purely quantitative perspective, gnumeric is the most accurate spreadsheet currently available. Excel combined with Mathematica via MathLink is the most comprehensive in terms of breadth of capabilities. An argument could be made that Siag would be closest to what you seem to be after. -- Never attach files. Snip unnecessary quoted text. Never multipost (though crossposting is usually OK). Dont change subject lines because it corrupts Google newsgroup archives. Is this message also being posted on newsgroups that are read through email programs ( like Outlook Express ) or only through Google? I am curious because I post through Google and therefore dont know if other users see it. > Is this message also being posted on newsgroups that are read through > email programs ( like Outlook Express ) or only through Google? I am > curious because I post through Google and therefore dont know if > other users see it. News postings can be read by anyone who has a news feed. For example, I use slrn, but there are many other programs available. Usenet news has existed for more than 20 years, long before anyone ever heard of Google. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. > Is this message also being posted on newsgroups that are read through > email programs ( like Outlook Express ) or only through Google? I am > curious because I post through Google and therefore dont know if > other users see it. > News postings can be read by anyone who has a news feed. For example, I > use slrn, but there are many other programs available. Usenet news has > existed for more than 20 years, long before anyone ever heard of Google. Perhaps theres more to manros question than one might think. If you post messages to newsgroups via The MathForum, for example, it will normally take several days now until the messages appear anywhere except at The MathForum itself. This very regrettable delay has virtually destroyed all of the geometry.* newsgroups, which are moderated by The MathForum. David Cantrell === Subject: Re: essence of FLT > The Barcelona conjecture: > Let c=(x+y+z)^p/(pxyz2^p) > for integer c,x,y,z and p prime greater than or equal to 5, the > Barcelona conjecture is that no solutions exist with gcd(c,xyz)=1 (no > c exist that shares no factor with x or y or z). I havent seen this conjecture before, but compare the Beal conjecture: www.math.unt.edu/~mauldin/beal.html The Beal conjecture is a stronger statement than FLT. Also, it seems to be less arbitrary-looking and more natural, at least to me. Ill let you know when I prove it :) LH === Subject: Re: Inertia: A mutual property of matter >All objects; bodies and masses of that material substance that we call matter >have one property in common: This property, called inertia, is their >indisposition, or inability to change their own motion. >They all depend on each other to supply the thrust; to force a change in _each >others_ motion: This change in speed and/or direction, is the acceleration; >which is directly proportional to the quantity of matter, or mass that they >each contain: >That is upon collision any two bodies exert a mutual equal and opposite >thrust; but the change in motion, or acceleration, that is imparted will be >directly proportional to the mass of each body; with the smaller lighter body >responding, or accelerating faster than the heavier more massive one: >This can be shown mathematically as: The mutual force [f] between two bodies >[m1 and m2], produces accelerations [a1 and a2] that are directly >proportional; so that [f = (m1)(a1) = (m2)(a2)]! SO: Now were back to the old cant do f = ma: If given m = 1 kilogram, and a = 1 m/sec^2 we get: f = 1 kg sec^2/m = 1 newton: But in classical physics doesnt 1 newton = 1 mkgs-2 ?? _Lets do it right_; you know, first since m = (f/a) = (w/g): Well have: f = ma :: (f/a)a = f, and w = mg :: (w/g)g = w! Force_f_ [and _w_] are fundamental: Without units! http://newsone.net/ -- Free reading and anonymous posting to 60,000+ groups other posts made through NewsOne.Net violate posting guidelines, email abuse@newsone.net === Subject: Re: Is this an NP complete problem? >> >> >> Students are supposed to do homework assignments themselves. Polite question, harsh answer. Fortunately many contributors > in this forum are not as rude. > Students asking about their homework are welcome, IMO. Fine, if thats the way you feel -- but if one of *my* students does this and I find them here getting answers for homework, Ill fail them. I have also sent notes to other algorithms/theory faculty I know when its obvious that its one of their students trying to get answers off the net. Most of them dont take very kindly to it. -- Thats News To Me! newstome@comcast.net === Subject: Re: Is this an NP complete problem? > Students are supposed to do homework assignments themselves. Helen === Subject: Re: Is this an NP complete problem? I am very grateful to your kinds words. Helen > Students are supposed to do homework assignments themselves. Polite question, harsh answer. Fortunately many contributors > in this forum are not as rude. > Students asking about their homework are welcome, IMO. > === Subject: Re: Is this an NP complete problem? I truely feel shame when I read your reply. I am very grateful to it. Helen For a peasant like me, this is undefined. All I can imagine an undirected > graph to be is a piece of squared paper with two axes and a line ßoaing > throught the air. But more seriously, are you talking about a four sided figure (curved lines > allowed) No because then it would be possible to make a straight cut through 3 or > even four lines. Or you could be talking about 4 parallel lines. > Please, the art of maths is to write it so that those of all specialisatons > can understand it, otherwise for all we know, you may be two memebers of a > mutual admiration society talking gibberish so you will appear cleaver. > Adding Formal Definition: A graph G is a pair (V,E), where V is a set of vertexes, > and E is a set of edges between the vertexes E = {{u,v} | u, v V}. If the > graph does not allow self-loops, adjacency is irreßexive, that is E = > {{u,v} | u, v V u v}. and saying a cut partitions the set of vertices would help tose who have not > been to the lectures. We must all be aware that new maths is being created far faster than any one > person can read it and that maths degree sylibi contain only a very small > subset of all that is maths. > -- Bruce Harvey > bruce@bearsoft.co.uk > The Alternative Physics Site > http://users.powernet.co.uk/bearsoft > Given an undirected graph, every edge in it will be in a specific > color. For example, there are 4 lines in the graph. Two lines are red. > One line is yellow. Another line is blue. Then there are totally 3 > kinds of colors in this graph. I further assume there are n cut sets > in the graph. In one cut set, the lines are either red or yellow. So > there are 2 kinds of color in this cut set. With the definition stated above, my objective is to find a cut set > that contains the least kinds of colors. I am wondering if this > problem is an NP complete problem. > Helen begin 666 member.gif > M1TE&.#=A0`+`( ``````/___RP`````0`+```#XR/J< OI1I[4!Y5Y4< > #%0`[ > ` > end begin 666 wedge.gif > M1TE&.#EA!P`(`( ``````/___RY! $```$`+ ```````@```(,C&&)J, (X)J>L*`#L` > ` > end begin 666 neq.gif > M1TE&.#EA@`*`/ ``/___P```Y! $`````+ `````*``H```(2A ^!&+KF > .7CNMV@A?9M ` > end === Subject: Breakthrough In Propulsion Physics. I have sent emails regarding Breakthrough In Propulsion Physics contents of which are given below, to reputed physicists. But in general, this message is addressed to all the physicists/scientists on planet earth. I believe, in these final monents, this is my duty. -Abhi. _____________________________________________________________ ___ I believe that I have a device through which we can generate unidirectional action force to propel any body without using reaction mass or propellant. This device is based on my Time Theory regarding origin of universe, cause of big bang, nature of time, space, energy, mechanism of gravity, magnetism, charge etc. http://www.geocities.com/actiondevice I am 34 year old Assistant serving with an insurance company with absolutely no scientific credentials or any kind of help at all. If I approach media with this device to bring it before world, I do not think that media reporters will take me seriously. After all, they are reporters, not physicists. As I have patented this device, I can always use commercial platform to bring this device before world. But it will be saddest moment of my life if I have to do so. Till this date, I tried on Internet through discussion forums, some emails so that some reputed physicist/scientific institution take notice of this invention and bring it before world from educational platform with justice it deserves. Because, I believe that this invention is going to change our way of looking at universe, course of physics and history. It will open gateway to entire universe in future for mankind. I will be thankful to you if you take initiative to communicate with me and bring this invention before world. I will wait until Xmas eve for your Action, NOT discussion. Please visit my homepage for mechanism of said Action Device.. http://www.geocities.com/actiondevice This message is also relayed to other reputed scientists, physicists and will be posted in physics discussion forums on Internet. -Abhi. Homepage: http://www.geocities.com/actiondevice Email: Abhijeet_B_Patil@hotmail.com Address:Abhijit B Patil , C/o: LIC of India, At/po/Tq: Shahada, Dist: Nandurbar (MS), India. PIN: 425409 Message relayed to: chris Isham c.isham@ic.ac.uk sandra faber, faber@ucolick.org owen gingerich, ginger@cfa.harvard.edu Alan Guth, guth@ctp.mit.edu Edward Kolb, rocky@fnas01.fnal.gov david latham, dlatham@cfa.harvard.edu Joel Primack, joel@ucolick.org Trinh Xuan Thuan, txt@virginia.edu Neil Turok, ngt1000@damtp.cam.ac.uk Steven Weinberg, weinberg@physics.utexas.edu John Barrow, J.D.Barrow@damtp.cam.ac.uk George Coyne, gcoyne@as.arizona.edu Bruno Guiderdoni, guider@iap.fr Marc millis, Marc.G.Millis@lerc.nasa.gov. Sergei Kopeikin, kopeikins@missouri.edu Tom van Flandern, tomvf@metaresearch.org Hideki Asada, asada@phys.hirosaki-u.ac.jp Stuart Samuel, samuel@thsrv.lbl.gov Clifford M. Will, cmw@wuphys.wustl.edu Steve Carlip, carlip@physics.ucdavis.edu. === Subject: Re: Breakthrough In Propulsion Physics. bots? Been drinking out of the Ganges or something? === Subject: Re: Breakthrough In Propulsion Physics. > I have sent emails regarding Breakthrough In Propulsion Physics > contents of which are given below, to reputed physicists. But in > general, this message is addressed to all the physicists/scientists on > planet earth. I believe, in these final monents, this is my duty. Ah, I see youve evaded the people that were out to kill you then. The addition of magnets is awe-inspiring. DaveL DaveL === Subject: Re: Breakthrough In Propulsion Physics. What you need to do now is give it a REALLY STRONG cup of tea, and it will go simultaneously through every point in the Universe. > I have sent emails regarding Breakthrough In Propulsion Physics > contents of which are given below, to reputed physicists. But in > general, this message is addressed to all the physicists/scientists on > planet earth. I believe, in these final monents, this is my duty. -Abhi. > _____________________________________________________________ ___ > I believe that I have a device through which we can generate > unidirectional action force to propel any body without using reaction > mass or propellant. This device is based on my Time Theory regarding > origin of universe, cause of big bang, nature of time, space, energy, > mechanism of gravity, magnetism, charge etc. http://www.geocities.com/actiondevice I am 34 year old Assistant serving with an insurance company with > absolutely no scientific credentials or any kind of help at all. If I > approach media with this device to bring it before world, I do not > think that media reporters will take me seriously. After all, they are > reporters, not physicists. As I have patented this device, I can always use commercial platform > to bring this device before world. But it will be saddest moment of my > life if I have to do so. Till this date, I tried on Internet through > discussion forums, some emails so that some reputed > physicist/scientific institution take notice of this invention and > bring it before world from educational platform with justice it > deserves. Because, I believe that this invention is going to change > our way of looking at universe, course of physics and history. It will > open gateway to entire universe in future for mankind. I will be thankful to you if you take initiative to communicate with > me and bring this invention before world. I will wait until Xmas eve for your Action, NOT discussion. Please > visit my homepage for mechanism of said Action Device.. http://www.geocities.com/actiondevice This message is also relayed to other reputed scientists, physicists > and will be posted in physics discussion forums on Internet. > -Abhi. > Homepage: http://www.geocities.com/actiondevice > Email: Abhijeet_B_Patil@hotmail.com > Address:Abhijit B Patil , C/o: LIC of India, At/po/Tq: Shahada, Dist: > Nandurbar (MS), India. PIN: 425409 > Message relayed to: > chris Isham c.isham@ic.ac.uk > sandra faber, faber@ucolick.org > owen gingerich, ginger@cfa.harvard.edu > Alan Guth, guth@ctp.mit.edu > Edward Kolb, rocky@fnas01.fnal.gov > david latham, dlatham@cfa.harvard.edu > Joel Primack, joel@ucolick.org > Trinh Xuan Thuan, txt@virginia.edu > Neil Turok, ngt1000@damtp.cam.ac.uk > Steven Weinberg, weinberg@physics.utexas.edu > John Barrow, J.D.Barrow@damtp.cam.ac.uk > George Coyne, gcoyne@as.arizona.edu > Bruno Guiderdoni, guider@iap.fr > Marc millis, Marc.G.Millis@lerc.nasa.gov. > Sergei Kopeikin, kopeikins@missouri.edu > Tom van Flandern, tomvf@metaresearch.org > Hideki Asada, asada@phys.hirosaki-u.ac.jp > Stuart Samuel, samuel@thsrv.lbl.gov > Clifford M. Will, cmw@wuphys.wustl.edu > Steve Carlip, carlip@physics.ucdavis.edu. === Subject: Re: Breakthrough In Propulsion Physics. > I have sent emails regarding Breakthrough In Propulsion Physics > contents of which are given below, to reputed physicists. But in > general, this message is addressed to all the physicists/scientists on > planet earth. I believe, in these final monents, this is my duty. > -Abhi. > _____________________________________________________________ ___ > I believe that I have a device through which we can generate > unidirectional action force to propel any body without using reaction > mass or propellant. This device is based on my Time Theory regarding > origin of universe, cause of big bang, nature of time, space, energy, > mechanism of gravity, magnetism, charge etc. So you basically refute Newtons 3rd law and conservation of momentum. You are saying that you, standing on the ground, with zero momentum, can suddenly attain momentum without an external force or mass reduction but only using internal forces to the system. Pretty spooky or Harry Pottery stuff. Do you have a prototype built? Warning, try this on the ground only.... === Subject: Re: Breakthrough In Propulsion Physics. > I have sent emails regarding Breakthrough In Propulsion Physics > contents of which are given below, to reputed physicists. But in > general, this message is addressed to all the physicists/scientists on > planet earth. I believe, in these final monents, this is my duty. > -Abhi. > _____________________________________________________________ ___ > I believe that I have a device There are several big problems here. The most obvious one is that you do not have such a device for sure. It would be very different if you had it. > through which we can generate > unidirectional action force to propel any body without using reaction > mass or propellant. This device is based on my Time Theory regarding > origin of universe, cause of big bang, nature of time, space, energy, > mechanism of gravity, magnetism, charge etc. > http://www.geocities.com/actiondevice > I am 34 year old Assistant serving with an insurance company with > absolutely no scientific credentials or any kind of help at all. If I > approach media with this device to bring it before world, I do not > think that media reporters will take me seriously. After all, they are > reporters, not physicists. > As I have patented this device, I can always use commercial platform > to bring this device before world. Now thats more useful! Just give us the patent number. Note: A patent goes through a review process, so that a patent usually has some quality. Harald === Subject: Re: Breakthrough In Propulsion Physics. I have sent emails regarding Breakthrough In Propulsion Physics > contents of which are given below, to reputed physicists. Milk, Coke-a-Cola, and coffee has been spewed in paroxysms of laughter at CRT, plasma, and LCD screens. Hey stooopid, 1) Space is homogeneous (translation) and isotropic (rotation). Linear and angular momenta are therefore conserved through Noethers theorem. 2) Translation in time is homogeneous. Energy is therefore conserved through Noethers theorem. 3) You are an ignorant bathetic ass - a credit to your race. 4) By posting e-mail addresses to Usenet you have fed a legion of with loathsome withered wog ass permanently barred from the Net. > I am 34 year old Assistant serving with an insurance company with > absolutely no scientific credentials or any kind of help at all. http://math.ucr.edu/home/baez/crackpot.html > Tom van Flandern, tomvf@metaresearch.org HA HA HA! YOU CANT EVEN SEPARATE THE FROM THE SHINOLA. Hey stooopid, 1) One begins by not contradicting any preceeding observation. 2) One proceeds by being educated in the discipline and its literature. 3) One then erects a mathematical model falsifiable by empirical test that is not equivalent to existing theory - additional predictions. Listen up, you insufferable ineducable jackass - physical theory is a series of internally self-consistent axiomatic systems. They contain no mistakes. You cannot use their content or structure to disprove themselves. There are only two routes to discrediting extant theory, 1) Make a falsifying observation. You are a bad joke. 2) Falsify a founding postulate. You are a bad joke. Uncle Al has the punchline, http://www.mazepath.com/uncleal/qz.pdf http://www.mazepath.com/uncleal/eotvos.htm -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Breakthrough In Propulsion Physics. >[...] >I believe that I have a device through which we can generate >unidirectional action force to propel any body without using reaction >mass or propellant. This device is based on my Time Theory regarding >origin of universe, cause of big bang, nature of time, space, energy, >mechanism of gravity, magnetism, charge etc. >http://www.geocities.com/actiondevice >I am 34 year old Assistant serving with an insurance company with >absolutely no scientific credentials or any kind of help at all. If I >approach media with this device to bring it before world, I do not >think that media reporters will take me seriously. After all, they are >reporters, not physicists. >As I have patented this device, I can always use commercial platform >to bring this device before world. ??? On the web page is says just Patent pending at Govt. of India Patent office. If youve actually patented it then you need to revise the page... >[...] >Message relayed to: >chris Isham c.isham@ic.ac.uk >sandra faber, faber@ucolick.org >owen gingerich, ginger@cfa.harvard.edu >Alan Guth, guth@ctp.mit.edu >Edward Kolb, rocky@fnas01.fnal.gov >david latham, dlatham@cfa.harvard.edu >Joel Primack, joel@ucolick.org >Trinh Xuan Thuan, txt@virginia.edu >Neil Turok, ngt1000@damtp.cam.ac.uk >Steven Weinberg, weinberg@physics.utexas.edu >John Barrow, J.D.Barrow@damtp.cam.ac.uk >George Coyne, gcoyne@as.arizona.edu >Bruno Guiderdoni, guider@iap.fr >Marc millis, Marc.G.Millis@lerc.nasa.gov. >Sergei Kopeikin, kopeikins@missouri.edu >Tom van Flandern, tomvf@metaresearch.org >Hideki Asada, asada@phys.hirosaki-u.ac.jp >Stuart Samuel, samuel@thsrv.lbl.gov >Clifford M. Will, cmw@wuphys.wustl.edu >Steve Carlip, carlip@physics.ucdavis.edu. ************************ David C. Ullrich === Subject: Re: A property of uncountable sets in R >I couldnt prove the following propostion, any help is welcome: >If S is an uncountable subset of R, then S contains a subset P such >that, if x and y are distintinct elements of P, then theres a z in P >between x and y. > Hint: remove every point which is in a closed interval of positive length > whose intersection with S is at most countable. But this doesnt necessarily lead to a set with the desired property. > For example, if S =[0,1] U [2,3], then those points form an empty set No: 0, 1, 2 and 3 will be removed (e.g. [1,2] is a closed interval of positive > length whose intersection with S has cardinality 2. Yes, youre right! (of couirse!) Amanda === Subject: Re: A property of uncountable sets in R >I couldnt prove the following propostion, any help is welcome: >If S is an uncountable subset of R, then S contains a subset P such >that, if x and y are distintinct elements of P, then theres a z in P >between x and y. First prove: LEMMA. If S is an uncountable subset of R, then there exists an > element x in S such that there are uncountably many elements of S > smaller than x, and uncountably many elements of S larger than x. Then you can use the Axiom of Choice to pick x_1 with that property; > then pick x_{21} < x_1 < x_{22}, x_{21} with the property that there > are uncountably many points of S smaller than x_{21}, and uncountably > many between x_{21} and x_1; etc. Keep making a choice of point in > between any two, and then take the union. That will be P. To prove the lemma, consider two sets: L = {x in S:there are only countably many elements of S smaller than x} > B = {x in S:there are only countably many elements of S larger than x} Then L is downward closed: if x in S, and y in S satisfies y in L; B is upward closed: if x in B and y in S satisfies x in B. Show that L is bounded above, and B is bounded below (hint: if L is > not bounded above, then every (-infty,n] intersects s for sufficiently > large integer n; this implies S is countable [prove it]). If L is not bounded above, then, for every natural n, there is z in L such that z >n. Since L is downward closed, this means the set I_n = {x in S : x< n} is countable. If y is in S, let m be the smallest integer > y. Then I_m = (-inf, m) intersection S is countable, which shows every element of S belongs to an open interval that contains at most finitely many elements of S. Therefore, S is countable (it has no condensations points). I think we can come to this same conclusion if we observe that S = Union (S /I_n), n=1,2,3...This equation shows S is the union of a countable collection of countable sets. Right? > Now prove that each of L and B are at most countable. E.g. if L is > empty, you are done. If it is not empty, let x_0 be its supremum. Then > for every integer n, (-infty,x_0-1/n) intersects S in at most > countably many elements, and L is either equal to the union of all > these, or is the union of all these plus x_0 (if the sup of L lies in > L). Do something similar with B. Well, theres nothing left to prove, you did it all. Therefore, S cannot be equal to the union of L and B. Pick any point > in S not in L union B, and you have the lemma. Sure. Now, can we use this result to prove that, if S is uncountable, then S has bilateral condensation points (not necessarily in S) and the set of the bilateral condensation points of S that are in S is uncountable? x is a bilateral condensation point of S if, for every eps>0, the intervals (x-eps, x) and (x, x+eps) intersect S in uncountably many elements. Amanda === Subject: Re: A property of uncountable sets in R Adjunct Assistant Professor at the University of Montana. >>I couldnt prove the following propostion, any help is welcome: >>If S is an uncountable subset of R, then S contains a subset P such >>that, if x and y are distintinct elements of P, then theres a z in P >>between x and y. >> >> First prove: >> >> LEMMA. If S is an uncountable subset of R, then there exists an >> element x in S such that there are uncountably many elements of S >> smaller than x, and uncountably many elements of S larger than x. >> >> Then you can use the Axiom of Choice to pick x_1 with that property; >> then pick x_{21} < x_1 < x_{22}, x_{21} with the property that there >> are uncountably many points of S smaller than x_{21}, and uncountably >> many between x_{21} and x_1; etc. Keep making a choice of point in >> between any two, and then take the union. That will be P. >> >> To prove the lemma, consider two sets: >> >> L = {x in S:there are only countably many elements of S smaller than x} >> B = {x in S:there are only countably many elements of S larger than x} >> >> Then L is downward closed: if x in S, and y in S satisfies y> in L; B is upward closed: if x in B and y in S satisfies x> in B. >> >> Show that L is bounded above, and B is bounded below (hint: if L is >> not bounded above, then every (-infty,n] intersects s for sufficiently >> large integer n; this implies S is countable [prove it]). >If L is not bounded above, then, for every natural n, there is z in L >such that z >n. Since L is downward closed, this means the set I_n = >{x in S : x< n} is countable for every n. > If y is in S, let m be the smallest >integer > y. Then I_m = (-inf, m) intersection S is countable, which >shows every element of S belongs to an open interval that contains at >most finitely many elements of S. Therefore, S is countable (it has no >condensations points). I think we can come to this same conclusion if >we observe that S = Union (S /I_n), n=1,2,3...This equation shows S >is the union of a countable collection of countable sets. Right? Yes; and thats easier. >> Now prove that each of L and B are at most countable. E.g. if L is >> empty, you are done. If it is not empty, let x_0 be its supremum. Then >> for every integer n, (-infty,x_0-1/n) intersects S in at most >> countably many elements, and L is either equal to the union of all >> these, or is the union of all these plus x_0 (if the sup of L lies in >> L). Do something similar with B. >Well, theres nothing left to prove, you did it all. Fair enough. >> Therefore, S cannot be equal to the union of L and B. Pick any point >> in S not in L union B, and you have the lemma. >Sure. >Now, can we use this result to prove that, if S is uncountable, then S >has bilateral condensation points (not necessarily in S) and the set >of the bilateral condensation points of S that are in S is >uncountable? x is a bilateral condensation point of S if, for every >eps>0, the intervals (x-eps, x) and (x, x+eps) intersect S in >uncountably many elements. Hmmm... You can certainly use the Lemma to prove that there exist bilateral condensation points, but it seems unnecessary: prove that if S is uncountable, then for every epsilon there exists an x in S, possibly depending on epsilon, such that (x-eps, x+eps) intersect S is uncountable. Now let x_1 be the point you find for eps=1/2. Then let x_2 be the point you find for eps=1/4, but for the set (x_1-1/2,x_1+1/2) intersect S. Then let x_3 be the point for eps=1/8, and the set (x_2-1/4,x_2+1/4) intersect S, etc. Then x_1,....,x_n,... is a Cauchy sequence, and so converges; the limit point will be a condensation point. -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Q: Latin square Given a Latin square, one can always transform it into a normalized or standard form through permuting rows and columns such that the first row and columm is given by {1, 2, ... n}. Is this normalized form unique for any given Latin square of arbitrary size n? If yes, could someone give a literature reference containing a proof? M. K. Shen === Subject: Re: Latin square > Given a Latin square, one can always transform it into > a normalized or standard form through permuting rows and > columns such that the first row and columm is given by > {1, 2, ... n}. Is this normalized form unique for any > given Latin square of arbitrary size n? If yes, could > someone give a literature reference containing a proof? Yes, for some discussion and references see: http://mathworld.wolfram.com/LatinSquare.html === Subject: Re: Latin square >> Given a Latin square, one can always transform it into >> a normalized or standard form through permuting rows and >> columns such that the first row and columm is given by >> {1, 2, ... n}. Is this normalized form unique for any >> given Latin square of arbitrary size n? If yes, could >> someone give a literature reference containing a proof? >Yes, for some discussion and references see: > http://mathworld.wolfram.com/LatinSquare.html The answer is no. Consider 1 2 3 4 5 2 1 4 5 3 3 4 5 1 2 4 5 2 3 1 5 3 1 2 4 Switch the first two rows and columns, and cyclically permute the third, fourth, and fifth rows and columns in that order. The resulting square is 1 2 3 4 5 2 1 5 3 4 3 5 4 1 2 4 3 2 5 1 5 4 1 2 3 -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Latin square > Given a Latin square, one can always transform it into > a normalized or standard form through permuting rows and > columns such that the first row and columm is given by > {1, 2, ... n}. Is this normalized form unique for any > given Latin square of arbitrary size n? If yes, could > someone give a literature reference containing a proof? Yes, for some discussion and references see: http://mathworld.wolfram.com/LatinSquare.html I see there mention about the Ônumber of normalized Latin squares of a given n, but there seems to be nothing dealing with the uniqueness issue stated above. M. K. Shen === Subject: integral over a disk Distribution: inet Heres a question that I already posted a while ago, to which I got several interesting answers (thread Ôdefinite integral in sci.math and sci.math.num-analysis). Nevertheless, I m still stuck with my practical problem, reason why I m giving it another go, now including sci.math.symbolic and sci.physics.electromag, and trying to specify the problem and the kind of solution I m looking for more clearly: The problem (its an electrostatics problem), is to integrate the function 1/R over a ßat, infinitely thin, circular disk where R represents the distance from the integration point on the disk to an arbitrary point P in space. Mathematically: F = int_0^{2*pi} { int_0^1 { 1 over{ sqrt{ D^2 + r^2 + rho^2 - 2*r*rho*cos{phi} }}} dr } dphi where the disk has radius 1, and it is centered at the origin of the standard cylindrical coordinate-system in the plane z = 0. The point P is at (r=rho, z=D, phi=0) This can be relatively easy reduced to a 1-D integral using Gauss theorem: F = int_L { (rho*cos{phi} - 1)*R over Q^2 } dl, a line integral over the circumference L of the disk, and Q equals R for z = 0. But then the problem starts. Mathematica gives me an expression of about 13 pages of increasingly complicated elliptical functions, that becomes indeterminate at both the integration limits. What I m looking for is a good approximation, that is, with a guaranteed relative error below, say 0.001, for any point P, which at the same time is not too computationally demanding, because its part of a numerical simulation code that depends on speed. Right now I m using numerical integration, and its way too slow (especially for small values of z,r). Alex === Subject: Re: integral over a disk Heres a question that I already posted a while ago, to which I got > several interesting answers > (thread Ôdefinite integral in sci.math and sci.math.num-analysis). > Nevertheless, I m still stuck with my practical problem, reason why I m > giving it another go, now including sci.math.symbolic and > sci.physics.electromag, and trying to specify > the problem and the kind of solution I m looking for more clearly: The problem (its an electrostatics problem), > is to integrate the function 1/R over a ßat, infinitely thin, circular > disk where R represents the distance > from the integration point on the disk to an arbitrary point P in space. Mathematically: F = int_0^{2*pi} { int_0^1 { 1 over{ sqrt{ D^2 + r^2 + rho^2 - > 2*r*rho*cos{phi} }}} dr } dphi where the disk has radius 1, and it is centered at the origin of the > standard cylindrical coordinate-system > in the plane z = 0. The point P is at (r=rho, z=D, phi=0) This can be relatively easy reduced to a 1-D integral using Gauss > theorem: F = int_L { (rho*cos{phi} - 1)*R over Q^2 } dl, a line integral over the circumference L of the disk, and Q equals R for z > = 0. > But then the problem starts. Mathematica gives me an expression of about > 13 pages of increasingly > complicated elliptical functions, that becomes indeterminate at both the > integration limits. What I m looking for is a good approximation, that is, with a guaranteed > relative error > below, say 0.001, for any point P, which at the same time is not too > computationally demanding, > because its part of a numerical simulation code that depends on speed. > Right now I m using numerical integration, and its way too slow > (especially for small values of z,r). > Alex Maple gives a series solution in rho and d (your D) and as follows: 3/16*rho^4*Pi/d^4- 1/2*rho^2*Pi/d^2+ (-2*ln(d)*Pi-1/4*rho^2*Pi+O(rho^6)-9/128*rho^4*Pi+2*Pi*ln(2))+ O(d^2) Sometimes it is possible to speed up the evaluation of an integral by subtracting off the near singular parts as identified by such expansions. Hope this helps, Jack Fearnley === Subject: Re: I WILL GET MY MONEY >>>| >| >|> Hi James. I checked your math - its wrong. I will >|> so that no one will have to read them. >| >| what, exactly, gives you the authority to do such thing? >> i said he could. >> and, who or what exactly, gives you the authority to do >> such thing? >> I said he could. >> and who made you the authority??? > > Sorry, but I dont have the authority to tell you that. But, who does? Apparently, I dont have the authority to be told that, either. > Im working on the assumption that, if I did have the authority, > someone would tell me. Hey, its working for everybody else! i want to talk to your supervisor! who is your supervisor? > Fortunately, I do have the authority to tell you I dont have > the authority to tell you that. and who gave you that authority? Unfortunately, I dont have the authority to tell you who gave > me the authority to tell you I dont have the authority to tell you > who gave me the authority to tell James D. that he could tell seems that you are abusing your power. if that is the case i will be calling for your resignation. > That could just be an oversight, though. Maybe you could check > back later? no, i want to get in touch your supervisor! > May I help the next customer? Jim Burns === Subject: Re: math error in this topology web page? said: >Is a set closed iff it has the above properties with respect to some >topology? No, on two grounds. First, set is not open or closed in an absolute sense, only with respect to a topology. Second, the properties you gave were properties of a topology, not properties of a subspace of a topological space. -- 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 === Subject: Re: math error in this topology web page? NOT! said: >He defines the sets in a topology T of a topological space X as >closed sets. There are several equivalent ways to define a topology; this is one of them. -- 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 === Subject: Log-probability scale I have a little problem. I have some sieve analysis data and I need them to be put in a graph so built: X axis: a log-probability scale plotting the cumulative percent Y axis: a log scale plotting the sieve size (micron) The matter is: can I do that with Microsoft Excel? If so, how can I? If Excel doesnt work, what kind of software can I use? I thank you all in advance. Best wishes, Andrea === Subject: Re: Final Rout of Synchronization Clocks in Relativity > ABSTRACT. The synchronization of clocks in Relativity has speculative > chatter, and this speculative chatter about Synchronizations of > clocks in Relativity has not ACTUAL TECHNICAL EMBODYING In CONCRETE > TECHNICAL DEVICES. The deductive Analysis of Surprising paradox of mythical so-called > pseudo of synchronization of clocks in the Relativity is given below: > > Sometimes claimants misquote or exaggerate to further their > own agendas. It is best to keep an open opinion until you have heard > from both sides of any story. -|Tom|- > Tom Van Flandern - Washington, DC - see our web site on replacement > astronomy research at http://metaresearch.org > The concept of clock synchronization confused me for the longest time. > I was trying to think of it in absolute terms of the twin paradox. As > a concept to be thought of deeply. > And the long antenna simulation proved it a far different thing. A gps can have the clock synchronized by the input of the correct > location. Without any satilite message!!!!!! A special kind of time difference is defined by special relativity. > It is not general relativities kind though!!! So the clock of special relativity is to be rememebered as only the > clock of Einstiens gedanken experiment. He defined a special dilemma in theory which the answer caused!!!! So think carefully of taking the gedanken experiment as a physical > experiment. It is to be a very, very, very special thought > experiment. Meaning it is a test of the theory in the school of classical theory. So when the train has two times. What causes the synchronization to > fail? And here the train has the relative speed of light never failing as > the cause of the gedanken experiment. That is its real purpose, while the lack of synchronized clocks is the > synthetic necessity to cause the relation of, never failing to reach > the same speed independent of the inertial reference frame. One relation is the theory, while the other causes the theory. This is the gedenkan experiment in abstract form. Real odd in form!!!! ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^ > Read the EPR paper. It uses the opportunity to define the gedanken > experiment abstractly. Never do the experiment, because it defines > the fool. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^ The theory of relativity was generated by remarkable mathematician Henry Poincare. Poincare has generated the _mathematical_ MENTAL chimera (Theory of Relativity), which one was and is till now a speculative Plague for physics of the Twentieth century: http://www.wbabin.net/comments/timo1.htm http://www.wbabin.net/ > So when the synchronized clock is introduced in special relativity it > is natures clock and never the experimenters clock!!!! Natures as a physical theory would cause to be identical to require > the solution. To allow the physical clock to follow this requirement would require > the synchronized clock to never be measurable! An absolute outcome of > special relativity is the meaning of clocks once separated are forever > unsynchronized. And here the frame of reference is the only means of inference of the > clocks difference. So, a complete theory is defined!!!!!!!! And back again to the synchronized clocks. All reference frames are > independent. Remarkable CRAZY mathematician Henry Poincare has generated the CRAZY _mathematical_ MENTAL chimera (Theory of Relativity), which one was and is till now a speculative Plague for physics of the Twentieth century. This Extremely Speculative Plague has stopped natural translational development of physics and has got it in DEADLOCK. > Complete in every fashion. Relativity has got development of physics in DEADLOCK, Complete in every fashion. === Subject: Re: seeking textbook recommendation for teaching > In a few months I will be teaching a course to undergraduate math majors >called Mathematical Structures. The catalog description is: A rigorous study of the mathematical structures which form the foundation >of higher mathematics. Set theory, logic, formal development of the number >systems from the natural numbers through the complex numbers, basic >algebraic structures (groups, rings, and fields), and elementary topological >concepts. Thats quite a lot for one course! Yes, I know. But the course will not treat any topics in depth. As far as the algebra goes, I doubt I will give them more then the definitions of the various structures and several examples. Same probably goes for the topology--and maybe continuous functions and homeomorphisms. Most of the students will be taking separate courses in algebra and topology after this course. > >The course is also supposed to introduce the students to the practice of >good mathematical writing and the construction of proofs. And each of those two by itself is quite a lot. Hence the word introduce I suppose. This course will just be additional practice in writing clear proofs. They also practice these skills in their other courses. It will be a >ten week course (which is supposed to be the equivalent of a normal >15 week semester course). Hey, why not. Since theres already impossibly too much stuff in it for > a 15 week course, theres little additional harm in having > impossibly*150% too much stuff. Well, to be fair, all of our ten-week courses cover what is traditionally done in a 15-week semester course. The course will meet for four seventy-minute periods a week. Students here take only three courses at a time, so they are expected to do a lot of work outside the classroom. I expect thats why you say supposed to a couple of times. Now > seriously, what do you ACTUALLY expect to accomplish in the available > time with the available students? It is IMO very important for your own > stress management to have realistic(ish) expectations. > Well, Im still formulating a plan, but I guess I want to give all of the ZFC axioms, use them to give a coding of the natural numbers. Then construct the integers. Then the rationals. Then the reals. Then the complex numbers. Along the way I will probably do things like define addition on the naturals and prove it is commutative and associative, and things like that. In addition to constructing these number systems, I want to expose the students to simple set theoretic constructions like functions, cartesian products, equivalence relations, etc. (Another idea is to teach the basic set theory, and then give the natural numbers by way of the Peano postulates and work from there.) Then Ill probably define groups, give examples, define subgroups, cyclic groups, maybe quickly get to Lagranges theorem. Then in one period, define rings and fields and give examples. Then Ill define topological space and give examples and tell them what topology is all about. Maybe Ill define continuous functions and homemorphisms, and give an example of homeomorphic spaces. I wont specifically take time to directly teach students to write good mathematical proofs. Ill do that as I go along by giving appropriate homework assignments for which Ill give feedback. And Ill be giving numerous examples of well-written proofs during the lectures. > It sounds like this is a new course. If so, that increases all risk > factors markedly. If not, you would definitely benefit from talking > with the previous instructors of this course about what went well and > what went badly. > Well its not a new course, and yes, of course Ive been talking to previous instructors. I just wanted some extra advice on textbooks (the last instructor didnt use a textbook), especially from logicians and set theorists. >If necessary I can teach the course without a text (I have extensive notes >on set theory and logic and the construction of the number systems from >various sources). Any thoughts on that? In my experience, having a text provides a security blanket for many > students (expecially if the instructor may have to go on medical stress > leave half way through...) Yes, I was thinking the same. Ive never taught anything where the students didnt have a textbook. I was wondering if anyone has had positive experiences doing so in an undergraduate course. > >Leonard (email defunct) Hmm, Email Defunct reminds me of Wire Paladin from the old Have > Gun, Will Travel TV series. === Subject: Re: seeking textbook recommendation for teaching Have a look at this course and its reading list: http://uzweb.uz.ac.zw/science/maths/courses/mth112.htm hth Guy Corrigall > In a few months I will be teaching a course to undergraduate math majors > called Mathematical Structures. The catalog description is: > A rigorous study of the mathematical structures which form the foundation > of higher mathematics. Set theory, logic, formal development of the number > systems from the natural numbers through the complex numbers, basic > algebraic structures (groups, rings, and fields), and elementary topological > concepts. > The course is also supposed to introduce the students to the practice of > good mathematical writing and the construction of proofs. It will be a > ten week course (which is supposed to be the equivalent of a normal > 15 week semester course). > A student will take this course after completing the calculus sequence, > linear algebra, and a couple of other courses, but before taking abstract > algebra or analysis. > Can anyone recommend a good text or two for my students? I realize that > one most likely cannot find a book covering set theory and logic and also > algebra and topology. I am most intersested in recommendations for > the set theory, logic, and proof writing parts of the course (which will > probably take up 80% of the course). > If necessary I can teach the course without a text (I have extensive notes > on set theory and logic and the construction of the number systems from > various sources). Any thoughts on that? > Leonard (email defunct) === Subject: Re: seeking textbook recommendation for teaching > Have a look at this course and its reading list: http://uzweb.uz.ac.zw/science/maths/courses/mth112.htm hth Guy Corrigall me a lot of ideas. Ill certainly be perusing the reading list. I only wish I had more time to plan such a course. -Leonard === Subject: Re: seeking textbook recommendation for teaching A Transition to Advanced Mathematics by Eggen , Smith , and St. Andre covers all of the topics you have mentioned except Topology. But you gain a few concepts from analysis. Really a nice book to learn proofs from. Also, How To Prove It is nice as well. > Have a look at this course and its reading list: > http://uzweb.uz.ac.zw/science/maths/courses/mth112.htm > hth > Guy Corrigall In a few months I will be teaching a course to undergraduate math majors > called Mathematical Structures. The catalog description is: A rigorous study of the mathematical structures which form the foundation > of higher mathematics. Set theory, logic, formal development of the > number > systems from the natural numbers through the complex numbers, basic > algebraic structures (groups, rings, and fields), and elementary > topological > concepts. The course is also supposed to introduce the students to the practice of > good mathematical writing and the construction of proofs. It will be a > ten week course (which is supposed to be the equivalent of a normal > 15 week semester course). A student will take this course after completing the calculus sequence, > linear algebra, and a couple of other courses, but before taking abstract > algebra or analysis. Can anyone recommend a good text or two for my students? I realize that > one most likely cannot find a book covering set theory and logic and also > algebra and topology. I am most intersested in recommendations for > the set theory, logic, and proof writing parts of the course (which will > probably take up 80% of the course). If necessary I can teach the course without a text (I have extensive notes > on set theory and logic and the construction of the number systems from > various sources). Any thoughts on that? Leonard (email defunct) === Subject: Re: seeking textbook recommendation for teaching In a few months I will be teaching a course to undergraduate math majors > called Mathematical Structures. The catalog description is: A rigorous study of the mathematical structures which form the foundation > of higher mathematics. Set theory, logic, formal development of the number > systems from the natural numbers through the complex numbers, basic > algebraic structures (groups, rings, and fields), and elementary topological > concepts. The course is also supposed to introduce the students to the practice of > good mathematical writing and the construction of proofs. It will be a > ten week course (which is supposed to be the equivalent of a normal > 15 week semester course). A student will take this course after completing the calculus sequence, > linear algebra, and a couple of other courses, but before taking abstract > algebra or analysis. Can anyone recommend a good text or two for my students? I realize that > one most likely cannot find a book covering set theory and logic and also > algebra and topology. I am most intersested in recommendations for > the set theory, logic, and proof writing parts of the course (which will > probably take up 80% of the course). If necessary I can teach the course without a text (I have extensive notes > on set theory and logic and the construction of the number systems from > various sources). Any thoughts on that? Leonard (email defunct) For the 80% in a similar kind of course, I had some success with and highly recommend: D. J. Velleman, How to Prove It, a structured approach, Cambridge U Press 1994. I found it carefully written. The students could read the book, and class discussions often got quite animated. Bob Pego === Subject: Re: seeking textbook recommendation for teaching For the 80% in a similar kind of course, I had some success with and > highly recommend: D. J. Velleman, How to Prove It, a structured approach, > Cambridge U Press 1994. I found it carefully written. The students could read the book, and > class discussions often got quite animated. serious consideration. -Leonard Bob Pego === Subject: Re: seeking textbook recommendation for teaching > In a few months I will be teaching a course to undergraduate math majors > called Mathematical Structures. The catalog description is: > A rigorous study of the mathematical structures which form the foundation > of higher mathematics. Set theory, logic, formal development of the number > systems from the natural numbers through the complex numbers, basic > algebraic structures (groups, rings, and fields), and elementary topological > concepts. > The course is also supposed to introduce the students to the practice of > good mathematical writing and the construction of proofs. It will be a > ten week course (which is supposed to be the equivalent of a normal > 15 week semester course). > A student will take this course after completing the calculus sequence, > linear algebra, and a couple of other courses, but before taking abstract > algebra or analysis. > Can anyone recommend a good text or two for my students? I realize that > one most likely cannot find a book covering set theory and logic and also > algebra and topology. I am most intersested in recommendations for > the set theory, logic, and proof writing parts of the course (which will > probably take up 80% of the course). > If necessary I can teach the course without a text (I have extensive notes > on set theory and logic and the construction of the number systems from > various sources). Any thoughts on that? > Leonard (email defunct) Hello Leonard, how about the book: Introduction to Mathematical Structures and Proofs by L. J. Gerstein, Springer-Verlag The index of the book looks like it covers all of your topics and this is a very nicely written and organized book, IMHO. I think it is worth a look. HTH, Flip === Subject: Re: seeking textbook recommendation for teaching Hello Leonard, how about the book: Introduction to Mathematical Structures and Proofs by L. J. Gerstein, > Springer-Verlag The index of the book looks like it covers all of your topics and this is a > very nicely written and organized book, IMHO. I think it is worth a look. Wow, I didnt think there would be a book that covered ALL the topics I -Leonard HTH, Flip === Subject: Re: Intuitive Group Theory - book suggestions? Groups and Symmetry, M.A. Armstrong (Springer, UTM) seems pretty friendly with lots of pictures. If he likes knots try out the program KnotPlot which can be freely downloaded here: http://www.pims.math.ca/knotplot/download.html Jim Buddenhagen -- To reply copy jbuddenh@REMOVEtexas.net to address bar and edit out REMOVE === Subject: Re: Intuitive Group Theory - book suggestions? >him on to branches of math so he can see whats out there. He likes >quirky number theory stuff, he loved Conways Book of Numbers, and he >loved the book Knots and Surfaces by Farmer and Stanford, which we >went through in a blitz - he did almost all of the exercises. I was >wondering whether someone knows of an introduction to group theory that >is intuitive enough that he can get a feeling for essential concepts >without too much formalism. I feel like something that had a good >presentation of why there are only a few groups of size smaller than >some limit, and what those groups are, for example, could capture his >imagination. (He likes to know what all the possibilities are in >various contexts.) The texts I know are too terse for a kid that age, >who will have time (and patience) to do the formal stuff later, but I >think it would be good for him to discover as soon as possible that he >loves math in a broad way - if indeed he does love math in a broad way. >Hes fine on simple set stuff and linear algebra. Are you sure that he needs an intuitive approach to group theory? One can consider examples after seeing formalism, and that is the time to do it. Why do you think that texts are too terse? Let him do the formal stuff as soon as he can, and then he can PROPERLY apply it later. Bright, and especially gifted, do not benefit from having to do it clumsily the first time, and then get the simple conceptions later. In fact, I do not think anyone benefits. As for age, my son was auditing graduate courses when in junior high. He did logic and algebra before starting first grade. Knot theory is more complex than formal group theory. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Intuitive Group Theory - book suggestions? >>him on to branches of math so he can see whats out there. He likes >>quirky number theory stuff, he loved Conways Book of Numbers, and he >>loved the book Knots and Surfaces by Farmer and Stanford, which we >>went through in a blitz - he did almost all of the exercises. I was >>wondering whether someone knows of an introduction to group theory that >>is intuitive enough that he can get a feeling for essential concepts >>without too much formalism. I feel like something that had a good >>presentation of why there are only a few groups of size smaller than >>some limit, and what those groups are, for example, could capture his >>imagination. (He likes to know what all the possibilities are in >>various contexts.) The texts I know are too terse for a kid that age, >>who will have time (and patience) to do the formal stuff later, but I >>think it would be good for him to discover as soon as possible that he >>loves math in a broad way - if indeed he does love math in a broad way. >>Hes fine on simple set stuff and linear algebra. Are you sure that he needs an intuitive approach to > group theory? One can consider examples after seeing > formalism, and that is the time to do it. Its an interesting question. Im not trying to start his formal foundational mathematical training here, Im trying to get him excited about things and have him feel confident about his intuition, which is excellent. At the moment this is going well because he *wants* to do it. Excessive formalism turns him off at the moment, and I think thats fine for now. If it was someones kid whose parents were paying me to turn into the best mathematician I could, I would agree with you (and Id also suggest the parents send him to someone else, very soon, rather than me). But this is just a friendly session that so far has been fun for him, and for now Im keeping it that way. > Why do you think that texts are too terse? Let him > do the formal stuff as soon as he can, and then he > can PROPERLY apply it later. Bright, and especially > gifted, do not benefit from having to do it clumsily > the first time, and then get the simple conceptions > later. In fact, I do not think anyone benefits. Informal and intuitive do not necessarily imply clumsy! > As for age, my son was auditing graduate courses > when in junior high. He did logic and algebra > before starting first grade. Thats great, though also slightly scary. If it can be done while maintaining a well-rounded social personality, all the better. > Knot theory is more complex than formal group theory. Definitely. But knot theory is Ôsexier for a kid, so even though its more complex, it would be easy for a bright kid to have a big appetite for it. Come on, very few kids could be expected to learn what the requirements of a group are with as much relish as mastering the very accessible proof that every map is 5-colourable, for example. Theres a Ôcool factor there that it would be great to find in an intro to group theory. suggestions! I will check out all of those books. Chaz === Subject: Maximum productive age (Was: Re: Intuitive Group Theory - book suggestions?) As for age, my son was auditing graduate courses > when in junior high. He did logic and algebra > before starting first grade. > much about it. Ive opened a chess book once about chess prodigies. One of them said that its not too bad if you start late, but probably not much later than the age of 7. I was wondering about this question regarding maths. Not so much as a general thought (would be interesting to know), but more in a personal sense (do I stand a chance?). Now, Im pretty sure a lot has been said about it, and Id appreciate any links and references. Broadly, the question is not whether I should study maths at my age (30), because Ill do that anyway (paying for courses and books, and sitting exams can be done at any age, as long as your ego allows you to fail). The question is more about productivity - does the brain dry out at some point and becomes blocked to abstract new ideas? Has anyone in this forum got his/her Masters degree in maths after the age of 30? PhD after the age of 40? First published paper at that age range? Is there any available statistics regarding this question (Im not saying statistics should determine what an individual should/shouldnt do, but it does give some indication). Im asking specifically about maths (I know its more than possible in other areas). And as a curiosity question - are there areas in maths which are known to be a young mans game, in that no old person (after the age of 25) has ever contributed to them? Once again - Im sure a lot has been said about it, Im looking more for references than for ßame wars (or brag wars). === Subject: Re: Intuitive Group Theory - book suggestions? I agree with Ken Pledgers two suggestions. Here are some more: 1. Mathematical Groups. Tony Barnard & Hugh Neill. Teach Yourself Books, 1996. 2. Abstract Algebra and Solution by Radicals. John & Margaret Maxfield. Dover Books, 1992. 3. Introduction to Group Theory. Walter Ledermann. Longman, 1973. 4. The Fascination of Groups. F J Budden. CUP, 1972. There are lots more good entry points into group theory and Galois Theory. I envy you, introducing a young person to this topic. hth Guy Corrigall > him on to branches of math so he can see whats out there. He likes > quirky number theory stuff, he loved Conways Book of Numbers, and he > loved the book Knots and Surfaces by Farmer and Stanford, which we > went through in a blitz - he did almost all of the exercises. I was > wondering whether someone knows of an introduction to group theory that > is intuitive enough that he can get a feeling for essential concepts > without too much formalism. I feel like something that had a good > presentation of why there are only a few groups of size smaller than > some limit, and what those groups are, for example, could capture his > imagination. (He likes to know what all the possibilities are in > various contexts.) The texts I know are too terse for a kid that age, > who will have time (and patience) to do the formal stuff later, but I > think it would be good for him to discover as soon as possible that he > loves math in a broad way - if indeed he does love math in a broad way. > Hes fine on simple set stuff and linear algebra. === Subject: Re: looking for a formula to derive these numbers > URL: http://www.research.att.com/projects/OEIS?Anum=A062775 if (moebius(n)!=0) A062775(n) = n^2 > else > (fill in the blank) > endif > seqfan-mailing list. But I think, Im not going deeper into the matter, except there is some more simplification on f(i). Gottfried Helms ------------------------------------------------------------- --------------- --------- Hi Seqfans - I came across the series A062775 and have some comments, and possibly enhancements. ||> ID Number: A062775 ||> URL: http://www.research.att.com/projects/OEIS?Anum=A062775 ||> ||> Sequence: 1,4,9,24,25,36,49,96,99,100,121,216,169,196,225,448,289, ||> 396,361,600,441,484, 529,864,725,676,891,1176,841,900,961, ||> 1792,1089,1156, 1225,2376,1369,1444,1521,2400,1681,1764,1849, ||> 2904,2475,2116,2209,4032,2695,2900 ||> ||> Name: Number of Pythagorean triples mod n; i.e. number of non-congruent ||> solutions to x^2 + y^2 = z^2 mod n. ||> ||> Comments: a(n) is multiplicative and for a prime p: a(p) = p^2. ||> ||>The starting number of this series is 1. which I came across by an email in sci.math by an anonymous sender: > Ive noticed that when the moebius function of n is either -1 or +1,then the > series term is n^2. However when the moebius(n) = 0,then the interesting set > of numbers that Ive given results. First comment: (on the description) Im not able to understand the description under Name: Name: (...) i.e. number of non-congruent solutions to x^2 + y^2 = z^2 mod n. I have implemented an excel-function to determine the solutions, and this is reproducing the series perfectly. I dont understand the non-congruent-term. The entries in the sequence are just the sum of all *possible* solutions to x^2 + y^2 = z^2 mod n.This can simply by brute force be determined by creating a twodimensional matrix, where all occuring moduli of x^2,y^2 mod n are determined, and then is checked how many solutions each combined entry (x^2+y^2) (mod n) for (z^2) (mod n) allows. So the term non-congruent is misleading here in my opinion, since it introduces an idea of using only a subset of the solutions. Second comments (on computing data without need of brute force) The sequence can be seen as a composition of a systematic set of much more simple sequences. First observation is: for certain indexes i, the entry a[i] is just the square of i. It can be seen very quickly, that the sequence of the is is just the sequence of the square-free integers. Lets call this sequence c. c = [1,2,3, 5,6,7, 10,11, 13,...] or its squares: the squares of this sequence give one sub-sequence of A062776 . A1 = [1,4,9, 25,36,49, 100,121, 169,...] Removing A1 from A, or more precisely removing c from N (sequence of integers>0) leaves a sequence X1 which was posted as part of the problem in the NG-mail. A062775: 1,4,9,24,25,36,49,96,99,100,121,216,169,196,225,448,289,396,36 1,600,441,484,5 29,864,725,676,891,1176,841,900,961,1792, c1=1*c= 1,2,3 5 6 7 10 11 13 14 15 17 19 21 22 23 26 29 30 31 x1= 4 8 9 12 16 18 20 24 25 27, 28 32 x1 was the sequence a posted in the query it is the sequence of composite integers, which contain squares or higher powers as factors. Now the poster asks, what are the values in A062775 at the positions, where x1 points to? Can these values be directly computed from their index similar to that of the first partial list A1? My attempt for a solution is as follows, but I did not succeed yet. Maybe someone else has an idea - or we keep this partial sequence as some basic. Lets accordingly to the OPs question build the second partial list from A062775, Ap: A062775: 1,4,9,24,25,36,49,96,99,100,121,216,169,196,225,448,289,396,36 1,600,441,484,5 29,864,725,676,891,1176,841,900,961,1792, A1 1,4,9, ,25,36,49, ,100,121, ,169,196,225, ,289, ,361, ,441,484,529, ,676, ,841,900,961, , Ap ,24, ,96,99, ,216, 448, ,396, ,600, ,864,725, ,891,1176, 1792, That corresponds to the same operation on the index-sequences: c1=1*c= 1,2,3 5 6 7 10 11 13 14 15 17 19 21 22 23 26 29 30 31 xp= 4 8 9 12 16 18 20 24 25 27, 28 32 At xp we can see, that we can it divide in two sublists again: xp= 4 8 9 12 16 18 20 24 25 27, 28 32 ------------------------------------------------------------- --------------- ---------------------------------------------------- c2= 4 8 12 20 24 28 c3= 9 18 27, c4= 16 32 c5= 25 where each sublist c(i) starts with i^2 and follows with arithmetic progression according to the basic sequence C of the square-free integers. This separation in sublists makes sense, since for each sublist, indexing the original A062775, its entries are also in simple arithmetic progression - just contamined with a factor, lets call that f(i). The whole list A062775 now can be written as a table, where the basic index is C, the sequence of square-free integers. In this notation the * and ^-operators mean a elementswise operation: A1: f(1)* 1 * 1 * C^2 = 1/ 1 * 1 * C^2 = 1 * [ 1,4,9, 25,36,49, , ,100,121,...] A2: f(2)* 4 * 4 * C^2 = 6/ 4 * 16 * C^2 = 24 * [ 1,4,9, 25,36,49, , ,100,121,...] A3: f(3)* 9 * 9 * C^2 = 11/ 9 * 81 * C^2 = 99 * [ 1,4,9, 25,36,49, , ,100,121,...] A4: f(4)*16 *16 * C^2 = 28/16 *256 * C^2 = 448 * [ 1,4,9, 25,36,49, , ,100,121,...] ... A(i): f(i)*i^2*i^2* C^2 The sequences A(i) are then simply mixed, using their indices k pointing with i^2*k into A062775 A062775 = { A1 , A2 , A3, ... } Example: A1 1,4,9, ,25,36,49, ,100,121, ,169,196,225, ,289, ,361, ,441,484,529, ,676, ,841,900,961, A2 ,24, ,96, ,216, ,600, ,864, 1176, A3 99, ,396, ,891, A4 448, 1792 A5 725 ... a(i) = ............. ... A062775: 1,4,9,24,25,36,49,96,99,100,121,216,169,196,225,448,289,396,36 1,600,441,484,5 29,864,725,676,891,1176,841,900,961,1792 For each sublist it is obvious, that they have simple arithmetic progression, depending on their number A(i) and on an additional factor f(i), which unfortu- nately is not so easy to derive from p. The sequence of the factors f(i) seem to be somehow the core of the whole sequence A062775. I did not succeed to determine a general formula depending on i. The first entries are f(i) = { 1/1 , 6/4 , 11/9, 28/16, 29/25, .... } i { 1 2 3 4 5,.... } Some simplifications can be made, so that a scheme can be seen. First, for all i=prime f(i) is i-1 1 f(i)= 1 + ---- * --- i i Second, for powers of 2 is 1 1 1 1 f(2^i) = 1 + --- + --- + ... + --- or f(2^i) = 2 - --------- 2 4 2^i 2^(i+1) Third For powers of primes the rule seems analoguous to that of the powers of 2, but I didnt verify that. Next: For composite is it is really difficult. I looks related to the group-orders or somehow. For i=6 it is interestingly 1 1 f(6) = 1 + --- + ---- 2 3 I think, it is reasonable to see the sequence of the fs somehow as the core of sequence A062775 (and possibly some relates). Using only the nominators of f would be an interesting, and somehow core-like entry in the database. An entry for OEIS could be f . ( 1,6,11,28,29,66,55,120,105,174,131...) or, possibly more sensible, only the difference to i^2 f ( 0,2, 2,12, 4,30, 6, 56, 24, 74, 10....) where we see the simple scheme for i=prime>2 i ( 3, 5, 7, 11 f ( 2, 4, 6, 10....) And I would like to find a general formula to determine f for the composite-indexed entries too, besides the brute force-approach. Gottfried Helms ------------------------------------------------------------- --------------- --- Post scriptum: Since the Number of Pythagorean triples mod n;(...) solutions to x^2 + y^2 = z^2 mod n is related to the series of square-free composites and primes, it was interesting, to see, what is about higher exponents. For the exponent of 3 there is an equivalent entry in OEIS, but not for 4,5, or higher. Also it no more true, that for primes i the entry in the analoguous series, say, A_3[i] is always i^2 - there are different classes of primes obviously. exp=3: E_3: (...) number of solutions to x^3 + y^3 = z^3 mod n primes p in E_3, which are *not* producing p^2 as entry: 7,13,19,31,37,43,61,67,73,79,97,103,109,127,139,151,157,163,18 1,193,199,211,2 23,229,241 Empirical observation: primes==1 mod 3 exp=4: E_4: (...) number of solutions to x^4 + y^4 = z^4 mod n primes p in E_4, which are *not* producing p^2 as entry: 5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,149,157,173,18 1,193,197,223,2 29,241, This sequence is not on OEIS Empirical observation: primes==1 mod 4 exp=5: E_5: (...) number of solutions to x^5 + y^5 = z^5 mod n primes p in E_5, which are *not* producing p^2 as entry: 11,31,41,61,71,... This sequence is not on OEIS Empirical observation: primes==1 mod 5 For E_3 there are some related sequences in OEIS, but thats again a lot of stuff... === Subject: Re: looking for a formula to derive these numbers Are you the Bart Goddard who was a graduate student at UNL in the late 80s? -- Christopher Heckman === Subject: lattice vector question Hi I have one question Let us say v is a lattice vector. a is a scalar. The vector I have is r = v/a now the only thing I know is r, and the lattice generator matrix, I need to find out whether v is a lattice point or not. Any good way to do that? === Subject: concavity of functions I am sure this is a very elementary question. Given the function f(x1,x2)=(x1,x2)^(3/4) show whether it is concave. First of all, I used the Hessian matrix of the function, since a negative semidefinite Hessian implies concavity. However, the Hessian turns out to be not negative semidefinite. In this case, the function can still be concave. I then tried using the definition of concavity (f(kx+(1-k)y)>=kf(x)+(1-k)f(y)). However, I dont succeed in analytically solving this inequality due to the exponent 3/4. Are there any other solutions? Or did I miss anything? Appreciate your help. Akshay === Subject: Re: concavity of functions I am sure this is a very elementary question. Given the function f(x1,x2)=(x1,x2)^(3/4) show whether it is concave. Ill assume you mean f(x1,x2)=(x1*x2)^(3/4) . First of all, I used the Hessian matrix of the function, since a negative > semidefinite Hessian implies concavity. However, the Hessian turns out to be > not negative semidefinite. In this case, the function can still be concave. > I then tried using the definition of concavity > (f(kx+(1-k)y)>=kf(x)+(1-k)f(y)). However, I dont succeed in analytically > solving this inequality due to the exponent 3/4. Are there any other solutions? Or did I miss anything? Appreciate your help. > Akshay Try x=(1,1), y=(2,2), k=1/2. Comment. If the Hessian is not negative semidefinite, it will tell you a direction to move that will defeat concavity. So I evaluated the Hessian at (1,1). It has a positive eigenvalue 3/8 with eigenvalue (1,1), so that told me which direction to move from the point (1,1) to get the counterexample. It also has a negative eigenvalue with eigenvector (-1,1), so we can also get a counterexample to convexity. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: concavity of functions > I am sure this is a very elementary question. Given the function f(x1,x2)=(x1,x2)^(3/4) show whether it is concave. Ill assume you mean f(x1,x2)=(x1*x2)^(3/4) . ... I took it to mean inner-product (the function being defined on R^n times R^n, or more precisely those (x1,x2) for which >= 0). But the principle is the same. Your counterexample works in n = 1, ergo the general proposition is false. --Ron Bruck === Subject: good stat book? Can someone recommend a good stat book? It needs to cover how to compute the exact confidence interval for the binomial distribution. (I dont need the normal approximation). === Subject: Re: good stat book? >Can someone recommend a good stat book? It needs to cover how to >compute the exact confidence interval for the binomial distribution. >(I dont need the normal approximation). There are few good statistics books; the one I recommend is Bickel and Doksum, second edition, much better than the first edition. However, do not expect to find an exact confidence interval for the binomial distribution there or elsewhere. If you are interested in that particular problem, send me email and I will forward it to someone who has recently worked on the problem. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: How do you prove.... Hi all, How do you prove that the minimum value of sin a cos b - cos a sin b never goes below -1 Yours faithfully, Euan === Subject: Re: How do you prove.... sin a cos b - cos a sin b = sin(a-b) :-) -- Maxi === Subject: Re: How do you prove.... > Hi all, How do you prove that the minimum value of > sin a cos b - cos a sin b > never goes below -1 Yours faithfully, > Euan sin a cos b - cos a sin b = sin(a - b) sin never goes below -1 === Subject: Re: question about limits > I interpreted the above statement as a polynomial expression. > Such as I could distribute the values and somehow apply a > reduction to that expression being able to derivate it. > For example, i can derivate `ax^2 + bx + c -> `2ax + b. I > wonder if I can do the same with this infinity expression. > I thought i could express `n! as a polynomial, p(n) of nth > degree. So, what would you consider the value of (1.5)! ? Remember that the derivative relates to the effect of *small* changes to the functions argument, not integral changes. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: question about limits i know that x^x > x!, because, x^x = x * x * x * x * x * ... * x > and > x! = x * (x - 1) * (x - 2) ... * 1 so, lim(x^x/x!) = oo (infinity) i know that 2x > x, because, 2x = x + x and x = x + 0 so, lim(2x/x) = oo (infinity) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Re: question about limits >> >> i know that x^x > x!, because, >> >> x^x = x * x * x * x * x * ... * x >> and >> x! = x * (x - 1) * (x - 2) ... * 1 >> >> so, lim(x^x/x!) = oo (infinity) >i know that 2x > x, because, >2x = x + x >and >x = x + 0 >so, lim(2x/x) = oo (infinity) Ah, but if x < 0 then 2x < x and lim(2x/x) = 0. === Subject: Re: question about limits > i know that x^x > x!, because, x^x = x * x * x * x * x * ... * x > and > x! = x * (x - 1) * (x - 2) ... * 1 so, lim(x^x/x!) = oo (infinity) >>i know that 2x > x, because, >>2x = x + x >>and >>x = x + 0 >>so, lim(2x/x) = oo (infinity) Ah, but if x < 0 then 2x < x and lim(2x/x) = 0. Actually my problem is in the next step, when I get `(oo)^(1/n). /lucas === Subject: Re: question about limits by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBHI3uV22984; >I came up with a question regarding limits today, >let n be real, >x = lim { n/(n!)^(1/) } n->inf >How can I solve this problem? I tried >LHosptals rule but I dont know >the answer of > d(x!) > ------ > dx >Could somebody please help me to answer these >two questions? > /lucas ... and you are too lazy to look it up, right? phil === Subject: Re: question about limits >>I came up with a question regarding limits today, >>let n be real, >>x = lim { n/(n!)^(1/) } n->inf >>How can I solve this problem? I tried >>LHosptals rule but I dont know >>the answer of >> d(x!) >> ------ >> dx >>Could somebody please help me to answer these >>two questions? >> /lucas ... and you are too lazy to look it up, right? > You want me to derivate an infinity expression? I bet you can do this better than me. :-) I just dont have the appropriate skill to answer that, and what im doing here is looking for an answer with people that knows much more than I do. I belive its not your case. /lucas === Subject: Re: question about limits > I came up with a question regarding limits today, > let n be real, > x = lim { n/(n!)^(1/) } n->inf > How can I solve this problem? I tried > LHosptals rule but I dont know > the answer of > d(x!) > ------ > dx > Could somebody please help me to answer these > two questions? > /lucas >> ... and you are too lazy to look it up, right? > You want me to derivate an infinity expression? I bet you can > do this better than me. :-) I just dont have the appropriate > skill to answer that, and what im doing here is looking for > an answer with people that knows much more than I do. I > belive its not your case. LHospital is a terribly bad idea in this case. Stirling formula (n! = n^ne^(-n)sqrt(2pi*n) (1+1/12n+...) ) should get you on the right track (and give you the answer : e); but you will get the same answer with the much worse interval n^n/e^(n-1) /lucas === Subject: partial order and total order Hi I am reading ttms algebra book it defines particial order as: Given a set T and a relation >= on T, if the relation >= satisfies the following conditions, then it is called partial ordering a) t1>=t1 for all t1 in T b) t1 >= t2 and t2>=t3 => t1>=t3 c) t1>=t2 and t2>=t1 => t1=t2 and defines the total order by Given a set T and a partial ordering >= on T, if the relation >= satisfies the following conditions (d), then it is called total ordering d) for any t1,t2 in T we always have either t1>=t2 or t2>=t1 I dont understand the difference between those two order === Subject: Re: partial order and total order The last condition seems to state that all of the elements of the set > Hi I am reading ttms algebra book > it defines particial order as: > Given a set T and a relation >= on T, if the relation >= satisfies > the following conditions, then it is called partial ordering > a) t1>=t1 for all t1 in T > b) t1 >= t2 and t2>=t3 => t1>=t3 > c) t1>=t2 and t2>=t1 => t1=t2 > and defines the total order by > Given a set T and a partial ordering >= on T, if the relation >= satisfies > the following conditions (d), then it is called total ordering > d) for any t1,t2 in T we always have either t1>=t2 or t2>=t1 > I dont understand the difference between those two order If I understand correctly, then if you consider only the 2nd and 3rd conditions for a moment and observe that they are propositions; i.e. they are more fully stated as being: b) IF t1 >= t2 and IF t2>=t3 THEN t1>=t3 c) IF t1 >= t2 and IF t2>=t1 THEN t1=t2 These conditions do not pre-suppose that the relation >= exists between any two elements of the set, only that IF the relationship >= holds in the manner described above then the rules hold. Condtion d states that the relation holds between EVERY combination of members in the set in the manner it (condition d) sets forth. Condtion a holds for all members of the set but only in for each member in relation to itself. i.e. for each t1, t1 >= t1, but it doesnt say that for each t2 (every other element of the set other than t1) t1 >= t2 or t2>= t1 (which is condition d). I think that one example of what it calls a partial ordering is within the set of complex numbers. While one can order real numers such that 5 >= 4, and one can order the imaginary numbers such that 5i >= 4i, this ordering does not state that 5i >= 4. (While you can say that the Magnitude of 5i is greater than or equal to the Magnitude of 4, you can also say the the Magnitude of -8 is is greater than or equal to the Magnitude of 5; But the way that the integers are normally ordered, -8 is less than or equal to 5 (it is to the left of 5 on the number line). A simpler example might be the set of the integers between 1 and 10, and a dog. One can define by statute (that is without explanation) the >= relation on a dog such that condition a holds, that is that a dog >= a dog. But if you dont define the full orderintg for the integers 1 through 10 that naturally exists to include a dog, then the set is only partially ordered. ???? === Subject: Re: partial order and total order >Hi I am reading ttms algebra book >it defines particial order as: >Given a set T and a relation >= on T, if the relation >= satisfies >the following conditions, then it is called partial ordering >a) t1>=t1 for all t1 in T >b) t1 >= t2 and t2>=t3 => t1>=t3 >c) t1>=t2 and t2>=t1 => t1=t2 >and defines the total order by >Given a set T and a partial ordering >= on T, if the relation >= satisfies >the following conditions (d), then it is called total ordering >d) for any t1,t2 in T we always have either t1>=t2 or t2>=t1 >I dont understand the difference between those two order You might like to think of a partial ordering as being:- a) Reßexive x R x, b) Antisymmetric - if x R y and y R x then x = y c) Transitive - if x R y and y R z then x R z Note that (b) and (c) say IF x >= y (etc...) However, elements are not necessarily comparable in a partial order. A total order guarantees that any pair of elements are comparable. -- Jeremy Boden === Subject: Re: partial order and total order > Hi I am reading ttms algebra book > it defines particial order as: Given a set T and a relation >= on T, if the relation >= satisfies > the following conditions, then it is called partial ordering > a) t1>=t1 for all t1 in T > b) t1 >= t2 and t2>=t3 => t1>=t3 > c) t1>=t2 and t2>=t1 => t1=t2 and defines the total order by > Given a set T and a partial ordering >= on T, if the relation >= satisfies > the following conditions (d), then it is called total ordering > d) for any t1,t2 in T we always have either t1>=t2 or t2>=t1 I dont understand the difference between those two order Take the set {1,2}, let T be the set of all subsets of T. So T itself has 4 elements. Let >= be set inclusion. Then >= is a partial order on T but not a total order. === Subject: Re: Old Wizards Never Die posted: > >> Destiny Matrix. >> >> >> >> All those minds of a united ÔTau field about ones Self as a charged >> ÔYod of a symbolic focus self-created could be spinning the electron >> womb dream at ÔHay(He?) crucifying ones self as a ÔVau child nailed >> down to imprint. All those minds cohered by a future script that becomes >> as one voice trembling a plucked Ôstring over a phone within the past >> as the daughter of synchronicity dances along the signal sent back to >> weave a ÔDestiny Matrix into making a skull crystalline at Hay! >> Its a bot. Ive written one a few years ago, you can have the Perl script for free if you want. >> >Wow, thats even more incoherent than Wollmann. -- mhm 27x12 smeeter #28 Usenet Valhalla Circle #19 & #21 CEO Alcatroll Labs Inc. === Subject: Re: Old Wizards Never Die > Destiny Matrix. All those minds of a united ÔTau field about ones Self as a charged >> ÔYod of a symbolic focus self-created could be spinning the electron >> womb dream at ÔHay(He?) crucifying ones self as a ÔVau child nailed >> down to imprint. All those minds cohered by a future script that becomes >> as one voice trembling a plucked Ôstring over a phone within the past >> as the daughter of synchronicity dances along the signal sent back to >> weave a ÔDestiny Matrix into making a skull crystalline at Hay! > Wow, thats even more incoherent than Wollmann. > Not just that - it even beats Danny Mean!! He still needs work to cross the Twonky high water mark, though. -- Dan Baldwin, unethical *by design* I have always thought that the reason Dinosaurs were so big is because of the dramatic difference in gravitational strength between that time period and now -Edmo the paleontologist Christ was just an enlightened person, not unlike me. -Edmo the humble Hail the un-alive === Subject: Re: Russell-like paradoxes permission for an emailed response. (but that doesnt stop the term from being meaningful, > IMHO). As a technical term, its perfectly meaningful. In the context of > discussions of Russells paradox, it is too often used as though > referring to some set theory used at the time, e.g. by Cantor. You dont seem to be very good at keeping track. As I said, the term is used to refer to many different things, and doesnt have a single well-specified reference in the literature. Now you seem to agree: it is too often used .... Thats exactly my point: it is used in a variety of different ways, with somewhat divergent meanings. Thomas === Subject: Re: Russell-like paradoxes > Now you seem to agree: it is too often used .... Thats exactly my > point: it is used in a variety of different ways, with somewhat > divergent meanings. If we use the phrase in the sense of non-axiomatized set theory, Russells paradox did not show the naive set theory of Cantor or Dedekind to be in the least inconsistent. If we use the phrase in the sense of the first order theory with extensionality and unlimited comprehension as its only axioms, we need to recognize that this is not a description of any set theory proposed by Cantor or Dedekind or any other pioneer of the subject. === Subject: Re: Russell-like paradoxes permission for an emailed response. Now you seem to agree: it is too often used .... Thats exactly my > point: it is used in a variety of different ways, with somewhat > divergent meanings. If we use the phrase in the sense of non-axiomatized set theory, > Russells paradox did not show the naive set theory of Cantor or > Dedekind to be in the least inconsistent. If we use the phrase in the > sense of the first order theory with extensionality and unlimited > comprehension as its only axioms, we need to recognize that this is > not a description of any set theory proposed by Cantor or Dedekind or > any other pioneer of the subject. However, the phrase *is* used in print to refer to both those things (and a variety of other shades of meaning as well). Which is what I said, if you recall, when you decided to try and score points. I wish you had the decency to say sorry, I was wrong. Thomas === Subject: Re: Russell-like paradoxes : > Now you seem to agree: it is too often used .... Thats exactly my : > point: it is used in a variety of different ways, with somewhat : > divergent meanings. : If we use the phrase in the sense of non-axiomatized set theory, That is NOT the definition. And you knew that, Torkel, even if TB/BSG didnt. : Russells paradox did not show the naive set theory of Cantor or : Dedekind to be in the least inconsistent. non-axiomatized set theory is a contradiction in terms. If they didnt know what their axioms where then whatEVER it was, it WASNT a theory. But that is not even the point: OF COURSE they knew what they considered axiomatic, EVEN if they didnt have a systematic notation for it yet. : If we use the phrase in the : sense of the first order theory with extensionality and unlimited : comprehension as its only axioms, we need to recognize that this is : not a description of any set theory proposed by Cantor or Dedekind or : any other pioneer of the subject. Thats ridiculous: Frege is a pioneer of the subject. -- --- Its difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America === Subject: Re: Russell-like paradoxes permission for an emailed response. This TOPS OFF my partygoing experience! Someone I DONT LIKE is talking to me about a HEART-WARMING European film.. > And Cantor certainly did, to the extent he is rightly the first > serious discoverer of Set Theory. Cantor did indeed create set theory. Since he did not introduce any > unrestricted comprehension axiom there is no basis for the idea that > he used or introduced naive set theory in your technical sense. Is there any suggestion anywhere in his work of restrictions on comprehension? === Subject: Re: Russell-like paradoxes > Is there any suggestion anywhere in his work of restrictions on > comprehension? Certainly. Furthermore, the work in set theory by Cantor was and remains perfectly good mathematics. You can save yourself some time by looking up earlier exchanges in the group on this topic. === Subject: Re: Russell-like paradoxes : : > Is there any suggestion anywhere in his work of restrictions on : > comprehension? : Certainly. Suggestion will NOT cut it. : Furthermore, the work in set theory by Cantor furthermore? Why is set theory more an further than restrictions on comprehension and the paradoxes that can arise if you dont restrict? Surely that is a KEY aspect of set theory. : was and remains perfectly good mathematics. the work? Thats a ridiculously ambiguous locution. What EXACTLY is the extension of that predicate? EVERY LAST piece of the work? If the WHOLE work was PERFECTLY good then that would imply he had NEVER made ANY mistake -- which is inherently ridiculous, but its what you said. : You can save yourself some time by looking up earlier exchanges in : the group on this topic. No, he cant. That will waste his time. Cantor alluded to his awareness of absolutely infinite multiplicities but he did not come up with any formal machinery for explaining why they werent sets. He noticed that calling them sets would lead to contradictions but he could not explain formally why any of his earlier set-formation techniques could avoid them. but dont have one: On the one hand a multiplicity can be such that the assumption that all of its elements are together leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as one finished thing. Such multiplicities I call absolutely infinite or inconsistent multiplicities. As one easily sees, the totality of everything thinkable, for example, is such a multiplicity; later still other examples will present themselves. When, on the other hand, the totality of elements of a multiplicity can be thought of without contradiction as being together, so that their collection into one thing is possible, I call it a consistent multiplicity or a set. But the point is, THIS is NOT set THEORY. Canotr himself didnt publish these other examples, and in any case, it is NOT inconceivable for them to exist as one thing -- it is just contradictory to CALL that thing a set -- if you call it a class, the contradiction goes away. Of course, it is re-constructible in analogous terms at the class level, but that just means it is re-avoidable by one more semantic ascent. In addition to not being aware of any of this, Cantor also did not have a formalism for it, and while the intuitions may well have constituted VERY good mathematics, Torkel Franzen is just ßat out LYING if he alleges that this take on inconsitent multiplicities was PERFECTLY good mathematics: good mathematics knows what axioms its using. -- --- Its difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America === Subject: Re: in the sequel? permission for an emailed response. > Yes. Sequel comes from the Latin sequella, meaning to follow. In > papers, sometimes below is used instead of in the sequel, but if > you are refering to something which is some way ahead (e.g., in > books), below is not normally used. Latin sequella actually that which follows. The verb is sequor (infinitive, sequi). Thomas === Subject: Re: Algebraic Topology and Distributed Computing In comp.theory Michael N. Christoff Algebraic toplogy (AT herein) seems to be a good way of formalizing > protocols in distributed systems (DS herein) (such as decision problems > like > Ôconsensus). Do you know about Petri nets, process algebras such as > CCS, CSP and the Pi-calculus, and more abstract > characterizations such as labelled transition systems (LTSs)? > If not, those would be important to know before plunging into > topology. There is even a widely-accepted language (LOTOS) > based on CCS and CSP that is used for formally defining > communication protocols. Moreover PROMELA, the language for > protocol definition which is used by the SPIN model checker, is > also based partly on CSP and is also widely used. > I actually have a book on CCS (that Ill be taking another look at), but if > Im not mistaken its based on an interleaving model of concurrency. You might take a glance F-Nets (or its language implementation, Software Cabling). These are not based on an interleaving model of concurrency, but on partial orderings. F-Nets is sort of a combination between Petri Nets, finite state machines, and dataßow (and/or functions, atomic transactions, TMs). > Also, I > am most interested in architectures that do not use shared memory in favour > of distributed Ôobjects with local hidden state. Software Cabling (SC) again fits this description. You can think of it as object-oriented, data-parallel, dynamically-reconfigurable F-Nets. > One of the main reason I > was interested in AT is that it has given some very elegant proofs of some > basic impossibility theorems (ie: impossibility of consensus with one faulty > process). Are there similar proofs based on CCS, etc... ? According to some references Ive seen, some properties are provable in AT that are not in the other systems. That implies that AT is good for something, but says nothing about whether it is good for what you want to do. If you want to work on basic theory, it looks like an interesting area. In going back and looking at some of the more fundamental work in distributed systems (e.g. Lamports work), I notice how little of it seems to actually come into play in the distributed systems world I inhabit. I do think it played/plays an important role, getting the world to shift its point of view from centralized to distributed, from total orders to partial orders, etc., but once those concepts are built into the underlying tools and paradigms, some of that basic theory has done its job and its time to move on to higher level issues. Logical clocks are primarily exploiting partial orders in fairly natural ways. I would imagine that relatively few people implement distributed voting for fault detection etc. Most complexity in distributed systems seems to involve correctness (ensuring proper orderings and event sequences/protocols to get the right answer/result) and efficiency/scalability, fault tolerance/recovery (rather than distributed fault detection), security, etc. I dont know much about AT, but of those topics, Im guessing that its best application would be in security (especially insofar as it can be likened to distributed fault detection). > Finally, on a > more personal level, I have always been interested in topology and group > theory (did a lot of research into the finite state automata / group theory > connection). I have also done a small amount of work with Petri nets. Are > these process algebras a prerequisite to learning AT for distributed > computing? I doubt that they would be a prerequisite, but they may help you to appreciate what AT can and cannot deliver compared to what many other people are looking at. Would be happy to hear more insights on AT myself, -- Dave ------------------------------------------------------------- ---- David C. DiNucci Elepar Tools for portable grid, dave@elepar.com http://www.elepar.com parallel, distributed, & 503-439-9431 Beaverton, OR 97006 peer-to-peer computing === Subject: Re: Algebraic Topology and Distributed Computing >> In comp.theory Michael N. Christoff Algebraic toplogy (AT herein) seems to be a good way of formalizing >> protocols in distributed systems (DS herein) (such as decision problems > like >> Ôconsensus). >> Do you know about Petri nets, process algebras such as >> CCS, CSP and the Pi-calculus, and more abstract >> characterizations such as labelled transition systems (LTSs)? > I actually have a book on CCS (that Ill be taking another look at), but if > Im not mistaken its based on an interleaving model of concurrency. Yes, and also handshake communication. Whether or not these are bad things depends on your application and your point of view. There doesnt seem to be anything inherently bad about them for the purpose of formalizing protocols. > Also, I > am most interested in architectures that do not use shared memory in favour > of distributed Ôobjects with local hidden state. Hmm... CCS processes do not share memory; your phrase distributed Ôobjects with local hidden state seems to me to describe CCS processes very well. > One of the main reason I > was interested in AT is that it has given some very elegant proofs of some > basic impossibility theorems (ie: impossibility of consensus with one faulty > process). OK, yes, if you are trying to prove a similar undecidability result then it would probably be a good idea to go with the formalism in which a lot of similar work has been done. I have to admit I didnt know about this work in algebraic topology. I looked at the paper you gave a URL for, and was a little surprised to see no references to process algebra, but perhaps there has been work going on in this area for a while independent of process algebra. > Are > these process algebras a prerequisite to learning AT for distributed > computing? Probably not. But again, depending on your application, you might find it useful to have a choice of whether to use AT or PA for your research. Using AT for something that PA is better suited for might be like using a sledgehammer to kill a ßy. Or rather, using a large, heavy, ornate ßyswatter to kill a ßy when the little wire-and-rubber thingie would work just as well. :-) --Jamie. (nel mezzo del cammin di nostra vita) andrews .uwo } Merge these two lines to obtain my e-mail address. @csd .ca } (Unsolicited bulk e-mail costs everyone.) === Subject: Re: function norms: which is the largest? >Consider Lp norms. Is it true that >for all - say continuous and integratable - functions > >manhaten norm <= euclidiean norm <= ... <= chebishev norm > >e.g. > >L1 <= L2 <= L3 <= ... <= Linf > >There is such relationship for finite vector spaces, >so is it true for the infinite spaces? For an infinite nonatomic measure (e.g. Lebesgue measure on R) > there is no inequality between the norms and no inclusion between > the spaces. > For a probability measure (e.g. Lebesgue measure on [0,1]) > ||f||_p is a non-decreasing function of p and so L_p is a subset > of L_q if p >= q. For a nice characterization of when L^p(mu) is contained in L^q(mu) no. 7 (1985) pages 485-487). -- A. === Subject: Re: generalized Euler angles and Haar measure on SO(n): 2 questions >>Before I spend any serious time trying to think this >>through by myself, I thought it would be easier to ask >>sci.math in congress assembled to do my homework for me. >>There are two, related, parts. > >>(1) Given the standard orthonormal basis e_1,...,e_n of >>n-dimensional Euclidian space E_n, for each pair (i,j) >>with 1=>of SO(n) consisting of those special orthogonal maps >>which fix e_k for k neither i nor j, and which rotate >>the plane spanned by e_i and e_j in the standard way. >>Fixing an enumeration of the pairs (i,j), we can thus >>parametrize a subset of SO(n) by n-choose-2 angles >>(i.e., real numbers modulo 2pi). For n=3, this is >>essentially the Euler angles parametrization (I >>think), and is (therefore) onto. Is it always onto? Yes, at least if you choose the enumeration properly: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ > SO(n) = T(1,2) T(1,3) T(2,3) ... T(1,n) ... T(n-1,n). > Prove by induction. > n=1 is trivial. > Given any member A of SO(n), it maps some unit vector v > to e_n = <0,...,0,1>^T. There is some B in > T(n-1,n) T(n-2,n) ... T(1,n) that maps e_n to v. > Then AB is a member of SO(n) that fixes e_n, > and thus is of the form [ C 0 ] > [ 0 1 ] > > with C in SO(n-1). By the induction hypothesis C is > in T(1,2) ... T(n-2,n-1), and then A = AB B^(-1) is in > T(1,2) ... T(n-2,n-1) T(1,n) ... T(n-1,n). I may be mistaken, but I think that the proof can be adapted to an ARBITRARILY given enumeration of the pairs (i,j): Use the fact that for all pairwise distinct indices i,j,k,l rotations in (i,j)-direction COMMUTE with rotations in (k,l)-direction to move all T(i,n)-factors to the very right of your enumeration. - Then it only remains to be proven that SO(n) = T(*,*) T(*,*) T(*,*) ... T(i1,n) T(i2,n) ... T(i{n-1},n) Now use the argument from above: Show that for any ordering of indices i1,...,i{n-1} and given A in SO(n) there is an element B in T(i{n-1},n) ... T(i2,n) T(i1,n) which maps the vector e_n to A^{-1} e_n. ... The rest will work as in Robert Israels proof above. I would like to add a remark which is ONLY DISTANTLY related to Lee Rudolphs second question concerning the pull-back of Haar measure on SO(n) to (S^1)^(n-choose-2) via this Euler angle map. - The question reminds me of the following similar problem: The space of real-valued polynomials of degree <=n can be identified with real projective space RP^n by mapping a_0 + a_1 z + ... + a_n z^n to the point with homogenous coordinates [a_0:a_1:...:a_n]. Multiplication of polynomials gives rise to various embeddings from products of lower-dimensional projective spaces to RP^n, and the space of polynomials of degree <=n with ONLY REAL zeros can in this way be identified with the image of RP^1 x RP^1 x ... x RP^1 (n times) in RP^n So the question of what percentage of polynomials of degree <=n has only real roots could be translated to what the measure of the image of this map is, which is such a simple question, but I find it quite hard to do ANY useful calculations about it. Thomas Mautsch