mm-178 === Subject: Re: Please read my preprint for a proof of Goldbach Conjecture erdosfan@yahoo.com (Hisanobu Shinya) Conjecture. It> consists of 8 pages and I would be happy to hear your comments.> Since I am almost convinced of my argument, any mistake only you> could 'nd will be extremely helpful for my pursuit of mathematics.> The paper could be found in> http://www.geocities.com/erdosfan/solution.htmlDe'nition 2.2: The de'nition of J'(k, p) doesn't make sense. J'(k,p) also depends on the numbers a and b. And you cannot de'ne that the number of elements of J'(k, p) is >= 2; this will only be true for certain choices of k, p, a and b. === Subject: Probablility and Monopoly type gameI was wondering if anyone has done any work on the following questions.Assume a game with a board like Monopoly except that all spaces are equallylikely (no redirecting spaces), every space is capable of charging rent, andall rents are equal if the space is owned.If one wants to be sure of someone landing on one of their spaces, having 12properties in a row would do it. This intuitively would be the way tomaximize the probability of an opponent landing on your properties at leastonce. However, what if your choice is between two groups of 6 properties andsome other two groups adding up to 12 (like 5 and 7)? Is there a generalrule for maximizing the probability of an opponent landing at least once ina given circuit of the board?Also, how do the odds vary as the distance from the block of propertiesincreases?And lastly, how would you group your properties to maximize the help and insight you can give on these questions,Jim Dobie === Subject: Re: Probablility and Monopoly type gameYour problem is not formulated carefully. If all spaces are equallylikely means they all have the same probability of being landed upon,then owning any set of 12 spaces is as good as any other. Thereference to 12 indicates that your opponent moves by rolling 2 dice.This opens the chance that some locations will be visited more oftenthan others. Finally, how long does the game last? At 'rst glance, Ithink that all locations will be visited eventually, but some may notbe visited in (say) 10 rolls of the dice.A way to solve for the occupancy probabilities is to set up a Markovchain that describes the location of the opponent. The diceprobabilities give the transition probabilities of the chain, and onecan (at least numerically) solve for the probability of landing ineach location by the end of n rolls. Including Go to jail or chancecards can be included with a little extra work.Dan Heyman> I was wondering if anyone has done any work on the following questions.> Assume a game with a board like Monopoly except that all spaces are equally> likely (no redirecting spaces), every space is capable of charging rent, and> all rents are equal if the space is owned.> If one wants to be sure of someone landing on one of their spaces, having 12> properties in a row would do it. This intuitively would be the way to> maximize the probability of an opponent landing on your properties at least> once. However, what if your choice is between two groups of 6 properties and> some other two groups adding up to 12 (like 5 and 7)? Is there a general> rule for maximizing the probability of an opponent landing at least once in> a given circuit of the board?> Also, how do the odds vary as the distance from the block of properties> increases?> And lastly, how would you group your properties to maximize the expected> income for one circuit give on these questions,> Jim Dobie === Subject: Re: Probablility and Monopoly type game> I all depends on your de'nition of a bunch and on what you want to> obtain. Usually, a bunch is between 2 and 30 numbers, but the algorithm should work if it's passed more numbers. The numbers will be in the range of a 32-bit signed integer (so really a 31-bit unsigned integer).All I want is the GCD; I don't care about the multipliers.-- My email address has an extra @ (spell it out) and an extra invalid. Please remove them if you are not a Bertha Thing blogs Big Bertha Thing progressCosmic Ray SeriesPossible Real World System Constructshttp://web.onetel.com/~tonylance/progress.htmlAccess page to 6K Web page Astrophysics net ring access siteNewsgroup Reviews including uk.transportThe Progress of Discontentby T. Warton(Written at Oxford in 1746) Half-hours With the Best AuthorsThe Chandos ClassicsEdited by Charles KnightPublished by Frederick Warne and Co. 1890 Big Bertha Thing besidesSome people think that Big Bertha is less suitable for the Have they heard the long name for CP Conf? Perspective. They think that is a better place to put Big Bertha.They must be besides themselves, which proves the thing. (C) Copyright Tony Lance 1998To comply with my copyright,please distribute complete and free of charge. Tony Lancejudemarie@big-bertha-thing.comBig Bertha Thing corporate1. What happened to the guys on the most wanted list?2. Corporate America opened 're with an industrial strength denial of service attack.3. It was complete with worm virus.4. 150K mailbox postings.5. Daily basis.6. Unlimited smtp mailboxes and names.7. Ownership of smtp mailbox provider.8. This is a proprietorial interest in Usenet newsgroup postings.9. Big Bertha is now on the list.10. A sustainable defense would be nice.11. When the guns fall silent, the battle will be over.12. There were enough shells for 42 months, without recycling.Big Bertha Thing binaryThe defense is of course binary; the old one two.You need two 'lters.The 'rst blocks mailbox entry and strips out the frommailbox address, for disclosure. It uses mailbox address or su'x to select postings.The second uses partial strings.Any portion of the mailbox address or subject line toselect postings.It ¤ags them as deleted and moves them to the trashcan.The trashcan auto-deletes before the mailbox 'lls up.Lastly you need vacation auto-reply and a mailing list,with an smtp mailbox.The replies from auto reply tell you which mailboxes areclosed.The mailing list replies con'rms which mailboxes are open.NB There may be another case with the same output=gplain === Subject: Little 4line code evening.First test at a RNG with a MotorolaMC68000 processor.First try was to code the Lewis, Goodmanand Miller algorithm from 1969, but thenrealized that a long division opcode wasnecessary, and MC68000 can only divideby words.So I gave up, and tried desperately witha 32bit long dividend described in abovementioned algorithm and a 16 bit numberas divisor, I chose 65533 that is nearlythe whole of 16bits.The algorithm here described is probablynot much random, and hasn't yet beenstudied or benchmarked respectly.Anyways I found the results good enoughto be included in my last retro scenedemo intro for Atari ST. So, I havedecided to include it here, and maybeget some advisings and/or proofs.The code I present here is in assemblerand can be easily compiled. It is only4 instructions long!It bases on the machine code of theMC68000 which delivers remainder in thehigh word, quotient in the low word ofa long word as result of an unsigneddivision.The only step inbetween is to rotatethis results around in the long wordby a 'xed amount. I chose 8 bits,but probably 7 or 23 are also goodvalues.saludos, Hermann Reserved.# RND1.s# An easy way ?RND1 move.l D,d0 divu d,d0 ror.l #8,d0 move.l d0,Ddc.l D 2147483647 ;dividenddc.w d 65533 ;divisor Hermann Samso http://members.tripod.com/so_o2 === Subject: Re: Little 4line The algorithm here described ...> ... maybe> get some advisings and/or proofs.You can get code to test (pseudo)random number generators, alongwith documentation, at http://csrc.nist.gov/rng/ http://www.fourmilab.ch/random/-paul--- Paul E. Black (p.black@acm.org) First test at a RNG with a Motorola> MC68000 processor.> First try was to code the Lewis, Goodman> and Miller algorithm from 1969, but then> realized that a long division opcode was> necessary, and MC68000 can only divide> by words.Recommended reading: D. Knuth, the Art of Computing, Vol. IINumerical and Semi-Numerical algorithms. The 'rst half of thebook discusses a lot of random generators, and a couple ofmore modern algorithms than this than can be easily ported tothe === Subject: Re: Topology BasicsShmuel said:>>He then shows how the metric can de'ne a topological space.> Presumably he de'nes the topology by taking the d-open sets as a> basis. >>That sounds right, but I don't see why it must be so. The>>de'nitions of T-limit points and d-limit points are so different. > If p is a d-limit point of S, p in U and U in T, then there is an> epsilon>0 such that B(p,epsilon) C U. This point eluded me for awhile: Why must every member of T containing p embrace some d-ball B(p,epsilon)? Then I went back and saw this was a d-type property that must carry over into the d-derived topology T. In fact, William Elliot used the same argument.These concepts seem to slip in and out of focus. I'll need more practice. But thank you for the help and for making me think.By the de'nition of d-limit,> that ball intersects S. But then U intersects S. Conversely, is p is a T-limit, then for every epsilon>0, B(p,epsilon)> in T, and therefor intersects S.> === Subject: = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f)> The form of the polynomial allows me to factor P(m) into> non-polynomial factors, and the factorization with those factors is> P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)> where the a's are roots of the following cubic:> a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m).> It's been a while since you took down your FLT proofpage, so I'd appreciate your answering some questions:1. m, f, and u are ordinary (rational) integers, right?2. f is a prime, right? Can f be 3?3. u is coprime to f in Z, right?4. What are you claiming about the divisibility properties of the a's? That exactly two of them are coprime to m? That exactly two of them === Subject: Re: JSH: My quick f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f) The form of the polynomial allows me to factor P(m) into non-polynomial factors, and the factorization with those factors is> P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) where the a's are roots of the following cubic:> a^3 + 3(-1+mf^2)a^2 - f^2(m^3 f^4 - 3m^2 f^2 + 3m).> It's been a while since you took down your FLT proof> page, so I'd appreciate your answering some questions:> 1. m, f, and u are ordinary (rational) integers, right?m and f must be algebraic integers to insure that the a_i arealgebraic integers. In his examples James chosesordinary integers. u is arbitrary (see below)> 2. f is a prime, right? Can f be 3?> f is arbitrary. The question is division in the algebraicintegers where primes do not exist. In his examples Jameshas chosen f to be a prime in the integers, usually 7. > 3. u is coprime to f in Z, right?>u can be chosen coprime to f, and must be so chosen forpart of the divisibility argument. However, u is super¤uous,and can be set to 1. > 4. What are you claiming about the divisibility> properties of the a's? That exactly two of them> are coprime to m? That exactly two of them> are divisible by f?Noting that a_1, and a_2 are 0 at m = 0 Jamesconcludes that the 'rst two terms of (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)(and hence a_1 and a_2) are divisible by f for all m. - William Hughes === Subject: Re: JSH: My quick math ordinary (rational) integers, right?> m and f must be algebraic integers to insure that the a_i are> algebraic integers. In his examples James choses> ordinary integers. u is arbitrary (see below)>I took the time (heaven only knows why) to look at theFLT work and was reminded of some dependencies thatwill apply once James specializes the general resultsto FLT.In particular, eventually the variables x, y, z willbe replaced by speci'c values such that x^3 + y^3 = z^3.Then, f will be a prime divisor of x and z - y andx = uf^j with u coprime to f. In addition, m willbe a solution to a linear congruence modulo apower of f. > Noting that a_1, and a_2 are 0 at m = 0 James> concludes that the 'rst two terms of> (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)> (and hence a_1 and a_2) are divisible by f for all m.> Which we know is not the case in general. In fact,it is guaranteed not be true if James' a-polynomialis irreducible (which it is for most === Subject: Re: factorizing integers in polynomial time? > Sonja Kisa to factorize integers in polynomial time? >It's for a school project. > I don't think students at the NSA's school are allowed to ask for help.I think they are, as long as they do not spread knowledge. Anyhow theanswer to this question depends on the person you ask. It can be no,or yes, but when I tell you I have to kill you.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: factorizing integers in polynomial time?> Sonja Kisa polynomial time?>It's for a school project.> I don't think students at the NSA's school are allowed to ask for help.> I think they are, as long as they do not spread knowledge. Anyhow the> answer to this question depends on the person you ask. It can be no,> or yes, but when I tell you I have to kill you.Very funny. But to answer the question, 'rst note that it is notclear-cut. Usually when asked to do a problem involving a parametern, what is meant is a polynomial in n. That is trivial. Just tryingall possible divisors up to sqrt(n) is subpolynomial in n, namelyn^{1/2}.But for this speci'c problem, what is meant is polynomial in thenumber of digits of n and that is far more exigeant. As far as is(publicly) known, there is no such algorithm. Of course, NSA mightknow and is not telling. If I knew how to do it, I might try to sellit to some dictator, but it is hard to see how I could do it withoutgiving too much away. Offerring it to NSA is another possibility, butthat would also be very dangerous, especially if they already know itand wish to conceal that fact. Anyway, it is certain that they areworking very hard on it. 63 28 EB DA E6 44 E5 5E EC F3 04 26 4E BF 1A 92X-Zippy-Says: HUMAN REPLICAS are inserted into VATS of NUTRITIONAL YEAST...renegedd@anglickan.org (The Revd Terence what the difference was between>Israel and Israel. > The former (in single quotes, if you please) is used by those who> don't acknowledge its existence; the latter is used by those who do.This is a curious usage. Normally we don't use anything special whenreferring to things that don't exist (for example, when speaking aboutunicorns, we don't put that word in quotes).And if it doesn't exist, I wonder what you think you are referring toanyhow. If you are just allergic to the word Israel, why not refer tothe Levant, or Palestine, or something like that?Apparently you confuse shouldn't exist with doesn't exist, whichseems like a dangerous prescription for === violence.ThomasSubject: Re: nilpotent, in Hebrew BSG) asked what the difference was between>>Israel and Israel. >> The former (in single quotes, if you please) is used by those who>> don't acknowledge its existence; the latter is used by those who do.>>This is a curious usage. Normally we don't use anything special when>referring to things that don't exist (for example, when speaking about>unicorns, we don't put that word in quotes).Is there an alternative word, in this case?>And if it doesn't exist, I wonder what you think you are referring to>anyhow. If you are just allergic to the word Israel, why not refer to>the Levant, or Palestine, or something like that?I do sometimes. But using the word 'Israel' helps to show disdain forits sympathisers.>Apparently you confuse shouldn't exist with doesn't exist, which>seems like a dangerous prescription for violence.The violence already exists. === Subject: Re: nilpotent, in message constitute permission for an emailed should be hard to use.renegedd@anglickan.org (The Revd Terence what you think you are referring to>anyhow. If you are just allergic to the word Israel, why not refer to>the Levant, or Palestine, or something like that?> I do sometimes. But using the word 'Israel' helps to show disdain for> its sympathisers.Oh, well, then, why not just say the disdain and leave insults out o't. Seems all rather rude. Especially in soc.culture.israel, whichI would presume is the place to discuss Israeli culture, not to attackIsraeli politics. And entirely out of place in sci.math, so I'll stopnow.Thomas === Subject: macdonald polynomialsWhat is a andde'nitions or examples... === Subject: Re: macdonald and>de'nitions or examples...Here is the URL for the original paper by MacDonald: ftp://ftp.maths.tcd.ie/pub/EMIS/journals/SLC/opapers/ s20macdonald.pdfA couple of other papers dealing with MacDonald polynomials are http://math.berkeley.edu/~mhaiman/ftp/nfact/msri.pdfand http://math.berkeley.edu/~mhaiman/ftp/newt-sf-2001/newt.pdfRob Johnson take out the trash before replying === Subject: Re: Is Binary BrainF*** Turing-complete?> I assume the numbers 0 to 100 inclusive come from the ASCII value> of 'c' plus 1?Correct. 1 and 5 are changed to ASCII '1' (49) and ASCII 'c' (99)respectively by:-adding 3-multiplying by 5/2-multiplying by 5-subtracting 1and the resulting values are then output. This is the chunk of codethat does it: [+++[<+++++>--]<[<+>>+++++<-]>-.[-]>]> I'm not nearly good enough at BF to tell whether your> program does what you say it does,I should put together a fully commented version of that UTM program.That's the weak point in the proof at present, from a publicperspective.> and even if I could do that I> wouldn't be good enough at math to tell whether that tag system is> Turing-complete.Minsky's Computation: 'nite and in'nite machines explains howtag-systems where 2 elements are removed each time can beTuring-complete. The restriction of the right-hand side to thosebeginning with a[n]a[n] just puts a lot of junk in the string, andslows down computation, besides implying that the string can never beshortened, thus cannot terminate because its length drops below 2.None of that interferes with Turing-completeness.But I'm willing to take your word for it if you'll> explain where the numbers 100 and 23 came from -- you don't use nearly> 23 different input characters, do you?There are 24 state-symbol combinations, and the UTM program stores thecombination in a cell at some points.-Daniel Cristofani. >++[<++++++++[<[<++>-]>>[>>]+>>+[-[->>+<<<[<[<<]<+>]>[>[>>]] ]<[>>[-]]>[>[- <<]+<[<+<]]+<<]<[>+<-]>>-]<.[-]>>]http://www.hevanet.com/ cristofd/brain/ === Subject: Re: A Class of Non-Halting TMs> of the steps this SM performs> that contains the following information:> Step, current state, input, and tape position.>> Essentially, you are using at technique commonly invoked> in a lot of proofs in complexity theory in which one counts> the total number of internal con'gurations of a to known methods.Can you give references?> For a machine with N states and uses at most M tape cells using an> alphabet of 'a' symbols, the number of possible tape con'gurations> is a^M, number of tape positions for the tape head is M, number of> possible states the machine can be in is N, so the total number of> con'gurations is N*M*(a^M). If a turing machine ever exceeds this> number of steps, then it MUST have repeated an internal con'guration> and so it will loop forever... and you could always keep a running> total to the max number of tape cells used.Less than an hour after I posted I found a mistake in my method.It is amazing how things that seem obvious become less obviouswhen you describe them to other people.I think the argument you give above may be false.Consider a TM that prints the 'rst ten milliondigits of PI and then halts.3.1415926535897932384626433832795...Now, assume this TM doesn't print one long list of digits.314951265853...a=[0,1,...,9]M=3Assume N=100.N*M*(a^M) = 100*3*(10^3) = 300,000We can assume that every 3 digit number between 000 and 999is written. (This may not be a good assumption. PI may not bethe best sequence of digits to choose. I could replace PI withanother sequence like 01011011101111011111...)Clearly, this TM executes more than 300,000 steps.So, If a turing machine ever exceeds this number of steps,then it MUST have repeated an internal con'gurationand so it will loop forever may not be true.> This is essentially what you are doing, except this is a bit stronger,> I think.> Also, I think your testing for non-halting depends on REPEATING itself,> but many (in fact most) non-halting machine histories will not repeat> themselves.>> (Aside: I say 'most' will not repeat themselves because of this> intuition: if you take a nonhalting machine history which eventually> repeats, you can take the step indexes as you are using for step 1, 2, 3,> etc, and build their history as step 5,6,7,8,9,10,11,5,6,7,8,9,10,> 11,5,6,... etc. Remove the commas and put a decimal point in front of the> sequence of numbers and you build a rational number. The number of such> repeatins histories, then, is mappable to a subset of rational numbers,> whereas the number of possible [in'nite-lengthed] histories is> uncountable.)Yes, this seems obvious.Unfortunately, I now have trouble believing proofs based onthe countability of the rationals vs the uncountability of the reals.> Question:> Are there non-halting SMs with a 'nite state> transition table that this method can't determine> are non-halting?>> Yup. Make a machine that will take a all-0-tape and write> 010110111011110111110... essentially, it counts in unary, separating> numbers with a 0-symbol. This is never have one of the 'loops' that> you are detecting for.I knew the answer must be yes.I was trying to construct such a SM.I have found several since yesterday.The ¤aw in my method is the assumption that the SMhas not tampered with the input tape outside of theloop I identify. The SM could write a 1 somewhereahead of the loop that will cause it not to loop.Consider this SM:State / Input / Operation / New State0 / 0 / 1 / 11 / 1 / R / 22 / 0 / R / 33 / 0 / R / 44 / 0 / R / 55 / 0 / 1 / 55 / 1 / L / 66 / 0 / L / 66 / 1 / 1 / 0All unde'ned state/input combinations halt.History TableStep) State / Input / Position / Op / New / Input Tape0) 0 / 0 / 0 / 1 / 1 / 0...1) 1 / 1 / 0 / R / 2 / 10...2) 2 / 0 / 1 / R / 3 / 10...3) 3 / 0 / 2 / R / 4 / 100...4) 4 / 0 / 3 / R / 5 / 1000...5) 5 / 0 / 4 / 1 / 5 / 10000...6) 5 / 1 / 4 / L / 6 / 10001...7) 6 / 0 / 3 / L / 6 / 10001...8) 6 / 0 / 2 / L / 6 / 10001...My method will say steps 7-8 repeat:7) x = 2-3 = -18) 2-0 : 1-0This SM does halt:9) 6 / 0 / 1 / L / 6 / 10001...10) 6 / 1 / 0 / 1 / 0 / 10001...11) 0 / 1 / HaltsTo avoid this problem I have to keep trackof the largest position written to by the SM.Let j be the current step.Let w = largest position written to by the SM before step j.Let i-j be a sequence of steps.Let P(i) be the position of the tape at step i.x=P(j)-P(i)I have to show that P(k)+x > w for some k: i <= k <= j.I think my method works with this additional assumption.Here is an example of a SM that my method can notshow is non-halting:State / Input / Operation / New State0 / 0 / 1 / 10 / 1 / 0 / 11 / 0 / R / 21 / 1 / R / 22 / 0 / 1 / 22 / 1 / L / 0History TableStep) State / Input / Position / Op / New / Input Tape0) 0 / 0 / 0 / 1 / 1 / 0...1) 1 / 1 / 0 / R / 2 / 10...2) 2 / 0 / 1 / 1 / 2 / 10...3) 2 / 1 / 1 / L / 0 / 110...4) 0 / 1 / 0 / 0 / 1 / 110...5) 1 / 1 / 0 / R / 2 / 010...Do steps 1-5 loop?1) x = 0-0 = 02) 1-0 : 1-1no loop6) 2 / 1 / 1 / L / 0 / 010...Do steps 3-6 repeat?3) x = 1-1 = 04) 0-1 : 0-0no loop7) 0 / 0 / 0 / 1 / 1 / 010...Do steps 0-7 repeat?0) x = 0-0 = 01) skip2) 1-0 : 1-1no loop8) 1 / 1 / 0 / R / 2 / 110...9) 2 / 1 / 1 / L / 0 / 110...Reduced history table: input/position0) 0 / 02) 0 / 14) 1 / 06) 1 / 17) 0 / 09) 1 / 1Steps 4-7 are unstable.Steps 3-9 can't be a loop.Steps 6-9 could repeat:6) x = 1-1 = 07) 0-0 : 0-1no loop10) 0 / 1 / 0 / 0 / 1 / 110...Steps 4-10 can not be a loop.11) 1 / 0 / 0 / R / 2 / 010...12) 2 / 1 / 1 / L / 0 / 010...Reduced history table: input/position0) 0 / 01) 1 / 02) 0 / 13) 1 / 14) 1 / 05) 1 / 06) 1 / 17) 0 / 08) 1 / 09) 1 / 110) 1 / 011) 0 / 012) 1 / 1Steps 8-11 are unstable.Steps 6-12 can't repeat.Steps 9-12 may repeat:9) x = 1-1 = 010) 0-1 : 0-0no loop0) 0 / 0 / 0 / 1 / 1 / 0...1) 1 / 1 / 0 / R / 2 / 10...2) 2 / 0 / 1 / 1 / 2 / 10...3) 2 / 1 / 1 / L / 0 / 110...4) 0 / 1 / 0 / 0 / 1 / 110...5) 1 / 1 / 0 / R / 2 / 010...6) 2 / 1 / 1 / L / 0 / 010...7) 0 / 0 / 0 / 1 / 1 / 010...8) 1 / 1 / 0 / R / 2 / 110...9) 2 / 1 / 1 / L / 0 / 110...10) 0 / 1 / 0 / 0 / 1 / 110...11) 1 / 0 / 0 / R / 2 / 010...12) 2 / 1 / 1 / L / 0 / 010...13) 0 / 0 / 0 / 1 / 1 / 010...14) 1 / 1 / 0 / R / 2 / 110...This SM can not halt because it has no halt state.It appears my method can not show that it repeats.(Doing all this by hand is a pain.)According to Jim Nastos' formula this SM must repeat:N = 3M = 2a = 2N * M * (a^M) = 24I may have to break down and write a programto see if this is true.Notice that starting at step 3 this SM repeats the states: 201.The input starting from step 3 goes: 111, 101, 110, 101.Even if this SM repeats steps 3-27, it will never repeatthe initial segment (steps 0, 1, and 2).As near as I can tell, there is no limit on how largethis initial non-repeating segment can be.Russell- 2 many 2 count === Subject: Re: Modulus % m))%m = (a + b)%m> where a,b,m are integers.Assuming your ns should be ms and % is the mod integeroperator...a = p*m + r; 0 <= r < m; r = a % m.b = q*m + s; 0 <= s < m; s = b % m.r + s = u*m + t; 0 <= t < m; t = (r + s) % m.t = ((a % m) + (b % m)) % m;a + b = (p + q + u) + t sot = (a + b) % m.-- Paul SperryColumbia, SC (USA) === Subject: Re: Modulus propertiesPaul > where a,b,m are integers.> Assuming your ns should be ms What's an imteger?:-)Phil-- Unpatched IE vulnerability: NavigateAndFind protocol historyDescription: cross-domain scripting, cookie/data/identity theft, command executionReference: http://safecenter.net/liudieyu/NAFjpuInHistory/ NAFjpuInHistory-Content.HTMExploit: http://safecenter.net/liudieyu/NAFjpuInHistory/ NAFjpuInHistory-MyPage.HTM === Subject: Re: JSH: Let's start now because for a while I believed that some of you were> *deliberately* lying about my work, but I'm starting to think that> it's far simpler: none of you can yet grasp it.> All these dumb people have somehow made it to professor positions and> such. So clearly they have been right about enough things before. You> have done only two things in your life and for both you're getting no> respect. Statistically that means you have far less -- one might say: no> -- evidence for you being more clever than other mathematicians here.> Maybe there is an even simpler explanation than the one you just> suggested.Unfortunately, at this point the conclusion which I feel best 'ts thedata is that my work is somehow beyond the capacity of mostmathematicians.Remember, I have contacts with mathematicians all over the world todraw my inferences from, including Barry Mazur, Ralph McKenzie, AndrewGranville, Andrew Odlyzko, among others.At 'rst I was puzzled by their reactions, later I thought they wererunning away in fear, now I think they were simply lost.James Harris === Subject: Re: JSH: Let's start this point the conclusion which I feel best 'ts the> data is that my work is somehow beyond the capacity of most> mathematicians.Finding value in your work is clearly beyond any mathematician.> Remember, I have contacts with mathematicians all over the world to> draw my inferences from, including Barry Mazur, Ralph McKenzie, Andrew> Granville, Andrew Odlyzko, among others.In the case of Andrew Odlyzko, we know that he thinks you are a worthless piece of ****, except that he expresses it more politely. > At 'rst I was puzzled by their reactions, later I thought they were> running away in fear, now I think they were simply lost.Your thoughts were wrong. === Subject: Re: JSH: Let's start againIn sci.math, James Harris for a while I believed that some of you were>> *deliberately* lying about my work, but I'm starting to think that>> it's far simpler: none of you can yet grasp it.>>> All these dumb people have somehow made it to professor positions and>> such. So clearly they have been right about enough things before. You>> have done only two things in your life and for both you're getting no>> respect. Statistically that means you have far less -- one might say: no>> -- evidence for you being more clever than other mathematicians here.>>> Maybe there is an even simpler explanation than the one you just>> suggested.> Unfortunately, at this point the conclusion which I feel best 'ts the> data is that my work is somehow beyond the capacity of most> mathematicians.> Your work *is* beyond the capacity of most mathematicians.You transcend logic, you jump to conclusions, you make daringleaps over objections which we mathematicians would have to pickour way carefully through, checking conclusions all along the way.(I have a BA so I'm a mathematician. So there :-P :-) )> Remember, I have contacts with mathematicians all over the world to> draw my inferences from, including Barry Mazur, Ralph McKenzie, Andrew> Granville, Andrew Odlyzko, among others.> At 'rst I was puzzled by their reactions, later I thought they were> running away in fear, now I think they were simply lost.And you're going to lead them to the Promised Land?How....interesting of you. :-)> James Harris-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: JSH: Let's start the conclusion which I feel best 'ts the> data is that my work is somehow beyond the capacity of most> mathematicians.Feelings aren't knowledge. You 'feel' invincible, omnipotent, infallible -- you are none of those things.The simplest explanation is that you are wrong. Your 'simple' explanation is that everyone else is wrongbut you. That's neither the simples explanation nor the one which 'ts the data.> Remember, I have contacts with mathematicians all over the world to> draw my inferences from, including Barry Mazur, Ralph McKenzie, Andrew> Granville, Andrew Odlyzko, among others.The inference anyone else would have drawn was that they had made errors.> At 'rst I was puzzled by their reactions, later I thought they were> running away in fear, now I think they were simply lost.Never thought that you might be wrong, eh? Here's a thought, try taking a copy of your above memo to aquali'ed psychiatrist for an interpretation and analysis. Better yet, hire a team of world classprofessional therapists -- preferably from Vienna.> James Harris -- resident Wacko.--There are two things you must never attempt to prove: the unprovable -- and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com === Subject: Re: JSH: Let's start againX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.html>> James Harris while I believed that some of you were>> *deliberately* lying about my work, but I'm starting to think that>> it's far simpler: none of you can yet grasp it.>>> All these dumb people have somehow made it to professor positions and>> such. So clearly they have been right about enough things before. You>> have done only two things in your life and for both you're getting no>> respect. Statistically that means you have far less -- one might say: no>> -- evidence for you being more clever than other mathematicians here.>>> Maybe there is an even simpler explanation than the one you just>> suggested.>>Unfortunately, at this point the conclusion which I feel best 'ts the>data is that my work is somehow beyond the capacity of most>mathematicians.>>Remember, I have contacts with mathematicians all over the world to>draw my inferences from, including Barry Mazur, Ralph McKenzie, Andrew>Granville, Andrew Odlyzko, among others.>>At 'rst I was puzzled by their reactions, later I thought they were>running away in fear, now I think they were simply lost.Hint: A rational person would see the possibility of a muchsimpler explanation - you're simply wrong.>James Harris************************David C. Ullrich === Subject: Re: JSH: Let's start againJames Harris conclusion which I feel best 'ts the> data is that my work is somehow beyond the capacity of most> mathematicians.What data? You've never done anything that was accepted, so there is noparticular reason to believe that you are more clever than everyoneelse. Or even remotely close to other people's level.V.-- email: lastname at cs utk eduhomepage: cs utk edu tilde lastname === Subject: Re: JSH: Let's start again> James point the conclusion which I feel best 'ts the> data is that my work is somehow beyond the capacity of most> mathematicians.> What data? You've never done anything that was accepted, so there is no> particular reason to believe that you are more clever than everyone> else. Or even remotely close to other people's level.> V.I understand your frustration, and I also understand what's compellingyou to push against my statements. What I need to understand is yourdeeper motivations, and how those affect your judgement.Each and every post I make now is part of an experiment.I wish to see now how you'll react now.James Harris === Subject: Re: JSH: Let's start again jstevh@msn.com > Unfortunately, at this point the conclusion which I feel best 'ts the> data is that my work is somehow beyond the capacity of most> mathematicians.> What data? You've never done anything that was accepted, so there is no particular reason to believe that you are more clever than everyone> else. Or even remotely close to other people's level.> V.> I understand your frustration, and I also understand what's compelling> you to push against my statements. What I need to understand is your> deeper motivations, and how those affect your judgement.> Each and every post I make now is part of an experiment.> I wish to see now how you'll react now.> James HarrisAnd we may speculate on JSH's deeper motivations. So far, greed and egocentricity seem primary. And they have corrupted his judgement woefully. === Subject: Re: JSH: Let's start againJames an experiment.What?? Again with the experiments? This is a MATH forum -- get it? This is 'sci.math' not'alt.crackpot.theories'.> I wish to see now how you'll react now.Readers will react as they usually do. If you commit math errors, they will post corrections. If youengage in vitriolic diatribes they will react as they deem appropriate. In any case, don't discount thepossibility that it is YOU upon whom experiments are being conducted!> James (Wacky) Harris--There are two things you must never attempt to prove: the unprovable -- and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com === Subject: Re: JSH: Let's start againJames Harris and I also understand what's compelling> you to push against my statements. Oh? Mind you, I don't do number theory. Never even taken a course in it.I do applied stuff, numerical linear algebra, 'nite elements. You don'taffect me in any way. I'm just marveling at how someone can waste yearafter year of his life in such a futile quest. I've never seriouslytried to understand your math, but all circumstantial evidence tells methat hte chances that you're right are pretty slim. Lots of -- provably-- clever people seem to agree on it.V.-- email: lastname at cs utk eduhomepage: cs utk edu tilde lastname === Subject: Re: JSH: Let's start againsee.sig@for.addy take bemusement to mean? === Subject: Re: JSH: Let's start bemusement> What do you take bemusement to mean?Main Entry: bemuse Pronunciation: bi-'my.9fzFunction: transitive verb1 : to make confused : BEWILDERV.-- email: lastname at cs utk eduhomepage: cs utk edu tilde lastname === Subject: Re: JSH: > 1 : to make confused : BEWILDER So what's bewildering you? === Subject: Re: JSH: Let's start againTorkel Franzen fact that no amount of evidence is swaying JSH's conviction that heis utterly brilliant. I mean, this is on the level of a psychologicalcondition, and it's absolutely fascinating to watch.V.-- email: lastname at cs utk eduhomepage: cs utk edu tilde lastname === Subject: Re: JSH: Let's start no amount of evidence is swaying JSH's conviction that he> is utterly brilliant. Why should you be bewildered? You've been observing JSH for yearsand years. === Subject: Re: JSH: Let's start again > At 'rst I was puzzled by their reactions, later I thought they were> running away in fear, now I think they were simply lost.Mr. Harris,has it EVER dawned on you that maybe, just maybe, it is you who is simplylost?Eh? === Subject: Re: Collatz theorem and generalizations 3X+1 -> 3X+D -> 5X +D -> AX+D / helpfull links?> Hey,> We all know the 3X+1 conjecture is unprooved to this moment,> But I was wondering if we know by proof that f.i. the 5X+1 ( or 7X+1, ...)> analogue problem has divergent sequences.> My guess is that such a proof could be 'simpler' and maybe is 'already out> there' but i don't seem ..Possibly beacuse they're not very interesting (too many loops and divergent sequences).5x+11 6 3 16 8 4 2 1 loop2 6 3 16 8 4 2 1 loop3 6 3 16 8 4 2 1 loop4 6 3 16 8 4 2 1 loop5 66 33 166 83 416 208 104 52 26 13 loop6 6 3 16 8 4 2 1 loop7 -> oo (still increasing after 10000 iterations)8 6 3 16 8 4 2 1 loop9 -> oo (Excel over¤ow)10 66 33 166 83 416 208 104 52 26 13 loop11 -> oo (Excel over¤ow)12 6 3 16 8 4 2 1 loop13 66 33 166 83 416 208 104 52 26 13 loop14 -> oo (Excel over¤ow)15 6 3 16 8 4 2 1 loop16 6 3 16 8 4 2 1 loop17 86 43 216 108 54 27 136 68 34 17 loop18 -> oo (Excel over¤ow)19 6 3 16 8 4 2 1 loop20 66 33 166 83 416 208 104 52 26 13 loop21 -> oo (Excel over¤ow)22 -> oo (Excel over¤ow)23 -> oo (Excel over¤ow)24 6 3 16 8 4 2 1 loop25 -> oo (Excel over¤ow)26 66 33 166 83 416 208 104 52 26 13 loop27 86 43 216 108 54 27 136 68 34 17 loop28 -> oo (Excel over¤ow)29 -> oo (Excel over¤ow)30 6 3 16 8 4 2 1 loopOf course, over¤ow in Excel doesn't prove it goes to in'nity. I checked 7and it was still diverging after 10000 iterations and had reached37672243973242126250189590466703862961447835126830831007 599129695435069333933396703808144674550910025424509863761448591 244515422412356038918081364299013432234704704481074957160467938 202386235343741024825179339738747129674395982794349586221549456 233187589166275985514375496038024432112321582081777456180822740 29215266318That's 319 digits, if there's a loop involved, it's a doozy.7x+1 wasn't much better. For the few numbers I checked, I only found a single8 4 2 1 loop, but most numbers seem to diverge to in'nity.> .> Does someone know of any proof of these generalizations or can help me with> some links for more informatioon about this generalizations?> Thx in advance> Kristof Clevers === Subject: Re: Real AnalysisDavid Ullrich proof>that f_n(x_n) -> 0. That _is_ true, after all, and it happens>that people 'nd simple proofs of things that previously seemed>hard. But does not seem hard to prove seems>to indicate you don't know exactly how to prove it, and it>seems hard to prove to the rest of us, precisely because we>do not know that f_n -> 0 uniformly. Do you have an actual>plan in mind for proving it?)>No, I don't have a plan (nor a proof). rich === Subject: Re: Real Analysis>(Or who knows, maybe you _can_ come up with a simple proof>>that f_n(x_n) -> 0. That _is_ true, after all, and it happens>>that people 'nd simple proofs of things that previously seemed>>hard. But does not seem hard to prove seems>>to indicate you don't know exactly how to prove it, and it>>seems hard to prove to the rest of us, precisely because we>>do not know that f_n -> 0 uniformly. Do you have an actual>>plan in mind for proving it?)>>No, I don't have a plan (nor a proof). >Let me take that back. I either have a unplanned proof, or a planned unproof. You all can decide.Given f_n:[0,1]-->[0,1], each f_n is continuous, and f_n(x)-->0 for x in[0,1]. Prove (by elementary means) that int(f_n(x),0,1)-->0.Proof:Note int(f_n(x),0,1)=f_n(x_n) for some x_n in [0,1]. Also note if f_n(x_n)=sup{ f_n(x) } then f_n(x)=f_n(y) for all x,y in [0,1]. Clearly, {f_n(x_n)} hassome limit point. Suppose L>0 is the largest one. There is some real numbers,with 0=s for in'nitely many n_i. For eachsuch n_i there is also some closed interval, I_n_i, containing x_n_i, withf_n[I_n_i]>=s and len(I_n_i)>0. Now let m_n_i be the midpoint of each I_n_i. The set {m_n_i} has a limit point L' and therefore L' is in in'nitely manyI_n_i. But then f_n(L') >=s for in'nitely many n, contradicting f_n(x)-->0. So the unique limit point of {f_n(x_n)} is 0, as required.rich === Subject: Re: Real with a simple proof>that f_n(x_n) -> 0. That _is_ true, after all, and it happens>that people 'nd simple proofs of things that previously seemed>hard. But does not seem hard to prove seems>to indicate you don't know exactly how to prove it, and it>seems hard to prove to the rest of us, precisely because we>do not know that f_n -> 0 uniformly. Do you have an actual>plan in mind for proving it?)>No, I don't have a plan (nor a proof). >>Let me take that back. I either have a unplanned proof, or a planned unproof. >You all can decide.>>Given f_n:[0,1]-->[0,1], each f_n is continuous, and f_n(x)-->0 for x in>[0,1]. Prove (by elementary means) that int(f_n(x),0,1)-->0.>>Proof:>>Note int(f_n(x),0,1)=f_n(x_n) for some x_n in [0,1]. Also note if f_n(x_n)=sup>{ f_n(x) } then f_n(x)=f_n(y) for all x,y in [0,1]. Clearly, {f_n(x_n)} has>some limit point. Suppose L>0 is the largest one. There is some real number>s,with 0=s for in'nitely many n_i. For each>such n_i there is also some closed interval, I_n_i, containing x_n_i, with>f_n[I_n_i]>=s and len(I_n_i)>0. Now let m_n_i be the midpoint of each I_n_i. >The set {m_n_i} has a limit point L' and therefore L' is in in'nitely many>I_n_i. But then f_n(L') >=s for in'nitely many n, contradicting f_n(x)-->0. >So the unique limit point of {f_n(x_n)} is 0, as required.Two points.1. if f_{n_i}(x_{n_i}) >= s then the closed interval, I_{n_i} containing x_{n_i} with f_{n_i}(x) >= s for all x in I_{n_i} may only contain x_{n_i}. This allows len(I_{n_i}) = 0. What you really want is for f_{n_i}(x_{n_i}) > s so that for some _open_ interval I_{n_i} we have f_{n_i}(x) > s for all x in I_{n_i} and len(I_{n_i}) > 0.2. Even assuming that we are considering intervals with positive length, that the midpoints of those intervals have a limit point does not imply that any of the intervals intersect. Consider the intervals { [(1/(2n-1),1/(2n)] : n = 1, 2, 3, ... }. 0 is the limit point of their midpoints, yet 0 is in none of them, not in'nitely many.Rob Johnson take out the trash before replying === Subject: Re: Real AnalysisX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.html>>(Or who knows, maybe you _can_ come up with a simple proof>that f_n(x_n) -> 0. That _is_ true, after all, and it happens>that people 'nd simple proofs of things that previously seemed>hard. But does not seem hard to prove seems>to indicate you don't know exactly how to prove it, and it>seems hard to prove to the rest of us, precisely because we>do not know that f_n -> 0 uniformly. Do you have an actual>plan in mind for proving it?)>No, I don't have a plan (nor a proof). >>Let me take that back. I either have a unplanned proof, or a planned unproof. >You all can decide.>>Given f_n:[0,1]-->[0,1], each f_n is continuous, and f_n(x)-->0 for x in>[0,1]. Prove (by elementary means) that int(f_n(x),0,1)-->0.>>Proof:>>Note int(f_n(x),0,1)=f_n(x_n) for some x_n in [0,1]. Also note if f_n(x_n)=sup>{ f_n(x) } then f_n(x)=f_n(y) for all x,y in [0,1]. I don't see what that last statement has to do with the proof, butit's true...>Clearly, {f_n(x_n)} has>some limit point. Suppose L>0 is the largest one. There is some real number>s,with 0=s for in'nitely many n_i. For each>such n_i there is also some closed interval, I_n_i, containing x_n_i, with>f_n[I_n_i]>=s Slightly informal notation but I'm pretty sure I know what you mean.Fine so far.>and len(I_n_i)>0. Now let m_n_i be the midpoint of each I_n_i. >The set {m_n_i} has a limit point L' and therefore L' is in in'nitely many>I_n_i. Whoops. How does it follow that L' is in in'nitely many I_n_i?(It doesn't, unless you explain why it does. It doesn't follow justfrom the fact that the I_n_i are intervals with midpointsapproaching L' (or some subsequence of midpoints approachingL'). You're overlooking a possibility like this: Supposethat a_1 < b_1 < a_2 < b_2 ... < 1 and a_n -> 1. Let I_n bethe interval (a_n, b_n). Then the midpoint of I_n tends to L' = 1, but L' is not in any of the I_n.)As has (ahem) been suggested, the problem is that we don'thave f_n -> 0 uniformly. That really is the problem, althoughit might not be quite so clear here: If we had len(I_n) >= d > 0for all n _then_ we could conclude that L' was in I_n_i. Andif f_n -> 0 uniformly it would follow that we _could_ obtainlen(I_n_i) >= d > 0. (That may not be obvious, but it'strue. It's true in spite of the more obvious fact that if wehad f_n -> 0 uniformly then it would follow that len(I_n) -> 0;this _is_ a proof by contradiction after all...)(But never mind the previous paragraph about how theproblem is that we don't have uniform convergence;it's clear in any case that regardles of what the problemis, the proof doesn't work.)> But then f_n(L') >=s for in'nitely many n, contradicting f_n(x)-->0. >So the unique limit point of {f_n(x_n)} is 0, as required.>>rich>************************David C. Ullrich === Subject: Re: Real Analysis>>Let me take that back. I either have a unplanned proof, or a planned>unproof. >>You all can decide.>>Given f_n:[0,1]-->[0,1], each f_n is continuous, and f_n(x)-->0 for x in>>[0,1]. Prove (by elementary means) that int(f_n(x),0,1)-->0.>>Proof:>>Note int(f_n(x),0,1)=f_n(x_n) for some x_n in [0,1]. Also note if>f_n(x_n)=sup>>{ f_n(x) } then f_n(x)=f_n(y) for all x,y in [0,1]. >>I don't see what that last statement has to do with the proof, but>it's true...>Clearly, {f_n(x_n)} has>>some limit point. Suppose L>0 is the largest one. There is some real>number>>s,with 0=s for in'nitely many n_i. For each>>such n_i there is also some closed interval, I_n_i, containing x_n_i, with>>f_n[I_n_i]>=s >>Slightly informal notation but I'm pretty sure I know what you mean.>Fine so far.>and len(I_n_i)>0. Now let m_n_i be the midpoint of each I_n_i. >>The set {m_n_i} has a limit point L' and therefore L' is in in'nitely many>>I_n_i. >>Whoops. How does it follow that L' is in in'nitely many I_n_i?>Because each I_n_i is *closed*? (That they exist follows from the note aboutsup{f_n(x_n)} above and the fact L is the largest limit point.)>> But then f_n(L') >=s for in'nitely many n, contradicting f_n(x)-->0. >>So the unique limit point of {f_n(x_n)} is 0, as required.rich === Subject: , Ronald Bruck David C. Ullrich book,which can be solved either by>>measure theory or by Lebesque Dominated Convergence theorem,but the>>author says it is too dif'cult to handle without these means .I think>>I need help on it.>>Let fn:[0,1]--->[0,1] are continuous functions and fn--->0 for each x>>in [0,1].Then Int(fn(x)dx,0,1)--->0!>>> First note Int(f_n(x)dx,0,1)=f_n(x_n) for some x_n in [0,1]. I think the>> claim>> is that {f_n(x_n)} has a unique limit point, L, and L=0. This does not>seem>> hard to prove. Perhaps I am wrong...>>Yes. You're wrong.>>It's a nice idea, but it won't work, because in order to conclude that>f_n(x_n) --> 0, you need the f_n to converge to 0 uniformly. (In fact,>this is more or less the de'nition of uniform convergence to 0. >Anyway, it's equivalent.)>>Others have given examples. Just consider a function f_n on [0,1]>whose graph consists of the two line segments joining (0,0) to (1/n,1)>to (1,0). Now, you say that int f_n(x) dx = f_n(x_n) for some x_n in>[0,1]. But why couldn't that x_n be 1/n, where f_n(x_n) = 1? >>Because f_n(0)-->0? Again, I don't really know.>>Well, I'm missing the point to Ron's example above, because those>f_n do not tend to 0. But regardless, in case you're still not sure,>his Yes. You're wrong and his explanation why were exactly>right.>>(Or who knows, maybe you _can_ come up with a simple proof>that f_n(x_n) -> 0. That _is_ true, after all, and it happens>that people 'nd simple proofs of things that previously seemed>hard. But does not seem hard to prove seems>to indicate you don't know exactly how to prove it, and it>seems hard to prove to the rest of us, precisely because we>do not know that f_n -> 0 uniformly. Do you have an actual>plan in mind for proving it?)I think that Ron Bruck left out a point in his example; I think that hemeant f_n to be the piecewise linear function connecting (0,0), (1/n,1),(2/n,0) [the missing link], and (1,0). These functions converges to 0at each point in [0,1], but they do not converge uniformly. As he says,it is not enough that there is a sequence {x_n} of points so that |1 f ( x ) = | f (x) dx n n | 0 nsince f_n(1/n) = 1 for all n.Perhaps I am missing something too.Rob Johnson take out the trash before replying === Subject: Re: Real one book,which can be solved either by>>measure theory or by Lebesque Dominated Convergence theorem,but the>author says it is too dif'cult to handle without these means .I think>>I need help on it.>>>>Let fn:[0,1]--->[0,1] are continuous functions and fn--->0 for each x>>in [0,1].Then Int(fn(x)dx,0,1)--->0!>>>> First note Int(f_n(x)dx,0,1)=f_n(x_n) for some x_n in [0,1]. I think the>> claim>> is that {f_n(x_n)} has a unique limit point, L, and L=0. This does not>seem>> hard to prove. Perhaps I am wrong...>>Yes. You're wrong.>>It's a nice idea, but it won't work, because in order to conclude that>f_n(x_n) --> 0, you need the f_n to converge to 0 uniformly. (In fact,>this is more or less the de'nition of uniform convergence to 0. >Anyway, it's equivalent.)>>Others have given examples. Just consider a function f_n on [0,1]>whose graph consists of the two line segments joining (0,0) to (1/n,1)>to (1,0). Now, you say that int f_n(x) dx = f_n(x_n) for some x_n in[0,1]. But why couldn't that x_n be 1/n, where f_n(x_n) = 1? >>Because f_n(0)-->0? Again, I don't really know.>Well, I'm missing the point to Ron's example above, because those>f_n do not tend to 0. But regardless, in case you're still not sure,>his Yes. You're wrong and his explanation why were exactly>right.>>(Or who knows, maybe you _can_ come up with a simple proof>that f_n(x_n) -> 0. That _is_ true, after all, and it happens>that people 'nd simple proofs of things that previously seemed>hard. But does not seem hard to prove seems>to indicate you don't know exactly how to prove it, and it>seems hard to prove to the rest of us, precisely because we>do not know that f_n -> 0 uniformly. Do you have an actual>plan in mind for proving it?)> I think that Ron Bruck left out a point in his example; I think that he> meant f_n to be the piecewise linear function connecting (0,0), (1/n,1),> (2/n,0) [the missing link], and (1,0). These functions converges to 0> at each point in [0,1], but they do not converge uniformly. As he says,> it is not enough that there is a sequence {x_n} of points so that> |1> f ( x ) = | f (x) dx> n n | 0 n> since f_n(1/n) = 1 for all n.> Perhaps I am missing something too.Nope, that's exactly what I was trying to do. Create a sliding hump. Problem was, my hump didn't slide :-(--Ron Bruck === Subject: Re: Real AnalysisX-DMCA-Noti'cations: that's exactly what I was trying to do. Create a sliding hump. >Problem was, my hump didn't slide :-(I hate it when that happens.>--Ron Bruck************************David C. Ullrich === Subject: Re: Set theory with maximum setShmuel said:>>In much the same way, you can develop number theory in my system.>> Adding axioms is not the same as using the existing axioms.>>Why should they necessarily be treated differently>by set theory?>> I don't know. They're not treated differently in ZF, but they are in> your set theory.>Group theory and number theory are treated in the same way in mysystem. Ifyou go to the Numbers menu, you will 'nd options for the axioms ofboth. Each will generate a standard premise that has been hard-codedin myprogram. If you don't like these axioms, you can formulate your ownpremisein the notation I have developed -- just click a button to bring upthePremise screen, and key in whatever you want.>Call them de'nitions if you like.>> Axioms are not de'nitions. De'nitions simply allow you to abbreviate> statments. Axioms change what you can prove.>>Have you tried to use my system?>> No. Please tell me what I need to do to prove results about Euclidean> and[1] Hyperbolic Geometry.List the axioms of any geometry here, and I will demonstrate. It'sjust amatter of notation. Include any axioms for real number arithmetic,etc. that you may require. Strictly speaking, you should construct therealnumbers from PA (my current project), but in a pinch you could simplyenterthem as axioms.In, e.g., ZF I can de'ne as many> structures as I want any not worry about con¤icting axioms, because I> don't add any axioms.If postulating Peano's Axioms gets you into trouble, I don't think ZFisgoing to help. PA is derivable in ZF after all.No, I'm sure that I could shoehorn that in to> your system, but is there a way to do it that is natural and not> clumsy?>Again, list the axioms here, and I will show you.DanVisit DC Proof Online at http://www.dcproof.com -- Free Download (newrelease today) === Subject: help with rings proofHi How to prove that a ring is commutativeif it has the property that ab=ca implies === Subject: Re: help with rings proof solrac140@hotmail.com (Carlos it has the property that ab=ca implies b=c when a!=0 (not equal to 0)?Think about aba.-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: help with rings proofGerry Myerson the property that ab=ca implies b=c when a!=0 (not equal to 0)?>>Think about aba.Oh, great. Now I've got Dancing Queen stuck in my head.Why am I facing my Waterloo?dave === Subject: Re: help with rings proofX-DMCA-Noti'cations: that a ring is commutative> if it has the property that ab=ca implies b=c when a!=0 (not equal to 0)?>>Think about aba.>>Oh, great. Now I've got Dancing Queen stuck in my head.If it's any consolation, this proves you're less of an old fogie thanI am - I got the joke, but Think about aba didn't make anythingstick in my head.>Why am I facing my Waterloo?>dave************************David C. Ullrich === Subject: Re: help with rings is commutative> if it has the property that ab=ca implies b=c so.....-- Paul SperryColumbia, SC (USA) === Subject: Re: Poincare %L;%tM$D+%zkQ$zp8f/vAx*mr6T79jgxh,SC!$,8.r%HBe}KZ)iMb$tB.Z,30 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21: none in the Poincare disk, if by square you> mean an equilateral and equiangular quadrilateral. The disk is a> model for the hyperbolic plane in which angle sums of polygons are> always less than what you are used to in Euclidean geometry. You> could construct a four sided polygon in which all the sides have the> same length but the angles can't all be the same.Huh? Of course they can all be the same. They just can't all be right.-- David Eppstein http://www.ics.uci.edu/~eppstein/Univ. of California, Irvine, School of Information & Computer Science === Subject: how to calculate the pdfThere are four positive random varialbe: X,Y,Z,A. The relationship betweenthem can be explained in the following 'gure and the simulation code.|<-------------X=x------------------------->||<----Y=y--- ------->||<-----------Z=z---------->|for i=1 to N generate a sample x of X from its p.d.f. f_X(t); generate a sample y of Y based on a policy and y must be no greater thanx; generate a sample z of Z based on a policy and z must be no greater thanx; if (ythem can be explained in the following 'gure and the simulation code.>|<-------------X=x------------------------->|>>|<----Y =y---------->|>>|<-----------Z=z---------->|>>for i=1 to N>> generate a sample x of X from its p.d.f. f_X(t);>> generate a sample y of Y based on a policy and y must be no greater than>x;>> generate a sample z of Z based on a policy and z must be no greater than>x;>> if (y> the sample value a of A is equal to y>> else>> a=z>> endif>>endfor>>where N is the number of sample values and the policy can be alternatively>viewed as the respective p.d.f.>>Then, how to compute the pdf of A based on the pdf of X,Y and Z. After>testing, it seems that A =min(Y,Z) is wrong. Is this because Y and Z are>constrainted by the same length of X ? But the samples of Y and Z are>independently generated. I am confusing about this. Could you please give>some suggestions??>As you have de'ned it, A = min(Y,Z). However, it does not seem to me that Y and Z are independent, though they may be conditionally independent given X. Exactly what are the policies for Y and Z, and how do these relate to X?-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: how be determined independently from Z but depend on the pdf ofX. For example, the sample value y of Y is uniformly distributed in [0,x]where x is the sample value of X.The pdf of Z can be determined independently from Y but depend on the pdf ofX. For example, the sample value z of Z is uniformly distributed in [0, 10 *x] where x is the sample value of X.That is, both of the sample values of Y and Z are reliant on the samplevalue of X. Then,1. are Y and Z dependent?2. if pdf of Y and Z are given, can we precisely obtain the pdf of A?3. any approximation algorithm to obtain the pdf of A from the pdf of Y andZ by considering that Y and Z are virtually >There are four positive random varialbe: X,Y,Z,A. The relationshipbetween>them can be explained in the following 'gure and the simulation code.>>|<-------------X=x------------------------->|>|<----Y=y---------->|>>| <-----------Z=z---------->|>>>for i=1 to N>> generate a sample x of X from its p.d.f. f_X(t);>> generate a sample y of Y based on a policy and y must be no greaterthan>x;>> generate a sample z of Z based on a policy and z must be no greaterthanx;>> if (y> the sample value a of A is equal to y>> else>> a=z>> endif>>endfor>>where N is the number of sample values and the policy can bealternativelyviewed as the respective p.d.f.>>Then, how to compute the pdf of A based on the pdf of X,Y and Z. After>testing, it seems that A =min(Y,Z) is wrong. Is this because Y and Z areconstrainted by the same length of X ? But the samples of Y and Z are>independently generated. I am confusing about this. Could you please give>some suggestions??> As you have de'ned it, A = min(Y,Z). However, it does not seem to me> that Y and Z are independent, though they may be conditionally> independent given X. Exactly what are the policies for Y and Z,> and how do these relate to X?>> --> Stephen J. Herschkorn herschko@rutcor.rutgers.edu> === Subject: Re: how to calculate Y can be determined independently from Z but depend on the pdf of>X. For example, the sample value y of Y is uniformly distributed in [0,x]>where x is the sample value of X.>>The pdf of Z can be determined independently from Y but depend on the pdf of>X. For example, the sample value z of Z is uniformly distributed in [0, 10 *>x] where x is the sample value of X.>>That is, both of the sample values of Y and Z are reliant on the sample>value of X. Then,>1. are Y and Z dependent?>2. if pdf of Y and Z are given, can we precisely obtain the pdf of A?>3. any approximation algorithm to obtain the pdf of A from the pdf of Y and>Z by considering that Y and Z are virtually independent??>As I said before, though Y and Z are *conditionally* independent gilven X, they are *not* in general independent (not even virtually). In previous posts, I have given formulae which determine the density of A. Here is another:P{A>a} = EP(Y>a, Z>a | X) = E[P(Y>a | X) P(Z>a | X)]= int(x=0..infty, int(w=a..infty, f(w|x)) int(w=a..infty, g(w|x)) h(x))where f(.|x) and g(.|x) are respectively the conditional densities of Y and Z given X=x and h is the marginal density of X. (Again, I assume that X, Y, and Z are jointly continuous, and that X is nonnegative.) Taking the derivative, the density of A at a isint(x=0..infty, h(x) [f(a|x) int(w=a..infty, g(w|x)) + g(a|x) int(w=a..infty, f(w|x))])-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Order and === Groups>Subject: Re: Order and Groups>Message-id: Van Jacques been posting on a subject where I am far>>from expert, and in sci.math there are many experts.>>There is ->nothing<- wrong about posting on a subject where you are>far from an expert. By all means, ask questions! That is one of the>good things about sci.math (and, I don't know about others, but part>of the reason I post and read sci.math is because answering these>questions helps keep me sane when I'm running into walls with my>research: it helps remind me that I do know something...)Gee, it usually reminds me that I know nothing.>In my defense, I am just learning algebra and number theory,>>and teaching myself. >>That's 'ne. The only thing is, if you don't know much about a>subject, and people are trying to tell you that you are incorrect, you>owe them at least to read their responses carefully and address their>points (at least, at 'rst).>>-- > == ==>It's not denial. I'm just very selective about> what I accept as reality.> --- Calvin (Calvin and Hobbes)> =>>Arturo Magidin>magidin@math.berkeley.edu--MensanatorAce of Clubs === Subject: Re: Order and Groups>Subject: Re: Order and GroupsArturo Magidin--got it (the spelling, i.e.) === Subject: Re: Final Rout of Synchronization Clocks in RelativityExpires: >> It is all explained by the fact that lioght speed is source dependent.>>Not according to the Sagnac experiment.>> That is a wrong interpretation. The sagnac effect DOES comply with source>> dependency.>>Go ahead and show how you get a 'rst order result from a>source dependent model. Last time we talked, you were unable>to do this.In the case of ring gyros, the internal re¤ection causes continuousin'nitesimal speed change. When integrated, this decribes what would beexpected under source dependency. In a four mirror interferometer, the angular change while the light is intransit, coupled with the velocity 'boost' given by the moving mirrors, fullyaccounts for the fringe changes.> It refutes relativity.>>Sorry, SR predicts the result precisely.Wishful thinking!You can twist anything if you have enough faith.>>George>Henri Wilson. www.users.bigpond.com/hewn/index.htm === Subject: Re: Final Rout <1074108859.6413.0@despina.uk.clara.net>>>>Not according to the Sagnac experiment.>> That is a wrong interpretation. The sagnac effect DOES comply withsource>> dependency.>>Go ahead and show how you get a 'rst order result from a>source dependent model. Last time we talked, you were unable>to do this.>> In the case of ring gyros, the internal re¤ection causes continuous> in'nitesimal speed change.The light is launched at c relative to the material of thering (bearing in mind the effect of refractive index) andsource-dependent (Ritzian) theory says it will continue todo so. There are no 'in'nitesimal speed changes' relativeto the material.> When integrated, this decribes what would be> expected under source dependency.When integrated, the time for a transit of the ring isindependent of the motion of the gyro so they shouldn'twork.> In a four mirror interferometer, the angular change while the light is in> transit, coupled with the velocity 'boost' given by the moving mirrors,fully> accounts for the fringe changes.Again if the light is launched at c relative to the perimeter,it approaches the 'rst mirror at c+v and the mirror is movingat v son that's c relative to the mirror in Ritzian theory.Whether you claim the re¤ected relative speed is the same asthe incident speed or that it is reset to c, the speed relativeto the mirrors is always c at each re¤ection whichever way youmodel it, it doen't work.George === Subject: Re: Final Rout of Synchronization >Aleksandr Timofeev skrev i melding>>> The above are merely statements of fact. The clocks are built>>> to run at 10.22999999543MHz prior to launch so that the signal>>> received once they are in orbit is measured as 10.23MHz on the>>> ground receivers.>>>> so that the signal received once they are in orbit is measured>>> as 10.23MHz on the ground receivers.>>>> Really??? >;^)>>>> The satellite is in a state of continuous motion always.>>> Why do you have forgotten about Doppler effect?>>It doesn't work quite like that.>>The P-code (military) pseudo random pattern is transmitted>>with 10.23Mbps, but this bit rate isn't normally measured>>from the ground.>>The important point is that the 10.23MHz signal is>>the frequency standard of the satellite clocks.>>That is, the satellite clock counts 10 230 000 cycles for>>each second it advances. The clock time is transmitted.>>(Coded as a bit pattern modulated on two carriers.)>>We know the frequency must be right since the satellite>>clocks stay in synch with the GPS time, not because>>the frequency is measured from the ground.>Paul>> It is all explained by the fact that lioght speed is source dependent.>>So source dependent speed of light will make clocks in GPS orbit>fasten by 38 us a day compared to ground clocks?>>Not even funny.>Just dumb.>>Paul>Paul, let me put you straight about this.Unless a GPS clock is directly overhead, it will have a velocity component wrtthe receiver. To cut a long story short, the doppler method used by the system confuses thiseffect with another that it tries to explain using relativity.Henri Wilson. www.users.bigpond.com/hewn/index.htm === Subject: Re: Final Rout >>>>Aleksandr Timofeev skrev i melding>>> The above are merely statements of fact. The clocks are built>>> to run at 10.22999999543MHz prior to launch so that the signal>>> received once they are in orbit is measured as 10.23MHz on the>>> ground receivers.>>>> so that the signal received once they are in orbit is measured>>> as 10.23MHz on the ground receivers.>>>> Really??? >;^)>>>> The satellite is in a state of continuous motion always.>>> Why do you have forgotten about Doppler effect?>>>>It doesn't work quite like that.>>The P-code (military) pseudo random pattern is transmitted>with 10.23Mbps, but this bit rate isn't normally measured>from the ground.>>The important point is that the 10.23MHz signal is>>the frequency standard of the satellite clocks.>That is, the satellite clock counts 10 230 000 cycles for>each second it advances. The clock time is transmitted.>(Coded as a bit pattern modulated on two carriers.)>We know the frequency must be right since the satellite>clocks stay in synch with the GPS time, not because>>the frequency is measured from the ground.>>>>Paul>>> It is all explained by the fact that lioght speed is source dependent.>>So source dependent speed of light will make clocks in GPS orbit>fasten by 38 us a day compared to ground clocks?>>Not even funny.>Just dumb.>>Paul Paul, let me put you straight about this.>> Unless a GPS clock is directly overhead, it will have a velocity component wrt> the receiver.> To cut a long story short, the doppler method used by the system confuses this> effect with another that it tries to explain using relativity.You amaze me, Henry.I thought you after all this time had at least a rough demo, it was very illustrating as your demos always are.Paul === Subject: Re: Final Rout of Synchronization Clocks the 'moron of the week' prize?>>I'll let you keep it, the GPS satellites are not 'near the>surface'. The paragraph you responded to said this:>The net effect of time dilation and gravitational redshift is that>>the satellite clock appears to run fast by approximately 38 [Micro]s per>>day when compared to a similar clock at rest on the geoid, including>>the effects of the velocity of rotation and the gravitational>>potential at the Earth's surface. ....>>and you replied:> Load of crap.>> If the orbiting clocks appear to run slow it is because they DO run slow.>>They don't appear to run slow, they appear to run fast.I don't give a stuff.If you use a pendulum clock, do you think it will still run fast?>... Let me put it in Newtonian terms to keep it>>simple, gravitational force varies as r^-2 while potential>>varies as r^-1. The change of rate varies as r^-1, not r^-2.>> That is because potential and force are quantities with different>dimensions,>> idiot.>>varies as indicates the dependency, not the complete equation.>You need to learn some basics. The change of coordinate rate>matches the the dependency of potential on altitude, it does not>match the dependency of the force.Whether or not TIME itself varies in a gravitational 'eld has nought to dowith how a clock will behave in free fall.>>> .....unless the 'tick-fairies' have been active again!>>>>If a satellite signal could be received throughout its orbit the>>>ground observer could count every tick emitted, so there are no>>tick fairies.>>>> It doesn't have to be seen over the whole orbit. The orbiting clock>(OC) knows>>> how many ticks it emits per orbit. The GO also knows, simply by reading>the>> OC's time.>>Sure, but I was agreeing with you anyway.>> Funny way to agree!>>Your words suggested you don't believe in tick fairies,>and I said there are no tick fairies, it seems obvious>to me I was agreeing.At least we agree on something. Tell Paul Andersen. >> The orbit itself provides a common clock.>>Sorry, it doesn't, the duration of the orbit is not the same>>measured on the ground as in orbit.>> Oh! really!>> You obviously have the IQ of a relativist.>> One ORBIT can be used to de'ne the basic TIME unit. The orbit IS the>duration.>> If it is measured to be different, then the clocks used to measure it are>> wrong.>>If you try to use the orbit as the clock, it is impossible>to measure the duration of the orbit, your suggestion is>self-referential so useless. You cannot measure any clock>by timing it against itself.You don't have to measure the duration of the ing orbit. The orbit DEFINESthe duration. You measure all other 'time intervals' as a fraction of the orbit.Both orbiting and ground observers can do the same.>>Nothing you have said con¤icts in any way with what was stated,>>>so what were you objecting to?>>> The claim that relativity is incorporated into the GPS system. It>isn't.>...>>Regardless of your views, the clocks are built with a correction>>term applied that is not included in ground-based clocks.>> Naturally, they are corrected for the free fall effect before they are>> launched. They are software 'ne tuned when in orbit.>>Exactly so the relativistic correction _is_ built in.It is NOT a relativistic correction. It is a correction to compensate for thefact that clocks circling the Earth in free fall experience mechanical changesas well as having their characteristics altered due to cutting the magnetic'eld.> The 'GR correction' amounts to an error of only 7cms per orbit anyway.>> That is regularly corrected out empirically.>>Bollocks, if the 38us per day applied to all the craft,>there would be no positional error but there would be a>time error of, guess what, 38us per day. GPS is used to>synchronise atomic clocks to a fraction of a nanosecond.>The system would be completely useless as a time source>in a few hours if the design didn't include the correction>predicted by GR.GPS is software corrected empirically every few hours.>>George>Henri Wilson. www.users.bigpond.com/hewn/index.htm === Subject: Re: Final Rout the relativistic correction _is_ built in.The correction is calculated using the General Theory of RELATIVITYand is veri'ed to be correct within the precision of the clocks, ca. 10^-12.But:> It is NOT a relativistic correction.The only parameters used in the calculation is the gravitational'eld of the Earth, and the speed and altitude of the clocks.But:> It is a correction to compensate for the> fact that clocks circling the Earth in free fall experience mechanical changes> as well as having their characteristics altered due to cutting the magnetic> 'eld.So relativity has OBVIOUSLY nothing to do with the correction.Keep it up, Henry. It's fun!Paul, amused === Subject: Re: Final Rout of Synchronization Clocks in <1074108349.6143.0@despina.uk.clara.net>>the satellite clock appears to run fast by approximately 38 [Micro]s per>day when compared to a similar clock at rest on the geoid, including>>the effects of the velocity of rotation and the gravitational>>potential at the Earth's surface. ....>and you replied:>>> Load of crap.>> If the orbiting clocks appear to run slow it is because they DO runslow.>They don't appear to run slow, they appear to run fast.>> I don't give a stuff.Obviously, so I guess there's no point in yet again pointingout the errors in the rest of your post way with what was stated,>>so what were you objecting to?>> The claim that relativity is incorporated into the GPS system. Itisn't.> ...>>Regardless of your views, the clocks are built with a correction>>term applied that is not included in ground-based clocks.>> Naturally, they are corrected for the free fall effect before they are>> launched. They are software 'ne tuned when in orbit.>>Exactly so the relativistic correction _is_ built in.>> It is NOT a relativistic correction. ...It is a correction which was calculated using the equations ofGeneral Relativity. Call it what you like, it is built into thecraft whatever you say and the system wouldn't work without it.George === Subject: Re: halting of given machine>>> 4. It doesn't halt and we can't prove that it doesn't.>> But if we can't prove that it doesn't halt, how do we know that it> doesn't halt ?> We don't, but it is a fact. Two simple cases : 1) a TM (with no input)> programed to halt if it 'nd a odd number not sum of two primes. We are all> willing to bet it doesn't halt, but we cannot prove it.> 2) a TM generating all proofs in ZFC (by Godel numbering in ascending> order) till it 'nds a proof of 0=1. Again, we are all willing to bet it> doesn't halt, and G.9adel did show that we will never be able to prove it.No, all unsolved mathematical problems belong to the recursive set. Sowilling to bet is not good enough, because someday, someone willsolve the Goldbach conjecture(either yes or no) and it will berecursive, if it will be recursive in 100 years, it is recursive now(that's what the bi-valent algorithm says). A = { 1 iff G.c. is true,0 iff G.c. is false } is a recursive set. Please correct me if I amwrong.Now the Godel argument is more interesting, if you say that Godel didshow that it is unprovable, what does it mean ? Can you pleaseelaborate on what was the unprovable thing === Subject: Re: halting of given machineAlex that it doesn't.>> But if we can't prove that it doesn't halt, how do we know that it> doesn't halt ?>> We don't, but it is a fact. Two simple cases : 1) a TM (with no>> input) programed to halt if it 'nd a odd number not sum of two>> primes. We are all willing to bet it doesn't halt, but we cannot>> prove it. 2) a TM generating all proofs in ZFC (by Godel numbering>> in ascending order) till it 'nds a proof of 0=1. Again, we are all>> willing to bet it doesn't halt, and G.9adel did show that we will>> never be able to prove it.>> No, all unsolved mathematical problems belong to the recursive set.Ha. You don't know what you are talking about. Poincare conjecture? And evenGoldbach is not recursive, but recursively enumerable. To stay in simplenumber theory : there exist an in'nity of twin primes is not evenrecursively enumerable (nor corecursively enumerable, of course).So> willing to bet is not good enough, because someday, someone will> solve the Goldbach conjecture(either yes or no)How do you know? Undecidable problems exists. What about the Continuumhypothesis? and it will be> recursive, if it will be recursive in 100 years, it is recursive now> (that's what the bi-valent algorithm says). A = { 1 iff G.c. is true,> 0 iff G.c. is false } is a recursive set. Please correct me if I am> wrong.You are.>> Now the Godel argument is more interesting, if you say that Godel did> show that it is unprovable, what does it mean ? Can you please> elaborate on what was the unprovable thing ?Have you never heard of Godel Uncompleteness theorems? For godsake... UseGoogle. Hint : ZF cannot proveconsis (ZF) *except* if ZF is === Subject: Re: halting of given machine> Ha. You don't know what you are talking about. Poincare conjecture? And even> Goldbach is not recursive, but recursively enumerable. To stay in simple> number theory : there exist an in'nity of twin primes is not even> recursively enumerable (nor corecursively enumerable, of course).> So willing to bet is not good enough, because someday, someone will> solve the Goldbach conjecture(either yes or no)> How do you know? Undecidable problems exists. What about the Continuum> hypothesis?But the burden of proof is on you. You say that there is a TM thatdoesn't halt on a speci'c input , but you just can't prove it.Then you tell me that you are right, since I can't proove otherwise.> and it will be> recursive, if it will be recursive in 100 years, it is recursive now> (that's what the bi-valent algorithm says). A = { 1 iff G.c. is true,> 0 iff G.c. is false } is a recursive set. Please correct me if I am> wrong.> You are.So you say set A is not recursive ? Can someone else please supportthis claim ? I think it is recursive (since its 'nite).>> Now the Godel argument is more interesting, if you say that Godel did> show that it is unprovable, what does it mean ? Can you please> elaborate on what was the unprovable thing ?> Have you never heard of Godel Uncompleteness theorems? For godsake... Use> Google. Hint : ZF cannot prove> consis (ZF) *except* if ZF is inconsistent.Please stay calm, I am trying to learn. For what I know incompleteness theorem states that if you take a setof rules,there are propositions you can't prove within that same system. It isnot related to our discussion, since I repeat, if you can't provesomething, you can't say that the TM halts (since there is nothing tosupport that claim),you _just can't prove it_. But if you go outside the system, you mayprove the wanted proposition, and since proving is an algorithm, thereis a TM that will either prove or dis-prove the proposition (maybe in100 years from now), so again such a proposition is recursive. === Subject: Re: halting the Godel argument is more interesting, if you say that Godel did>> show that it is unprovable, what does it mean ? Can you please>> elaborate on what was the unprovable thing ?>> Have you never heard of Godel Uncompleteness theorems? For godsake... Use> Google. Hint : ZF cannot prove> consis (ZF) *except* if ZF is inconsistent.>Not to be picky, but I'd try Godel Incompleteness theorems, rather than uncompleteness. You'll fond a lost more hits... === Subject: Re: halting of given machine> First of all thank you all, I am beginning to understand this issue> better.> 4. It doesn't halt and we can't prove that it doesn't.>> But if we can't prove that it doesn't halt, how do we know that it> doesn't halt ?Good question!> It seems that knowing that it doesn't halt must be a series of logical> steps that shows you that it doesn't halt. This reasoning is the> proof that it doesn't halt. So how it is that it doesn't halt and> you can't prove if it doesn't ?You might be interested in a thread I started called:A Class of non-halting TMs.I describe a method of determining that a TM doesn't haltand then give examples of TMs that can't be shownto halt using this method. Of course, I must use adifferent method to show the TM doesn't halt.The easiest way to prove a TM doesn't halt is toshow that it doesn't have a halt state.> Can you give me an example of something speci'c that doesn't halt,> but you can't prove it ? Remeber that I had asked my question> regarding a speci'c machine on a speci'c input, and my argument with> my teacher was about him saying (and you have also agreed), that you> don't know if the machine halts,> but you have an algorithm (the bi-valent one).I give examples that won't work with my method.You might have to come up with one of your ownto baf¤e your teacher.Russell- 2 many 2 count === Subject: Jacobian for doing some derivations that involve Jacobian for transformation, and I have encountered the following problem, which leads me to suspect that the expression I obtained after performing the Jacobian transformation is incorrect. I very much appreciate whoever can give me some help and assistance on this. Below is the detailed derivation so that you know how I arrived at my current result, and my question comes after it. I truly appreciate your correspondence and please send your reply to eyh5@ece.cornell.edu . I have a function f(m,n) that describes the joint probability density function (jpdf) of two random variables m and n, and is written as the following: f(m,n)=-1/(2*b)^4*m*n-1/(2*b)^3*m+1/(2*b)^3*n+1/(2*b), [1]where b is a non-zero positive constant (it's just used as a parameter here), and the ranges of m and n are: 0 <= m <= 2*b, -2*b <= n <= 0 [2]Now, I de'ne two new variables x and y such that x=sqrt(m^2+n^2), y=arctan(n/m) [3]You might recognize this is very much like the change from Cartesian to polar coordinate system (in fact, the way m and n are de'ned indicates we are operating in the fourth quadrant of the Cartesian system). So, to solve for m and n in terms of x and y, we can write the following: m=x*cos(y), n=x*sin(y) [4]What I want to do is to 'nd the joint pdf of x and y, denoted as g(x,y). Since this is a case of one function with two variables, I go ahead and use the Jacobian for transformation by applying its de'nition: g(x,y)=f(m,n) * abs(J(x,y)) =f(x*cos(y), x*sin(y)) * abs(J(x,y)) [5]where abs(x) denotes the absolute value of x, and J(x,y) is the Jacobian. f(x*cos(y), x*sin(y)) is pretty straightforward and is given by: f(x*cos(y), x*sin(y)) = [6] -1/(2b)^4*x^2*sin(y)cos(y)-1/(2b)^3*x*(cos((y)-sin(y))+1/(2b)^ 2J(x,y) can also be pretty easily found by using its de'nition, and is given as: J(x,y) = x [7]Note that by way of its de'nition in [3] above, x is always positive. The joint pdf of x and y is therefore:g(x,y)=f(x*cos(y), x*sin(y)) *abs(x) [8] =-1/(2b)^4*x^3*sin(y)*cos(y)-1/(2b)^3*x^2*(cos((y)-sin(y))+1/( 2b)^2*xthe ranges for x and y are: 0 <= x <= 2*sqrt(2)*b, 3*pi/2 <= y <= 2*pi [9]Now I want to verify that I've arrived at the correct g(x,y) (with the correct bounds for x and y). To do that, I 'rst take the double integral of Eq. [1], with the bounds of m and n as de'ned in [2]. The result is 0.25. Now, I take the double integral of Eq. [8], with the bounds of x and y de'ned in [9], and the result obtained is 0.1852. There is this 0.07 difference between the two. I'm really puzzled by this discrepancy since, if I understand correctly, the Jacobian for transformation is supposed to preserve the probability before and after. So I should expect 0.25 as the result of the double integral of [8], right? But it's not. I went over my steps a couple times and am now pretty certain as far as carrying out the double-integral computations, I didn't make any mistakes. In fact, both my computation by hand and by Matlab give me the same result, 0.1852. And I also double-checked Eq. [1] to make sure for all values of m and n, f(m,n)>=0, which is a valid condition as a pdf (actually, this is a partial pdf for it occupies but one of the four quadrants; hence the value 0.25). So where could this error arise from? I would really appreciate if someone can take a look at my derivations and see if there's anything wrong or that I've overlooked when doing the Jacobian for transformation. Please send your reply to === Subject: Re: yes , Edward Hua <= n <= 0 [2]> Now, ....> m=x*cos(y), n=x*sin(y) [4]> .... > the ranges for x and y are:> 0 <= x <= 2*sqrt(2)*b, 3*pi/2 <= y <= 2*pi [9]> .... After just skimming your calculations I noticed that you've replaced integration over a square by integration over a quadrant of a circle. Could that be the problem? Kenj Pledger. === Subject: Re: Archimedes, need some help with my history essay.> Does anyone know what Archimedes corollary is that was included in his> Measurement of a Circle? Or have any ideas how i could 'nd it, as i'm> having dif'cultity 'nding it in the books i have got.Look for the Great Books of the Western World. This is a 50 or sovolume of translations into English of classic books of thewestern world.One volume is partially devoted to the works of Archimedes.Other volumes contain the works of Euclid, Apollonius, Ptolemy,Newton, and Fourier.-- Bill Hale === Subject: Largest Rectangle in a TriangleFollowing Largest Rectangle in a Quadrilateral posted here afortnight ago, one tends to ask a simpler question: What is thelargest rectangle inside a triangle ABC? Let BC be the largest side.It would appear that perpendiculars dropped onto BC from mid-points ofAB and CA (P and Q), line PQ and portion of BC projected from PQ makethe largest rectangle. It has half the area of the given triangle.Does the rectangle area increase sometimes by rotating this 'gurekeeping three corners on the sides of triangle and bringing the fourthcorner inside the triangle? inertia? is it measurable?how?If R is the ring of integers of a number 'eld K, and S is the ring o'ntegers of an extension L of K, then a prime ideal p in R is saidto remain inert in L if pS is a prime ideal too (i.e., does not factoras a product of more than one prime ideal).Even if pS does factor, one still says that a prime ideal q dividingpS in S has a degree of inertia the dimension of S/q over R/p(denoted by f(q|p)). S/q and R/p are both 'nite 'elds, with a naturalinclusion of R/p into S/q, so f(q|p) is 'nite. That would be how onewould measure inertia. Generalizations are surely possible.Or did you for some reason post your question to sci.math whenyou actually meant to post it to sci.physics?Keith Ramsay === Subject: Re: Con(PA) proved it. Is a sketch|of his proof readily available?||Was Gentzen's proof better in some sense?I just did a web search. It appears Ackermann proved the consistencyof arithmetic with open induction (induction only on quanti'er-freeformulas) in his thesis under Hilbert, using a new epsilon-substitutionmethod. I also see a reference to a paper by Ackermann provingthe consistency of all of PA using his epsilon-substitution method,but in 1940, after Gentzen's proof. So are you sure he proved theconsistency of PA before Gentzen did?I know that cut-elimination gained a lot in popularity relative to theepsilon-substitution method eventually. Not having worked seriouslywith either one, I don't know why. But I would assume this hassomething to do with why Gentzen's proof is better liked today.Keith Ramsay Genzten proved Con(PA), Wilhelm Ackermann proved it. Is a sketch> |of his proof readily available?> |> |Was Gentzen's proof better in some sense?> I just did a web search. It appears Ackermann proved the consistency> of arithmetic with open induction (induction only on quanti'er-free> formulas) in his thesis under Hilbert, using a new epsilon-substitution> method. I also see a reference to a paper by Ackermann proving> the consistency of all of PA using his epsilon-substitution method,> but in 1940, after Gentzen's proof. So are you sure he proved the> consistency of PA before that cut-elimination gained a lot in popularity relative to the> epsilon-substitution method eventually. Not having worked seriously> with either one, I don't know why. But I would assume this has> something to do with why Gentzen's proof is invalid to be removed if you're e-mailing me. === Subject: Re: Con(PA) again.George Cox Genzten proved Con(PA), Wilhelm Ackermann proved it. You have in mind Ackermann's 1940 proof? According to Szabo, Ackermann[1940, 1950-1953] adapted Gentzen's method of using trans'niteinduction to proofs which use Hilbert's epsilon-symbol. For a paperexplaining the technicalities, seewww.logic.at/people/moser/publications/ ackermann050901.pdf(Before Gentzen, Ackermann had a supposed proof of the consistency ofanalysis, but this turned out to be incorrect.) proved it.> You have in mind Ackermann's 1940 proof? According to Szabo, Ackermann> [1940, 1950-1953] adapted Gentzen's method of using trans'nite> induction to proofs which use Hilbert's epsilon-symbol. For a paper> explaining the technicalities, see> www.logic.at/people/moser/publications/ Ackermann had a supposed proof of the consistency of> analysis, but this turned out to be incorrect.)Maybe I'm you're e-mailing me. === Subject: Re: Transversal Intersection creating the standard notation>for transversal intersection (unicode U02ADB) which looks like a cap with a>mid through it. Is there any token for this symbol in TeX? If not, how>would I glue together a cap and a mid so as to reproduce it.(i) Have you checked The Comprehensive LATEX Symbol List?[*](ii) You'd better ask this on comp.text.tex[*]BTW: the answer is there, IMHO. More precisely I can see a symbolthat very much Binary Relations...Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore Aursand on comp.lang.perl.misc === Subject: did you 'nd this? I knew it must have beenincluded somewhere but no sites include it in the common set of symbols.Anyway, recently encountered the problem of creating the standardnotation> for transversal intersection (unicode U02ADB) which looks like a capwith a> mid through it. Is there any token for this symbol in TeX? If not,how> would I glue together a cap and a mid so as to reproduce it.>> e-mailing me. === Subject: Re: Transversal Intersection this? In TeXnicCenter's Online Help. (TeXnicCenter being a free TeX for MSWindows.)> I knew it must have been> included somewhere but no sites include it in the common set of problem of creating the standard> notation> for transversal intersection (unicode U02ADB) which looks like a cap> with a> mid through it. Is there any token for this symbol in TeX? If not,> how> would I glue together a cap if you're e-mailing me. === Subject: Can someone check this sigma algebra which contains the sets [0,1/4),(1/4,1/2),[1/2,3/4),[3/4,1).So I let S be such sigma algebra..my question is how many elements must S have? is it 2^4? the same as the power set? First i compute the complements of [0,1/4),(1/4,1/2),[1/2,3/4),[3/4,1) and then take unions of allthese intervals..here's my result..can someone check it?:S={empty set, [0,1], [0,1/4) , (1/4,1/2), [1/2, 3/4), [3/4,1),[1/4,1],[0,1/4] Union [1/2,1], [0,1/2) Union [3/4,1] , [0,3/4),[0,1/2]-{1/4},[0,1/4) Union [3/4,1) , (1/4,3/4) , (1/4,1/2) U [3/4,1),[1/2,1], [0,1/2] U [3/4,1] } === Subject: Re: Can someone check this sigma algebra??Carlos sigma algebra which contains the >sets [0,1/4),(1/4,1/2),[1/2,3/4),[3/4,1).>>So I let S be such sigma algebra..>my question is how many elements must S have? is it 2^4? >the same as the power set? >First i compute the complements of >[0,1/4),(1/4,1/2),[1/2,3/4),[3/4,1) and then take unions of all>these intervals..here's my result..can someone check it?:>>S={empty set, [0,1], [0,1/4) , (1/4,1/2), [1/2, 3/4), [3/4,1),>[1/4,1],[0,1/4] Union [1/2,1], [0,1/2) Union [3/4,1] , [0,3/4),>[0,1/2]-{1/4},[0,1/4) Union [3/4,1) , (1/4,3/4) , (1/4,1/2) U [3/4,1),>[1/2,1], [0,1/2] U [3/4,1] }>It looks like you want the smallest sigma-algebra which contains the given subintervals. If you meant that X = [0,1) and the last subinterval is [3/4, 1), or that X = [0,1] and the last subtinerval is [3/4, 1]), then the intervals are in a one-to-one correspondence with the singletons of 4 = {0,1,2,3}; hence there are |P(4)| = 2^4 = 16 sets in the sigma-'eld. However, if you really meant that X = [0,1] and that the last subinterval is [3/4, 1), then {1} must be in your sigma-'eld, whence there are 2^5 = 32 sets in the smallest sigma-'eld.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: TnVHEsZVYeTDhyvpoRZm-oDsDLCSqsrJGOK4RbZmNBnADInngZSTkmStephen have the interval X=[0,1]>>Find a sigma algebra which contains the>>sets [0,1/4),(1/4,1/2),[1/2,3/4),[3/4,1).>>Of course you could equally well take the standard Borel algebra (generatedby all intervals) or the sigma-algebra of all Lebesgue-measurable sets oreven the sigma-algebra consisting of all subsets of X, i.e. the power set.-- Just because you're paranoidDon't mean they're not after youreverse my forename for mail! === Subject: Re: absolute in'nity> in other words, this is a strict extension of zfc.> the main idea is that in russell's paradox, if S={x in U: x is not an element of x}, then the truth value of the statement S is in S is the third truth value. the details are in my paper.> Quantum entities are described as probability distributions, which areattributes of an underlying phase space, where theproperties-attributes such as spin and charge are not thedistributive entities, being the attributes of a coherent wavefunction. It is this wave-distribution property that then decoheresinto the ostensible wave function collapse, as waves becomestanding-spherical-wave resonances, which are condensations of spaceitself. The continual collapse-condensation of space intomatter-energy is the continual change, i.e. the property calledtime. The spherical waves, or probability distributions arerepresented by the Schrodinger wave function, psi.The continual intersection and collapse of probability distributions,also known as quantum phase entanglement, is a continual increasing ofthe total combined information of the universal wavefunction itself.Information density. With more information, more complex structurescan be created.Quantum mechanics leads us to the realization that all matter-energycan be explained in terms of waves or probability distributionscontaining information. In a con'ned region(i.e. a closed universe ora black hole) the waves exist as STANDING WAVES In a closed system,the entropy never decreases.The analogy with black holes is an interesting one but if there isnothing outside the universe, then it cannot be radiating energyoutside itself as black holes are explained to be. So the amount o'nformation i.e. quantum states in the universe is increasing. Wesee it as entropy, but to an information processor with hugecomputational capabilities, it is compressible information.The information density of the universal system must be increasing.The increase of information density is analogous to a pressuregradient.[density 1]--->[density 2]--->[density 3]---> ... --->[density n][<-[->[<-[->[U]<-]->]<-]->]Intersecting wavefronts = increasing density of spacelike slicesAs the wavefronts intersect, it becomes a mathematical computation:2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n According to conventional theories, the surface area of the horizonsurrounding a black hole, measures its entropy, where entropy isde'ned as a measure of the number of internal states that the blackhole can be in without looking different to an outside observer, whomust measure only mass, rotation, and charge. Another theory statesthat the maximum entropy of any closed region of space can neverexceed one quarter of the area of the circumscribing surface, with theentropy being the measure of the total information contained by thesystem.S' = S_m + A/4So the black hole theorists came to realize that the informationassociated with all phenomena in the three dimensional world, can bestored on a two dimensional boundary, analogous to the storing of aholographic image.Since entropy can also be de'ned as the number of states, thatof the universe must always increase, the next logical step is torealize that the spacetime density, i.e. the information encodedwithin a circumscribed region of space, must be increasing in thethermodynamic direction of time.Of course, thermodynamic entropy is popularly described as thedisorder or randomness in a physical system. In 1877, the physicistLudwig Boltzmann de'ned entropy more precisely. He de'ned it inin a system can be con'gured, while still looking like the samemacroscopic system. For example, a system such as a gas cloud, onewould count the ways that the individual gas molecules could bedistributed, and moving.In1948, mathematician Claude E. Shannon, introduced today's mostwidely used measure of information content: entropy. The Shannonentropy of a message is the number of binary digits, i.e. bitsneeded to encode it. While the structure, quality, or value, of theinformation in Shannon entropy may be an unknown, the quantity o'nformation can be known. Shannon entropy and thermodynamic entropyare equivalent.The universal laws of nature are explained in terms of symmetry. Thecompleted in'nities, mathematician Georg Cantor's in'nite sets,could be explained as cardinal identities, akin to qualia[Universally distributed attributes] from which 'nite subsets, andelements of subsets [quantum decoherence-wave function collapse] canbe derived.Completed in'nities, called alephs are distributive in nature,similar to the way that a set of red objects has the distributiveproperty of redness[qualia]. Properties, or attributes like red arenumbers in the sense that they interact algebraically according to thelaws of Boolean algebra. Take one object away from the set of redobjects and the distributive number red still describes the set. Thedistributive identity[attribute] natural number or real numberdescribes an entire collection of individual objects.The alephs can be set into a one to one correspondence with a propersubset of of themselves. The in'nite Cantorian alephs are reallydistributive[qualia].Yet, if we have a 'nite set of 7 objects, the cardinal number 7 doesnot really distribute over its individual subsets. Take anything awayfrom the set and the number 7 ceases to describe it[wave functioncollapse-condensation into speci'c localization].Symmetry is analogous to a generalized form of self evident truth, andit is a distributive attribute via the laws of nature, beingdistributed over the entire system called universe. A strati'cationof Cantorian alephs with varying degrees of complexity. Lesscomplexity = greater symmetry = higher in'nity-alephs. So the highestaleph, the absolute-in'nity distributes over the entire set calledUniverse and gives it identity.The highest symmetry is a distributive mathematical identity[also atotal unknown but possibly analogous to a state of nothingness].This fact is re¤ected in part, by the conservation laws.So an unbound-in'nite-potentia and aconstrained-'nite-bound-actuality, are somehow different yet thesame. The difference and sameness relation is a duality.Freedom(higher symmetry) and constraint-complexity-organizationalstructure(lesser symmetry) form a relation that can be described by aninvariance principle.On a ¤at Euclidean surface, the three angles of a triangle sum to 180degrees. On the curved surface of a sphere, the three angles add up tomore than 180 degrees. On the hyperbolic surface of a saddle they sumto less than 180 degrees. The three types of surface are notequivalent.There is a rotational invariance for a triangle, that seems to holdfor the three types of surface though.ABC = BCA = CAB CBA = BAC = ACBAccording to Einstein, and the CTMU of Langan, www.ctmu.org , spaceand time are modes by which we think, and not conditions in which welive. Space becomes abstract, a relation that is perceptual andmental, where distance interval between two points becomes a mentalperception.[ abstract representation]--->[semantic mapping]--->[representedsystem]An abstract representation is exactly that, abstract. It is not aspace, or time, but is instead a product of consciousness, or a mentalconstruct. Topologically it is equivalent to a point. The abstractdescription contains the concrete topology. Likewise, the concretecontains the abstract.A duality. A point contains an in'nite expanse of space and time? Could it be, that the absolute in'nity, is actually a dimensionlesspoint? Or more correctly, an in'nitesimal?Universe? = Zero?On one level of strati'cation, two photons are separate. On anotherlevel, of strati'cation, the photons have zero separation.Instantaneous communication between two objects, separated by adistance interval, is equivalent to zero separation[zero boundary]between the two objects.According to the book Gravitation, chapter 15, geometry of spacetimegives instructions to matter telling matter to follow the straightestpath, which is a geodesic. Matter in turn, tells spacetime geometryhow to curve in such a way, as to guarantee the conservation ofmomentum and energy. The Einstein tensor[geometricfeature-description] is also conserved in this relationship betweenmatter and the spacetime geometry. Eli Cartan's boundary of aboundary equals zero.A point can be de'ned as an in'nitesimal. The Topological spacesare de'ned as being diffeomorphism invariant. Intersecting cotangentbundles[manifolds] are the set of all possible con'gurations of asystem, i.e. they describe the phase space of the system.concrete localizations.http://en.wikipedia.org/wiki/Singleton_set[quote ][b]In mathematics , a singleton is a set with exactly one element. Forexample, the set {0} is a singleton. Note that a set such as {{1,2,3}}is also a singleton: the only element is a set (which itself ishowever not a singleton).A set is a singleton if and only if its cardinality is 1. In theset-theoretic construction of the natural numbers the number 1 isde'ned as the singleton {0}.[/b][/quote]The problem with set theory appears to be the primitive concept of : inside / outsideA or not-A Set theory is based on 2-valued logic of course. Since the absolutein'nite is the power set of everything that exists, ergo, an elementcannot be removed from it by de'nition, hence the talk of the removal of elements becomes wishful thinking. A fruitless exercise infutility, for the imagination. Also, the problem arises where, if,an element such as a singleton, is removed from the absolutelyin'nite, and it[absolute in'nity] somehow becomes less than itsprevious absolute condition, that seriously implies that the absolutein'nity is totally non-replenishable, but, actually, if theabsolute is generalized as the Platonic state of affairs known astotal nothingness, then it no longer has this unprecidented weaknessinherited from ZF set theory which is dooomed to die a slow andagonizing death, as die hard academians continue to ¤og the dead ZFhorse for all it's worth.[quote][b]Structures built on singletons often serve as terminal objects orzero objects of various categoriesThe statement above shows that every singleton S is a terminal objectin the category of sets and functions. No other sets are terminal inthat category.Any singleton can be turned into a topological space in just one way(all subsets are open). These singleton topological spaces areterminal objects in the category of topological spaces and continuousfunctions. No other spaces are terminal in that category.Any singleton can be turned into a group in just one way (the uniqueelement serving as identity element These singleton groups are zeroobjects in the category of groups and group homomorphisms . No othergroups are terminal in that category.[/b][/quote]An excellent idea: http://www.mmsysgrp.com/stefanik.htm[quote][b]Thesis: Scienti'c objectivity is best characterized by the concept o'nvariance as explicated in category theory than the concept of truthas explicated in mathematical logic.[/b][/quote]interaction AND conscious observers? === Subject: How to decide if a serie/collection exists??Hello to you all,I don't know the english names zo I will just call it .... a serie.I've got to proof that the next serie exist and I've got to determine thesumresult.Sn = (2^n + n) / (2*2^n + 3*n)I started ....To decide if it exists, the limit (n-> inf) should exist ........ shouldthis be the limit of Sn or should you take te limit of the sum????.......Please === Subject: Re: How to decide if a serie/collection exists??* kg.engbersen@zonnet.nl> Hello to you all,> I don't know the english names zo I will just call it .... a serie.> I've got to proof that the next serie exist and I've got to determine the> sumresult.> Sn = (2^n + n) / (2*2^n + 3*n)> I started ....> To decide if it exists, the limit (n-> inf) should exist ........ should> this be the limit of Sn or should you take te limit of the sum????> .......> Please explanation (with answer).Not from me. This looks clearly like a home work, so you have to showsome hard work yourself. What have you done so far? What do you knowabout limits? Do you have some handy rules to use?-- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i'.uio.no http://www.i'.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: How to decide if a serie/collection exists??Hello Jon,Indeed it is part of an assignment for a study (bachelor degree) which I amfollowing at the moment. (I haven't had Mathematics in 10 years)I think I know Limits very well, but I'm not very skilled in series .....that's why I ask for an explanation instead of a simple answer. (so I canmake other assignments my self)I thought about the limit because I remembered something like this.if lim(n->inf) Sn exists ..... the serie exists.if I try this ..... this limit will result in 1/2 ....... looks OK ..... butthis is not the sum of the whole serie (try some values and take the sum).That's why I want to know if my way of using the limit of the Sn is corrector should this be of the sum???How to decide the sum ??KJon Haugsand schreef in bericht> * kg.engbersen@zonnet.nl> Hello to you all,>> I don't know the english names zo I will just call it .... a serie.>> I've got to proof that the next serie exist and I've got to determinethe> sumresult.>> Sn = (2^n + n) / (2*2^n + 3*n)>> I started ....> To decide if it exists, the limit (n-> inf) should exist ........should> this be the limit of Sn or should you take te limit of the sum????> .......>> Please explanation (with answer).>> Not from me. This looks clearly like a home work, so you have to show> some hard work yourself. What have you done so far? What do you know> about limits? Do you have some handy rules to use?>> -- > Jon Haugsand> Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i'.uio.no> http://www.i'.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: How to decide if a serie/collection exists?? Subteam skilled in series .....> that's why I ask for an explanation instead of a simple answer. (so I can> make other assignments my self)> I thought about the limit because I remembered something like this.> if lim(n->inf) Sn exists ..... the serie exists.> if I try this ..... this limit will result in 1/2 ....... looks OK ..... but> this is not the sum of the whole serie (try some values and take the sum).Sn = (2^n + n) / (2*2^n + 3*n). You think the limit of Sn is 1/2? Then you do not know limits very well. (Put away the Matlab; it's destroying your mind.) You need some serious help with series. It's a huge waste of time to attempt these problems without a good understanding of the basics. === Subject: Re: How to decide 8bitThe World Wide Wade know Limits very well, but I'm not very skilled in series .....> that's why I ask for an explanation instead of a simple answer. (so I can> make other assignments my self) I thought about the limit because I remembered something like this.> if lim(n->inf) Sn exists ..... the serie exists.> if I try this ..... this limit will result in 1/2 ....... looks OK ..... but> this is not the sum of the whole serie (try some values and take the sum).> Sn = (2^n + n) / (2*2^n + 3*n). You think the limit of Sn is 1/2? Then you > do not know limits very well. (Put away the Matlab; it's destroying your > mind.) You need some serious help with series. It's a huge waste of time to > attempt these problems without a good understanding of the basics.Looks to me like lim(S_n, n = in'nity) = 1/2.-- Paul SperryColumbia, SC (USA) === Subject: Re: How to decide if a serie/collection + n) / (2*2^n + 3*n). You think the limit of Sn is 1/2? Then you > do not know limits very well. (Put away the Matlab; it's destroying your > mind.) You need some serious help with series. It's a huge waste of time to > attempt these problems without a good understanding of the basics.> Looks to me like lim(S_n, n = in'nity) = 1/2.Yes, you and subteam are quite right. I carelessly misread 3*n as 3^n. My apologies to subteam for the comment on limits. === Subject: Re: How to decide if a serie/collection exists??Apol. acceptedBut I don't understand the thing you said to me on the other newsgroup Do not do this. (sending messages to more groups) It prevents people over here from seeing what is written overthere, and vice-versaWhy is this ......? Do you read all you news together???(Changing the messagetitle would then solve this????)The World Wide Wade schreef in> Paul Sperry 3*n). You think the limit of Sn is 1/2? Thenyou> do not know limits very well. (Put away the Matlab; it's destroyingyour> mind.) You need some serious help with series. It's a huge waste oftime to> attempt these problems without a good understanding of the basics.>> Looks to me like lim(S_n, n = in'nity) = 1/2.>> Yes, you and subteam are quite right. I carelessly misread 3*n as 3^n. My> apologies to subteam for the comment on limits. === Subject: Re: about series I wouldn'thave asked the question!!!!!!!!!!!!!Or is this newsgroup only for people who already know everthing !?Who's waste of time is it .... mine or yours .........Just searching for some help with it ....The World Wide Wade schreef in> Sn = (2^n + n) / (2*2^n + 3*n). You think the limit of Sn is 1/2? Then you> do not know limits very well. (Put away the Matlab; it's destroying your> mind.) You need some serious help with series. It's a huge waste of timeto> attempt these problems without a good understanding of the basics. === Subject: Re: How Jon,>>Indeed it is part of an assignment for a study (bachelor degree) which I am>following at the moment. (I haven't had Mathematics in 10 years)>>I think I know Limits very well, but I'm not very skilled in series .....>that's why I ask for an explanation instead of a simple answer. (so I can>make other assignments my self)This is very basic series stuff so you should maybe pick up some bookthat covers them before trying to solve any problems.>I thought about the limit because I remembered something like this.>>if lim(n->inf) Sn exists ..... the serie exists.>if I try this ..... this limit will result in 1/2 ....... looks OK ..... but>this is not the sum of the whole serie (try some values and take the sum).What happens when you sum together an in'nite number of terms thatare very close to 1/2? === Subject: Re: How to decide if a serie/collection exists??Hello Toni,I picked up a book ..... looked at the basics .... understand the basics.........but can't 'nd the right starting point to solve this one.I already saw that if take several values (e.g. n=0,1,2) ....the sum isalready >1/2 ....... so lim(n->inf)Sn doesn't give me the sumvalue I waslooking for.Greets KimballToni Lassila >>Indeed it is part of an assignment for a study (bachelor degree) which Iam>following at the moment. (I haven't had Mathematics in 10 years)>>I think I know Limits very well, but I'm not very skilled in series .....>that's why I ask for an explanation instead of a simple answer. (so I can>make other assignments my self)>> This is very basic series stuff so you should maybe pick up some book> that covers them before trying to solve any problems.>>I thought about the limit because I remembered something like this.>>if lim(n->inf) Sn exists ..... the serie exists.>if I try this ..... this limit will result in 1/2 ....... looks OK .....but>this is not the sum of the whole serie (try some values and take thesum).>> What happens when you sum together an in'nite number of terms that> are very close to 1/2? === Subject: Re: How to decide if a serie/collection exists??* kg.engbersen@zonnet.nl> Hello Jon,(Please put quotes in front of your own text.)> I think I know Limits very well, but I'm not very skilled in series .....> that's why I ask for an explanation instead of a simple answer. (so I can> make other assignments my self)> I thought about the limit because I remembered something like this.> if lim(n->inf) Sn exists ..... the serie exists.> if I try this ..... this limit will result in 1/2 ....... looks OK ..... but> this is not the sum of the whole serie (try some values and take the sum).A series is a sum. Let An=1, thus lim(n->inf) An clearly exists andis equal to 1. But the series, i.e. the sum over An as n->inf doesnot exist.The terms must converge to 0 if the limit of the sum is to exist,quite fast also.> That's why I want to know if my way of using the limit of the Sn is correct> or should this be of the sum???> How to decide the sum ??Try to 'nd out what lim(n->inf) Sn is.-- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i'.uio.no http://www.i'.uio.no/~jonhaug/, Phone: +47 22 95 21 52 === Subject: Re: How to decide if a serie/collection exists??Hello Jon,lim(n->inf) Sn = 1/2 (manually of with Matlab)A serie where the next value is the last value + some other value: An+1 = An+ d (the difference d = An+1 - An)....... is for n-> inf always inf .... sum does not exist.But a serie where you have the next value is the last value times aconstant: An+1= An * r ( r = An+1 / An) ......... n-> inf .... sumonly exist when |r| < 1.How to decide of Sn belongs to the 'rst one or the second one or somethingcompletely different. (and what is the r??)(I'm trying to split Sn .... but no succes)GreetsKimballJon Haugsand schreef in bericht> * kg.engbersen@zonnet.nl> Hello Jon,>> (Please put quotes in front of your own text.)>> I think I know Limits very well, but I'm not very skilled in series.....> that's why I ask for an explanation instead of a simple answer. (so Ican> make other assignments my self)>> I thought about the limit because I remembered something like this.>> if lim(n->inf) Sn exists ..... the serie exists.> if I try this ..... this limit will result in 1/2 ....... looks OK .....but this is not the sum of the whole serie (try some values and take thesum).>> A series is a sum. Let An=1, thus lim(n->inf) An clearly exists and> is equal to 1. But the series, i.e. the sum over An as n->inf does> not exist.>> The terms must converge to 0 if the limit of the sum is to exist,> quite fast also.>>> That's why I want to know if my way of using the limit of the Sn iscorrect> or should this be of the sum??? How to decide the sum ??>> Try to 'nd out what lim(n->inf) Sn is.>> -- > Jon Haugsand> Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i'.uio.no> http://www.i'.uio.no/~jonhaug/, Phone: +47 22 95 21 52> Den Ende Konrad Den Ende ( (n+1)5^n ) ]>>Proving that the series converges is pretty simple but how>do i go about computing it's value? I suppose it starts with>extending the fraction somehow but that's how far i get...Consider the function oo --- 1 n f(x) = > --- x --- n+1 n=0Your sum is just f(3/5). Compute the derivative of xf(x): oo --- n (xf(x))' = > x --- n=0 1 = --- 1-xSolving for f(x), and noting that f(0) = 1, gives f(x) = -log(1-x)/xSo f(3/5) = 5/3 log(5/2) which is about 1.52715122.Rob Johnson take out the trash before replying === Subject: Re: How to compute this sum?En el escribi.97:> sum[ n=0 , n=oo , (3^n) / ( (n+1)5^n ) ]>> Proving that the series converges is pretty simple but how> do i go about computing it's value? I suppose it starts with> extending the fraction somehow but that's how far i get...Ln(1 + x) = Sum((-1)^nx^n/n, n, 1, inf) ==>-Ln(1 - x) = Sum(x^n/n, n, 1, inf) = x*Sum(x^(n-1)/n, n, 1, inf) = x*Sum(x^n/(n+1), n, 0, inf) ==>S(x) = Sum(x^n/(n+1), n, 0, inf) = -Ln(1 - x)/x (all if |x| < 1)Then your sum is S(3/5) = -Ln(1 - 3/5))/(3/5) = (5/3)Ln(5/2) ~= 1.527151219...-- Best (Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: NG!>> And to you Abhi, forget the device and have a ___________________________________________>> Mere Ghar Ka Seedha Sa Itna Pataa Hai> Ye Ghar Jo Hai Chaaron Taraf Se Khula Hai> Na Dastak Zaruri, Na Aavaz Dena> Mere Ghar Ka Darvaaza Koi Nahin Hai> Hain Deevaren Gum Aur Chhat Bhi Nahin Hai> Badhi Dhoop Hai Dost> Kadhi Dhoop Hai Dost> Tere Aanchal Ka Saaya Churake Jeena Hai Jeena> Jeena Zindagi, Zindagi> O Zindagi Mere Ghar Aana> Aana Zindagi Zindagi Mere Ghar Aana> _______________________________________________>> I am still alive and back in comfort of my own room from where all this saga began in March 2000.>> I got new beautiful computer, listening beautiful songs right now in> calm night.>> Happy new year to all of you!>> I am still in Action...>> -Abhi.> Don't you realize that They are watching you through> the new computer....Some lines for my beloved Action Device..____________________________________________Naino Se Behte Ashqo Ke Dharo MeinHumne Tujhko Dekha Chand Sitaro MeinVirhaki Agni Mein Pal Pal Tapti HaiAb To Sasein Teri Mala Japti HaiTere Liye Tere LiyeTere Liye Is Duniya Ka Har Sitam Hai Gawara SanamHo Har Sitam Hai Gawara SanamTere Naam Hum Ne Kiya Hai Jeevan Aapna Sara SanamHo Jeevan Aapna Sara Sanam________________________________________Design of Action begun..-Abhi. >> Happy X'mas to all Sci.* NG!>> And to you Abhi, forget the device and have a good time.>> ___________________________________________>> Mere Ghar Ka Seedha Sa Itna Pataa Hai> Ye Ghar Jo Hai Chaaron Taraf Se Khula Hai> Na Dastak Zaruri, Na Aavaz Dena> Mere Ghar Ka Darvaaza Koi Nahin Hai> Hain Deevaren Gum Aur Chhat Bhi Nahin Hai> Badhi Dhoop Hai Dost> Kadhi Dhoop Hai Dost> Tere Aanchal Ka Saaya Churake Jeena Hai Jeena Jeena Zindagi, Zindagi> O Zindagi Mere Ghar Aana> Aana Zindagi Zindagi Mere Ghar Aana> _______________________________________________>> I am still alive and back in comfort of my own room from where all this saga began in March 2000.>> I got new beautiful computer, listening beautiful songs right now in calm night.>> Happy new year to all of you!> I am still in Action...>> -Abhi.> Don't you realize that They are watching you through> the new computer....> Some lines for my beloved Action Device..> ____________________________________________> Naino Se Behte Ashqo Ke Dharo Mein> Humne Tujhko Dekha Chand Sitaro Mein> Virhaki Agni Mein Pal Pal Tapti Hai> Ab To Sasein Teri Mala Japti Hai> Tere Liye Tere Liye> Tere Liye Is Duniya Ka Har Sitam Hai Gawara Sanam> Ho Har Sitam Hai Gawara Sanam> Tere Naam Hum Ne Kiya Hai Jeevan Aapna Sara Sanam> Ho Jeevan Aapna Sara Sanam> ________________________________________> Design Apocalypse has begun..> -Abhi.To patent or not to patent..This is NOT the question. I am NOT going to patent this design. As I have said earlier, actions, thought process, minds of people onplanet earth were controlled, are being controlled by that supremeforce. Now I have this design which is simpler than earlier designs.It could have been discovered accidentally. But that supreme force hasput theory of probability aside. Absolutely everything, except*nothing*, has been setup. We are living in Matrix.Now beyond doubt it is clear that things, circumstances in my life, mymind, thought process was controlled and still being controlled bythat supreme force.I was seeing light ¤ashing before my eyes in last 4-5 months. Now Iam sure that it was not illusion. It was that supreme force, God. OK,things are going right at least at this moment. But there are fewthings, which have gone horribly wrong in my life, and of course,there are questions in our human life, answers of which we are tryingto 'nd out from thousands of years.I have theory and proof, *how* He created this universe from Vacuum,Zero. But I just want to know *why* He has created this universe. Ihave theory but I want proof.I have design of this Action Device through which we can generateunidirectional force to accelerate spacecraft towards stars, galaxiesusing just energy, no need of any propellant. On Earth, we cansuspend things in air, vacuum and anyone can ¤y, pilot thesevehicles.Look at starry sky in silent night. Look at silent blue sky up there.Everything is going to change.I am not going to patent this design. I am not supposed to patent thatSupreme Force, God. This Action Device belongs to poor, developingcountries on planet earth and I want India to have Veto Power,permanent member of Security Council in UN. If not, I will exercise myright to remain silent throughout my life and entire mankind will feelvibrations of my silence forever. If God and entire mankind think thatI can not halt this design of Action device Alone, Anytime.Anytime on or === Subject: Re: instructions, script to enter in this new character>>on universal stage. I have done lot of rehearsal in real life and I>>am aware of my melodramatic tragedy. That is why I am shouting that,>>those rehearsals were setup, a plot. I was controlled to do those>>rehearsals in real life.>>In those rehearsals, I never tried to commit suicide. This is why I>>can't die. So I will have to play this character.>Whatever I do, it is already scripted anyway.>>-Abhi.>>>> Why is it always about you?>>Because I am Zero, feeling of zeroness, cause of creation of this universe.>>Because I am truth.You misspelled TRVTH.--Dr.Postman USPS, MBMC, BsD; Disgruntled, But UnarmedMember,Board of Directors of afa-b, SKEP-TI-CULT member #15-51506-253.You can email me at: TuriFake(at)hotmail.comShake it like a polaroid picture. - Andre 3000 of Outkast === Subject: this design of Action device Alone, Anytime.>> Anytime on or don't believe you'll do any such thing. In the real world thereare things like limitations, and you seem very averse to consideringanything other than you're own ideas. You'll start, hit a snag and thenquit. Why? Because it represents the necessity of accepting thepossibility you may be wrong.If you do manage to build your mystical machine, please feel free to ¤y onby my house and thumb your nose at me.Namaste'O' === Subject: Re: Turning triangles of dots by GzCf3rZ-QeDAxcf3eo1pNONagHKpG3Ir06pbW1++AddY9+CJ-svv0iJohn R you'r wrong: >>> O >> X X >> X X X >> X X X X >> O X X X O >> O O X X O O > Darn. I thought my reply might be oversimplifying the problem. >I see it as an optimization problem of the following form:search minimal s_x ...for rows=2 s0 = 2*0 + 1for rows=3 s0 = 2*0 + 3 s1 = 2*1 + 0for rows=4 s0 = 2*0 + 6 s1 = 2*1 + 1for rows=5 s0 = 2*0 + 10 s1 = 2*1 + 3 s2 = 2*3 + 0for rows=6 s0 = 2*0 + 15 s1 = 2*1 + 6 s2 = 2*3 + 1for rows=7 s0 = 2*0 + 21 s1 = 2*1 + 10 s2 = 2*3 + 3 s3 = 2*6 + 0for rows=8, s0 = 2*0 + 28 s1 = 2*1 + 15 s2 = 2*3 + 6 s3 = 2*6 + 1for rows=9, s0 = 2*0 + 36 s1 = 2*1 + 21 s2 = 2*3 + 10 s3 = 2*6 + 3 s4 = 2*10+ 0Where in the formula sx = 2*a + ba is the sum of dots of the two lower triangles, which are movedand b is the sum of dots of the top triangle which is moved. b . . a . aSolutions for minimal overall-sum of dots: rows ! rows_a! rows_b ! sum_a ! sum_b ! overall_sum ----------------------------------------------------- 2 ! 0 ! 1 ! 0 ! 1 ! 1-------------------------------------------------- 3 ! 1 ! 0 ! 1 ! 0 ! 2 4 ! 1 ! 1 ! 1 ! 1 ! 3 5 ! 1 ! 2 ! 1 ! 3 ! 5-------------------------------------------------- 6 ! 2 ! 1 ! 3 ! 1 ! 7 7 ! 2 ! 2 ! 3 ! 3 ! 9 8 ! 2 ! 3 ! 3 ! 6 ! 12-------------------------------------------------- 9 ! 3 ! 2 ! 6 ! 3 ! 15I extend this just by eyeball-inspection.... 10 ! 3 ! 3 ! 6 ! 6 ! 18 11 ! 3 ! 4 ! 6 ! 10 ! 22(Hmm, too lazy now to put it into a formula... ;-) )Gottfried Helms === Subject: Re: The absolute truth> The consistent way to say, There is no absolute truth>> You said it already. Why do you need to say anything more?>> (How about this one?) ;o)>> THIS STATEMENT IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUE STATEMENT.>>> This statement is false if your 'rst statement is true.>> It seems you have been tortured by locking you up in a round room and> telling you that can only piss at a corner.>> Clari'cation for You:>> There is no absolute truth> How about There is no absolute truth ?> Is it propositions p (p is not absolutely true) then, (q isnot absolutely true) ..for any q.(including There is no absolute truth)Even if we were in possession of an absolute truth, how could we know thatit is absolutely true?that contains all truths.i.e. there is no method of deciding absolutely.It's not the case that all (apparent) questions have answers.It makes no sense to ask a question where there cannot be an answer.The relativity of truth is that, truth is meaningful only with respect to aparticular method of deciding it.Truth is that which can be shown to be the case within some method === Subject: Re: The absolute truth> Of course not.It was attempt to 'nd a non-explicit way to state consistently thereis no absolute truth.The post frame=right&th=3f55bc43b6983a5a&seekm=vvgbgvfrh8909a% 40corp.supernews.com#link21suggested the elegant way to destroy my hope ;o(-------------THE STATEMENT THE STATEMENT THIS STATEMENT IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUE STATEMENT IS WRONG IS WRONG-------------There is another way to look at this:If there is a way to state consistently there is no absolute truthOR there is no way to do so, the both scenarios cause the sameconclusion about impossibility of absolute truth. In case of the 'rstscenario it is true because it is expressed explicitly, in case ofsecond scenario - because even statement about non-absoluteness is notabsolute that indicates the deep internal consistency ofnon-absolute assumption.I am not sure it sounds === Subject: Re: The absolute truth The consistent way to say, There is no absolute truth (How about this one?) ;o)> THIS STATEMENT IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUE STATEMENT.> George Buyanovsky> If there were no truth, neither would there be any lies. If reality> exists, then its existence is true, thus proving that truth exists. If> reality doesn't exist, how could we be deceived into experiencing it?> How can there be deception unless that deception masks a reality about> which we're being deceived? Either way, some kind of reality must> exist, thus proving something true.> One Day Tesshu, the famous swordsman and zen devotee, went > to Dokuon and told him triumphantly he believed all that exists > is empty, there is no you or me, and so on. The master, who had > listened in silence, suddenly snatched up his long tobacco pipe > and struck Tesshu's head. The infuriated swordsman would have > killed the master there and then, but Dokuon said calmly, > Emptiness is quick to show its anger, isn't it? > Forcing a smile, Tesshu left the room.> (Soul Food -- Stories to Nourish the Spirit and the Heart> Ed. Jack Korn'eld & Christina Feldman)> Accept the terrible truth that all is illusion. All being One, > wherever you go, there you are. But that doesn't have to ruin > things. The Gnosis of illusion and what to do about it leads> the seeker to Moksha-- experiential knowledge of the liberation> from dualistic bondage. The choice after Moksha is your own. > Return to source? To nothingness? Or choose the road for the > sake of the undiscovered country and the experience of experience> for the sake of itself. > Accept the terrible truth and return to yourself. Your path> after that crossroads is your own choice. =~)> Students achieving oneness, will move ahead to twoness.> (Woody Allen)> We shall not cease from exploration.> And the end of all our exploring> Will be to arrive where we started> Knowing the place for the 'rst time.> (T.S. Eliot)So we agree then.- Tue === Subject: Re: The absolute truth> buyanovsky@attbi.com (George Buyanovsky)> The consistent way to say, There is no absolute truth (How about this one?) ;o)>> THIS STATEMENT IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUE STATEMENT.>> George Buyanovsky>> If there were no truth, neither would there be any lies. If reality> exists, then its existence is true, thus proving that truth exists. If> reality doesn't exist, how could we be deceived into experiencing it?> How can there be deception unless that deception masks a reality about> which we're being deceived? Either way, some kind of reality must> exist, thus proving something true.> Tue true!Absolutely! :-)- Tue === Subject: Re: Linear Algebra QuestionX-DMCA-Noti'cations: the vector space structure>> of 'xed rank (k), (n x n) matrices ?>> The set of all nxn matrices of rank k with sum and multiplication by a>scalar>> is not a vector space. Indeed, the sum of 2 matrices of rank k is not, as>a>> general rule, a rank k matrix. (take a matrix A that has rank k. -A has>rank>> k too, and A + (-A) = 0, so it has rank 0).>>I think that it can be deduced from>my question that I have already known>that fact. Certainly didn't look that way to me.>I was asking how can you>still consider it as a vector space,>say by quotienting by the 'right' equivalence>relation (or is that impossible) ?Well, for example you could say any two matricies of rankk are equivalent. Then they become the vector space {0}.That's probably not the answer you want, but it's hard tosee what other answer there is. Usually when whatever/~is a vector space it's because whatever is _already_ avector space, and then the addition in whatever/~ isde'ned using the addition in whatever. But the matriciesof rank k simply do not form a vector space - I have noidea why you think that there is a vector space structurehidden in there somewhere.(Of course the question is not precise enough to givea precise answer, until you specify what sort of additionyou had in mind for these equivalence Ullrich === Subject: Re: Continuity Question - BasicX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.html> _is_ continuous.>>>>You haven't been reading elementary calculus texts recently, have>you...???No.>Of course a mathematician will say it is continuous. But their>de'nition may not agree with the de'nition in all elementary texts.I was aware of that. If he'd asked whether calculus books wouldcall this function continuous I wouldn't have spoken up, cuz Iwouldn't have known the answer.A much worse example: Say as usual (i) when a function is de'nedby a formula we take the domain to be the set of all reals for whichthe formula makes sense, (ii) we say a function is continuous if itis continuous at every point of its domain. Then the function f(x) = 1/xis continuous.************************David C. Ullrich === Subject: Re: Continuity Question - BasicDavid C. Ullrich Say as usual (i) when a function is de'ned> by a formula we take the domain to be the set of all reals for which> the formula makes sense,I'll also assume, in that case, that you're intending the range to bea subset of R. But even then there can be differences of opinion aboutthose reals for which the formula makes sense. As a simple example:If we were asked for the implied domain of f(x) = Sqrt(x^2 (x-1)),I would hope that everyone would say that it is {0} U [1, +oo). Butsuppose that we were asked for the implied domain of g(x) = |x| Sqrt(x-1).I suspect that some would say that 0 is in the implied domain of g (andalso that f and g are the same function), while others would say that 0is not in the implied domain of g (and thus that f and g are differentfunctions).BTW, is there a better term than implied domain for what I was talkingabout above?David Cantrell === Subject: Re: Continuity Question - BasicX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.html function is de'ned>> by a formula we take the domain to be the set of all reals for which>> the formula makes sense,>>I'll also assume, in that case, that you're intending the range to be>a subset of R. But even then there can be differences of opinion about>those reals for which the formula makes sense. As a simple example:>If we were asked for the implied domain of>> f(x) = Sqrt(x^2 (x-1)),>>I would hope that everyone would say that it is {0} U [1, +oo). But>suppose that we were asked for the implied domain of>> g(x) = |x| Sqrt(x-1).>>I suspect that some would say that 0 is in the implied domain of g (and>also that f and g are the same function), while others would say that 0>is not in the implied domain of g (and thus that f and g are different>functions).If we were talking about actual math we'd need to be a lot morecareful with all of this, or better yet simply throw out the notionof implied domain. I think we're talking about the way terminologyis used in a typical calculus class - in a context like that there areno complex numbers, so the domain does not include 0.I hope so, anyway.>BTW, is there a better term than implied domain for what I was talking>about above?>>David Cantrell************************David C. Ullrich === Subject: Re: Request for clari'cation of some simple factsX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.html> korovyev@rambler.ru unproved proposition (the proof is said>> to be obvious) which states that function f: X subset R -> R is>> differentiable at point a in X if and only if there exist A = lim_{x>> -> a+} ((f(x) f(a)) /(x-a)), B = lim_{x -> a-} ((f(x) f(a))>> /(x-a)) and A = B. I do not understand this, given that the textbook>> de'nition of the derivative of function f at point a does not require>> a to be interior. Does the statement of the theorem imply that the>> condition of point a being interior is implied in the de'nition of>> the derivative or I miss something? In short, what must the derivative>> de'nition look like for this theorem to hold true?>>> 2) Which conditions are needed so that differentiability at a point>> would imply differentiability on some interval embracing the point?>>> 3) What are the most common and accepted precise de'nitions of the>> conditions of a function to be continuous at a point and to be>> differentiable at a point? Consider identity function f: [0,1] -> [0,1]. Is it continuous at 0? Is it differentiable at 0? (according to>> the de'nitions you then X does not have any interior >points (as a subset of R), but functions from X to R can still have >derivatives.According to whom? Where I come from functions have derivativesat interior points of their domains.>On the other hand, The function f(x) = sqrt(x^3) from the non-negative >reals to the reals clearly has a derivative at x = 0. This is not clear to me.>So that the if-and-only-if fails, at least for many de'ntions of >derivative.>>The precise wording of the de'nition of derivative in your text may be >critical to the issue.************************David C. Ullrich === Subject: Re: Request for clari'cation of some simple factsx-no-archive: yes the set of rationals, then X does not have any>> interior points (as a subset of R), but functions from X to R can>> still have derivatives.>> According to whom? Where I come from functions have derivatives> at interior points of their domains.The rationals are a perfectly good metric space in their ownright. Nothing stops one from trying to de'ne continuity anddifferentiability on that space. (Virgil appears not to have noticedthat the rationals can be regarded as a space by itself; there areindeed interior points.)OTOH, I've never seen anybody bother with it--and the loss of leastupper bounds means that lots of interesting theorems will === Subject: Re: Request for clari'cation of some simple facts David C. Ullrich > 1) In my textbook there is an unproved proposition (the proof is said>> to be obvious) which states that function f: X subset R -> R is>> differentiable at point a in X if and only if there exist A = lim_{x>> -> a+} ((f(x) f(a)) /(x-a)), B = lim_{x -> a-} ((f(x) f(a))>> /(x-a)) and A = B. I do not understand this, given that the textbook>> de'nition of the derivative of function f at point a does not require>> a to be interior. Does the statement of the theorem imply that the>> condition of point a being interior is implied in the de'nition of>> the derivative or I miss something? In short, what must the derivative>> de'nition look like for this theorem to hold true?>>> 2) Which conditions are needed so that differentiability at a point>> would imply differentiability on some interval embracing the point?>>> 3) What are the most common and accepted precise de'nitions of the>> conditions of a function to be continuous at a point and to be>> differentiable at a point? Consider identity function f: [0,1] ->>> [0,1]. Is it continuous at 0? Is it differentiable at 0? (according to>> as the set of rationals, then X does not have any interior points (as a subset of R), but functions from X to R can still have >derivatives.> According to whom? Where I come from functions have derivatives> at interior points of their domains.But using the topology on Q induced by R, all points of Q are interior points. Or am I missing something?>On the other hand, The function f(x) = sqrt(x^3) from the non-negative >reals to the reals clearly has a derivative at x = 0. > This is not clear to me. D = Dom(f) is the set of non-negative reals, [0, +oo). In the topology on D induced by R is 0 an interior point of D or not? My topology is a bit rusty.In any case, f(x) = sqrt(x^3) has a one-sided derivative at x = 0.>So that the if-and-only-if fails, at least for many de'ntions of >derivative.>>The precise wording of the de'nition of derivative in your text may be >critical to the issue.> ************************> David C. Ullrich === Subject: Re: Request for clari'cation of some simple factsX-DMCA-Noti'cations: there is an unproved proposition (the proof is said>to be obvious) which states that function f: X subset R -> R is>differentiable at point a in X if and only if there exist A = lim_{x>-> a+} ((f(x) f(a)) /(x-a)), B = lim_{x -> a-} ((f(x) f(a))>/(x-a)) and A = B. I do not understand this, given that the textbook>de'nition of the derivative of function f at point a does not require>a to be interior. Does the statement of the theorem imply that the>condition of point a being interior is implied in the de'nition of>the derivative or I miss something? In short, what must the derivative>de'nition look like for this theorem to hold true?I dn't know what your textbook says - possibly the author disagreeswith me, possibly there's a word left out that he meant to include,possilby you misunderstood some detail. But _I_ would certainlysay that for f to be differentiable at a point that point must bean interior point of the domain of f.(Or to put the same thing differently: I would only speak offunctions de'ned in _open sets_ as being differentiable or not.)>2) Which conditions are needed so that differentiability at a point>would imply differentiability on some interval embracing the point?I really don't think there are any. A function can be very strangein a heighborhood of a point and still be differentiable at thatpoint; to make it nice in a neighborhood you're going to haveto assume it's nice in a neighborhood.>3) What are the most common and accepted precise de'nitions of the>conditions of a function to be continuous at a point and to be>differentiable at a point? As has come up recently in another thread, calculus books andmathematicians seem to have different de'nitions here. Ifyou're trying to understand what it says in the book you needto use the de'nitions in the book, not the most commonde'nitions. But regarding the de'nitions that I think are standardamong mathematicians:>Consider identity function f: [0,1] ->>[0,1]. Is it continuous at 0? Yes.>Is it differentiable at 0? (according to>the de'nitions you Ullrich === Subject: Re: Request for clari'cation of some simple sha1:3KYH79l0QWqfV2kSzM3WqhNAIyg=David C. Ullrich given that the textbook de'nition of the>> derivative of function f at point a does not require a to be>> interior...>> I dn't know what your textbook says - possibly the author disagrees> with me, possibly there's a word left out that he meant to include,> possilby you misunderstood some detail. But _I_ would certainly say> that for f to be differentiable at a point that point must be an> interior point of the domain of f.Same here. But some textbooks apply the terms left differentiableand right differentiable to the cases that only the left- or right-handed limits exist.> (Or to put the same thing differently: I would only speak of> functions de'ned in _open sets_ as being differentiable or not.)In boundary value problems, one is also interested in whetherfunctions and their derivatives can be continuously extended to theboundary. One oddity introduced by this is that the geometry of theboundary plays a part as well. Some PDE researchers ignore boudarybehavior almost entirely, while others specialize in === Subject: Unable to imagine 3DIn a spherical triangle with dihedral angles A,B,C and side common toA,B angles subtending angle c at sphere center,we haveCos C= - Cos A Cos B + Sin A Sin B Cos c . This very same relationship would result if we take two taper wedgeswith taper angles [(A + 3Pi/2),B] and rotate one prismatic wedge blockwith respect to the other as follows: We place the second wedge bottomplane on the 'rst wedge top plane,make a planar contact and rotatesecond wedge by rubbing/rotating it through an angle c in theinterface plane. Non contacting edges of these two wedges (consideronly edges containing angles [(A + 3Pi/2),B])now make a skewed angle(Pi/2- C) exactly .I am unable to imagine the identity of these two situations yieldingthe same formula. If the blocks visualization from the above is notclear enough,I could supply a link to drawing. TIA. === Subject: Re: why i can't use oe to open this group?> why i can't use oe to open this group?> is the url math.sci?I use oe to open this group. It is sci.math === Subject: Re: why i can't use oe to open this group?X-DMCA-Noti'cations: is the url math.sci?>url for newsgroup?like ftp sites start with ftp and web pagesstart with http.>This group is sci.math************************David C. Ullrich === Subject: Re: why i can't use oe to open this group?A N Niel scribbled the following:> group?>> is the url math.sci?-- /-- Joona Palaste (palaste@cc.helsinki.') ------------- Finland ---------- http://www.helsinki.'/~palaste --------------------- rules! --------/The question of copying music from the Internet is like a two-barreled sword. - Finnish rap artist Ezkimo === Subject: Re: Embarrassingly simple question has me Olympiad questions, and in one> of the solutions came across this little equation:>> x^2 + x + 1 = 0 (i)>> For fun, I started to push things around a bit, instead of just going> with x = (-b +- sqrt(b^2 - 4ac))/(2a):>> x(x + 1) = -1> (x + 1) = -1/x (ii) <-- I though it would be OK to divide by x,> since x<>0>> Substituting (ii) back into (i):>> x^2 + (x + 1) = 0>> x^2 - 1/x = 0>> x^3 - 1 = 0 <-- again, multiplying by x, since x<>0>> (x - 1)(x^2 + x + 1) = 0 (iii)>>> Somehow in my little manipulations I introduced a new factor onto the> original equation, but I can't 'gure out how. This seems pretty> ridiculous, but I'd appreciate another pair of eyes looking at it and> letting me know where the (x - 1) factor popped in.>> What you have proved was that>> x^2 + x + 1 = 0 => (x - 1)(x^2 + x + 1)>> and this is obviously correct. It would still be correct if x - 1 was> replaced by anything else. But note that your computations do not prove> the reverse implication: you cannot devide by x - 1 since you are not> assuming that x <> 1.Oh, sure you are, since the original equation came up because you arelooking for the non-trivial solutions of x^3-1=0.Jon Miller === Subject: Re: Calculating new position of two wheeled current position of the robot,: current angle of the robot,: distance the left wheel has traveled since the last frame,: distance the right wheel has traveled since the last frameThen hard luck. It can't be done. Think about the position change of the robots resulting from a full revolution of both wheels ifa) those revolutions happen simultaneouslyb) the right wheel turns once, then the left wheel turns once.Ian === Subject: Re: Optimization Problem> I am looking for some web sites and references that deal with> algorithms of optimization that permit to arrange geometric shapes in> order to uniformly cover a surface (example of problem : How to> optimize the parameters of a sprayer in order to have an uniform> thickness coating?). Have you some informations about these a real world problem that I encountered a few days ago that> might be what you're looking for.>> I work for a meteorology website. We are currently undergoing some changes> and want to promote our website even more. As a result, we considered> looking at making rain gauges. Being an environmental website, we want to> use the least amount of material, but we'd like it to hold 50 cc o¤iquid.> What dimensions do you make the gauge?You make it a sphere? Cool, how do I get one?Jon Miller === Subject: Re: Optimization Problem> I am looking for some web sites and references that deal with> algorithms of optimization that permit to arrange geometric shapes in> order to uniformly cover a surface (example of problem : How to> optimize the parameters of a sprayer in order to have an uniform> thickness coating?). Have you some informations about these sorts of> problems?>> problem that I encountered a few days agothat> might be what you're looking for.>> I work for a meteorology website. We are currently undergoing somechanges> and want to promote our website even more. As a result, we considered> looking at making rain gauges. Being an environmental website, we wantto> use the least amount of material, but we'd like it to hold 50 cc of> liquid.> What dimensions do you make the gauge?>> You make it a sphere? Cool, how do I get one?>> Jon Miller>>We were working with cylinders. We probably won't do this promotion, but wewere messing with it a few days ago.-- David MoranChief MeteorologistOklahoma Storm Team === Subject: Re: number of chess positions> Number of positions...> After exactly two pawn moves:> ... after two moves by the same pawn: 16> ... after two pawns each moving 1 step: 28> ... after two pawns each moving 2 steps: 28> ... after two pawns, one moving 1 and one moving 2: 56> TOTAL: 128> After a pawn move and a non-pawn move:> ... after a3: 4> ... after a4: 6> ... after b3: 6> ... after b4: 6> ... after c3: 6> ... after c4: 7> ... after d3: 12> ... after d4: 13> ... after e3: 15> ... after e4: 15> ... after f3: 4> ... after f4: 5> ... after g3: 6> ... after g4: 6> ... after h3: 4> ... after h4: 6> TOTAL: 121> After a knight move and a rook move: 4> After two moves by same knight: 11> After two moves by different knights: 4> TOTAL: 19> Grand total: understand it!> the 16 is ok, but how do you get the 28?> I think after the two moves of the pawn you go back> to the initial boardposition, right? How do i get up> to 28 different positions by moving 2 pawns, each one step?> complicated stuff...:-(> 28 is found by what mathematicians call 8 choose 2. Look up> combinations, or binomial coef'cients to understand. > But this is easy to count. We have the following combinations of> two pawns to move one square each (named after their 'le):> ab> ac> ad> ae> af> ag> ah> bc> bd> be> bf> bg> bh> cd> ce> cf> cg> ch> de> df> dg> dh> ef> eg> eh> fg> fh> gh> 28 possibilities.> Note that this is 7+6+5+4+3+2+1.> -Leonard (email defunct)ok, is see!so i have to setup the initial boardposition after each move, right? === Subject: Re: Bizarre math positions> On 03 Jan analogy doesn't work. There are lots of ways to factor>> an expression. If I said, What is the form of the>> factorization of (125*x^3 + 75*x^2 + 45*x + 7)?, most people> might think I wanted a factorization as a polynomial in x.> That, however, is not what you have done with your expression.>> You have factored it as a polynomial in 5. That is why it>> is important to say that you *assumed* a factorization of>> the speci'ed form. It is a natural and even necessary>> part of the discussion.>>Let's say you assume that irrationals are not integers.>>Yet someone else tells you, hey, why assume that, as they're just not?>Don't you think a *rational* response might be to go, oh, yeah, you're>right?>> Again not a valid parallel. This argument is so goddamned silly> and petty. I have little patience with arguments about semantic trivia. > If you don't like my saying that you assumed a factorization of the > form given, you can replace assumed by speci'ed. That conveys the > same meaning: that there were lots of other ways it could have been > speci'ed (or assumed for that matter).So why are you arguing? Can you see my point?A *reasonable* person can see how the word assume has conotations,which don't apply if something is known to be true, like why assumethat irrationals are not rational, or why assume that 3 is a factor of6?What I'm exploring is how deep your problem in this area goes.Will you *ever* be able to acknowledge what most people take forgranted? >Now then, what if instead, you acted defensive and argued endlessly>about your use of words where there was an implication that something>might not be true by use of the word assume, which *reasonable*>people could see.>>Is there any possibility that you might concede that saying assume>for a truth might be loaded or could look like you're trying to imply>that something might not be true?>>> And here, it's a distraction. You are trying desperately to focus> on a minor wording issue to take attention away from the fact>> that you have not dealt with the mathematics.>I'm curious about how you manage to spend so much time ratherobviously ignoring the truth, twisting things a certain way, yet>acting as if you're aboveboard no matter how often you're caught.>>Your defensive reaction here with the word assume is a case in>point.>>You basically refuse to be rational, and refuse to concede even minor>points, arguing endlessly as if you're always right, no matter what,>or how much evidence is produced to challenge your world view.>> No, you're simply wrong. You assumed a factorization of a> certain form. You could have assumed many other forms. It> is appropriate to note what you did and to convey the impression> that the choice involved a decision on your part. By no means> was I implying that your assumption was wrong. It's not.> There IS a factorization of that form. I was just describing that> that was the form you were specifying..Your original usage may have just been a minor mistake, but my pointis that you can't accept correction, or at a minimum, in the interestof moving forward, shift your wording on what you yourself claimed isa minor point.Your *actions* speak volumes as you've gone on now through severalreplies, 'ghting over what you claim is a minor point, only to nowjust push the previous position! That's what I was checking on, as itlooks to be some deeply ingrained pattern of behavior.You have demonstrated unreasonableness to a high degree. It's uselessto present a rational position to you contrary to what you're saying,as you have demonstrated that you will simply reject it, hold on toyour own point of view, and eventually just repeat your originalposition, now matter what has come before.Therefore, it's pointless to discuss anything with you, unless someoneis willing to agree with you at the outset, as you will just continuewith your own position, no matter what.James Harris === Subject: Re: Bizarre math positions> On > [inane drivel snipped]> Your original usage may have just been a minor mistake, but my point> is that you can't accept correction, or at a minimum, in the interest> of moving forward, shift your wording on what you yourself claimed is> a minor point.> Your *actions* speak volumes as you've gone on now through several> replies, 'ghting over what you claim is a minor point, only to now> just push the previous position! That's what I was checking on, as it> looks to be some deeply ingrained pattern of behavior.> You have demonstrated unreasonableness to a high degree. It's useless> to present a rational position to you contrary to what you're saying,> as you have demonstrated that you will simply reject it, hold on to> your own point of view, and eventually just repeat your original> position, now matter what has come before.> Therefore, it's pointless to discuss anything with you, unless someone> is willing to agree with you at the outset, as you will just continue> with your own position, no matter what.> This is transparent pompous B.S. designed to provide you witha super'cial excuse to avoid talking about the mathematics. There was in fact one central problem with my previous post: I did not provide an adequate answer to your comment regarding the existence of the algebraic integers A, B, and C. I claimed they exist but I did not specify what they were or even give an existence proof. If I had been you, I would have pounced on that like a rooster on a June bug as evidence that I could not back up my claims. You were too focussed on your stupid semantic issue and trying to 'nd an excuse to stop replying to the math (not that you ever did). Dik Winter has also claimed to have a way of de'ning A, B, and C but has recently backed off from his original de'nition (see his post in the thread 'Mathematical consistency, courage') and provided another one. Below I give an existence proof for A, B, and C. I refer to themas Dik has done before, i.e. A = w1 = w1(x), B = w2 = w2(x), and C = w3 = w3(x). All are dependent on x. In general when x <> 0,they are not, needless to say, equal to 7, 7, 1. = factor P(x)/49 in the form [*] P(x)/49 = (c1*5 + d1)*(c2*5 + d2)*(c3*5 + d3), where c1, c2, c3, d1, d2, and d3 are all algebraic integers. Note that this is a factorization as a polynomial in 5,not of course as a polynomial in x. Note that d1*d2*d3 = 7. Note that c1 and d1 are coprime, as are c2, d2 and c3, d3. This is true because in general P(x)/49 has integer coef'cients and is primitive. Now de'ne w1 = d2*d3, w2 = d1*d3, and w3 = d1*d2. Now go back to the Harris factorization of P(x) itself: [**] P(x) = (a1*5 + 7)*(a2*5 + 7)*(a3*5 + 7). Clearly 7 = d1*w1 = d2*w2 = d3*w3. Now you know that -7/a1 and -d1/c1 are roots of the same polynomial, a^3 + 3*(-1 + 49*x)*a^2 - 49*(2401*x^3 - 147*x^2 + 3*x). Therefore a1 = 7*c1/d1 = c1*(7/d1) = c1*d2*d3 = c1*w1. Similarly, a2 = c2*w2 and a3 = c3*w3. Finally, note that w1 = gcd(a1, 7) = gcd(c1*d2*d3, d1*d2*d3). Similarly, w2 = gcd(a2, 7), and w3 = gcd(a3, 7). All of these things - w1, w2, w3, c1, d1, ..., a1, b1, ... are algebraic integers that are dependent on x, and should be written as w1(x), ..., c1(x), ..., a1(x), ... etc. Thus w1, w2, and w3 give you the factors to go from the Harrisfactorization [**] to the Magidin-Mckinnon factorization [*], andof course w1*w2*w3 = (d2*d3)*(d1*d3)*(d1*d2) = (d1*d2*d3)*(d1*d2*d3) = 7*7 = 49. And all the coef'cients in the Magidin-Mckinnon factorization arealgebraic integers, which is what you wanted in the 'rst place. QED. = The Magidin-Mckinnon result is constructive, i.e., with enoughwork, for any given integer x, you can compute the answers. Similarlythe gcd function in the algebraic integers. Actually carrying outthe calculations however is admittedly dif'cult. Nora B.> James Harris === Subject: Re: Bizarre math positions jstevh@msn.com anything with you, unless someone> is willing to agree with you at the outset, as you will just continue> with your own position, no matter what.> James HarrisJSH obviously avoids knowing himself.Doesn't this sound like something that everyone else has said about JSH? === Subject: Re: Bizarre math valid parallel. This argument is so goddamned silly> and petty. I have little patience with arguments about semantic trivia. > If you don't like my saying that you assumed a factorization of the > form given, you can replace assumed by speci'ed. That conveys the > same meaning: that there were lots of other ways it could have been > speci'ed (or assumed for that matter).> So why are you arguing? Can you see my point?> A *reasonable* person can see how the word assume has conotations,> which don't apply if something is known to be true, like why assume> that irrationals are not rational, or why assume that 3 is a factor of> 6?> What I'm exploring is how deep your problem in this area goes.> Will you *ever* be able to acknowledge what most people take for> granted?Most people can't do the math so what they take for granted is irrelevant here. Why does JSH keep on going about things which are of no mathematical import, unless he is unable or unwilling to face the mathematical issues directly? === Subject: Re: Bizarre math positions> On 03 Jan personality,psychology or social interaction or your inane hairsplittingabout semantics. If you cannot address the math, don't bother to post at all. Nora B. === Subject: Re: Bizarre work. There are lots of ways to factor>> an expression. If I said, What is the form of the>> factorization of (125*x^3 + 75*x^2 + 45*x + 7)?, most people>> might think I wanted a factorization as a polynomial in x.>> That, however, is not what you have done with your expression.>> You have factored it as a polynomial in 5. That is why it>> is important to say that you *assumed* a factorization of>> the speci'ed form. It is a natural and even necessary>> part of the discussion.>>Let's say you assume that irrationals are not integers.>>Yet someone else tells you, hey, why assume that, as they're just not?>>Don't you think a *rational* response might be to go, oh, yeah, you're>right?>> Again not a valid parallel. This argument is so goddamned silly> and petty. I have little patience with arguments about semantic trivia. > If you don't like my saying that you assumed a factorization of the form given, you can replace assumed by speci'ed. That conveys the > same meaning: that there were lots of other ways it could have been > speci'ed (or assumed for that matter).> So why are you arguing? Can you see my point?Nora sees your point.Do you see Nora's point?Are you reading her replies?You keep giving examples like Assume sqrt(2) is irrational andNora has repeatedly pointed out that such an example is notat all similar to the case being discussed. This indicates thatyou are not understanding what the discussion is about, whetheror not you are correct and Nora is wrong.> A *reasonable* person can see how the word assume has conotations,> which don't apply if something is known to be true, like why assume> that irrationals are not rational, or why assume that 3 is a factor of> 6?How many explanations and examples does Nora have to give you toindicate that 3 is a factor of 6 is not a good example of what isbeing discussed?The statement 3 is a factor of 6 is always true. However, thisis not the situation that we are in.Here are some more, better examples of what is being discussed:1) Let a deck of cards be shuf¤ed. With the assumption thatthe top card is the three of clubs, then there are 26 redcards in the remaining portion of the deck.2) With the assumption that 30 is factored in the form 60 = 2*5*6,then there are an odd number of factors in the factorization.The point is that the top card of a shuf¤ed deck need not bethe three of clubs. Likewise, the factorization of 60 canbe in other forms, like 60 = 2*2*3*5 (where there are nowan even number of factors in the factorization).> What I'm exploring is how deep your problem in this area goes.> Will you *ever* be able to acknowledge what most people take for> granted?It is not clear to me that you are right. As with Nora, I agreethat you might be right, but even if you are right, you are notexplaining it very well: you are giving cases where most peopleagree that the phrase assuming would be misleading, and avoidingthe cases where the problem area is (where the statement isnot a fact true in all situations). >Now then, what if instead, you acted defensive and argued endlesslyabout your use of words where there was an implication that something>might not be true by use of the word assume, which *reasonable*>people could see.>>Is there any possibility that you might concede that saying assume>for a truth might be loaded or could look like you're trying to imply>that something might not be true?>>> And here, it's a distraction. You are trying desperately to focus> on a minor wording issue to take attention away from the fact>> that you have not dealt with the mathematics.>I'm curious about how you manage to spend so much time rather>obviously ignoring the truth, twisting things a certain way, yet>acting as if you're aboveboard no matter how often you're caught.>>Your defensive reaction here with the word assume is a case in>point.>>You basically refuse to be rational, and refuse to concede even minor>points, arguing endlessly as if you're always right, no matter what,>or how much evidence is produced to challenge your world view.>> No, you're simply wrong. You assumed a factorization of a> certain form. You could have assumed many other forms. It> is appropriate to note what you did and to convey the impression> that the choice involved a decision on your part. By no means> was I implying that your assumption was wrong. It's not.> There IS a factorization of that form. I was just describing that> that was the form you were specifying..> Your original usage may have just been a minor mistake, but my point> is that you can't accept correction, or at a minimum, in the interest> of moving forward, shift your wording on what you yourself claimed is> a minor point.> Your *actions* speak volumes as you've gone on now through several> replies, 'ghting over what you claim is a minor point, only to now> just push the previous position! That's what I was checking on, as it> looks to be some deeply ingrained pattern of behavior.> You have demonstrated unreasonableness to a high degree. It's useless> to present a rational position to you contrary to what you're saying,> as you have demonstrated that you will simply reject it, hold on to> your own point of view, and eventually just repeat your original> position, now matter what has come before.> Therefore, it's pointless to discuss anything with you, unless someone> is willing to agree with you at the outset, as you will just continue> with your own position, no matter what.> James Harris === Subject: Re: >> The analogy doesn't work. There are lots of ways to factor>> an expression. If I said, What is the form of the>> factorization of (125*x^3 + 75*x^2 + 45*x + 7)?, most people>> might think I wanted a factorization as a polynomial in x.>> That, however, is not what you have done with your expression.>> You have factored it as a polynomial in 5. That is why it>> is important to say that you *assumed* a factorization of>> the speci'ed form. It is a natural and even necessary>> part of the discussion.>>Let's say you assume that irrationals are not integers.>>Yet someone else tells you, hey, why assume that, as they're just not?>>Don't you think a *rational* response might be to go, oh, yeah, you'reright?>> Again not a valid parallel. This argument is so goddamned silly> and petty. I have little patience with arguments about semantic trivia. > If you don't like my saying that you assumed a factorization of the form given, you can replace assumed by speci'ed. That conveys the > same meaning: that there were lots of other ways it could have been > speci'ed (or assumed for that matter).> So why are you arguing? Can you see my point? Nora sees your point.My point is that using assume for a fact is misleading. > Do you see Nora's point?a certain way, no matter what.> Are you reading her replies?Yup.> You keep giving examples like Assume sqrt(2) is irrational and> Nora has repeatedly pointed out that such an example is not> at all similar to the case being discussed. This indicates that> you are not understanding what the discussion is about, whether> or not you are correct and Nora is wrong.I picked that example quite deliberately as in fact factorizations aretautologies.I hope you understand that is true.For instance, x^2 + 2x + 1 = (x+1)(x+1)is an identity or what I call a tautology.It's just true.Now then, what if I say, assume that x^2 + 2x + 1 has x+1 as a factor?I've replied to you here as *your* reply shows what I see is adeliberate attempt to deny basic facts, in a social need to promote aperson you see as a member of your group.The fact is that saying assume for a truth is just not the way mostpeople operate. > A *reasonable* person can see how the word assume has conotations,> which don't apply if something is known to be true, like why assume> that irrationals are not rational, or why assume that 3 is a factor of> 6?> How many explanations and examples does Nora have to give you to> indicate that 3 is a factor of 6 is not a good example of what is> being discussed?How much time do I have to take to show that it is?First, do you accept that a factorization is a truth? > The statement 3 is a factor of 6 is always true. However, this> is not the situation that we are in.Yes it is, as I've repeatedly explained, as a factorization is atruth. > Here are some more, better examples of what is being discussed:> 1) Let a deck of cards be shuf¤ed. With the assumption that> the top card is the three of clubs, then there are 26 red> cards in the remaining portion of the deck.But what if the top card is NOT the three of clubs?Your example allows falsity. That is, implied with the wordassumption usually there is the possibility that something might notbe true.It's that implication that I'm faulting in the statement by NoraBaron.What if I assume that Nora Baron is a female? By saying that, am Inot also pointing out that possibly that poster is a man masqueradingas a woman? > 2) With the assumption that 30 is factored in the form 60 = 2*5*6,> then there are an odd number of factors in the factorization.That is a specious example given the actual statements by NoraBaron.More like that poster's statement would be something like, assume that30 has the factorization 2*3*5. > The point is that the top card of a shuf¤ed deck need not be> the three of clubs. Likewise, the factorization of 60 can> be in other forms, like 60 = 2*2*3*5 (where there are now> an even number of factors in the factorization).And my point is that implicit with the word assume there's thepossibility that what is assumed is *false*, while withfactorizations, a factorization is a truth, so it can't be false.Now the poster Nora Baron further claimed that it was a minor point,but repeatedly argued that supposedly minor point through severalreplies.My major point is that Nora Baron is NOT reasonable, but simply hasa desire to push one point of view, without the willingness to concedeeven minor points.If you're dealing with such a person, there's no reason to debatethem, as the only thing they will accept is your acceptance of theirclaims, so what's the point of debating with them?James Harris === Subject: Re: Bizarre math positionsJSH's Bizarrre math position seems to be that the math is not as important as one's attitude, particularly one's attitude about JSH's claims.He is tha only one whose _math_ position is the least bit bizarre in any of his threads (though there are some bizarre math positions in threads to whichJSH has not coontributed).In sci.math postings, one's positions on non-mathematical issues should be entirely irrelevant, and omitted, however much that might constrain JSH's volubility. === Subject: Re: Bizarre math positionsX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.html>> [...]>>Therefore, it's pointless to discuss anything with you, unless someone>is willing to agree with you at the outset, as you will just continue>with your own position, no matter what.Sounds like you're 'nally starting to realize she's right about themath...>James Harris************************David C. Ullrich === Subject: Re: Bizarre math positionsX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.html>[...]>>As for your claim about ways to factor 49, that's just something>you're making up.>>What I *say* is that the constant terms go from being 7, 7 and 22 to>being 1, 1, and 22, when 49 is divided off, which is an easily>veri'able fact.Why don't you _prove_ it if it's so easy, instead of simply continuingto _assert_ it?Just for fun, I'm going to read Nora's mind: the reason she thinksyou're saying something about factoring 49 when you say whatyou say you say is that (i) _if_ there were only one way to factor49 then what you say you say would follow, (ii) she can't imaginewhy _else_ you think what you say you say is true.So _prove_ that the new constant terms must be 1, 1, and 22already. >>[...]************************David C. Ullrich there an explicit example of a function f: O->F (E, F are two Banachspaces, O is an open set of E) such that there is one point a in O, df(a) isa continuous isomorphism E->F, but f^(-1) is not differentiable at point a ?Note that such an example would require that both E and F are in'nitedimensional, and that df(a) is not bicontinuous.--J.S === Subject: Re: Example in differential there an explicit example of a function f: O->F (E, F are two Banach> spaces, O is an open set of E) such that there is one point a in O, df(a) is> a continuous isomorphism E->F, but f^(-1) is not differentiable at point a ?> Note that such an example would require that both E and F are in'nite> dimensional, and that df(a) is not bicontinuous. Julien assumed E and F are Banach spaces; let's say they're only> normed vector spaces (the point of my question was to show explicitely that> df(a) can be a continuous isomorphism, and yet f^(-1) is not differentiable> at point a).Let E = l^2 and de'ne f((x_n)) = (x_n/n). Set F = f(E). Then f is a linear continuous 1-1 map of E onto the normed linear space F. We have df(a) = f for every a in E, yet f^(-1) is continuous nowhere in F, hence differentiable nowhere in F. === Subject: Re: Example in differential calculusX-DMCA-Noti'cations: http://www.giganews.com/info/dmca.htmlF (E, F are two Banach>spaces, O is an open set of E) such that there is one point a in O, df(a) is>a continuous isomorphism E->F, but f^(-1) is not differentiable at point a ?Presumably you meant to ask for the inverse to fail to be differentiable at f(a).>Note that such an example would require that both E and F are in'nite>dimensional, and that df(a) is not bicontinuous.Actually an example doesn't require that, nor does it require thatE and F be in'nite-dimensional. There is a map f:R^2 -> R^2 whichis differentiable at the origin, such that f(0) = 0, df(0) is theidentity, but such that the inverse is not differentiable at 0;it's easy to give such an example, for example arranging thatthe range of f is not even a neighborhood of 0!Probably that's not what you meant either, but if not maybeyou should re-state the question, including all the conditionsyou have in mind.************************David C. Ullrich === Subject: Re: Example in differential calculus David C. Ullrich which> is differentiable at the origin, such that f(0) = 0, df(0) is the> identity, but such that the inverse is not differentiable at 0;> it's easy to give such an example, for example arranging that> the range of f is not even a neighborhood of 0!No need to go to R^2 for that: On R^1, just let f jump back and forth between x and x - x^2 in an appropriate manner. === Subject: Re: Example in differential an explicit example of a function f: O->F (E, F are two Banach> spaces, O is an open set of E) such that there is one point a in O, df(a) is> a continuous isomorphism E->F, but f^(-1) is not differentiable at point a ?> Note that such an example would require that both E and F are in'nite> dimensional, and that df(a) is not bicontinuous.> So df(a) is continuous and linear and bijective? Then it isbicontinuous by the closed graph theorem. === Subject: Re: Example in differential calculus> So df(a) is continuous and linear and bijective? Then it is> bicontinuous by the closed graph theorem.I shouldn't have assumed E and F are Banach spaces; let's say they're onlynormed vector spaces (the point of my question was to show explicitely thatdf(a) can be a continuous isomorphism, and yet f^(-1) is not differentiableat point a). === Subject: Re: Dif'cult Math Problem Addiction (What should I do?)>> I am obssesively addicted to working on hard problems, especially like> the 3x+1(see http://www.cecm.sfu.ca/organics/papers/lagarias/.) Every> time, I look at any mathematical text, or do anything else I get drawn> back to hard mathematical problems. I could easily spend hours, days,> weeks, or even years working on these problems.>> [...]I'd say your best bet is to try and make it an unbreakable rule to stopworking on a hard problem each time you think you have, or may have,made some progress, even a small step or idea. Then put it aside untilthe next day.That way you'll perhaps assuage the natural compulsion to go on hammeringaway at the problem, and you'll probably 'nd that over time you make moreoverall progress on it anyway, added to which you'll have more time foryour 'bread and butter' === Subject: Re: Dif'cult Math obssesively addicted to working on hard problems, especially like> the 3x+1(see http://www.cecm.sfu.ca/organics/papers/lagarias/.) Every> time, I look at any mathematical text, or do anything else I get drawn> back to hard mathematical problems. I could easily spend hours, days,> weeks, or even years working on these problems.Sorry I didn't respond to this earlier. I overlooked it somehow.Something is an obsession (in the OCD sense) only if thinking about itimpairs your ability to function on a day to day business. The linebetween a passion and an obsession is a 'ne one. If it ruins your life,it's an obsession. If you can make a career out of it and get a PhD inmathematics, it's a passion. I personally obsess about formal logicproblems. I don't like numbers. They are too real. They make me uneasy. Rather than 'ghting it, I'm going for a career in academic philosophy. I suppose it would be a pain in the ass if you wanted to sell shoes for aliving, but why in god's name would you want to do that?-- To announce that there must be no criticism of the president or that we areto stand by the president right or wrong, is not only unpatriotic andservile, but is morally treasonable to the American public. --Theodore Roosevelt === Subject: Re: Dif'cult Math Problem Addiction :>> I am obssesively addicted to working on hard problems, especially like>> the 3x+1(see http://www.cecm.sfu.ca/organics/papers/lagarias/.)Been there, done that. Spent 6 months investigating this problem, instead ofworking in my research area. But then again, I'm easily distracted. I alsoworked on the Riemann Hypothesis, but actually managed a couple ofpublications.--irascible since 1957 === Subject: Re: Corrupt science as intellectual crime, or, The Emperors New Clothes ?? Adjunct Assistant Professor at the sure about math not qualifying as science, however. I think that it>>may. What is the de'nition of a science ? Results must be veri'able,>>falsi'able, and reproducible. I think that math satis'es these three>>better than any physical science, and it does so with complete precision and>>an exactness which is unique to math. If you know something that I dont>>please post.>>Where are the experiments? They are called conjectures and proofs of special cases of conjectures.> Is the scienti'c method taught in math classes? Why is this relevant to whether or not mathematics, as practiced,employ the scienti'c method or not?>Ever try to verify a number is not computable experimentally? Why are there>colleges of science *and* math? There are? I usually see math in the College of Arts and Sciences.-- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.edu === Subject: Re: Corrupt science as intellectual crime, or, The Emperors New Clothes math not qualifying as science, however. I think thatit>may. What is the de'nition of a science ? Results must be veri'able,>>falsi'able, and reproducible. I think that math satis'es these three>>better than any physical science, and it does so with complete precisionand>>an exactness which is unique to math. If you know something that I dont>>please post.>>Where are the experiments?>> They are called conjectures and proofs of special cases of conjectures.Yes - essentially experiments in abstract.You experiment with conjectures, etc, to determine if they are valid withinthe framework of a given abstract model. So what. I call that an experiment.> Is the scienti'c method taught in math classes?>> Why is this relevant to whether or not mathematics, as practiced,> employ the scienti'c method or not?If it does not employ the scienti'c method, then JSH's conjectures might aswell be taught at Harvard or Yale. Would you want that ? No. Math woulddisintegrate without the scienti'c method.>Ever try to verify a number is not computable experimentally? Why arethere>colleges of science *and* math?>> There are? I usually see math in the College of Arts and Sciences.The distinction is made because there is a big difference between someonelike Stephen Hawking and someone like Thomas Edison.Universities have always tried to accomidate both types people to somedegree.Also, making math a distinct dept. removes it from the various social andpolitical in¤uences which can affect the 'ndings of researchers andothers.> => It's not denial. I'm just very selective about> what I accept as reality.> --- Calvin (Calvin and Hobbes)> =>> Arturo Magidin> magidin@math.berkeley.edu> === Subject: Re: Corrupt science as intellectual crime, or, The Emperors New Clothes ?? Adjunct Assistant Professor at the University of Montana.Toilet Seat taught in math classes?>> Why is this relevant to whether or not mathematics, as practiced,>> employ the scienti'c method or not?>>If it does not employ the scienti'c method, then JSH's conjectures might as>well be taught at Harvard or Yale. Would you want that ? No. Math would>disintegrate without the scienti'c method.I think you have me confused. I happen to think that mathematics isscience and uses the scienti'c method, though the way in which mathis usually presented hides this.-- =It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.edu === Subject: Re: Corrupt science as intellectual crime, or, The Emperors New Clothes the scienti'c method taught in math classes?>> Why is this relevant to whether or not mathematics, as practiced,>> employ the scienti'c method or not?>>If it does not employ the scienti'c method, then JSH's conjectures mightas>well be taught at Harvard or Yale. Would you want that ? No. Math would>disintegrate without the scienti'c method.>> I think you have me confused. I happen to think that mathematics is> science and uses the scienti'c method, though the way in which math> is usually presented hides this.I agree. I think that your input.> => It's not denial. I'm just very selective about> what I accept as reality.> --- Calvin (Calvin and Hobbes)> =>> Arturo Magidin> magidin@math.berkeley.edu> === Subject: Re: Corrupt science as intellectual crime, or, The Emperors New Clothes ??>>I'm not sure about math not qualifying as science, however. I think that it>may. What is the de'nition of a science ? Results must be veri'able,>falsi'able, and reproducible. I think that math satis'es these three>better than any physical science, and it does so with complete precision>and>an exactness which is unique to math. If you know something that I dont>please post.>Where are the experiments? >>They are called conjectures and proofs of special cases of conjectures.>True enough. I suppose experiments can also be used to formulate conjectures.> Is the scienti'c method taught in math classes? >>Why is this relevant to whether or not mathematics, as practiced,>employ the scienti'c method or not?>It isn't, really. I just assumed that if the scienti'c method was a vitalpart of mathematics it would be a vital part of math courses. Bad assumption. Mathematics, as practiced, is clearly different from mathematics, as taught. Why?>>Ever try to verify a number is not computable experimentally? Why are there>>colleges of science *and* math? >>There are? I usually see math in the College of Arts and Sciences.>There are. Although not as many a I 'rst thought.rich === Subject: Re: Corrupt science as intellectual crime, or, The Emperors New Clothes ?? Adjunct Assistant Professor at the University of Montana.R3769 qualifying as science, however. I think that it>>may. What is the de'nition of a science ? Results must be veri'able,>>falsi'able, and reproducible. I think that math satis'es these three>>better than any physical science, and it does so with complete precision>>and>>an exactness which is unique to math. If you know something that I dont>>please post.>>Where are the experiments? >>They are called conjectures and proofs of special cases of conjectures.>>True enough. I suppose experiments can also be used to formulate conjectures.Exactly. You do special cases, you do a bunch of examples, you test anumber of speci'c instances, and you make the conjecture. Sometimes,the conjecture is immediately followed by proof. Sometimes there is alarge gap between one and the other in time. Often, parts of theconjecture are proven, others not. The main difference between mathematics and other sciences is that thestandard of proof required in mathematics is far more stringent thanthat required in, say, physics. > Is the scienti'c method taught in math classes? >>Why is this relevant to whether or not mathematics, as practiced,>>employ the scienti'c method or not?>>It isn't, really. I just assumed that if the scienti'c method was a vital>part of mathematics it would be a vital part of math courses. Bad>>assumption. Mathematics, as practiced, is clearly different from mathematics, as taught. >Why?Because mathematics places such a premium on proof. Just like inphysics you don't describe all the experiments that you attempted todesign but was unable to, and you only describe the experiments thatwere actually performed, if you 'nd a proof for a theorem you do notusually go through explaining all the conjectures you made whichproved to be false, or all the speci'c examples you worked out(especially if you have a proof for the general case).Nonetheless, you will sometimes see a textbook (or more frequently, aor discussing possible guesses followed by counterexamples, or guessesfollowed by proofs that need only to 'x certain details on theguesses, and so on.Because mathematics does have an ultimate arbiter (proof), it is notnecessary in most instances to discuss the prior steps in thescienti'c method as you do in empirical sciences, where no amount ofveri'cation is enough to establish truth. There is also atradition to polish proofs, which usually tends to draw them away fromthe experimentation or the thought process that led to theproof. Gauss used to say that the thought process that led to a proofwas like the scaffolding in construction: once you have the building(the proof/theorem), you take away the scaffolding. (This tendency seems to be eroding some, in that more and more papersare taking the time to describe the intuition behind certainparticularly technical proofs).But if you read papers that discuss evidence for conjectures, or thatpropose conjectures, you will see much of the same sort of things thatyou see in physics and other empirical sciences.-- =It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidinmagidin@math.berkeley.edu === Subject: Re: Corrupt science as intellectual crime, or, The Emperors New Clothes ??> However, based on> my sensate human experiences in this world, I 'nd it absolutely and> patently absurd in the extreme to say the least, that someone couldclaim> to> have collected data about things like bigfoot, UFO's, alien abductions,> Wolfman (for e's g), when it is rather obvious that these items cannotbe> studied due to their failure to even exist.>> Well, that argument was once made about rocks falling from the sky.>> Suppose, for example, that I published information regarding the average> height, weight and IQ of a sample population of ghosts, and I made the> claim> that the average IQ was higher and standard deviation was exactly 1/2 ofa> similar sized sample from a population of angels.>> You dont have a problem with that ? If this turned up in the MAA Math> Magazine or somewhere, and it was presented as factual data - you are> telling me that you would not question how I took the samples ? Where I> found such populations in the 'rst place ?>> Such things show up in word problems all the time. It's a cute way to> inject some levity into a problem that is really simply about the> relationships between things.>> As Hilbert said, the important thing to remember about geometry is thatit's> just as true if you replace point, line and plane with table, chair, and> beer stein.>> Without rewriting the question, or rewording the statement, I give youmy> claim.> I claim that if you do statistics on ivalid data, then your results> are> invalid by de'nition, regardless of whether the derived solutionscome out correct or not.>> Yabbut. In real life, data are not really valid or invalid, simply moreor> less correct. If your data pretty much re¤ect reality, then your> conclusions will again pretty much re¤ect reality as well. To 'nd out> how well, you need to do research on robustness of statisticaltechniques.> Some things are much like walking a tightrope -- one little slip andyou're> history. Other things are more like walking -- a little trip and nothing> much happens (usually).>> The concept of robustness has also found its way into arti'cial> intelligence and lots of other engineering applications. Again, it's the> idea that you can recover from incomplete data or errors introduced> somewhere in the process.>> Also, we don't really care if you are talking about manufacturing widgetsor> sighting ghosts. What matters mathematically is the process. It's well> known that applying the right process to the wrong data generatesnonsense,> as A N Niel said at the beginning of this few observations. First, let merecall an issue relating to dangling chads, where a certain set of datawas drawn into serious question, and sealt a serious blow to the verycredibility of elections in the US as far as I'm concerned. The point being, however, that trials in which discrete decisions arebeing recorded such as in an election - how does it come out so screwed up?? And if we are having thes issues with discrete data from a poll booth,then how much more questionable is psychological data where EVERYTHING issubjective ?? My concern here is not to attack statistics - which is soundtheoretically, but rather the complete misapplication of statistics bypsychology. The only thing in life that I compare this to is that there is nothingwrong with a sportscar. The car is theoretically sound. But the operator isa madman, and he slams the car into a brick wall. This is what psychologydoes with statistics. They crash it into a brick wall with their stupidityand their erroneous subjective idiocy. I will demonstrate elsewhere, thattheir usage of statistics is false.My main concern here, is to raise the issue of data and conclusions. It maysound pedantic, btu I am statistics upon bad data is not a sound statistical practice.If you have incorrect data to work with, you cannot produce conclusionswhich are correct. It is impossible to derive real world truths based uopndata which is corrupted.Now - con'dence levels notwithstanding, I am not talking about errors dueto approximation or levels of con'dence.Here's an example of what I am after. I am hired by Ford Corp to do a statistical analysis of a sample of Fordautomobiles. Unfortunately, all I have is a parking lot full of Chevy's andBMW's. Can I perform the task which is required of me based on the populationthat I have been given ?? Can I turn in my results to Ford, an analysis ofFord vehicles, if I do not have any Ford cars to study ?? Is it valid tostudy a BMW or a Chevy and turn in the results as if having studied a Ford ? === Subject: Re: distance of a point to a polygonThe trivial answer is to compute the distance for each eadge separately andchoose the minimum.But, if you have a constant polygon and several points, you could probablydo faster at cost of space for storing the polygon's additional data or atcost of time needed to order the polygons data in some way, which is doneonly once. What I am thinking of is some arrangement of edges in pre-sortedarrays or tree-like structures so that for a 'xed given polygon you get adecision tree so that the number of quieries (hence, running time) is low.Maybe you could use the fact that in order to decide between edge A iscloser and edge B is closer, there exist one line (in rare cases 2 lines)which are at the same distance to both, so that the query (left/right ofthis line) could distinguish between them.Look into books about computational n-sided polygon. How can i tell how> far p is from any point (either vertex or edge) of the polygon?>> === Subject: Re: JSH: thought I'd talk about it in detail. Here>are some headers === so you can 'nd the post:>>Subject: Re: Mathematical consistency, courage>>In his post Decker claimed to mirror my argument using a quadratic>instead of a cubic, where he has>>(5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) >>where his a's are roots of >>a^2 - (x - 1)a + 7(x^2 + x).>>Checking at x=0 reveals that the actual constant terms of the>factorization are 7 and 2, where Decker picked a_1(0) = 0 at x=0.>>Now then, consider what happens if you divide both sides of>(5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) >by 7, as then you end up with something like>(5b_1(x) + 1)(5b_2(x) + 2) = 25x^2 + 30x + 2>>where the b's are roots of some unknown quadratic, though the 'rstand last coef'cients ARE known:>>b^2 + ? b + (x^2 + x).>>Now then, it's just a quadratic people. SOME mathematician in all the>world should be able to give what the middle coef'cient is, right?>>>>Right. The middle coef'cient in this case is>> (-3x + sqrt(-7x^2 - 8x))/4 Well I put that up kind of curious that someone might 'll it in, but> you failed badly here Decker, as my 'rst check was at x=1.> a^2 - 7(1^2 + 1) = a^2 - 14, so a=+/-sqrt(-14).> Now then, dividing off 7, should then give b=+/-sqrt(-2).>> No. See below.> But with your claims, you get> b^2 + (-3 + sqrt(-15))b/4 - 2 = 0.> Right, that's what I get. Here's how. Your speci'cation> was that you wanted two functions, b_1(x) and b_2(x)> which satisfy> P(x)/7 = 25x^2 + 30x + 2 = (5b_1(x) + 1)(5b_2(x) + 2) [1]> and which are roots of> b^2 + C(x)b + (x^2 + x) [2]> Where C(x) is to be found.> I remind you that those are your requirements, as stated above.> [Note in passing that there are in'nitely many other b's> one could pick, but in each case you'd have a different> polynomial for which they would be roots.]> At any rate, since [2] is de'ned by> (b - b_1(x))(b - b_2(x)) = b^2 + C(x)b + (x^2 + x)> we must have b_1(x)b_2(x) = x^2 + x> and> -(b_1(x) + b_2(x)) = C(x). Expand the RHS of [1] and we have> 25x^2 + 30x + 2 = 25b_1(x)b_2(x) + (2b_1(x) + b_2(x))(5) + 2> = 25(x^2 + x) + (2b_1(x) + b_2(x))(5) + 2> so, rewriting the LHS and subtracting 2 from both sides we have> 25(x^2 + x) + 5x = 25(x^2 + x) + (2b_1(x) + b_2(x))(5)> so> 2b_1(x) + b_2(x) = x> from which we have> b_2(x) = x - 2b_1(x)> Since we have your requirement on the product of the b's> we must have x^2 + x = b_1(x)b_2(x) = b_1(x)(x - 2b_1(x))> so (b_1(x))^2 - xb_1(x) + (x^2 + x) = 0, which we> can solve for b_1(x):> b_1(x) = (x + sqrt(-7x^2 - 8x))/4Nope. You dropped a 2, as you should have2b_1(x)^2 - x b_1(x) + x^2 + x = 0.I was curious to see if anyone on sci.math would correct your or evennotice your mistake.At this time, I'm curious to know if anyone read Decker's post, andthought he was right, or noticed his mistake.Or did no one else besides me bother to read his post?James Harris === Subject: Re: JSH: Rick Decker's = b_1(x)(x - 2b_1(x))>>so (b_1(x))^2 - xb_1(x) + (x^2 + x) = 0, which we>>can solve for b_1(x):>> b_1(x) = (x + sqrt(-7x^2 - 8x))/4> Nope. You dropped a 2, as you should have> 2b_1(x)^2 - x b_1(x) + x^2 + x = 0.>Quite right, I dropped a coef'cient of 2. Everything else iscorrect, however. === Subject: Re: JSH: Rick Decker's example> James b_1(x)(x - 2b_1(x))>>so (b_1(x))^2 - xb_1(x) + (x^2 + x) = 0, which we>>can solve for b_1(x):>> b_1(x) = (x + sqrt(-7x^2 - 8x))/4> Nope. You dropped a 2, as you should have> 2b_1(x)^2 - x b_1(x) + x^2 + x = 0.>> Quite right, I dropped a coef'cient of 2. Everything else is> correct, however.Yes, despite dropping the 2, you correctly solved for b_1(x), but thepoint here is that now it's clear that in general b_1(x) is not analgebraic integer function!!!That jumped out at me with that dropped 2 added back.Now then, do you understand why that's signi'cant?James Harris === Subject: Re: JSH: Rick Decker's exampleJames Harris = b_1(x)b_2(x) = b_1(x)(x - 2b_1(x))>>so (b_1(x))^2 - xb_1(x) + (x^2 + x) = 0, which we>>can solve for b_1(x):>> b_1(x) = (x + sqrt(-7x^2 - 8x))/4>Nope. You dropped a 2, as you should have>>2b_1(x)^2 - x b_1(x) + x^2 + x = 0.>>Quite right, I dropped a coef'cient of 2. Everything else is>>correct, however.> Yes, despite dropping the 2, you correctly solved for b_1(x), but the> point here is that now it's clear that in general b_1(x) is not an> algebraic integer function!!!> That jumped out at me with that dropped 2 added back.> Now then, do you understand why that's signi'cant?> Yup. In fact, that underscores the point I've been makingall along--you can't in general split 7 as (in Nora's terms)7, 1 and expect to have a factorization in algebraic === Subject: Re: what is this matrix operation?> This is a special case of a Kronecker Product. Ask How to describe the following matrix operation formally?>> A big matrix, A, if we divide it into blocks>>> A=[A11 A12 A13;> A21 A22 A23]>> There A11, A12, A13, A21, A22, A23 are blocks... now I want to multiplyeach> block with a scalar C11, C12, C13, C21, C22, C23,(these C's are scalar> numbers) how to describe this operation mathematically and formally?>> Result=[A11*C11 A12*C12 A13*C13;> A21*C21 it is a variant ofKronecker product... Can you explain more?The result is the same size as A, (since the C's are scalars...)If it is a Kronecker product, the size of the result should besize(A)*size(C)...So this is not a Kronecker product...Best,-Walala === Subject: Re: what is this matrix operation?Hello Walala,ok, it isn't a special case of the Kronecker product, nor is ita generalization of it - I just was too hasty.Unfortunately I don't know another special case of a Kronecker Product. Ask all,>>How to describe the following matrix operation formally?>>A big matrix, A, if we divide it into blocks>A=[A11 A12 A13;> A21 A22 A23]>>There A11, A12, A13, A21, A22, A23 are blocks... now I want to multiply> each>block with a scalar C11, C12, C13, C21, C22, C23,(these C's are scalar>numbers) how to describe this operation mathematically and formally?>>Result=[A11*C11 your answer... But I cannot see that it is a variant of> Kronecker product... Can you explain more?> The result is the same size as A, (since the C's are scalars...)> If it is a Kronecker product, the size of the result should be> size(A)*size(C)...> So this is not a Kronecker product...> Best,> -Walala> === Subject: Re: how to design iterative optimization algorithms to solve matrix equation? Any books?>> help me!>>In this problem, I need to construct some matrices which satisfy the>following matrix equation:>(( A * A )V1 + ( B * B) V2) V = (D * D)>>where * denotes Kronecker product, A, B, V1, V2, V are unknownmatrices>that needs solving; they are all square. Some structure needs to beimposed:>I have certain pattern for A and B; and V1, V2, and V are required to be>diagonal... Matrix D is given...>The task is to 'nd the best approximation of the above-mentioned A, B,V1,>V2 and V... I have been thinking about this for long time... can anybody>give me some hints?>I guess it is dif'cult to 'nd closed form analytical solution..>> How dii'cult is it - it seems to me that the problem has an awful lot> of structure.>>. How to>design iterative algorithm to let computer search for the answer? It is>really hard... please help me!>> You are trying to solve the nonlinear system of equations>> (( A * A )V1 + ( B * B) V2) V - (D * D) = 0>> If the problem is not too large use Newton's method. If the problem> is too big for that look for a canned program that will solve> nonlinear systems without questions:a) (( A * A )V1 + ( B * B) V2) V - (D * D) = 0These * are Kronecker products not the normal multiplier, how to designNewton's iteration algorithm for this equation?b) The worst thing is that this problem is not continuous; the elements in Vcan be arbitary; but the elements in A, B, V1, V2 need to be integer...hence I lost completely how to do iterative algorithm for these kind ofmix-integer-continuous much,-Walalla === Subject: Re: Powerset topology product topology {0,1}^S or actually the> power-set topology is the largest topology assuring the assigned limits.>> You have shown that, in this topology, all the topological limits are> set-limits, no? Something to ponder: If you start throwing in more> open sets, do you lose some of these convergences?>Possibly, but when it's the largest assuring designated limits ofsequences, a 'ner topologies will lose some of those convergences.> Another puzzle I'm musing upon is if a topology assuring the assigned> limits can be enlarged to a 1st countable topology while still assuring> the assigned limits. This I think likely not possible.>> Do you really mean enlarge? The coinduced topology is already the> largest. Also, it seems to me that to make a space 'rst countable,> you have to *remove* open sets, not add them, no? (Not sure about> that one.) For example, all sequences converge to every point in the> indiscrete topology, which is 'rst countable. Thus, you could use as> a subbase the union of all 'rst countable topologies where the> sequences converge to the designated limits. Trouble is, you get even> more convergent sequences than before.Is 'rst countable a subbase property? No.A 'rst countable subbase topology is stronger property than a'rst countable base topology; the later being equivalent toa 'rst countable topology.Thus, am I right there's no largest 'rst countable topology?No, the discrete topology is the largest.The nascent feature being that both the indiscrete and the discretetopology are 'rst countable and somewhere in between is the realm of thenot 'rst countable. Thus to chose the smallest or the largest likelydepends where in the middle one starts.For example when I combined requirement for 'rst countable with a secondproperty such as assuring designated limits, then it appears, tho thesecond (a subbase) property has a largest topology preserving itself, whenin combination with 'rst countable, the sup of the topologies with bothproperties will preserve the second property but may not be 'rstcountable. === Subject: Re: Trivial question on notationJgissw2 scribbled the following:> Been a number of years since I took math - I was trying to remember how the> notation, O(p^n), and o(p^-n), if that is correct, ran. That is, suppose you> have a series of powers in, say, which are at least as great as n, or smaller> than -n. How does that go?> John GWAFAIK it is something like:O(something) means smaller than or equal to somethingo(something) means smaller than somethingTheta(something) means equal to somethingOmega(something) and omega(something) mean greater than or equal tosomething, and greater than something, but I forget which is which.-- /-- Joona Palaste (palaste@cc.helsinki.') ------------- Finland ---------- http://www.helsinki.'/~palaste --------------------- rules! --------/ === Subject: Re: Trivial question on notationJoona I means smaller than or equal to something> o(something) means smaller than something> Theta(something) means equal to something> Omega(something) and omega(something) mean greater than or equal to> something, and greater than something, but I forget which is which.I hope the original poster doesn't take this response too seriously,as it is fairly inaccurate, egf(n) = O(p^n) means f(n)/p^n is bounded, at least for n >= Nf(n) = o(p^n) means f(n)/p^n -> 0 as n -> infty-- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ietel: +353-86-2336090, +353-1-2842366s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: distances sha1:Ob7sRMnn7FLOH/APYrVJjnXqFFA=Here is an interesting question that I came acrossgiven N points on a two dimensional plane, one can de'ne a distance matrix A = [a ] of order N x N such that a is the distance between ij ijthe point i, and j.The question is given this matrix can a set of points be given thatwill satisfy the given distance matrix in an ef'cient manner?abi === Subject: Re: distances between n points> Here is an interesting question that I came across> given N points on a two dimensional plane, one can de'ne a distance > matrix A = [a ] of order N x N such that a is the distance between> ij ij> the point i, and j.> The question is given this matrix can a set of points be given that> will satisfy the given distance matrix in an ef'cient manner?Yes, provided a set of points exists. Google for Euclidean and embeddingfor more results. You can de'ne P(1) = (0, 0), and P(2) = (r, 0), where r = a(1,2). Now, for i = 3, 4, ..., N, solve the equations distance ((x,y), (0,0)) = a(1,i) and distance ((x,y), (r,0)) = a(2,i),where distance ((x1,y1), (x2,y2)) = sqrt ((x1-x2)^2 + (y1-y2)^2)to give two solutions. If i = 3, you can arbitrarily choose one of them. Otherwise, check your solutions to make sure that distance((x,y), P(j)) = a(i,j) for j = 3, 4, ..., i-1. (You may have to throwout one of them.) If there is a solution (x,y) which works, set P(i) = (x,y)and repeat until i = N. -- Christopher Heckman === Subject: Re: across>given N points on a two dimensional plane, one can de'ne a distance >matrix A = [a ] of order N x N such that a is the distance between> ij ij>the point i, and j.>>The question is given this matrix can a set of points be given that>will satisfy the given distance matrix in an ef'cient manner?Well, if you could 'nd such a set of points, you could translate thewhole thing so that point #1 is at the origin, right? The question isto decide where the other N-1 points should be. The information you'vegot determines the lengths of the N-1 vectors v_j from the origin tothese other points, but it also determines all the inner productsbetween these vectors: < v_i, v_j > = (a_{ij})^2 - (a_{i1})^2 - (a_{j1})^2.So these points you're looking for, or I should say these vectors you'relooking for, can be thought of as columns in an (N-1)x(N-1) matrix Vfor which the matrix product (V^t)(V) is something you can readilycalculate from your A; it's another (N-1)x(N-1) matrix I'll call B.I'm assuming the data with which you began are symmetric; that willmake B a symmetric matrix.Now, on the one hand, a matrix B which can be written in the form(V^t)(V) will have to be non-negative semide'nite. So you can checkyour matrix B and see whether it satis'es this condition -- if not,you can't 'nd the points you're looking for. In essence, what thiscondition is checking is the triangle inequality -- you can't for example'nd three points with a_{12} = 5, a_{23} = 6, a_{13} = 20.On the other hand, if B is positive de'nite, then it has a Cholesky decomposition, which is not hard to compute: that is, thereis a lower-triangular matrix C with B = C C^t. The transpose ofthis matrix C is exactly the matrix V you're looking for, thatis, you can read off the locations of the N-1 points from the rowsof C. This particular construction will perform a naturalnormalization for you: after you've put one point at the origin, youcan arrange to have the next on the x-axis, the next in the xy-plane,and so on.Note that this process assumes that by point you mean point in R^N;that will in general be necessary when the distances a_{ij} atrandom. If you meant point in R^3, then a necessary condition isthat B have rank 3. (And you can ALWAYS embed your points into a space with a non-Euclidean metric; the signature of the matrix B shows the signature you need.)It's a little more subtle to handle a borderline situation whereB is only semide'nite, or when the data are noisy and you can'tquite decide whether the matrix is de'nite or not. In that caseyou can use a Singular Value Decomposition to see how far B isfrom being a de'nite matrix of such-and-such a rank; if most ofthe singular values are very small you can project down to theselow-rank matrices and decompose them, that is, the SVD allows usto write B = sum lambda_i (u_i)(u_i)^t ; if all but k of thelambda_i are roughly zero, and those k are positive, then youcan replace B by the sum over just those k summands and willget a matrix representing the distances among some set of pointsR^k (just read off the N rows of the matrix whose columns aresqrt(lambda_i) (u_i) ; the u_i should be an orthonormal familyof vectors.)davePS- No, that just happened. (Well, OK, I tweaked the second halfof the paragraph to make it keep happening.) === Subject: Re: distances between n pointsAbi question that I came across> given N points on a two dimensional plane, one can de'ne a distance> matrix A = [a ] of order N x N such that a is the distance between> ij ij> the point i, and j.> The question is given this matrix can a set of points be given that> will satisfy the given distance matrix in an ef'cient manner?The solution may not exist. Consider the 3x3 matrix0 5 15 0 11 1 0 === Subject: Re: distances between n question that I came across>> given N points on a two dimensional plane, one can de'ne a distance>> matrix A = [a ] of order N x N such that a is the distance between>> ij ij>> the point i, and j.>> The question is given this matrix can a set of points be given that>> will satisfy the given distance matrix in an ef'cient manner?>>The solution may not exist. Consider the 3x3 matrix>0 5 1>5 0 1>1 1 0Obviously you have to de'ne the matrix suitably, i.e.:i) a_ij >= 0 for all i, jii) a_ii = 0 for all iiii) a_ij = a_ji for all i, jiv) a_ik <= a_ij + a_jk for all i, j, kThe question is, if all these norm qualities are ful'lled does thatguarantee a solution exists in R^2 (I think Dave Rusin showed that itdoesn't) and if so can we always 'nd it without going through allpossible combinations of point symmetries (which I assume is not easyfor large n). === Subject: Re: distances between n pointsToni Lassila points i and j]>Obviously you have to de'ne the matrix suitably, i.e.:>>i) a_ij >= 0 for all i, j>ii) a_ii = 0 for all i>iii) a_ij = a_ji for all i, j>iv) a_ik <= a_ij + a_jk for all i, j, k>>The question is, if all these norm qualities are ful'lled does that>guarantee a solution exists in R^2 (I think Dave Rusin showed that it>doesn't)Yes, in case I wasn't clear:(1) Condition (ii) is needed to place one point into R^0 and have a_ii be the distance from the point to itself.(2) Conditions (i) and (iii) are needed to place two points into R^1 and have a_ij be the distance from one point to another.(3) Condition (iv) is needed to place three points into R^2 consistent with the distances given. (It's the triangle inequality).(4) There is an additional constraint among the six distances between points in a set of four which is necessary and suf'cient for the four points to be isometrically embedded into R^3. You could call it the tetrahedral inequality, and we did, once, in this newsgroup. (It can be interpreted as stating that the thing which ought to compute the volume of a tetrahedron is in fact non-negative.)(5) Further inequalities are needed to ensure that any set of 've points can be embedded into R^4, etc. These are equivalent to the conditions you would check to show that a matrix (I called it B in a prior post) is positive-de'nite: you check a 4x4 submatrix, then a 5x5 submatrix, etc. If you want all N points to lie in R^(N-1) then you want these to be all positive (well, positif I suppose.)If you want to go further and require that the embedded points lie inR^k for some k>in the Cantor set, |f(x) - f(y)| <= |x - y|.> Whoops, |f(1/3) - f(0)| = |1/2 - 0| = 1/2. (Also note that any f satisfying > |f(x) - f(y)| <= |x - y| takes sets of measure 0 to sets of Santos === Subject: Re: US Map Graph Theory Problem?Robert list would be good, to clarify for example whether you>count corner adjacencies (like UT and NM, or AZ and CO), and whichunderwater boundaries (as between HI and AK, MN and MI, or RI and NY)>you treat as adjacencies. Without specifying a list you probably>won't get useful answers.> HI and AK??? There's a heck of a lot of international water between> them. Or are you claiming American sovereignty over the whole North> Paci'c?I'm not making that claim -- Russia probably would object -- but merely wanted the OP to give his or her adjacency list. On the three cases above, I would say no, yes, and maybe.The maps I looked at show a Minnesota - Michigan boundary within Lake Superior, and don't indicate if Rhode Island and New York have a boundary in Long Island Sound.-jiw <-> (|| Az( y c z -> x e z )| /| Ez( x e z / ~(y c z) ) ) )Do you mean x e y instead of a e b in the 'rst line?[...]|Well, it is not a great theory. But, it does have some relation to|apartness.I don't have much to say about it, partly because I don't seewhat the motivation is. Is it supposed to be some kind ofset theory? Inventing your own home-brew set theory is 'ne,but without anything more speci'c you intend to accomplishwith it, it's not liable to go anywhere.Keith Ramsay === Subject: Re: Who Proved In'nite Series Can Have A Finite Sum? Estela Beslerzewski in'nite series, be a paradox itself?> Or> A sum of in'nite series, would then classify an 'in'nite series', in fact a > 'nite series, since a sum can be made.> Not if the real numbers behave as advertized. The Cauchy construction of the real numbers guarantees that for every Cauchy sequence there is a real number. If the partial sums of an in'nite sum form a Cauchy sequence, q.v. , there MUST be a unique real number represented by that sequence. === Subject: Re: Who Proved In'nite Series Can Have A Finite Sum?In other word's, you can't work with real numbers, if your goal is to 'ndsomething that is infact in'nite.By in'nite, there is no end.> Estela determining a sum for a in'nite series, be a paradox itself? Or> A sum of in'nite series, would then classify an 'in'nite series', infact a> 'nite series, since a sum can be made.> Not if the real numbers behave as advertized. The Cauchy construction of> the real numbers guarantees that for every Cauchy sequence there is a> real number. If the partial sums of an in'nite sum form a Cauchy> sequence, q.v. , there MUST be a unique real number represented by that> sequence. === Subject: Re: Who Proved In'nite Series Can Have A Finite Sum? Estela Beslerzewski in'nite series, be a paradox itself?> Or> A sum of in'nite series, would then classify an 'in'nite series', in> fact a 'nite series, since a sum can be made.>>> Not if the real numbers behave as advertized. The Cauchy construction of the real numbers guarantees that for every Cauchy sequence there is a> real number. If the partial sums of an in'nite sum form a Cauchy> sequence, q.v. , there MUST be a unique real number represented by that> sequence.> In other word's, you can't work with real numbers, if your goal is to 'nd> something that is infact in'nite.> By in'nite, there is no end.Quite the reverse. If you want to work with the reals, you have to have a basic set that is already uncountably in'nite., and is the set of real numbers is based on an in'nite set of in'nite sets (Dedekind cuts or equivalence classes of Cauchy sequences). === Subject: Re: Who Proved was hard to write; it should be hard to use.Shmuel (Seymour PM, tb+usenet@becket.net (Thomas Bushnell, BSG) said:>So the concept of in'nite physical extension has no meaning? > Again, you are challenging things that I didn't write instead of> things that I did.I'm a little confused. Perhaps you took that to be a rhetoricalquestion, but it was not. Rather, I'm seeking clari'cation of yourobjection, because I don't yet understand it.One way that in'nite was used, in particular one way in which itwas asserted by the ancients or the medievals that the actualin'nite couldn't exist was in arguing that in'nite physicalextension cannot exist. Their arguments for this position do notgenerally depend on any particular metaphysics or curiousunderstanding of in'nite, but rather upon an argument about whatthey thought was physically possible.Physics changes. In a Newtonian world, it doesn't seem so crazy tothink there could be an in'nitely extended physical thing. But in aclosed relativistic world, it is once again taken to beimpossible--though of course for a different reason than that given bythe ancients. GR does not settle the question, but for all we know,the next theory (perhaps reconciling GR and QM) will also imply thatthe universe must be closed. (In which case, in'nite physicalextension would be impossible.)Now I don't think that it's a good idea for anyone to start insistingthey know whether in'nite physical extension is possible abstractedfrom any particular physics. But the arguments given by the ancientsor the medievals against the notion *are* arguments from their bestphysical theories. Of course, we have vastly better physicaltheories, which don't seem to have settled the question very welleither.In all this, it does seem clear to me that there *is* a reasonablequestion: is it possible for there to be something which is in'nitein physical extension? Now this was one of the things that the ancients and medievals meantby the term. Is this one meaningless in your opinion? If not, isthere some other objection that you would raise? And if it ismeaningless, what distinguishes it from current discussions aboutwhether the universe is open or closed?Thomas === Subject: Re: Who Proved 63 28 EB DA E6 44 E5 5E EC F3 04 26 4E BF 1A 92X-Zippy-Says: Now that I have my ``APPLE,'' I comprehend COST ACCOUNTING!!Shmuel (Seymour J.) Metz attributed to me something that I did not write; I returned the> favor.At this point, I've lost track. If I attribute something to you whichyou didn't write, then it was by accident or from some confusion, andI apologize. Can you tell me what it was, since I have looked over,and can't 'nd it (presumably, because I'm still confused aboutwhatever it was)?>You seem to say X has no meaning when all>you should say is I didn't understand when you said X.> No. I write X has no meaning when X has no meaning. If you really> meant Y, please clarify.Curious. How do you distuingish between when you don't understand themeaning of a thing and when you know it has no meaning at all?I gave a fairly detailed description of different usages o'n'nite, and my understanding of how the word was used by many ofthe ancients and medievals in contexts relevant to the discussionhere. Is your objection that I have misunderstood them? In whichcase, I would be very interested to see some of the texts that give adifferent understanding, so that I can correct my mistakenimpression. Or is it your objection that the meaning as I havedescribed it is really no meaning at all? I gave a fairly detailedpost, which you didn't see 't to answer in any kind of detail, so I'mafraid that I don't see which part of it you thought was incorrect, orwhy you thought it was incorrect. My snail-like brain is sometimes unable to see my mistakes when all Iam told is there is a mistake there. Can you please point it outmore slowly using more sentences?Thomas === Subject: Re: Who Proved In'nite Series replies to this message constitute permission for an emailed EC F3 04 26 4E BF 1A 92X-Tom-Swiftie: RTFM, Tom informedShmuel at 08:14 PM, tb+usenet@becket.net (Thomas Bushnell, BSG) said:What connection between irrational>numbers and in'nity do you think the Greeks might have noticed?> The method of exhaustion comes to mind.I'm unsure what speci'cally you are referring to here. Can youidentify please where the Greeks spoke of irrational numbers (that is,of numbers they knew were irrational) in connection with the method ofexhaustion, or where they notice such a connection?They usually speak of irrational numbers as being incommensurate, oroccasionally as having no ratio, but I can't think of any case wherethey connect them to exhaustion or the like.Thomas === Subject: Re: >> It looks like George beat me to the essential idea. Mine is simpler,> but there is conceptual common ground.>> [...]> If sheer sustained effort and commitment guaranteed results, I don't> doubt that Jack Sarfatti (and JSH) would be up there with Newton and> Einstein. Here's an image of Jack: http://www.thinking-allowed.com/1jsarfatti.html Jack Sarfatti, PhD, is President of the Internet Science Education Project.Google for Internet Science Education ProjectHit 1: Jack Sarfatti -- Physics/Consciousness Research Group - PCRGSo he's president of his own self-invented ¤ight of fancy.Vain crossposted to certain groups in GNUS?If I could bin anything crossposted to sci.physics.relativity I'd be a happy man. Second to that, anyone got any good subject-line regexps to ensure I see even less Sarfatti-related nonsense than I currently do?(/Sarfatti/ is a start certainly)Phil -- Unpatched IE vulnerability: WebFolder data InjectionDescription: Injecting arbitrary data in the My Computer zoneReference: http://msgs.securepoint.com/cgi-bin/get/bugtraq0305/13.html Carmody John R Ramsden [Hammond....?] beat me to the essential idea. > Mine is simpler, but there is conceptual common ground.> Here's an image of Jack:> http://www.thinking-allowed.com/1jsarfatti.html> :< /) :< /) :< /) :< /) mmmhh :<|) :<|) :-) hahah :-) :-) :-) :-)) wwwhhooooaahh :-))) hahahah...hahah.....hohhhhhhohho..hahahah.... uuhhhhhaaahhahahah....wohahhh..hahaha.....wohahhh..hahaha..... hahaha.......hahah.....hohhhhhhohho..hahahah....wohahhh.. hahaha.....hahaha....hahahah...hahah.....hohhhhhhohho.. hahahaaaa........whoahhhhhhhh........hohohoh....... uuhhhhhaaahhahahah....wohahhh..hahaha.....hahaha.......hahah.. ... :-)) :<) :-))) ;>)) hohhhhhhohho..hahahah....wohahhh..hahaha.....hahaha....hahahah ...hahah.....hohhhhhhohho..hahahah....wohahhh..hahaha..... hahaha.......hahah.....hohhhhhhohho..hahahah....wohahhh.. hahahaaaa........whoahhhhhhhh........hohohoh....... uuhhhhhaaahhahaha.....:-)) :<) :-))) ;>)) hohhhhhhohho..hahahah....wohahhh..hahaha....hahaha.......hahah .....hohhhhhhohho..hahahah.........:-)) :<) :-))) ;>)) hahaha....hahahah...hahah.....hohhhhhhohho..hahahah....wohahhh ..hahaha.....hahahaaaa........whoahhhhhhhh........hohohoh..... ..uuhhhhhaaahhahaha.......hahah.....hohhhhhhohho..wohahhh.. hahaha.....hahaha....hahahah...hahahhohhhhhhohho..hahahah.... wohahhh..hahaha.....hahaha............:-)) :<) :-))) ;>)) hohhhhhhohho..wohahhh..hahaha.....hahaha....hahahah........... ..ts,ts,ts................has Gergy, Gorgy or Ge/orgy Hammond ever posted or urled his pix? Does anyone know?ahahahaha.......ahahahanson === Subject: Re: What is The universe is composed of desire. The purpose of the universe istwofold: 1) Excrement. 2) Orgasms.HIV/AIDS demonstrates the inadvisability of combining the two.--Uncle Alhttp://www.mazepath.com/uncleal/qz.pdfhttp:// www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) === Subject: Re: What is The Universe made of?> Jack of desire. The purpose of the universe is> twofold:>> 1) Excrement.> 2) Orgasms.>> HIV/AIDS demonstrates the inadvisability of combining the two.>> --> Uncle Al> http://www.mazepath.com/uncleal/qz.pdf> http://www.mazepath.com/uncleal/eotvos.htm> (Do something naughty to physics)so the big bang was one huge orgasm? Hmmmm..... === Subject: Re: What is The Universe made purpose of the universe is>> twofold:>> 1) Excrement.>> 2) Orgasms.>> HIV/AIDS demonstrates the inadvisability of combining the two.>> -->> Uncle Al>> http://www.mazepath.com/uncleal/qz.pdf>> http://www.mazepath.com/uncleal/eotvos.htm>> (Do something naughty to physics)>>so the big bang was one huge orgasm? Hmmmm.....Even God gets off every once in a while.--Dr.Postman USPS, MBMC, BsD; Disgruntled, But UnarmedMember,Board of Directors of afa-b, SKEP-TI-CULT member #15-51506-253.You can email me at: TuriFake(at)hotmail.comShake it like a polaroid picture. - Andre 3000 of Outkast === Subject: Re: What is The desire. > The purpose of the universe is twofold:> 1) Excrement.> 2) Orgasms.> HIV/AIDS demonstrates the inadvisability of combining the two.> Uncle Al> http://www.mazepath.com/uncleal/qz.pdf> http://www.mazepath.com/uncleal/eotvos.htm> (Do something naughty to physics)> so the big bang was one huge orgasm? Hmmmm.....Didn't you understand what Al said:Al was there, watched the orgasm, got naughty & took a huge .It's that simple. No Hmmmm.....ahahaha.....ahahahanson === Subject: calendars of 6 year intervals; calendar of treesI collect pretty calendars and knew that one year I could revisit anold calendar and hang it up to use for that new year. I had to wait 6years because now I am re-using a 1998 calendar for year stock of 6 calendarsto cover every future year.I collect calendars that have a botany interest. The 1998 calendar isfrom Sierra Club showing for January a Bloodroot from Illinois. It isa beautiful ¤ower of luminescent white and in the middle a yellowthat is almost gold.What I need for my collection of calendars are SierraClub typepictures of pretty and outstanding trees.I have crop patterns calendar but I need calendars showing trees.Archimedes Plutoniumwhole entire Universe is just one big atom where dotsof the electron-dot-cloud are galaxies === Subject: Re: calendars of 6 year intervals; calendar of trees