mm-224 === Subject: Re: polysigned numbers> Yes, four-signed numbers are three dimensional in the traditional> RxRxR sense. Ts comes simply as a result of general summation, or> interpretation involves a tatrahedral set of poles coming out of the> origin.I agree that the four-signed domain is probably more new and> interesting. There is a natural product and so the ability to perform> the mandelbrot mapping on it exists. I am not aware of what ts> product's equivalent is for the cartesian space. It may be new, and> since it maps to the complex numbers for three-signed we could> consider it to be of value. The trouble that I am having is in the> transformation back to cartesian dimensions. Ts involves decomposing> a tetrahedron into its three-dimensional components. It is just> triginometry but I am struggling with it. Perhaps you will be able to> do it. I would put the # pole right on the x-axis. Then the - pole in> the x-y plane, then the last two wherever they fall. There seem to be> some arbitrary choices in ts transform that didn't exist back in> three-signed. Anyhow, I tnk we should look at it grapcally as a> tetrahedral since that is the symmetrical mapping of four poles in> RxRxR.There are a few maps that could be gotten working without a transform> and just analyzing for example the minus-pound (-,#) plane. There are> six of these graphs. I am coding up general sign math now. Once I have> that I should be able to graph these and gher signs too.I have done the mandelbrot mapping for Y-space(three-signed) and it> looks a whole lot like the standard complex one, as it should since> the math works out to be equivalent, though I did not know it until> after I did the graph. I have been more focused on physics thoughts> and trying to stretch ts math towards a source of stability.I am very happy to talk to someone who appreciates and understands> ts construct. I'm still not sure that someone else hasn't done ts> math already. There is a guy really close to it with terplex numbers> but I don't tnk it's the same. He seems to be using reals as their> basis wch confuses me.> I've done some work on the four-signed math ts weekend. It looks like the key to making it work is carefully defining a reduced form. Under certain definitions of reduced from, n-signed math exists in R^(n-1), Under other definitions, n-signed math is isomorpc to C. I need to finish some more details, then I'll let you know what I come up with. I tnk that tnking in terms of n-tuples will make tngs easier when dealing with general n-signed math. Otherwise you may need to === SINntp-Posting-Host: apps.cwi.nlIt would have been better when it had been factually correct. > A metallic prototype of the chosen unit of mass was made, and is >called the Ôkilogram'; the French name for weight.It never has been a French name for weight. It is derived from the oldGreek who used the gramme to denote a certain amount of weight or mass(in those times the distinction was not made) equal to 1/8 drachme. > Herein lies the problem: > Is the kilogram a measure of weight, or a measure of mass? OR doesn't it >really matter?It is a measure of mass, but the general public uses it for both weight andmass because it is not interested in the distinction. Note moreover thatto weigh sometng can both mean determine the weight or determine themass. > The gram >and kilogram of mass are commonly called weights, and are inconsistently >defined in various texts and dictionaries:Dictionaries are not there to prescribe what is correct, but to describehow sometng is used. When by some ßuke a large percentage of thepopulation starts to use the gramme as a unit of speed, than, for thedictionaries, it becomes a unit of speed. > A kilogram weighs 9.806 newtons (about 2.2 lb), on Earth. Its mass >(w/g) is 9.806 newtons sec^2/9.806 meters = (reduces to) 1 newton >sec^2/meter (= f/a). Ts connection may be rewritten, so that (9.806 >newtons sec^2/9.806 meters) x 1 meter/sec^2 = (reduces to) 1 newton; wch >is a fundamental unit because all of the other units fundamental unit. Byts view 1 acc = 9.806 m/s^2 is a arbitrarily, or otherwise, ... taken as fundamental, >and force as derived, ..., because it's the other way around!!Why is the one fundamental and the other derived? > Better yet, for the time being at least, our existing >foot-pound-second system - with decimals thereof - should be retained, >and/or reinstituted. Ts system, where a pint's a pound the whole world >around, has been, and still is, quite successful.The world is small. Only the USA and Liberia, I tnk. In the UK alreadya long time a pint was not a pound, and that was when they did use pintsand pounds. (I am not old enough to have been in the UK when there wasa pint that also was a pound.) In almost all of Europe there was noconnection of the (local equivalent) of pint with the (local equivalent)of pound. > So who *cares* if the >foot was the length of one of some king's feet.Well, in the Netherlands it was not. Unless it was both a huge king and adwarf. Before standardisation to metric there were a few hundred differentfeet in use in the Netherlands. (O, our first king is from 1815...) > Its a more convenient >length than one ten millionth of the distance from the Earth's equator to >one of its poles anyway.Convenience is in the eye of the beholder. > [And the foot too, could be defined as the >distance light travels in a second.]But it is. The inch is defined to be 2.54 cm exactly. The latter isdefined in terms of light-speed and the foot is defined in inches.The values are slightly different when you use purveyor's feet. > Finally, the decimal system is applicable to any numerical measure, >including the foot, the pound, the degree of arc, and the second. The >metric systems, with their individual names (prefixes) for each and every >decimal place, have no special claim to it.Indeed. Finally sometng that is correct.-- === to write up a proof, and I'm using the fact if a function> f: R -> R> is differentiable everywhere, then the derivative f' is Borel measurable.Is there an easy way to do ts?f'(x) is the === another question about circular sectors :)continued...There may be two solutions:For example take Area= .7854 and L =.7071 , input data for a quartercircle.In Mathematica, Plot[ { (1-Cos[th])/th - L^2/Area,L/Sin[th/2]},{th,1.5,Pi }];Apart from the expected quarter circle of radius 1, you also get === Theorem/Andrew Wiles>...can anyone recommend a good book with regard to Wiles'>attempt to solve the problem over 7 years?For the sincere and dedicated amateurs: Fermat's Last Theorem for Amateurs(currently on sale in the Springer Yellow sale) and Invitation to the Mathematics of Fermat-Wiles(written by the individual who first realized that> elliptic curves were an essential step towards the solution)Both these are by what I believe to be respected individuals> writing serious mathematics and are not just popularized near-fiction.The same goes for Alf van der Poorten's book, Notes on Fermat's Last === will not as easy to lie in future!Here's 5 peoples posts to verify that mind reading technology is already here.When you tnk, you are not silent, a radar can pick up your thoughtsJUST LIKE SPEACH and sound them out.Because I'm the truman I get it constantly.100,000 people in townsville australia know for certain that 100% clear mind readingis possible, all my neighbours listen to my every thought every day.its the most deous torture possible to constantly have you're thoughts playedback to you audibly and be FORCED to answer truthfully every passing remark,like I do every time I go out.Your voice box gets a trace stimulus of every thought you tnk, makes a smallnoise just like speaking, it can be picked up. They can play with the timing,they can hear compressed phonetics of whole sentences you are about totnk, and tell you your thought a second before you are aware of it.HercI CANNOT LIEEEEor another Jim Carrey, Majestic costarring Laurie HoldenExbit A: http://tinyurl.com/fuf8 she looks exactly like Laurie HoldenExbit B: http://tinyurl.com/fuf2 government has spied on them clearlyin between the release dates ofThe Truman Show : 1998 : Jim CarreyMajestic : 2002 : Jim Carrey and Laurie HoldenI'm from Townsville and YOU ARE the Truman!http://tinyurl.com/iky5I was in Townsville over the weekend, and I heard m.Very spooky!http://tinyurl.com/iky8>phone someone in Townsville, half of you must know someone there,>every day I go out people say THERES THE TRUMANI'm in Townsville. We're sick of you.http://tinyurl.com/iky9http://tinyurl.com/iky4You rule Truman!>Do you know if the truman is living in Townsville?I've === Borel measurable derivatives>> Hey all,I'm trying to write up a proof, and I'm using the fact if a function>> f: R -> R>> is differentiable everywhere, then the derivative f' is Borel>> measurable. Is there an easy way to do ts?f'(x) is the limit of n (f(x+1/n) - f(x)).--Ron Bruck> bloody helli'm an idiot. for the info, dude.I should've seen that....... === coverage area of a few circles?In my simulation, N circles with the same radius r are randomlyplaced. Let P_i denote the center of circle i. For any i, p_i lieswitn the coverage range of at least one other clicle, i.e. at leastone other circle contains p_i. How to calculate the total coveragearea of the N overlaped circle? The method should be easy to beimplemented by programming for simulation.Any comments is welcome. Leng === future! Here's 5 peoples posts to verify that mind reading technology is already here. When you tnk, you are not silent, a radar can pick up your thoughts JUST LIKE SPEACH and sound them out. Because I'm the truman I get it constantly. 100,000 people in townsville australia know for certain that 100% clear mind reading> is possible, all my neighbours listen to my every thought every day. its the most deous torture possible to constantly have you're thoughts played> back to you audibly and be FORCED to answer truthfully every passing remark,> like I do every time I go out. Your voice box gets a trace stimulus of every thought you tnk, makes a small> noise just like speaking, it can be picked up. They can play with the timing,> they can hear compressed phonetics of whole sentences you are about to> tnk, and tell you your thought a second before you are aware of it. Herc> I CANNOT LIEEEE or another Jim Carrey, Majestic costarring Laurie Holden> Exbit A: http://tinyurl.com/fuf8 she looks exactly like Laurie Holden> Exbit B: http://tinyurl.com/fuf2 government has spied place them clearly> in between the release dates of The Truman Show : 1998 : Jim Carrey> Majestic : 2002 : Jim Carrey and Laurie Holden I'm from Townsville and YOU ARE the Truman!> http://tinyurl.com/iky5 I was in Townsville over the weekend, and I heard m.> Very spooky!> http://tinyurl.com/iky8phone someone in Townsville, half of you must know someone there,>every day I go out people say THERES THE TRUMAN> I'm in Townsville. We're sick of you.> http://tinyurl.com/iky9 http://tinyurl.com/iky4> You rule Truman!Do you know if the truman is living in Townsville?> I've been hearing stuff, yeah> http://tinyurl.com/p0w3An infinite number of exbits constitutes merely evidence -- not proof.- -The second greatest error in reasoning is mistaking evidence for proof. The greatest ismistaking testimony for evidence.--Democracy: The triumph of popularity over === quantity of matter Physical mass is an aggregation - via gravitation and other centripetal> forces - of the substances comprising any object; body, and or mass of> material matter; wch causes these accumulations to have inertia, and/or> heft; the property of matter whereby it becomes more obvious that it> requires greater> (net) force to change the velocity of an accumulation as it becomes> larger: Here on Earth Galileo found that the rate of change in the velocity of any> body free falling at Earth's surface was about [s/t = 16'/sec] - half of> ÔNewton's' acceleration of free fall [g] - and furthermore - in effect -> he> found that the force restraining ts change in velocity of free fall from> continuing toward Earth's center is the mutual force exerted between the> body and Earth's terra firma; wch force is commonly measured with> weight-scales: On Earth, the ratio of ts weight-force [w], divided by> the> rate of change in velocity [s/t = 16'/sec] that it restrains, is a> constant [wt/16' = wt/s]; to be known hereafter, as one half of the> body's> gravitational mass [g/2], and/or inertia. On any similar planet, such as the moon, ts _ratio_ will still be equal> to> half of any body's mass, and/or inertia!> In everyday use, a body's mass [m] is commonly confused with its weight> [w]; A practice wch must cease immediately; for the sake of physics! According to newton's second law weight is the product of mass [m] and the> gravitational acceleration of free fall [g]: Ts erroneous formula w = ma> is a special case of f = ma, and like f = wa/g, must be written as w = fg/a:> All because inertial mass m = f/a, and gravitational mass m = w/g; where it> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a => w/g. Cheeze, what have I got to do, write a book for youse people(:^?I doubt if it is necessary for you to distinguish between Ômass' and Ôweight'for any physicist. What's your problem?--There are two tngs you must never attempt to prove: the unprovable -- and theobvious.--Democracy: The triumph of popularity over === students in a 10-15 minute talk Well, for (1), all you need is a series. ßy's speed and that time, work out the distance.I expect most people here have heard it, but I'll pass on the followinganecdote:Someone presented Feynman with that problem [bug ßying between twoapproacng vecles], and he of course solved it very quickly. The otherguy said, would you believe some people solve it with series! To wchFeynman responded, === problem with ring of A.I. then I can assure you it's a battle you will lose. The US Army has proven itself time and time again, and I dare you to> reply to me on that playing field. I tnk you're a ing coward. I dare you to attack me as an Army> veteran. Come on sthead. Come at me for my military service. Do your best.US ARMY VETERANI don't know why you're so proud of being part of an organization that kills(mostly) innocent people for the sake of commercial U.S. interests: maybeit's a reßection on your general style of resolving conßict.BTW, you weren't accidentally exposed to agent orange? I hear it makesveterans go ballistic (if you'll pardon the pun), whenever their militaryservice is === quite get the question (fractional iteration?), but> folks should know that the universality of the M-set,> that is the recurrence of mini-bugs or cardioids,> at every level of magnification, is just an artifact> of the ßoating-point ops (IEEE-755, -855, I tnk).> ts was (really/partially) confirmed by monsieur M,> when he glroriously begged my (only) technical question> at a talk for a general audience.>It's quite simple to disprove your claim by setting the roundingmethod, wch you can do in hardware on the Pentium (and manyother processors as well) to all it's values and seeing what changes,or doesn't change, in calculations. FP doubles are good for severaltens of thousand of iterations, down to an area of 10E-10 or so,before the precision gives out. The cartoids are visible === length of Gamma function>But I also wonder though, for the family of function where the arc length>from 0 to x *is* the same as its function value in x-1, so :f(y-1) = int,0_y(sqrt(1+Df(x)^2) dx)Differentiating, you getf'(y-1) = sqrt(1 + f'(y)^2)or f'(y-1)^2 = 1 + f'(y)^2If g(y) = f'(y)^2, ts has general solutiong(y) = h(y) - y where h is periodic with period 1.and thusf(x) = int_0^{x+1} sqrt(1 + h(t) - t) dt = int_{-1}^x sqrt(h(t)-t) dtwch satisfies the original equation. For example, with h(t) = 1you get f(x) = (4 sqrt(2) - 2 (1-x)^(3/2))/3.Of course tngs get interesting when g(t) === proposal for Herc)>Herc,Just so we all know what we are attempting here, can you please>clarify your claimed powers. How about we just drop the whole tng down a well, and then fill thewell with concrete?--V.G.People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper StickerSarcasm is === === Sets Three Houses Ablaze>Subject: Re: Electrical Storm Sets Three Houses Ablaze:>I have some very direct experience with the initiator of ts thead and for>that reason would like to know a little about ts forum. >I made way directly here via Google and so have no idea who you are, what you>discuss and why Mr. Daryl Shawn Kabatoff's tolerated, Don't be an ass. === === Combination/Permutation Question...>Subject: Re: Combination/Permutation Question...Image that you have 10 balls.>Permutation : The way(order) you can take them out from a box.>Example:> 1,2,3,4,5,6,7,8,9,0> 0,1,2,3,4,5,6,7,8,9> ..................> 4,3,2,1,5,8,6,7,9,0Combination with no repeating: The way(order) you can take only m of them out>and m<10.>Example: m=3 > 1,2,3> 5,3,7> 0,8,2> .....Combination with repeating: The way(order) you can take only m of them out>and m<10 and you have to return ts ball back before take out another one .>Example: m=3 > 1,2,3> 4,7,7> 1,1,1> === Hey, look! It will not as easy to lie in future!> Here's 5 peoples posts to verify that mind reading technology is already here. When you tnk, you are not silent, a radar can pick up your thoughts JUST LIKE SPEACH and sound them out. Because I'm the truman I get it constantly. 100,000 people in townsville australia know for certain that 100% clear mind reading> is possible, all my neighbours listen to my every thought every day. its the most deous torture possible to constantly have you're thoughts played> back to you audibly and be FORCED to answer truthfully every passing remark,> like I do every time I go out. Your voice box gets a trace stimulus of every thought you tnk, makes a small> noise just like speaking, it can be picked up. They can play with the timing,> they can hear compressed phonetics of whole sentences you are about to> tnk, and tell you your thought a second before you are aware of it. Herc> I CANNOT LIEEEE or another Jim Carrey, Majestic costarring Laurie Holden> Exbit A: http://tinyurl.com/fuf8 she looks exactly like Laurie Holden> Exbit B: http://tinyurl.com/fuf2 government has spied on me so clearly> in between the release dates of The Truman Show : 1998 : Jim Carrey> Majestic : 2002 : Jim Carrey and Laurie Holden I'm from Townsville and YOU ARE the Truman!> http://tinyurl.com/iky5 I was in Townsville over the weekend, and I heard m.> Very spooky!> http://tinyurl.com/iky8 >phone someone in Townsville, half of you must know someone there,every day I go out people say THERES THE TRUMAN> I'm in Townsville. We're sick of you.> http://tinyurl.com/iky9 http://tinyurl.com/iky4> You rule Truman! >Do you know if the truman is living in Townsville?> I've been hearing stuff, yeah> http://tinyurl.com/p0w3 An infinite number of exbits constitutes merely evidence -- not proof.>that's why we use limits to reason what point it makesHerc> - -> The second greatest error in reasoning is mistaking evidence for proof. The greatest is> mistaking testimony for evidence.> --> Democracy: The triumph of popularity over principle.> --> === test proposal for Herc)Herc,Just so we all know what we are attempting here, can you please>clarify your claimed powers. How about we just drop the whole tng down a well, and then fill the> well with concrete?> --that's no way to vanilla a === an equation, how do i go about finding the equation ifit is unknown (and the equation must not contain any irrational numbers)?i've used the quad.formula and worked backwords on some but what if theroots are sqrt(2) + sqrt(3)....any === give a short talk (15-20 minutes) to our> new mathematics students, and I find myself at a bit of> a loss as to what I am going to say. I should talk about> sometng mathematical, maybe some application, but the> technical part should be minimal. Ideally, it should> demonstrate some mathematical point or somehow prepare> and motivate them for the courses. It would also be> nice if there were some story attached to it to grip> their attention. However, anytng entertaining and> mathematical would be acceptable. Any ideas?>a couple of years ago my brother an I both studying as maths teachers would havehad 20 emergency fun maths puzzles to fill up half lessons.The bird ßying between two trains has a good anecdote, some students ofVon Neumon asked m after a lecture one day the puzzle how far doesthe bird ßy, and after a couple seconds thought he said Ô20 miles'. Thestudents said oh you know the trick and he answered, Ôno I just summedthe infinite series'.And / or you could talk about storic mathematicians, notably that most ofthem lead tragic lives. Pythagorus, Newton, Turing, ... there's many othersgetting all sorts of terrible tngs happen, === Four Color TheoremWhat I have been trying to say is that if c(G)=5, G cannot beplanar!The only minimal counter-example to the FCT is K5!The conjecture that there exists a 5-chroma graph may be recolored to4-chroma is false.Yes I am confused! I am confused as to why anyone would argue sopassionately and so ineffectually in favor of a false FCT. I amconfused as to why I am considered crazy for wanting the FCT to betrue.Let H be any subgraph of G, where G has n vertices and H has n-1vertices. Then, the description of H seems to imply that the deletionof Ôany' vertex from G will make c(H)<=4.But ts interpretation is generally false and is valid only forn=5!!! I intended ts to mean that only K5 could be an mc-e to the FCT. Noneof the responses have convinced me to change my mind. Although, someof the responses really confused me!Perhaps, I erred in not offering a more substantial proof for theconjectureG = K5? Probably because I don't have one as yet. === <3F81D1ED.5070806@rutcor.rutgers.edu> In general, if at least one of A and B are infinite, then the> cardinality of A U B is the same as the cardinality of the larger> set. One can establish ts fact without the axiom of choice. That surely depends on how you define infinite. If you mean> infinite = Dedekind infinite, then I agree.I don't. In the first place, if AC fails, then there are incomparablecardinals, whether or not there are any Dedekind-finite infinitecardinals. In the second place, without AC you can't prove that2^(aleph_0) and aleph_1 are comparable; if they are incomparable, thentheir sum is greater than either of them. In the trd place, if x isa Dedekind-finite infinite cardinal, then y = x + aleph_0 is aDedekind-infinite cardinal, and y + y > y.> But to me, infinite means larger than any natural number,That is the standard meaning.> and with that === Class of computable functions> I took a class in basic foundations and computability last year, but> there is a question that has been bothering me. Suppose I want a Ôlarge' computably enumerable collection of functions> f_i : N --> N with the following properties:> 1. Each primitive recursive function is included.> 2. Each f_i is total by construction.> 3. Given i and n in N, there is a computable function time: N x N -- N wch tells me that the value of f_i(n) will take at most time(i,n)> to compute by a turing macne or equivalent.> [ÔTake as long as you want, but PLEASE tell me when you will be> done!']> 4. The function time is computable in ÔAckermann + constant' time,> or at least it's behavior is boundedly nasty in some sense. :) I feel rather queasy about the prospects for ts, but it has been> bothering me for too long. The basic theme is ts: I want to know> worst case scenario computation time. Will I not get much besides> primitive recursive functions? Rex Butler PS I've heard the term Ôstrongly computable.' Is ts related?take a look at http://members.ozemail.com.au/~chess3/tm1.htmlIts 10 lines of code (actually a 5 state Turing Macne) and there is noproof if it terminates. Only 10 lines and no value for f_i.In theory your class could be all well defined and well behaved functions,but in practice they will only be trivial.Say you are nesting loops, then the invariants at each level would have tobe combined into parallel equations, you have to solve all enumerationsof possible equations, and show they are satisfied for each loop to terminate.Even then the class of functions has no scope, here is a tiny 3 state TMhttp://members.ozemail.com.au/~chess3/tm2.html you can construct it easyenough but how do you categorise what === function it is? ~ binary counter.HercSubject: Re: Length of go around a circumference. Eg> If I wanted to go around a piece of pipe 3 1/2 in diameter, with 1/4 ßatbar, how long should I cut the bar. Ts is considering the fact that I can form it completely round>C = 2 * pi * r = pi * d, where C = circumference of the circle,pi = 3.14159..., r = radius, d = === construction (doubling a point): Let X be a compact Hausdorff space and p be a non-isolated point of X.> Take some q not in X and define X' = X+{q}, where the base in q is the> set of all {q}+U{p}, U is a neighbourhood of p; the topology of X is> the original one. Then any two neighbourhoods of p and q in X' have> non-empty intersection, but they have Hausdorff compact neighborhoods. What's the compact Hausdorff nhood of q?If U is a compact Hausdorff neighbourhood of p, then {q}+U{p} ishomeomorpc to U in the topology of X'. In fact we may take U=X.Simeon> Could anyone give me an example of a topological space X such that every point of X has a Hausdorff compact === newsgroup useless?> The number of replies is the thread's rating. Apparently, post's quality has> notng to do with the rating. Now, face it: the JSH show has the ghest> rating in sci.math. That's what makes it so sad - that the rantings of that pathetic loony> seem to be stuff that so many contributors to sci.math are so interested> in. I'm puzzled why ts bothers you so much. The huge majority of posts on sci.math are about mathematics. It's easy to skip the other === Re: Test |-|erc (was Re: A test proposal for Herc)> ie, you post. Someone responds. The content of their post > can be matched to their name.Lets try some reverse engineering here, what would someone whose nameis Hung Too Long post? > pretty good, the magic is inducing the response, Got news for you, Herky, magic is just a pretty synonymn for parlortrick.> so there's no divining necessaryThere is no divining parlor tricks.> its just analytical work to figure out the name, > anyone can do it once the posts are available, but since> you've set it up so I have to deduce them myself divine is apt. > for ts claim.Yeah, well deduce MY name, k00k. As usual, you will either not respond or will reply evasively andoff-topic.--Grand Inspector of the K00kfinder GeneralSkepticult Member 518-27581-876For Entertainment Purposes Only. - DisclaimerWe're not laugng with you. - Skepticult CreedIn my life, I have prayed only one prayer in asking for divine favor: ÔO Lord, make my enemies ridiculous.'And God granted it. - VoltaireAgainst stupidity, the Gods themselves contend invain. - The Postman Syndrome, === question,> I tnk that it would have helped (at least me) tremendously to have> included ts statement in the proof. That if an open set, (N) is in the> exterior of a set (E), then N cannot contain any limit points of E.> I tnk ts is not quite obvious enough to be left out of the proof> that bar{E} is closed.I agree, and I tnk it's an oversight on === math textbook (Rosen) I'm reading about sequences like ts: a_n = sum_{1<=i<=k} c_i * a_{n-i}where the c_i Ô s are constants. The sequences that satisfy an equation likets constitute a k-dimensional vector space V, since the first k terms in anysequence determine all the remaining terms. The equation above can easily befiddled into a polynomial equation whose solutions are exactly the numbers x forwch a_n=x^n is a solution of the original equation. The book states, withoutproof, that if the polynomial equation has no repeated roots, then V is spannedby the set of vectors (1,x,...x^{k-1}) for wch x is a solution of thepolynomial equation. A wle ago, when I first skimmed through the chapter, Itried to tnk of a reason why that set of vectors should necessarily beindependent, but I came up empty. Now one of my homework questions is to provethe analogous theorem, but without the hypothesis of no repeated roots. Doesanybody have a good nt about how to crack ts nut, or, failing that, areference? === Re: Cardinality of sets>> In general, if at least one of A and B are infinite, then the>> cardinality of A U B is the same as the cardinality of the larger>> set. One can establish ts fact without the axiom of choice.>> That surely depends on how you define infinite. If you mean>> infinite = Dedekind infinite, then I agree.> I don't. In the first place, if AC fails, then there are incomparable> cardinals, whether or not there are any Dedekind-finite infinite> cardinals. In the second place, without AC you can't prove that> 2^(aleph_0) and aleph_1 are comparable; if they are incomparable, then> their sum is greater than either of them. In the trd place, if x is> a Dedekind-finite infinite cardinal, then y = x + aleph_0 is a> Dedekind-infinite cardinal, and y + y > y.>> But to me, infinite means larger than any natural number,> That is the standard meaning.>> and with that definition, I disagree with your conclusion.I was just trying to understand under what conditions the statement mightbe true. Obviously, I didn't tnk it through.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.Also, when are the hour, minute and second hands positioned so that>they divide the clockface in three === Minimal Graph, Four Color Theorem for recommending the Diestel book! Chapter 5 is all about colourings. The place where I'm getting the feeling> that there is a disconnect in terminology is that when we say a graph is> 5-chromatic, it just means that some colouring with 5 colours is possible> (and no fewer than 5). The specific colouring you use to establish ts is> unimportant and is not considered to permanently colour the graph and all> its subgraphs (you might call ts a graph with colouring to distinguish> it from just a graph).There could be more than one colouring (and in fact there likely is more> than one). So unlike a bipartite graph where if it is bipartite then the> colouring is basically unique (assuming connectedness), a 5-partite graph> doesn't necessarily have a uniquely well-defined partition into 5 classes.> Those vertices that do not belong in only one specific partition maybe summarily discarded. Then the partitioning will be uniquelydefined.> That's why it doesn't make too much sense to talk about whether the sizes> of the five partitions must be such-and-such. When we delete a vertex> and ask if the resulting graph is 4-chromatic, the colouring needn't share> any common ground with the 5-colouring in the larger graph.> There is a common ground. Vertices that were adjacent before removinga vertex are still adjacent afterwards. So 4-chroma coloring ispartially the same as the 5-chroma coloring.> Certainly it is possible to formulate the idea of a minimal 5-chromatic> graph in a way you might find more pleasing. For instance, we could say> that it's equivalent to a graph that is 5-chromatic in such a way that> for every vertex v there exists a 5-colouring of the graph in wch v> is the sole vertex with the colour blue.> The only 5-chroma graph that I am likely === mulitple urns question>I have a probability question I could use some help with. I'll use>the urn/ball model to make it more general. Assume that I have n>urns, each with a different proportion of colored balls in it. I know>the probability P_n(c) of selecting each color of ball from each urn. >I will select one ball from each urn, for a total of n unordered balls>in the selected set S. Now, given a particular reference set of>unordered balls T, how do I determine the probability that S will>match it? For example, if I have 10 urns, how might I determine the>probability that I will choose 3 red, 2 blue and 5 green balls?If P_j(c) is the probability of selecting colour c from urn j, the probability of getting n_1 balls of colour 1, ..., n_k balls of colour kis the coefficient of x_1^n_1 x_2^n_2 ... x_k^n_k in product_{j=1}^n (P_j(1) x_1 + === students in a 10-15 minute talk > Well, for (1), all you need is a series. ßy's speed and that time, work out the distance. I expect most people here have heard it, but I'll pass on the following> anecdote:> Someone presented Feynman with that problem [bug ßying between two> approacng vecles], and he of course solved it very quickly. The other> guy said, would you believe some people solve it with series! To wch> Feynman responded, what's the other way?:-)Here (USA), it's almost always (i.e., every time I've ever heard it) toldabout Von Neumann mself: he solves the problem very quickly, exclaims,Ah! Yes, it is 150 miles! or whatever, and, when the curious onlookersask m how he did it so quickly, he gives a blank look and replies,I summed the series.I never heard the algebraic approach referred to as the Von Neumannapproach === roots of an equation, how do i go about finding the equation if>it is unknown (and the equation must not contain any irrational numbers)?>i've used the quad.formula and worked backwords on some but what if the>roots are sqrt(2) + sqrt(3)....any nts?If r_1, r_2, ..., r_n are all the (complex) roots of a polynomial in x, then the polynomial is a constant === a quantity of matter> Physical mass is an aggregation - via gravitation and other centripetal>> forces - of the substances comprising any object; body, and or mass of>> material matter; wch causes these accumulations to have inertia, and/or>> heft; the property of matter whereby it becomes more obvious that it>> requires greater>> (net) force to change the velocity of an accumulation as it becomes>> larger:>> Here on Earth Galileo found that the rate of change in the velocity of any>> body free falling at Earth's surface was about [s/t = 16'/sec] - half of>> ÔNewton's' acceleration of free fall [g] - and furthermore - in effect ->> he>> found that the force restraining ts change in velocity of free fall from>> continuing toward Earth's center is the mutual force exerted between the>> body and Earth's terra firma; wch force is commonly measured with>> weight-scales: On Earth, the ratio of ts weight-force [w], divided by>> the>> rate of change in velocity [s/t = 16'/sec] that it restrains, is a>> constant [wt/16' = wt/s]; to be known hereafter, as one half of the>> body's>> gravitational mass [g/2], and/or inertia.>> On any similar planet, such as the moon, ts _ratio_ will still be equal>> to>> half of any body's mass, and/or inertia!>> In everyday use, a body's mass [m] is commonly confused with its weight>> [w]; A practice wch must cease immediately; for the sake of physics!>> According to newton's second law weight is the product of mass [m] and the>> gravitational acceleration of free fall [g]: Ts erroneous formula w = ma>> is a special case of f = ma, and like f = wa/g, must be written as w = fg/a:>> All because inertial mass m = f/a, and gravitational mass m = w/g; where it>> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a =>> w/g.>> Cheeze, what have I got to do, write a book for youse people(:^?I doubt if it is necessary for you to distinguish between Ômass' and Ôweight'>for any physicist. What's your problem?With physicists, it's often the opposite problem--imaginingdifferences when they don't exist. Both of those words, of course,are ambiguous words with several different meanings.What Dense Donny is talking about is tngs like the net weight ofmy bag of sugar, 4 lb (1.81 oz) right on the label. Of course, netweight is not a physics concept in the first place. Second, whenevernet weight is used ts is always the very same tng as mass inphysics jargon--but that synonym is not legal on those labels, the lawrequires either the spelled out word weight or the particularabbreviation wt on the labels.If you are a physicist, of course, you can choose not to call tsquantity weight. But what often happens instead is that they stillcall it weight but misapply a definition of weight wch isinappropriate for the context. That simply is not an acceptablealternative.Just remember that your choosing not to call it weight does not meanthat someone else is making some error if they do call it weight--wehave a prior claim to ts word by over 750 years over the physicistswho recently borrowed it and often use it with a different meaning.I really have a hard time understanding how so many people can be soGod-awful stupid as to tnk that when we buy and sell goods byweight, we'd want to measure some quantity that varies with location.We should not do so; we do not do so; we have never done so.I'm sure you wouldn't tnk twice about somebody calling a troy ouncea unit of weight. However, those troy units of weight, unlike theiravoirdupois cousins, and unlike grams and kilograms, have neverspawned a unit of force of the same name. They are always units ofmass.-- Gene Nygaardhttp://ourworld.compuserve.com/homepages/Gene_Nygaard/ It's not the tngs you don't know what gets you into trouble. It's the tngs you do know that just ain't so. Will === one of the agreed point of reference, i.e. decimal dot.> Therefore the question to divide by zero is irrelevant.> How to divide by the point of reference indicator?I'm afraid that I don't understand what you meanYou should only be afraid when === is a quantity of matter> Physical mass is an aggregation - via gravitation and other centripetal> forces - of the substances comprising any object; body, and or mass of> material matter; wch causes these accumulations to have inertia, and/or> heft; the property of matter whereby it becomes more obvious that it> requires greater> (net) force to change the velocity of an accumulation as it becomes> larger:>> Here on Earth Galileo found that the rate of change in the velocity of any> body free falling at Earth's surface was about [s/t = 16'/sec] - half of> ÔNewton's' acceleration of free fall [g] - and furthermore - in effect -> he> found that the force restraining ts change in velocity of free fall from> continuing toward Earth's center is the mutual force exerted between the> body and Earth's terra firma; wch force is commonly measured with> weight-scales: On Earth, the ratio of ts weight-force [w], divided by> the> rate of change in velocity [s/t = 16'/sec] that it restrains, is a> constant [wt/16' = wt/s]; to be known hereafter, as one half of the> body's> gravitational mass [g/2], and/or inertia.>> On any similar planet, such as the moon, ts _ratio_ will still be equal> to> half of any body's mass, and/or inertia!> In everyday use, a body's mass [m] is commonly confused with its weight> [w]; A practice wch must cease immediately; for the sake of physics!>> According to newton's second law weight is the product of mass [m] and the> gravitational acceleration of free fall [g]: Ts erroneous formula w = ma> is a special case of f = ma, and like f = wa/g, must be written as w = fg/a:> All because inertial mass m = f/a, and gravitational mass m = w/g; where it> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a => w/g.>> Cheeze, what have I got to do, write a book for youse people(:^?>>I doubt if it is necessary for you to distinguish between Ômass' and Ôweight'>>for any physicist. What's your problem?With physicists, it's often the opposite problem--imagining>differences when they don't exist. Both of those words, of course,>are ambiguous words with several different meanings.What Dense Donny is talking about is tngs like the net weight of>my bag of sugar, 4 lb (1.81 oz) right on the label. Of course, netcorrection--that's a typo; make that 4 lb (1.81 kg)Those pounds, of course, are legally defined as 0.45359237 kg exactly.Neither pounds force nor kilograms force are legal for use in themarketplace in selling goods by weight.>weight is not a physics concept in the first place. Second, whenever>net weight is used ts is always the very same tng as mass in>physics jargon--but that synonym is not legal on those labels, the law>requires either the spelled out word weight or the particular>abbreviation wt on the labels.If you are a physicist, of course, you can choose not to call ts>quantity weight. But what often happens instead is that they still>call it weight but misapply a definition of weight wch is>inappropriate for the context. That simply is not an acceptable>alternative.Just remember that your choosing not to call it weight does not mean>that someone else is making some error if they do call it weight--we>have a prior claim to ts word by over 750 years over the physicists>who recently borrowed it and often use it with a different meaning.I really have a hard time understanding how so many people can be so>God-awful stupid as to tnk that when we buy and sell goods by>weight, we'd want to measure some quantity that varies with location.>We should not do so; we do not do so; we have never done so.I'm sure you wouldn't tnk twice about somebody calling a troy ounce>a unit of weight. However, those troy units of weight, unlike their>avoirdupois cousins, and unlike grams and kilograms, have never>spawned a unit of force of the same name. === McKenzie, problem with ring of A.I. 7e3Z2dkQEBC5ubm60r3HAAACeUlEQVR4nIXTQW+ bMBQAYChLfcWzUq5pipUr7lPptUNGXBMC8zUZ qnt1aNz392c7pGXSpvkSyV/ee37POFr/Y0X/g56rxd+ gb2IQC8X79brdLtovaBgACIsRMVTr4hN4 Ah54xy0rtdbjFRCAupizIhUDB69X2IHAGKCrxyrxcJyA76CQKwZn2ckEXK6h/ YwoOld/zJUMEYds ghMIYhj0kjehxqGYFacgalk1PpM+jBOkvg0othFR4bjDVHyjQx+FZ+ b29fMEe114SIC6UB/RXoCX enAhGQORAdxp/TaN5N63lNqGJZACO2n9/ QL81qfVQjEXAkBKPyoPm7BfPqFLJ0qBt1fwmfTAFiYW MbAC1WmCnyEitcYImtI07fY3FwglBuwMUiqoiLv98gL+ 4PrNSmXdWGJK7Q4D8JBpWeXGxi6CpqOJ QoOrkOksKzMaRinN7LsI8OChsGOl+ogiE8SeaIAP9/ 9BRIs1xzgzESWteQ7gDgWQpmRdmcgtNFYJ / 5W4CWpRRsTyHE2UBgifjzvUQMGMOfYYO1CI7wFWfhwsWXSNq0qQcMQkgBvhADvI rOrduRArpW7D Rßb0gjHvO5kpUhX93h3DJ3v9VB+S8pMulW1kis73aAqh/ LpBYalbA2qiLZKP16G+KHLowIYojSm UTys+/ LrogaFVFTKpN2P45qT6UX58qrbEFcil81h9tQSfVSyIZ0v3zzNYEMPVtbR2e3XL 8v547w/ uDymk7wfXx7n8PBscrm1fW/G1a8/gKiz3GJeYbt5ncMJ+ Y1ckVxi19zMwaIcXXUreb6yc1gQ9ypr tLKSCueQjQ4qxFzKXo0zwNDbxv/U2ynkN+Dj/ sha1:mB6vm4C42vE7NAHhzOBFWKAq2QI=> I don't know why you're so proud of being part of an organization that kills> (mostly) innocent people for the sake of commercial U.S. interests: maybe> it's a reßection on your general style of resolving conßict.> BTW, you weren't accidentally exposed to agent orange? I hear it makes> veterans go ballistic (if you'll pardon the pun), whenever their military> service is questioned.I don't know why you work so hard to be an idiot. What complete andutter garbage you write here. You trying to make James appear areasonable man?Though, it is a cute and new accusation about agent orange and thesymptoms it causes. Ludicrous, sure, but it could possibly work in aHollywood movie.-- [T]here's no point in telling any of you what mathematicians I'm inemail contact with, just like there's no point in going into detailabout my contacts in a major news organization. [...] [P]olitedisinterest is what I've found. === blackboard bold?Another golden name-dropping opportunity. :-)|(I tnk I will not act on s implicit|suggestion that one might ask Serre; but if anyone else has |the chutzpah to do so, I don't doubt sometng useful might |be learned from the experiment.)suggested I show to Serre, who was visiting at the time, and he was niceabout critiquing it. I had denoted a finite field with q elements byblackboard-bold F with q as a subscript. He told me not to use blackboard-bold symbols in print, as they were merely a technique to approximate boldsymbols on the blackboard. So that much at least is definitively settled.;-)Keith RamsayP.S. Too bad I didn't tnk to ask when === know the roots of an equation, how do i go about finding the equation if> it is unknown (and the equation must not contain any irrational numbers)?> i've used the quad.formula and worked backwords on some but what if the> roots are sqrt(2) + sqrt(3)....any nts?There may not be any equation having only the roots you want and still have all rational coefficients, but if you are allowed to have extra roots to aceve rational coefficients, you can take any algebraic number and all its conjugates as roots of a polynomial.For example, the conjugates of r1 = sqrt(2) + sqrt(3) are r2 = -sqrt(2) + sqrt(3), and r3 = sqrt(2) - sqrt(3), and r4 = -sqrt(2) - sqrt(3), so that (x-r1)(x-r2)(x-r3)(x-r4) = x^4 - 10*x^1 +1 is a polynomial with integer coefficients.Ts can always be done in theory, but can be difficult in practice, especially where roots other than square roots are involved or more than 2 === about finding the equation if> it is unknown (and the equation must not contain any irrational numbers)?> i've used the quad.formula and worked backwords on some but what if the> roots are sqrt(2) + sqrt(3)....any nts?You form the polynomial whose roots are the conjugates of the given root eg.(x - sqrt(2) - sqrt(3)) . (x - sqrt(2) + sqrt(3)) . (x + sqrt(2) - === Re: I am a freshman of CST,wch book of maths should I read first?> I am a freshman of computer science and technology.> I came across a problem.That is I don't know wch book should I read> first,wch should I read second......> I have three books right here.They are , Maths>,.> If you know,please help me! Change your major to basket weaving. What do any of the newsgroups have to do with CS?-- Memory Hole: The only al Qaeda cell uncovered intact wasoperated by Israel. -- The Iron Webmaster, === Certainly it is possible to formulate the idea of a minimal 5-chromatic>> graph in a way you might find more pleasing. For instance, we could say>> that it's equivalent to a graph that is 5-chromatic in such a way that>> for every vertex v there exists a 5-colouring of the graph in wch v>> is the sole vertex with the colour blue.>>The only 5-chroma graph that I am likely to find pleasing is K5! Fair enough, but there are other minimal 5-chromatic graphs besides K5even if you aren't pleased by them :). For example, glue two regularpentagonal cones together at the base to get a polyhedron with 7 vertices,and form the natural adjacency graph on those vertices (of course, we geta planar graph). Then add one more edge joining the apex vertices.The resulting graph is 5-chromatic, but removing any vertex, no matterwch one, always gives a graph that is 4-chromatic (and also planar,if I'm not mistaken). Personally I find it just as pleasing as K5 :). -- === = a+ib and its derivates as dz/dt = da/dt + idb/dt.Now, putting r = sqrt(a^2 + b^2), how to obtain the dr/dt ? Thanxs === is a new perspective on ts question. Please check out the post by> Tomas, http://mathforum.com/discuss/sci.math/m/77228/77277New? That's === generator polynomial both for your valueable insets. It is === me an example of a function (R=real numbers) f:R-->Rsuch that for any a,b,c in R, there exists an x in R such that a Having a complex number z = a+ib and its derivates as dz/dt = da/dt + i> db/dt.> Now, putting r = sqrt(a^2 + b^2), how to obtain the dr/dt ? Thanxs === codes are well-known codes wch are also maximumdistance separable (MDS), i.e. the minimum amount of redundancy isneeded to correct errors ( => dmin = 2t+1, where t equals the amountof correctable errors).RS codewords have length n, where n equals 2^m - 1 and m defines theGalois field GF(2^m) in wch the code polynomial is defined. Thus,the Galois Field have 2^m distinct elements. Will the amount ofdistinct elements in the Galois Field play a role in the maximumlength of MDS codes' codewords of a specific Galois Field? very much, === a ressource for tabulated moreobscure quantities like unknotting or bridging number(also for links if possible)? (Should go to at least8-9 crossings) -- Hauke Reddmann <:-EX8 Private email:fc3a501@math.uni-hamburg.deFor our chemistry === sectors :)A math wiz I am not, but when I run your numbers backwards, I get L =1.4142.Using your first answer where theta = 90 deg (or Pi radians) and radius = 1: L = 2 (r * sin(theta/2)) = 1.4142And, your second where theta = 180 deg (Pi radians) and radius = 1/sqrt(2) =0.7071: L = 2 (r * sin(theta/2)) = 1.4142By the way, I agree that there are two solutions...> continued... There may be two solutions:> For example take Area= .7854 and L =.7071 , input data for a quarter> circle.> In Mathematica, Plot[ { (1-Cos[th])/th - L^2/Area,L/Sin[th/2]> },{th,1.5,Pi }];> Apart from the expected quarter circle of radius 1, you also get a> === semi-circle radius 1/sqrt(2).Subject: Re: Topology question === Re: Topology question > A general construction (doubling a point):> Let X be a compact Hausdorff space and p be a non-isolated point > of X. Take some q not in X and define X' = X+{q}, where the base > in q is the set of all {q}+U{p}, U is a neighbourhood of p; the > topology of X is the original one. Then any two neighbourhoods of > p and q in X' have non-empty intersection, but they have Hausdorff > compact neighborhoods. >> What's the compact Hausdorff nhood of q? >If U is a compact Hausdorff neighbourhood of p, then {q}+U{p} is >homeomorpc to U in the topology of X'. In fact we may take U=X.Ok, I agree.>> Could anyone give me an example of a topological space X such that>> every point of X has a Hausdorff compact === Professor McKenzie, problem with ring of A.I.>I have no idea what forum to post ts in so here it goes,i need a formula that will give me the answer to >(1*1)+(2+2)+.....(999*999)>I do not tnk that you have provided enough terms in ts series toenable us to guess the intended general term.Do you perhaps mean(1*1)+(2+2)+(3*3)+(4+4)+(5*5)+(6+6)+ ... === x^8-x^7+29x^2+29 Revisitedz^7-7z^6-2763z^5-19523z^4+1946979z^3+34928043z^2+ 119557031z-3247^2=0MAPLE tells me ts has cyclic Galois group of order 7. Thusits roots may be ordered z_0, ..., z_6 in such a way thata^7 = b in Z[zeta]where zeta = exp(2 pi I/7) anda = sum_{j=0}^6 z_j zeta^jis a Galois resolvent.I calculated the roots numerically to 50 dp and tried various orderings.I came across one wch gavea^7 = -12392836399104 + 20856562728960 zeta + 26834412666880 zeta^2 -13437774020608 zeta^3 + 2576491954176 zeta^4 + 25860547313664 zeta^5I am 99.999999% confident that ts is exactly true. Ts meansthat if we call ts b, and its Galois conjugates b_1, ..., b_6 then7 z_0 = 7 + b_1^(1/7) + ... + b_6^(1/7)etc.(as long as we take the correct seventh roots :-) === convinced ts line of research is of much use for practical> factorization, but it is of some interest to find hard instances for> SAT solvers (ts was the motivation of the Horie-Watanabe ISAAC paper> mentioned earlier in ts thread, a version of wch is also at> .I wonder if SAT solvers might be of use for factorization.I have noticed that the factorization to SAT conversionsare not optimized for SAT solvers.For example, using Purdom and Sabry's generator to factor 9produces a SAT instance with 22 variables and 73 clauses.Factoring 9 can be converted to a Boolean formula with as few as 4 inputvariables (two 2-bit multiplicands).Most of the extra variables are intermediate results that are definedfrom the multiplicands or carry bits. These extra variables reduce thenumberof clauses, but increase the number of variables. Most SAT solvers are muchmoresensitive to the number of variables than they are to the number of clauses.All the factors of a 2N bit number can be found using an N-bit x (N+1) bitmultiplication circuit.There exists a Boolean formula with N+1 inputs thatwill find allfactors of an N-bit number.Ts N+1 bit circuit probably can't be generated in polynomial time, butthat may not matter since the circuit can be reused to factor any N bitnumber.Assume we have a 3-bit x 2-bit multiplication circuit.There are 5 input variables and 4 output variables.Let A,B,C, and D be the CNF formulas for the four output bitsand let ~A, ~B, ~C and ~D be the inverse of these formulas (again in CNF).Let D be the formula for the most significant output bit and A be theformulafor the least significant output bit.Any 4-bit number can be factored by ANDing four of these eight formulastogether:Factors of 9 = (D & ~C & ~B & A)Factors of 15 = (D & C & B & A)Factors of 11 = (D & ~C & B & A) wch has no solution.Most of the individual formulas of a multiplication circuit don't changeas we increase the number of input variables.The M'th output bit of a multiplication circuit is determined by the M loworderbits in the multiplicands. For example, the formula for A is the alwaysdeterminedby the lowest order bit of the two multiplicands.If we interleave the input bits of the multiplicands then the formula forthelower order output bits will not change as we create larger multiplicationcircuits.Number the inputs for the 3x2 bit multiplier:54321Let the first multiplicand = 4,2 and the second multiplicand = 5,3,1.The formula for output A = (1 AND 2), ~A = (~1 OR ~2).The formula for A will be the same in a 4 x 3 multiplier circuit using tsinterleaving representation.Creating a 51 x 50 bit multiplier circuit would require considerableprocessing, butonce one had the fomulas for such a circuit (and their inverses), one couldfactor any 100 === functionComplete Moron je v sporo.8filu ...> can anyone show me an example of a function (R=real numbers) f:R-->R> such that for any a,b,c in R, there exists an x in R such that a and f(x)=c?>Let x = dThen f(x) = x === Could you give me some examples of terplex values please.> Our product rule matches but I believe your values carry two signs> don't they?I apologise for the delay in responding - I've been upgrading mysystem to compensate for my downgrading brain.In Terplex, {a,a,a} and {a,b,-a-b} are zero-sized. Multiplying ordividing by triples with these characteristics constrains the resultto a sub-algebra in wch that size is always zero.The triple {a,b,c} can be written a+ b'J +c'J'J, where ÔJ^3=1. a,b,care in a field, and so can be real or complex, etc.; they have thesigns from the field, such as the complex signs {+,i,-,-i} as well asthe signs ÔJ & ÔJ'J. Terplex can also be written in polar form,{a+b+c,((a-b)^2+(b-c)^2+(c-a)^2)/2, ArcTan[2a-b-c,Sqrt(3)(c-b)]},where the second term is a squared radius and the trd a polar angle.On multiplication, the first two terms multiply and the angles add.> Also you have stated that these terplex values fit in between the> reals and the complex numbers. The construction I am using produces> the complex numbers on the three signed stage.snipYou are re-developing a 3-phase description of the complex plane -sometng that I taught to electrical engineering students in 1948! InTerplex, 1, ÔJ, and ÔJ'J are orthogonal directions that can beprojected onto the complex plane. Projecting them destroys theirinteresting properties.Algebras work over fields (e.g. real, complex, quaternion, octonion)that have their own signs. Terplex is just one of the manyconservative algebras that introduce another set of signs such as ÔJ,wch are probably better thought of as directions. Real & complexnumbers work well at human-scale problems, but other systems aremore appropriate to the quantum and cosmological scales. I fear thatyour approach does not break out of the complex strait-jacket.> I don't understand the symbolic system above. They are very long. the> Ô=1' part seems redundant. Perhaps sometng got lost in the font> translation to my macne. I'm not seeing anytng double struck. Are> the letters s, d, n, o, g, h, p, I, J, Y, k, l, m important?The Ôd (etc) terms are written that way to satisfy the Groups.Googleconvention that material should be printable on simple equipment,wch precludes double-struck letters. My choice of letters is:-'d 12 dozen; Ôn 9 nine; Ôo 8 octal; g'7 seventh letter'h 6 hex; Ôp 5 penta; Ôi 4 standard nomenclature;'j & Ôk 3 & 2 following Ôi; Ôm &,n 2 minus and negative;'Y 3 symmetry. The -1 terms completed the definitions. I tried tomake the symbols easily remembered; the result is messy but mnemonic.Roger Beresford.Confusion worse confounded. Milton.That explanation still leaves me confused, === number HELP> dr/dt=(a.da/dt+b.db/dt)/2rwrong Having a complex number z = a+ib and its derivates as dz/dt = da/dt + i> db/dt.> Now, putting r = sqrt(a^2 + b^2), how to obtain the dr/dt ? Thanxs TLan easy way to derivation is to differentiate the expressionr^2 = x^2 + y^2 on both sides, giving:2r dr/dt = 2x dx/dt + 2y dy/dt, or:dr/dt = (x dx/dt + y dy/dt)/r = (x === [so-and-so]> I agree that Professor R. M. is a very bright person. My opinion> is based on personal acquaintance over a period of several> years; he was my Ph.D. advisor at U.C. Berkeley.is he as bright socially? crackpot's report indicates otherwise...btw, what did the berkeley math phd do for you?>> Interesting question. What is your purpose in asking such a question? It> sounds pretty rude to me (especially given what you also say below).>> seems that about 20 out of 23 known of so-and-so's students>> turned duds thus far.Ts is even more interesting. On what do you base ts statement? I> hope you aren't relying too much on the Mathematical Genealogy Project. > Their information is not really updated (eventually they may get pretty up> to date, but not anytime soon).To make a wild guess, I would guess that you looked up Ralph McKenzie up> on the Project website, found he didn't have many grandcldren (according> to the not up-to-date Project), and somehow concluded that s students> were duds, whatever that means.If you really care that much, a better way to proceed is to look Ladkin and found> he's doing very well as a Professor at Bielefeld (I should add ts is a> real accomplishment in case you were wondering).http://www.rvs.uni-bielefeld.de/~ladkin/So I know it's bad form to follow up to one's one post (using the samename at least ;-0), but I just realized that one of McKenzie's studentswas Richard Thompson! Of the Thompson groups fame. I thought s namelooked familiar but I thought I was just confusing m with the grouptheorist Thompson.I don't know what your precise definition of dud is (although itapparently has sometng to do with how many students one produces), butjust s early work with groups definitely makes me === rate m as a non-dudfor sure.Subject: Re: how to find limit n->0 ?(1-n^2)^(1/3) + n^(3/2) = ((1-n^2)^(1/3) + n^(3/2)) n^(2/3) / n^(2/3) = ((1/n^2 - 1)^(1/3) + n^(5/6)) n^(2/3) -> === CST,wch book of maths should I read first?> I am a freshman of computer science and technology.> I came across a problem.That is I don't know wch book should I read> first,wch should I read second......> I have three books right here.They are , Maths>,.> If you know,please help me! Change your major to basket weaving. What do any of the newsgroups have todo> with CS?Piss off, racist.-- Felony case 02-CR-0617 9/1/03: Oregon Department ofJustice V. Raymond Ronald Karczewski, Defendant.The defendant's name is === roots of an equation, how do i go about finding the equation if> it is unknown (and the equation must not contain any irrational numbers)?> i've used the quad.formula and worked backwords on some but what if the> roots are sqrt(2) + sqrt(3)....any nts?Let x = sqrt(2) + sqrt(3)x^2 = 5 + 2*sqrt(6)x^2 - 5 = 2*sqrt(6)(x^2 - 5)^2 = x^4 - 10x^2 + 25 = Ca.96estroA Coru.96a === Maximum distance seperable codes?> Reed-Solomon codes are well-known codes wch are also maximum> distance separable (MDS), i.e. the minimum amount of redundancy is> needed to correct errors ( => dmin = 2t+1, where t equals the amount> of correctable errors).RS codewords have length n, where n equals 2^m - 1 and m defines the> Galois field GF(2^m) in wch the code polynomial is defined. Thus,> the Galois Field have 2^m distinct elements. Will the amount of> distinct elements in the Galois Field play a role in the maximum> length of MDS codes' codewords of a specific Galois Field? very much,> VersfeldIf I understood you correctly, the answer is Ôlikely yes'. A Ôstandard' exampleis the following. Assume that your alphabet is a field F of q elements, andassume that you are interested in linear MDS codes C defined by two checkequations, i.e. the check matrix H of C has two rows, and your code shouldhave minimum distance 3 (MDS means that the code is at the Singleton bound). Let n be the length of your code. Then any two of the n columns of H mustbe linearly independent. For otherwise, you have two columns that are scalarmultiples of each other. Consequently you get codewords of weight twocontradicting the MDS property. But any vector in F^2 is a scalar multipleof either (0,1) or (1,x) for some x in F. Thus the maximum number of pairwiselinearly independent vectors is q+1 (there were q choices for x above). Thuswe arrive at the inequality n < q+2.MacWilliams & Sloane lists the maximum length of a k-dimensional MDS-code (over F)as an open research problem. In another research problem they conjecture (lookup their exact wording) that the maximum length is q+1 except for the casesk=3,k=q-1, where q=2^m, when the answer should be === numbers escribi.97 en el it spelled correctly?in>Spanish is Criva de Erat.97stenes), and I use it in conjunction withdividing>only by odd numbers.Ehh.... In Spanish, the correct spelling is ÔCriba de Coru.96a === Deep Thoughts # 1: A new limitation to the human mind 7e3Z2dkQEBC5ubm60r3HAAACeUlEQVR4nIXTQW+ bMBQAYChLfcWzUq5pipUr7lPptUNGXBMC8zUZ qnt1aNz392c7pGXSpvkSyV/ee37POFr/Y0X/g56rxd+ gb2IQC8X79brdLtovaBgACIsRMVTr4hN4 Ah54xy0rtdbjFRCAupizIhUDB69X2IHAGKCrxyrxcJyA76CQKwZn2ckEXK6h/ YwoOld/zJUMEYds ghMIYhj0kjehxqGYFacgalk1PpM+jBOkvg0othFR4bjDVHyjQx+FZ+ b29fMEe114SIC6UB/RXoCX enAhGQORAdxp/TaN5N63lNqGJZACO2n9/ QL81qfVQjEXAkBKPyoPm7BfPqFLJ0qBt1fwmfTAFiYW MbAC1WmCnyEitcYImtI07fY3FwglBuwMUiqoiLv98gL+ 4PrNSmXdWGJK7Q4D8JBpWeXGxi6CpqOJ QoOrkOksKzMaRinN7LsI8OChsGOl+ogiE8SeaIAP9/ 9BRIs1xzgzESWteQ7gDgWQpmRdmcgtNFYJ / 5W4CWpRRsTyHE2UBgifjzvUQMGMOfYYO1CI7wFWfhwsWXSNq0qQcMQkgBvhADvI rOrduRArpW7D Rßb0gjHvO5kpUhX93h3DJ3v9VB+S8pMulW1kis73aAqh/ LpBYalbA2qiLZKP16G+KHLowIYojSm UTys+/ LrogaFVFTKpN2P45qT6UX58qrbEFcil81h9tQSfVSyIZ0v3zzNYEMPVtbR2e3XL 8v547w/ uDymk7wfXx7n8PBscrm1fW/G1a8/gKiz3GJeYbt5ncMJ+ Y1ckVxi19zMwaIcXXUreb6yc1gQ9ypr tLKSCueQjQ4qxFzKXo0zwNDbxv/U2ynkN+Dj/ sha1:Klr5jb960A8ywMH6SklBPyHcOhY=>> 1. Mathematics is the science in wch we make sometng out of>> notng. Wrong. Mathematics is built on the 13 Axioms, wch are not notng.Wch 13?-- And yes, for those who tnk that just maybe I did find a short proofof Fermat's Last Theorem, and THE prime counting function, if Isucceed at what I'm working on now world economy === What to tell students in a 10-15 minute talk>Also, when are the hour, minute and second hands positioned so that>>they divide the clockface in three equal sectors?The short answer is never.I was aware of the result. The interesting bit is tnking ofdifferent mathematical solutions to solving it, wch leads us to thetopic of having multiple ways to reach a given === Gamma function is very nice. It was proved (byHadamard?) that it does not satisfy any ordinary differentialequation in the form of a polynomial in the variables andderivatives, equated eventually be reduced to either 0 or an> elementary function by repeated differentiation.> By nice function, I mean a function that can be written on a (big)> sheet of paper using only:> 1) The integers, pi, and e> 2) The operations of +,*,^, and log> 3) The sine function.> By elementary function, I mean sin and === <>sSHfTy;{Dhe&:+?b`9fUj5A~$gIYlYT0/$-asR-K~3S3[]q.R3YSmpR|$- GiZp>UN2a}!Fmw+%h}YL`!h_XXr5Q>_nGsY2_> can anyone show me an example of a function (R=real numbers) f:R-->R> such that for any a,b,c in R, there exists an x in R such that a and f(x)=c?CompleteSo you want a function that takes every value on every interval.Try defining it in terms of the decimal representations of the numbers.Taking every other digit, and tngs like that.-- G. A. Edgar === relations> In my discrete math textbook (Rosen) I'm reading about sequences like ts: a_n = sum_{1<=i<=k} c_i * a_{n-i}where the c_i Ô s are constants. The sequences that satisfy an equation like> ts constitute a k-dimensional vector space V, since the first k terms in any> sequence determine all the remaining terms. The equation above can easily be> fiddled into a polynomial equation whose solutions are exactly the numbers x> for> wch a_n=x^n is a solution of the original equation. The book states,> without> proof, that if the polynomial equation has no repeated roots, then V is> spanned> by the set of vectors (1,x,...x^{k-1}) for wch x is a solution of the> polynomial equation. A wle ago, when I first skimmed through the chapter, I> tried to tnk of a reason why that set of vectors should necessarily be> independent, but I came up empty. Now one of my homework questions is to prove> the analogous theorem, but without the hypothesis of no repeated roots. Does> anybody have a good nt about how to crack ts nut, or, failing that, a> reference? Needless to say, all help is mucho appreciado.Peaceone word: Vandermonde.[I tnk it's one === anyone show me an example of a function (R=real numbers) f:R-->R> such that for any a,b,c in R, there exists an x in R such that a and f(x)=c? Complete In terms of mappings, you are looking for a function that mapsevery (non-empty open) interval onto all of R. I can give you a function that maps every such interval ontoall of [0,1]. You can modify it to satisfy your requirement. Take the binary expansion of x = (integer ßoor of x) + 0.(b_1)(b_2)(b_3)...(where b_j are 0's or 1's.)Exclude expansions with tails consisting of all 1's.Then f(x) = lim sup[as n->infinity] === <>sSHfTy;{Dhe&:+?b`9fUj5A~$gIYlYT0/$-asR-K~3S3[]q.R3YSmpR|$- GiZp>UN2a}!Fmw+%h}YL`!h_XXr5Q>_nGsY2_,> The problem I had was that I thought that any proof in Rudin (at least> from chapter 2!) would be _complete_ in the sense that the sequence of> statements should imply the hypothesis without absolutely any recourse> obvious-ness other than elementary set logic.Why would you tnk that? Rudin's texts are known for their concinnityand concision, not for detailed proofs.-- G. A. Edgar === odds calculation problema few weeks ago, i made an observation in alt.sports.baseball.mn-twins aboutwhat seemed to be a rarity: all three teams in baseball's american leagueclinched their respective divisions on the same date. my exact remark wasi bet ts has never happened before.several others agreed and then tried to pin down the probability of tshappening in any given season. at ts point the discussion turned a heatedargument over the probability issue, with widely disparate answers--we'retalking answers not even in the same universe. worlds apart.anyway, i'm posting ts problem here hoping that a few knowledgeable folkswill be willing to take a crack at it. wle it helps to have a knowledgeof how competition among baseball clubs works mathematically, i don'tbelieve thorough understanding of the sport is required. it seems to bemore of a math issue than a baseball issue.here's the problem, specifically:there are three divisions in the american league. every season, each one ofthese divisions will be clinched by one team on a certain date. ts is amathematical certainty.past baseball story tells us that dates before september are extremelyrare for teams clincng. in fact, the later days of september, roughly the15th through the 28th, are where the likelihood of teams clincng is thegreatest.i hope i've done an adequate job of spelling out the problem. what are theapproximate odds that all three divisions will be === wanted to do linear regression with the least median square methodin Maple. But the result I got had two unknown parameters:y=a0 +a1*x1+(linear combination of a0, a1)*x2+(linear combination ofa0, a1)*x3(the as were called myparams).Any idea what ts means?I dont tnk that they are just free parameters, because when I didthe plot for some values of a0 and a1 (the ones I got from leastsquares regression or random) the fit was really not good.And anyway (x1,x2) and (x1,x3) should form straight lines for such aresult, wch is not the case.Am I seeing anytng wrong here?, any help will === of matterCut< I doubt if it is necessary for you to distinguish between Ômass' and'weight'> for any physicist. What's your problem?>The problem isn't mine: It's that any physicist doesn't tnk that there_is_ a problem.Here I am attempting to prove the obvious, and all the wle democracy isallowing crooked popularity to triump over === possible reason researchers have sed away from> the factoring->SAT problem. it turns out that even a very> good satisifiability algorithm running on these instances> is apparently much slower than the worlds best factoring> algorithms. but there are two questions here: - is the O(f(n)) growth poorer? ts could just be due to a large> constant delay factor in SAT algorithms vs the straight or direct> factoring algorithms.It seems more likely to me that the growth is asymptotically poorer and> not just poorer by a constant factor. Good SAT solvers still have times> like c^n (with c very close to 1) for random instances, wle NFS is> more like c^(n^{1/3}). The instances one gets from factoring are of> course ghly nonrandom but I don't see any reason for expecting a> general-purpose SAT solver to be able to exploit their structure in the> deep ways that NFS does.> Moreover ... it should be easy to decide the satisfiability of thoseformulas (since Primes is in P) but it should be hard to find asatisfying assignment, since the satisfying assignment yields a factor(and that is supposed to be hard). Since all SAT solver I know, yield asatisfying assignment if the formula is satisfiable they are notoptimal.Jochen === ts newsgroup useless?http://www.giganews.com/info/dmca.html> The number of replies is the thread's rating. Apparently, post's quality has>> notng to do with the rating. Now, face it: the JSH show has the ghest>> rating in sci.math. That's what makes it so sad - that the rantings of that pathetic loony>> seem to be stuff that so many contributors to sci.math are so interested>> in. I'm puzzled why ts bothers you so much. The huge majority of posts on >sci.math are about mathematics. I'm glad you said that, because I wanted to but hesitated to do so because after all I'm one of the guys who's part of the problemaccording to various posts here.Seems to me that an even huger majority of the _threads_ areabout mathematics, making it even easier to filter out the crapif you want.>It's easy to skip the other threads, >assuming a moderately decent === God (long)Thread my OP in sci.math referencing PT's much earler post. Quoting from PT'sreply:>I am suprised that anyone reads that stuff from so far back.I tripped over it and took the time to look.>I had psychological problems at the time and was a crank even>if I was unaware of it at the time.So do I but I take a _lot_ of meds, about twice as much as I could afford withevery penny I've got, rent and peanut butter be damned. I've been accused ofgoing over the line. We write in order to be more aware of our thoughts and arenever fully aware of anytng. I agree with:>Even talking about the contents of a thought >is meaningless if there are *no* *individual* *thoughts*.That's what I hate about therapy. It's so limiting. And what I like aboutUsenet. What is written is know to be only a representation of a thought. Ifthe representation be clear and the thought of a certain class, we have atheorem. Big deal. Keep writing, I say.>Its over with and>hopefully I'll never return to crankdom.It could happen to me in just a couple of days. I'd like to try goingdrug-free. I really would. In a safe place like a hospital, wch costs somethousands of dollars per day. If I just stopped taking ts stuff, I might behanging from the light fixture when my smooce got home. I know that wouldhurt her, but that doesn't matter when you're strung out.>I don't>want to renounce my old views completely.Yes. Keep the old friends and make some new ones.>I don't even know if math could>approach a topic like ts because math talks about>tngs that don't really exist.The latest theory in physics is that reality is all and only that wch we willeventually know to exist through observation, measurement, and math. Now, Imust say I read that in Scientific American, wch is not a peer-reviewedjournal, but rather an excellent popular publication.>Everytng in>the phenomenal world never really *is* because it>undergoes change, and math speaks of absolutely existing>entities. Well, that is interesting and I tnk I have been working on sometng similar.psi(r1, r2, ... rn, t)inßuence is a constant in the wave function, even a nonuniform potential orfield, even a time varying field. Well not exactly a constant, but the psi isnot a function of that variable.So we find that the probability of the system being in a state approacng nowspecific r1, r2, ... rn, at exact time t is| psi (r1, r2, ... rn, t) |^2 dv1 dv2 ...dvndifferential, not even integral calculus, ts goes to a limit of zero at dv1or dv2 or ... dvn = 0. One zero makes it all zero. So the actual value cannever be known.Now, if 1/2 * h<= delta x * delta p , the Heisenberg uncertainty,than we might find thatdv1 * dv2 <= kh or some damn tng and maybe, if we introduced a timetolerance or accuracy with wch we can measure time, then _maybe_dv1 * dt <= kh or some damn tng anddv2 * dt <= kh etc.Ts changes the probability wave function from describing the likelyhood thatdv2 etc, _at_ time t, to ts:r2 witn dv2, and etc, _at_ time t but now _witn_ time tolerance dt.Wch implies _to me alone so far_ that yes, Virginia, the wave function isknowable to a tolerance, since we no longer consider dv1 = 0, or dt = 0 or anydifferential = 0.I wish I had the ability to copy ts to sci.physics.research. Wait, I do!Where's that Administriva? Maybe after I post ts, I'll stick that to the wallhere with my monther's telephone number.>Indeed, that's probably not a failing of math.>Its a failing of intellect itself when we divide the world>into beings. Yes, I tnk it is.>At any rate either conciousness moves in time wle>what is perceived is static, or what is perceived>changes wle conciousness remains (paradoxically) fixed.(spelling?) wch I am not qualified to write about, understanding it only justa little bit, even though I took more LSD in 1979 than anyone with my brainchemistry and genetic story ever should, that is to say, more than none. Itook about ten ts.I may have deeply hurt and violated the privacy of a good friend of my freshmanroomate. One day, I will have the courage to ask, and then I will know.Can you imaging a $2.50 t of acid and speed costing you $1400 a month for therest of your life? I can.Can you imagine going without the proper treatment for twenty years because youdidn't have decent insurance? I can.>ndu plosophy>claims that conciousness >(i.e. me and you) is absolutely fixed>and I consider that to be a likely proposition.>(i.e. the prime unifier = you, now and forever).Yes. = you and / or me>I apologise to the readers of sci.math for having>started ts crankery long ago. Oh, please, I don't tnk we're out of charter, do you? Nobody has posted an OTwarning.> I am cross posting ts post>to alt.plosophy and I would appreciate>it if all replies to ts post were posted there>(and not to sci.math), since ts subject is>not mathematical in nature.Once ts post appears in both sci.math and alt,plosphy, due to a peculiarityof my newsreader, we can discuss it there, and I believe my newsreader willfollow your diversion to that newsgroup after then. If not, then I will contactmy news service.In my newsreader I see two blue links when I read your post. The links are notblue in your quotable original message text, nor in ts window.>Sorry for not reading and digesting your own reply.>It really isn't up my alley right now, and I do not>mean to imply that your ideas are uninteresting>or not worthy of consideration.. If any sci.math or alt.plosophy reader would like to hear whycounting nuts and bolts seems to be a lot like finding the one reason foreverytng, whether it's God or sometng else, I will write more.I use the quantity matrix for a shopping list or even to make a couple ofquesadillas, and my Mom loves those!You see it combines the notions of couting and process in a matrix with all theusual matrix advantages like fast multiplcation. Compared to an algorithm, it'sblindingly fast. If I can show it is univeral (with an extension) than we havean instantaneous (quantum) universal computer! In that case universal would beless than the full capacity of an as-yet-only-theoretical quanum computer, andtherefore more realizable.The only operations in the quantity matrix are counting, sequencing, addition,multiplication or a type of division, and assembly. A state macne issomewhat similar and is known to be universal.The quantity matrix in algorithmic form includes sequencing, multiplebrancng, and the operations above, but it doesn't include logical operations,a good point to try for an extension. That is, why have two quantity matrixes,say for the 14 and 15 cubic foot refrigerators of modular design?Just have a branch: If size is small then do ts. Otherwise, do ts.Everytng else is the same. Mechanical engineers do that all day long, andthat's why we decent applicances. A little more energy used than in, sayEurope, but energy is more ghly subsidized here than it is there, and yes,Virginia, it is subsidized everywhere.In my favorite, Mathcad, an if statement is a function or in the extension(Pro/Plus version) can be a program branch. It usually returns on value byevaluating an expression if the condition is true, and another if false, eitherof wch can be an if. But it returns a value.In the Mathcad programming language, wch has seven primitives that mustalways be selected from a pallete, never typed, not even the assignmentoperator, and wch allows any nesting required and show it particularlyclearly, the if can evaluate any one of a number of expressions if theirconditionals are true, applying them in sequence, or do an otherwise. It canalso execute programs, wch are expressions in Mathcad.You can return anytng you want from a program, a single value, more than one,vectors, and matrices, in any combination, even vector- and matrix- valuedvectors, matrices, and numbers.What fun I've had exploring FLT in that environment!Yours, Goncz (at aol dot com)Replikon Research, Seven Corners, VA1200+ original posts === certainly read sci.math.moderated but who would moderate it? >> It would be a lot of hard work.It wouldn't be so hard to read a message and decide whether to >post it. And having decided not to post it, to return the message>to the author with the (first) offending sentence ghlighted and>a pointer to the website with the posting guidelines.The REAL hard work would be in the fielding of the the challenges>to the moderator's decisions. If the moderator was strict and did>s work well, the posting-decisions work would be minor (but a>daily chore, no doubt) since posters would quickly learn what was>acceptable and abide by the rules. The moderator, by being consistant>and fair, could keep s own work minimal.It's when some dingbat engages m in an argument about whether>s post should have gone through and starts wning about free>speech and starts posting libelous crap about the moderator and>s fascist policies on sci.math and (as in one case I know) actually>takes the moderator to court, that tngs get out of hand. (The>idiot was laughed out of court, of course, but what a pain for the>moderator.) One buffoon on the christianity with penis-enlargement ads >with active embedded scripts, wch, upon opening, of course, >signed everyone up for even more reminds me of a mathematicalproblem recently raised by a member of my department, who shall remainanonymous. Suppose you (applies to male readers only!) had taken advantageof every offer of the above type received, and all had worked as way to Construct a context free grammer for ts.. I believe its one more a than b. Like aba, bbbaaaa etc. How would I prove it using the Induction method?L = {w | n (subscript a)(w) = n === Theorem|Yes I am confused! I am confused as to why anyone would argue so|passionately and so ineffectually in favor of a false FCT. I am|confused as to why I am considered crazy for wanting the FCT to be|true.No, you are confused because you read badly. Nobody has been arguing thatthe four color theorem is false. Nobody!When in the proof of the four color theorem, it is assumed temporarilythat there exists a planar graph with chromatic number >4, that assumptionis being made in order to do a proof by contradiction. In a proof bycontradiction, one makes a temporary assumption in order to show eventuallythat the assumption must have been false. So the only reason anyone hasbeen considering the assumption that there exists such a graph, has beenin order to explain the proof wch eventually shows that such a graphcan't exist after all.Arturo Magidin and I have both explained ts rather key point before, butas far as I can see you just skimmed past both explanations. I urged youto deal with any qualms you might have with proof by contradiction first.If you keep ignoring our explanations, and instead keep pretending thatwe're arguing so passionately and so ineffectually for sometng thatwe're not arguing for at all, then there's not much point in continuing.It isn't necessary to phrase the proof as a proof by contradiction. Itcould be rephrased to make it a direct proof by induction on the numberof vertices of the graph. But such a rephrasing wouldn't affect the contentof the proof, and isn't needed to make the proof constructive or anytnglike that either.Keith === Visiting Assistant Professor at the University of Montana.What I have been trying to say is that if c(G)=5, G cannot be>planar!You know, you ->really<- should drop the stupid exclamation signs. Itmakes you look like a raving loon.The claim that if c(G)=5 then G cannot be planar IS the 4-colortheorem. You cannot ->assume<- it if you are trying to ->prove<- the 4color theorem.>The only minimal counter-example to the FCT is K5!No, K5 is NOT a counterexample to the Four Color Theorem, because the4 color theorem states that any ->planar<- graph can be colored withat most 4 colors in such a way that no two adjacent vertices share thesame color.>The conjecture that there exists a 5-chroma graph may be recolored to>4-chroma is false.There is no such conjecture.>Yes I am confused! I am confused as to why anyone would argue so>passionately and so ineffectually in favor of a false FCT. Nobody is. > I am>confused as to why I am considered crazy for wanting the FCT to be>true.You are being considered crazy because your posts read like you arecrazy. Let H be any subgraph of G, where G has n vertices and H has n-1>vertices. Then, the description of H seems to imply that the deletion>of Ôany' vertex from G will make c(H)<=4.Ts is true if G is a minimal counterexample for the 4 Color Theorem.>But ts interpretation is generally false and is valid only for>n=5!!!The triple exclamation points make you look like a raving loon. Sostart by removing them.Then note that the original argument started by ->assuming<- that theFCT is ->false<-, from wch we deduce that if ts is the case, thenamong them, there is one with the least number of vertices. Call n thenumber of vertices of ts HYPOTHETICAL counterexample. Then, by thedefinition of n, any graph with fewer than n vertices must be4-colorable. In particular, if you took ts HYPOTHETICAL example G,and removed one vertex, then the resulting graph would be 4-colorable.What exactly are you having trouble understanding about the aboveargument? Try to answer without using a ->single<- exclamation point.>I intended ts to mean that only K5 could be an mc-e to the FCT.K5 is NOT even a hypothetical a counterexample to the FCT, because K5is not planar.> None>of the responses have convinced me to change my mind. Although, some>of the responses really confused me!If the responses have been confusing, I must say that it is becauseYOU are a very confusing fellow.>Perhaps, I erred in not offering a more substantial proof for the>conjecture>G = K5? Probably because I don't have one as yet. There is no such conjecture. The existence of a minimal counterexampleG (wch must be PLANAR) is assumed as part of a just very selective about what I accept as reality. --- Calvin === > Zero is the the one of the agreed point of reference, i.e. decimaldot. Therefore the question to divide by zero is irrelevant. How to divide by the point of reference indicator? I'm afraid that I don't understand what you mean You should only be afraid when you *do* understand what Tapio means. --> === a feeling that the question I pose is silly( I have never>>studied set theory rigorously) - Suppose there are sets A and B such>>that there exits a bijective function f:A -> B. Also assume that the>>cardinality of A and B is uncountabily infinite. Now suppose I>>construct a set C = B U D, st that cardinality of D is finite( D is>>not a subset of B). Then it seems to me that the cardinality of C is>>different from A (even though they are both uncontabily infinite)>>because I am not able to create a function g:A->C, st g is understanding ts problem.>In general, if at least one of A and B are infinite, then the >cardinality of A U B is the same as the cardinality of the larger >set. One can establish ts fact without the axiom of choice.On the contrary, it is equivalent to the axiom of choicethat there is a larger set. However, without the axiom ofchoice, it can be the case that even if one is larger thanthe other, the cardinality of the union is larger.In the question, if B is infinite but not transfinite, soit is uncountably infinite but contains no countable set, Cis strictly larger than B, even though B is strictly largerthan === mass is an aggregation - via gravitation and othercentripetal forces - of the substances comprising any object; body, and or mass of material matter; wch causes these accumulations to have inertia,and/or heft; the property of matter whereby it becomes more obvious that it> requires greater (net) force to change the velocity of an accumulation as it becomes> larger:> Here on Earth Galileo found that the rate of change in the velocity ofany body free falling at Earth's surface was about [s/t = 16'/sec] -half of ÔNewton's' acceleration of free fall [g] - and furthermore - ineffect -> he found that the force restraining ts change in velocity of free fallfrom continuing toward Earth's center is the mutual force exerted betweenthe body and Earth's terra firma; wch force is commonly measured with weight-scales: On Earth, the ratio of ts weight-force [w], dividedby> the rate of change in velocity [s/t = 16'/sec] that it restrains, is a constant [wt/16' = wt/s]; to be known hereafter, as one half of the> body's gravitational mass [g/2], and/or inertia.> On any similar planet, such as the moon, ts _ratio_ will still beequal> to half of any body's mass, and/or inertia!> In everyday use, a body's mass [m] is commonly confused with itsweight> [w]; A practice wch must cease immediately; for the sake of physics! According to newton's second law weight is the product of mass [m] andthe> gravitational acceleration of free fall [g]: Ts erroneous formula w =ma> is a special case of f = ma, and like f = wa/g, must be written as w =fg/a:> All because inertial mass m = f/a, and gravitational mass m = w/g; whereit> follows that inertial mass f/a is equal to gravitational mass [w/g]: f/a=> w/g. Cheeze, what have I got to do, write a book for youse people(:^? I'm sure that people would stand in line for blocks === useless?http://www.giganews.com/info/dmca.html>>[...] One buffoon on the christianity group got mad at the>>moderator active embedded scripts, wch, upon opening, of course, bring the thread back on-topic, ts reminds me of a mathematical>problem recently raised by a member of my department, who shall remain>anonymous. Suppose you (applies to male readers only!) had taken advantage>of every offer of the above type received, and all had worked as claimed.You're assuming we haven't, and that if we had they wouldn't work.>Estimate how long it would be.One learns not to step on === newsgroup useless?>>I'm puzzled why ts bothers you so much. The huge majority of posts>>on sci.math are about mathematics. I'm glad you said that, because I wanted to but hesitated to do so > because after all I'm one of the guys who's part of the problem> according to various posts here.Well, as I tnk about it more, the social aspect ofthe newsgroup is part of its appeal also. My original outlinefor the strict moderation would leave the atmosphere prettyantiseptic, after all.We have to feel like we're sitting around the math lounge chatting and if we want to go off on tangents then we can ifwe feel like it. As someone said, a moderated group was triedbefore, and almost no one participated. Nice argument Areyou going to Phoenix ts year? I never go to those silly tngs. Waste of time. Waste of time! I'm giving a short course!So you're one of those idiots padding s vita by pretending tobe an expert on a non-topic? What a bozo! ÔBozo'? I hope thatclown sues you for copyright infringement! It's not Ôcopyright'it's Ôtrademark', Krusty.......After all, most of us are nerds, and ts may be our onlysocial life....;-)So perhaps the _real_ problem is that the math lounge has2000 people in it, and the cross conversation requires a bitmore subtle etiquette than a normal math lounge with only 5 or10 people. Several people have commented that the filtering systems shouldwork fine. Sure they do, if you continuously update them.Maybe some semi-official header guidelines (that is some sortof netiquette for sci.math) that aids in efficient filteringwould be the tng to persue. Violators could be politely informedof the conventions, and if they choose to ignore them, then there'snotng we can do about it, except ignore them, but that should besufficient.I guess having JSH in certain headers was a step in that direction,but I don't really have any good ideas as to what header guidelinesshould be. I'm just tnking === O.P. may well be a troll. Believe me, I am not. I am just expressing despondency about the status> quo nowadays. I have been reading sci.math, on and off, for some 14 years> now, and it has gotten steadily worse. With all the trolls, morons,> lunatics and otherwise deranged contributors to ts group so prevalent> today, going through new headers every day has become a painful exercise. Why? Do you talk to the nntp-server via the raw protocol?Just use a decent news-reader and ignore === tell students in a 10-15 minute talk>Also, when are the hour, minute and second hands positioned so that>they divide the clockface in three equal sectors?>>The short answer is never.I was aware of the result. The interesting bit is tnking of> different mathematical solutions to solving it, wch leads us to the> topic of having multiple ways to reach a given mathematical result.So if you tnk of the three hands as rotating unit vectors,their sum will never be zero. So here's another problem. Assignangular velocities to the three unit vectors so that the mimimumlength of their sum is as large as === (0,1) is continuous. Does f have to have a fixedpoint? If it was f : [0,1] ---> [0,1] or f : [0,1] ---> (0,1), then yes.Any === perhaps the _real_ problem is that the math lounge has>2000 people in it, and the cross conversation requires a bit>more subtle etiquette than a normal math lounge with only 5 or>10 people. Nah. The _real_ problem is that the math lounge has 2000people in it, of whom a number (probably not out of proportion to that in the general population) have the bad qualities of mindthat made life hell for Nash and those around m, withouthaving the good qualities of mind that made Nash a mathematician of note.That is: it's not the nerds, it's the === C Bond>>Will, you're tnking of a question wch has correct answer 1/3. Ourquestion has correct answer 1/2.There is a coin ßip sequence whereas you can end with three equallylikely outcomes. It isn't defined by Two coins were ßipped and atleast one is a tail or Two coins were ßipped and at least one is ahead.Whether those statements were made, generated, spoken, or written,they have a meaning.Two coins were ßipped, the sample space is HH, HT, TH, or TT. Of thetwo statements, the Ôheads' statement and the Ôtails' statement, weknow that at least one is true, maybe both.We can make neither, prior to some kind of inspection. THE GENERATORof our statement COULDN'T either.Our statement describes a storical event. Two coins were ßipped,then there was an Ôinspection', the statement WAS made.When the Ôheads' question and the Ôtails' question have equal answers,> the correct answer is 1/2. The secret is in the prejudice, not some> Ômagic' in the at least one is statement.> The secret is in correctly setting up the sample space. S={HH,HT,TH,TT} > does not accurately reßect the situation unless you consider the pairs > to be {(called)(uncalled)}. Since ts is not the way you appear to be > tnking about it, the sample space will be S={(first)(second)(called)} > = {HH1,HH2,HT1,HT2,TH1,TH2,TT1,TT2}>With HH, the at least one is a head statement is true.With TT, the at least one is a tail statement is true.With HT, or TH, either statement would be true.One statement WAS made. It's story. > Bill wins on exactly 4, Jane wins on 4, so each has a probability of > winning of 1/2, and Jane's bet should equal Bill's.The problem with all ts is the following:Your original question was Two coins were ßipped and at least one is a > tail. What are the chances for two tails?In ts case the sample space is S={HH,HT,TH,TT}.> at least one is a tail={HT,TH,TT}.> two tails={TT}.> You are correct, the sample space WAS HH, HT, TH, TT. Then thestatement and now it is HT, TH, TT, or HH, HT, TH, depending uponwch statement was made.Where you err. Prior to the statement, the four were equally likelyevents. After the statement, the three wch are left are no longerequally likely.> P(two tails | at least one is a tail)> = P(two tails and at least one tail) / P(at least one tail)> = (1/4) / (3/4) = 1/3> Ts is true, but it does not represent our question:Our Question:P(two tails |the at least one is a tail statement was made)= P(two tails and at least one tail) / P(the at least one tail wasmade)= (1/4) / (0+1/2*1/4+1/2*1/4+1/4) = 1/2= (1/4) / (1/2) = 1/2> Here's the key. These two problems are *not* equivalent. By adding the > betting, and the assumption that someone is looking at *one* coin and > calling it, rather than looking at both, you have changed the problem > from what was originally intended.Here, Will, you get sloppy with your language.I said that there is no evidence in our problem statement that thewriter of the question knew the outcome of both coins.You said that I make the assumption that she Ôlooked' at one. There isa difference.I said that our statement can have been generated by onlyinvestigating one, wch would definitely have correct answer 1/2.I also say, and correctly so, that the coin ßip sequence with correctanswer 1/3, cannot be generated without investigating both coins. Youmust add evidence, in the problem statement, of knowledge of bothcoins. To make the tails question answer 1/3, TT must have beenßipped around. TT cannot be eliminated by only investigating onecoin.Therefore, to assume your question, you must add information to ourproblem statement. I can design situations wheras our question hascorrect answer 1/2. You need more information. You assume too much.You are tnking a question. Our problem statement is === (0,1) --> (0,1) is continuous. Does f have to have a fixed> === limitation to the human mind> 1. Mathematics is the science in wch we make sometng out of>> notng. Wrong. Mathematics is built on the 13 Axioms, wch are not notng. Wch 13? --> And yes, for those who tnk that just maybe I did find a short proof> of Fermat's Last Theorem, and THE prime counting function, if I> succeed at what I'm working on now world economy as you know it will> be gone. -- branches out.There is a guy who is really a fundamentalist religious preacher callingmself Paniagua (Bread Ôn water)He got nicknamed sometng like Taco Bill.I hope === pointsf(x)=x^2gs> Suppose f : (0,1) --> (0,1) is continuous. Does f have to have a fixed> point? If it was f : [0,1] ---> [0,1] or f : [0,1] ---> (0,1), then yes. Any thoughts? === Professor R. M. is a very bright person. My opinion>> is based on personal acquaintance over a period of several>> years; he was my Ph.D. advisor at U.C. Berkeley.is he as bright socially? crackpot's report indicates otherwise...btw, what did the berkeley math phd do for you?> Interesting question. What is your purpose in asking such a question? Itsounds pretty rude to me (especially given what you also say below).> seems that about 20 out of 23 known of so-and-so's students> turned duds thus far.>Ts is even more interesting. On what do you base ts statement? Ihope you aren't relying too much on the Mathematical Genealogy Project. Their information is not really updated (eventually they may get pretty upto date, but not anytime soon).To make a wild guess, I would guess that you looked up Ralph McKenzie upon the Project website, found he didn't have many grandcldren (accordingto the not up-to-date Project), and somehow concluded that s studentswere duds, whatever that means.If you really care that much, a better way I looked up Peter Ladkin and foundhe's doing very well as a Professor at Bielefeld (I should add ts is areal accomplishment in case you were wondering).http://www.rvs.uni-bielefeld.de/~ladkin/ === continuous. Does f have to have a fixed> point? No.>Why? (Should have asked before)> -- > Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html> Needless to say, I had the last laugh.> Alan Partridge, _Bouncing Back_ (14 === is continuous. Does f have to have afixed point? No.> Why? (Should have asked before)Nevermind. Gillian posted a counterexample. I will tnk about ts more. Robin. -- > Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html> Needless to say, I had the last laugh.> Alan Partridge, _Bouncing Back_ (14 === (0,1) is continuous. Does f have to have a fixed> point? If it was f : [0,1] ---> [0,1] or f : [0,1] ---> (0,1), then yes. Any thoughts? MarcCan you make up a continuous function [0,1] === ... it should be easy to decide the satisfiability of those>formulas (since Primes is in P) but it should be hard to find a>satisfying assignment, since the satisfying assignment yields a factor>(and that is supposed to be hard). Since all SAT solver I know, yield a>satisfying assignment if the formula is satisfiable they are not>optimal.But the general problem of producing satisfying assignments is at mostlinearly harder than deciding satisfiability, since you can use O(n)applications of a SAT decider to generate a satisfying assignment, ifone exists.-- Daniel A. Jim.8enez djimenez@cs.utexas.eduAssistant Professor djimenez@cs.rutgers.eduDepartment of Computer ScienceRutgers === useless?http://www.giganews.com/info/dmca.html>[...]Several people have commented that the filtering systems should>work fine. Sure they do, if you continuously update them.If you continuously update them, right. That's really a _lot_of work with a decent newsreader.Like take Agent, for example. I download the new headers.Years ago I'd just sit there for a few minutes with a fingeron the N key, moving to the next new post and ttingEnter if I wanted to read it. These days I sit there with onefinger on the N key and one on the I key - when a wackythread appears (in some other group where I don't wantto see the wacky threads, not here) I t the I key insteadof the N key, and I'm never bothered by that thread again.Wch is to say that simultaneously scanning for tngsyou want to read and killfiling tngs you don't want toread in the future is _no_ harder than just browsingthrough the headers, if you're using a decent newsreader.You must have internet access - Agent has a free version,so I don't see what the problem is.Well, you do need to be able to find the N key and theI key. That can be tricky at first, but you only have todo it once per session; just leave your fingers there.>Maybe some semi-official header guidelines (that is some sort>of netiquette for sci.math) that aids in efficient filtering>would be the tng to persue. Violators could be politely informed>of the conventions, and if they choose to ignore them, then there's>notng we can do about it, except ignore them, but that should be>sufficient.I guess having JSH in certain headers was a step in that direction,>but I don't really have any good ideas as to what header guidelines>should be. I'm just tnking outloud.You're certainly doing _sometng_ out loud - hard to say whethertnking is the right word, since the problems you're complainingabout are so easy to fix. Luckily ts is not a moderated group,so you're free to bounce ideas off us just like JSH === Ablaze> :> I have some very direct experience with the initiator of ts thead and for that reason would like to know a little about ts forum. > I made way directly here via Google and so have no idea who you are, what you discuss and why Mr. Daryl Shawn Kabatoff's tolerated, much less taken up.> ,> RayHe isn't tolerated, he's merely ignored (for the most part). There isn'tmuch else those of us outside of Canada can some of s stuff to the Saskatoonpolice (they thanked me and promised to investigate, and I never heardanytng else from them), but lately I've been ignoring m as justanother nutcase. (I keep hoping, though, that some of s victims willget together and have m prosecuted. He *really* ought to be lockedaway -- permanently, if possible.)Oh, and to answer your other question: Ts newsgroup is for thediscussion of mathematics -- *real* mathematics, not the sort of insaneidiocy Kabatoff promulgates.-- Wayne Brown | When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.e^(i*pi) = -1 -- === zero...??>>A (potentially) stupid question :)>>What would happens if someone discovers a self-consistent new algebra>>of numbers in wch we could divide by zero?? revolutionnary?>>interesting but not useful?>>The universe would explode.>> Not implode?No, it would implode if someone were to multiply by zero.>(Wch of course _has_ been done. Ts proves that what>appears to be the universe is just an illusion - it imploded>long ago.)Well, that bursts my bubble.You really have to explain _everytng_ to some people. I wanted to reply with a really good smartarse remarkbut I could not tnk of one that was good enough; it shouldhave shades of mathematical ßavors entwined in the line. You'll just have to settle for the paltry line === Re: Fixed points>> Suppose f : (0,1) --> (0,1) is continuous. Does f have to have a fixed>> point?>> No.Why? (Should have asked before)'cos there's a continuous function (indeed lots of them) f:(0,1) -> (0,1)with no fixed point. === ...What's the size of a largest subset of vertices on the 120-cell({5,3,3}) such that no two vertices in the subset are joined byan edge?I have reason to suspect that the number is an integer square,and the computer searches I've done come tantalizingly close,but I haven't yet been able to conclude that I've found a maximalset.Suggestions? and === software ?What is the best software for solving differentialequations? Is there anytng better than Matlab --that at least won't crash and necessitate a rebootof my computer when I give it a differential equationit can't === pointsCan you make up a continuous function [0,1] -> [0,1] whose only> fixed points are 0 and/or 1?f(x) = x^2 should do it. f(0) = 0 and f(1) = 1 and f(x) < x at all points in === ALL headers otherwise we will be unable to process your complaint properly.I need to find an algorithm that can produce a unique non-predictable 12digit (0-9) number for any given 12 digit number. Ts is to be used tocreate a unique barcode on a ticket that cannot be predicted. It is notrequired that the original seed number be computed from the resultingbarcode, so some form of one-way hasng function would be acceptable.Any help in ts problem would be === you, Mr. or Ms. anonymous reposter troll?Of course, we all know that Sthead hasn't learned even one tinytng about either algebra or physics or metrology in the intervening6 1/2 years, so it might be Dense Donny mself reposting ts. Henormally reposts bits of it every day or so anyway, so the whole tngwouldn't be any big surprise. The really sad tng is that besidesnot learning anytng in all these years, he's forgotten how prettythat little church in the valley was. Maybe he's just a robot with afew blown circuits.Gene Nygaardhttp://ourworld.compuserve.com/homepages/Gene_Nygaard/= === ==Subject: Re: Numeric one-way hash function produce a unique non-predictable 12> digit (0-9) number for any given 12 digit number. Ts is to be used to> create a unique barcode on a ticket that cannot be predicted. It is not> required that the original seed number be computed from the resulting> barcode, so some form of one-way hasng function would be acceptable.> Any help in ts problem would be appreciated.>The simplest way is to encrypt the first number using AES or3DES. You will have to convert the result from binary to decimal.Any extra digits can be thrown away. Change the key variableregularly and keep the old ones secret.Random key variables can be generated by rolling dice.Andrew for ts.. I believe > its one more a than b. Like aba, bbbaaaa etc. How would I prove it using > the Induction method?L = {w | n (subscript a)(w) = n (subscript b)(w) + 1}> Smells like homework to me, especially since it's about thattime in the term when ts would be covered. Here's a nt:Can you find a CFG for the language over {a, b} that consistsof all strings with an equal number of as and bs? If so, call thatlanguage L_e. Then what would the strings in your L look likein terms of L_e or perhaps (L_e)*? Once you come up witha CFG, it's not hard to prove by induction that it does indeedgenerate the language === coverage area of a few circles?> In my simulation, N circles with the same radius r are randomly> placed. Let P_i denote the center of circle i. For any i, p_i lies> witn the coverage range of at least one other clicle, i.e. at least> one other circle contains p_i. How to calculate the total coverage> area of the N overlaped circle? The method should be easy to be> implemented by programming for simulation.You can calculate the intersection of any two circlesanalytically:http://mathworld.wolfram.com/ Circle-CircleIntersection.htmlHowever, your problem might involve a large number of these,and in addition you need to calculate overlaps of 3 circles,4 circles, etc.It seems to me your best bet (reasonable accuracy in reasonabletime) is Monte Carlo integration: Generate some large number (M) of points uniformly in an area (A) that covers all your circles. Count how many (Mc) are covered by your circles.Estimate the coverage area as A*(Mc/M).Ts should be pretty quick if you === ÔCore Error' argument ...> what a tour de force, not that I'm able to read it.> has James actually said what is supposed> to fix the ring?He hasn't even said what's broken.Just that there exist numbers that should be in thering, but aren't because they don't fit the definition ofalgebraic integers. He hasn't been pinned down on why theyshould be (though it has sometng to do with factoring),or an example of such a number that === me an example of a function (R=real numbers) f:R-->R>such that for any a,b,c in R, there exists an x in R such that aand f(x)=c?Complete>That description resembles the Intermediate Value Theorem and it characterisesfunctions. Many functions can be characterised by ts theorem. Check acollege algebra === need to find an algorithm that can produce a unique non-predictable 12>> digit (0-9) number for any given 12 digit number. Ts is to be used to>> create a unique barcode on a ticket that cannot be predicted. It is not>> required that the original seed number be computed from the resulting>> barcode, so some form of one-way hasng function would be acceptable.>> Any help in ts problem would be appreciated.> The simplest way is to encrypt the first number using AES or> 3DES. You will have to convert the result from binary to decimal.> Any extra digits can be thrown away. Change the key variable> regularly and keep the old ones secret.But shouldn't the bar codes be unique?Your procedure can generate duplicate bar codes.When the bar codes must be unique you could use thefollowing algorithm. Take a 40 bits block cipher(you can build ts with the Luby-Rackoff construction).Now encrypt the ticket number, when the result is>= 10**12 you re-encrypt the result as many timesas necessary to get a value that is < 10**12.greetings,Ernst === GAP doesn't answer all your questions, just ask. I'vedug outmy own decades-old notes on the groups of order 16 and 32,andnow have presentations, permutation reps, conjugacyclasses,subgroup structure, automorpsms, etc., for orders 16 and 32,if they aren't too hard to find. Email them to me if theyaretoo much to post.I recently studied group theory for the first time(I used Hungerford's *Algebra* mainly), and found itfascinating,but I only know ts introduction to groups.I had to return the lib. books on groups, and don't havea text of my own yet.I just got GAP, and don't know how to use it yet.If anyone could tell me some of the commands I should use toget info on groups with 16,24,32 elements, (as well as whatthe commands mean), I would be grateful.BTW, could you define a perfect group for me.Also, GAP speaks of polycyclic and finitely presentedgroups.I hope to find an intro to groups on the web (with some definitions), and downloadit.Van === functionOriginator: grubb@lola>>can anyone show me an example of a function (R=real numbers) f:R-->R>>such that for any a,b,c in R, there exists an x in R such that a>and f(x)=c?>>Complete>That description resembles the Intermediate Value Theorem and it characterises>functions. Many functions can be characterised by ts theorem. Check a>college algebra textbook. Um, no. As stated, the poster is asking for a function f:R->R wch is ontoas a function from each subinterval. Clearly such a function cannot becontinuous and is not likely to be in a college algebra textbook.--Dan === certainly doing _sometng_ out loud - hard to say whether> tnking is the right word, since the problems you're complaining> about are so easy to fix. Luckily ts is not a moderated group,> so you're free to bounce ideas off us just like JSH is...I haven't been complaining at all. Just chatting with the original complainer. Everyone knows what the really, really,absolutely true problem is, however, and it is the insufferablerudeness of several posters. But like you say, === extension proof>Given:>1.- F is a field (not necessarily Borel)>2.- u is a measure on F>3.- G is the minimal Borel Field containing F.I really don't see what sense the terminology > minimal Borel field makes. Maybe you meant> that G is the minimal sigma-field containing F?That's what I meant. In the material I have read, the terms BorelField, Sigma Field, and Sigma Algebra appear to mean the same,that is a collection of sets closed under complementation andcountable unions. Is ts not correct?>4.- v is a measure on G.>5.- v and u agree on F.>6.- v and u are sigma-finite on F.It can be proved that v is the unique extension of u from F to G. >Apparently 6.- is a sufficient but not necessary condition for ts>uniqueness. Can someone please indicate the necessary condition and>outline how the proof would then proceed? Many I doubt that there _is_ a simple necessary and sufficient condition. Can you please indicate what is the importance of tsmeasure extension subject in terms of further development of measureand probability theory? In other words, is understanding of itimportant in terms of understanding new material down the === Antidiagonal, Infinity at 07:58 PM, raf@tiki-lounge.com (Ross A. Finlayson) said:>I tnk it's not worthless to consider the reals as a sequence of>points. It's worthless because it is meaningless. You can't define a successorfunction compatible with the ordering.>Is an infinitesimal some ghost of a departed quantity, as Berkeley>suggests in s treatise damning the foundations of the integral>calculus as infinitesimal analysis of what Newton called the ßuents>and their ßuxions?That depends on what you mean by infinitesimal. Bishop Berkeley'sargument was valid as applied to Newton's writing, but would beinapplicable to definitions in terms of germs or Nonstandard Analysis.>Is instead the infinitesimal 1/x for x=oo equal to dx?The question is meaningless, unless you have definitions forinfinitesimal and for 1/oo. Even then it is meaningless unless yourdefinition is such that 1/oo is an infinitesimal.>I'm still tnking that f(x)=1 for irrational x and f(x)=0 for>rational x that f is everywhere discontinuous. That much is correct.>What that describes is>that between each pair of any two rationals is at least one>irrational and between any two irrationals is at least one rational.That part is wrong; it describes no such tng.>What I propose is that given any>rational that the value greater than it and less than any other>greater is irrational, There is no such number, as several different people have shown you.>Obviously enough then under ts axiom Adding such an axiom to the standard axioms would yield aninconsistent axiom system, and all statements would be provable. Itwould not be an axiom system for the real numbers.>That's like calling the endpoints,>points adjacent to only one instead of two other points in the>continuous sequence of the reals, of the open interval (0,1)>irrational.I don't know what you're trying to say, but that sentence ismeaningless. It contains an assumption contrary to fact.>Is the evaluation of the integral of f(x) dx on [0,2] equal to 1? No.>I've queried before If there are infinite integers are there>infinite integers?, the answer is not immediately necessarily, no.No. The answer is that If P then P is always true.>because I say so.That doesn't constitute a proof.>Mathematics is exact. FSVO exact that doesn't match the way you are trying to use it.>With some level of exactitude I can note that>there are ten times as many real numbers on [0,10) as there are on>[0,1). You can note that Red Fuming Nitric Acid is a healthful beverage, butfew will take your word for it. The statment that you noted doesn'thave any meaning.>A good example comes from the theoretical random uniform probability>distribution over the reals.There is no such animal. There are various distinct probabilitymeasures on the reals.>Imagine that any number of>the real numbers is as likely to occur as a sample as any other,I can't imagine it, because the statement has no meanining.>Along with that there is a similar convenient distribution over the>interval from zero to ten. Convenient to whom?>Select ten samples, say the>sample set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, a representative>sample.What makes that a representative sample? Why isn't {.01, .02, .03,.04, .05, .06, .07, .08, .09, .1} a representative sample.>Only one of the ten samples, 1/10'th of the samples, is in the>interval of one tenth length. All ten of my samples are in that interval.>That exactly parallels the fact that>1/10 is one tenth.No, it parallels the fact that you didn't take a random sample, butinstead picked a specific sequence.>If you have a problem with that, imagine you have a random uniform>probability distribution over {0,1}. That casts no light on the point.>If you don't know the answer or the question, ßip a coin.Flipping a coin does not answer Mathematical questions.>Now then, assume a uniform random distribution over the set {0, 1}>can be sampled by ßipping a coin. Why would I assume that, when the first sample would require aninfinite amount of time?>Now, consider that the point in question is showing that half of the>reals of [0,1) are in [0,1/2), and the other half are in [1/2,1). You keep using that expression as though it had any meaning. You'vebeen asked repeatedly to define, in Mathematical language, what youmean by it. So far you have refused to do so.>Half of the real numbers in the interval [0,1) are less than one>half,That has as much meaning as saying that half are red and half areblue. You have not defined how to compare or measure sets of reals.>Anyways it's pretty clear that it is possible to describe the>propensity of reals What is a propensity of reals ?>Half the reals are positive, and the other half are negative. Meaningless.>All that might seem patently obvious. It might. But actually it's patently nonesense.>That's because it is.Then define your terms and produce a proof. You can't, because you'rejust throwing buzzwords around.>What I want to figure out are tngs like what is the probability>of a real being a rational, or an algebraic irrational or a>transcendental irrational? You can't, because you haven't defined what the terms mean. Firstdefine them, then worry about their values in specific cases.>I have a conception that the probability>of a real being a rational is one-half, I derive that notion from>the consideration that the rationals and irrationals alternate on>the real number line,You can derive any conclusion from a false assumption. You've beenshown half a dozen times that there can be no such alternation,because you can produce intermediate points by taking an average.>How about ts: a+b is not defined,Then you're not talking about the real numbers, because in the realnumbers a+b is always defined, (a+b)/2 is always defined, and if a(a+b)/2 is not defined, and it's approximately a.How can it be approximately anytng if it's not defined?>In a way that's about considering the set of the elements of the>reals with none of the operations defined on the reals, except < as>they are totally ordered, but perhaps even not that, where density>and order would imply sometng contrary to what I am trying to>infer.You seem to be saying that if we apply the term reals to sometngother than reals, with properties different from those of the reals,that it's properties would be different. That's obvious, butirrelevant; what is at issue is the properties of the ordered fieldknow as the reals.>I search for modulus of precision and am unclear of its meaning.If you want to invent terms, then you are responsible for definingtheir meaning.>Similarly half of the subsets of N contain zero,Meaningless. at 07:15 PM, raf@tiki-lounge.com (Ross A. Finlayson) said:>There are a variety of definitions of measure.K3wl! What's yours?>Consider your example about the topological property of the>rationalsIt doesn't exist. Again you are slinging buzzwords around. You need tolearn some Topology before you can make any rational references totopolgical properties.>I'm aware of logic bend the considerations that there are many>more irrationals than rationals. I just want to ignore it>temporarily to consider the rationals and irrationals alternating.When you ignore Logic then you're no longer discussing Mathematics.>If you consider the measure of the reals [0,1] to be one, I totally>agree.I suspect that he considers the measure of [0,1] to be unspecified. Isuspect that he considers the measure of [0,1] to be 0, given theright measure.>Allow yourself to consider that the measure of the rationals on>[0,1) is 1/2.Sure: use a measure in wch {0} and {1} have measure 1/2 and any setnot containing 0 or 1 has measure 0.>What are the verifiable real-world results of assuming the rationals>to be measure zero? If you use a uniform distribution then you don't have to prove it;it's a standard theorem.>With what analytical results does it disagree?That depends on the measure you pick.>What was the function bijecting the reals and irrationals again? There are many.-- Unsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === useless?X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 02:32 PM, No One said:> I guess that the time has come for me to ditch ts group>altogether. It seems to have been taken over by deranged individuals>exposing their misguided ideas over and over again (Mr. Harris being>the best representative of ts increasingly vocal group,) impervious>to any kind of logic reasoning; by people trying to get the group to>do their homework for them, and by plain idiots who post the most>irrelevant babbling (like ts moron who posts numerological stuff,>wle changing s profile frequently.) Kooks are endemic to Usenet. Your filter is your friend. Actually,sci.math is in good shape compared to some, even in the sci.*erarchy.-- Unsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === theory, logic - wch comes first? <1xUcb.14254$O85.6040@pd7tw1no> <3f79e264$7$fuzhry+tra$mr2ice@news.patriot.net> tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 01:50 PM, mattias_wikst71@hotmail.com (Mattias Wikstr?m) said:>G.9adel did not show ts. He showed that for a given sound formal>system in wch the statements about natural numbers can be>formulated, there are statements that can neither be proven nor>disproven.Google for G.9adel number. He represented statements as naturalnumbers. The theorems he proved about statements were also theoremsabout the integers representing them.-- Unsolicited bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === Sum Of Some Divisors> Take an n-by-n-grid, n>= 3. Place the integers 2 to (n^2 +1) into the grid,> one DISTINCT integer per grid-square, so that: If s(k,j) = a grid-square (ie. an element of an> n-by-n matrix), then> (for all k and j where n >= k >= 3 and n >= j >= 1)> s(k,j) = (any divisor >= 2 of s(k-1,j)) +> (any divisor >= 2 of s(k-2,j)),and> (for all k and j where n >= k >= 1 and n >= j >= 3)> s(k,j) =(any divisor >= 2 of s(k,j-1)) +> (any divisor >= 2 of s(k,j-2))>[...] an n=3 example is:> 5 3 8> 2 6 4> 7 9 10> Is there an n=4 example??There appear to be 2 basic solutions and of course their transposes: 5 2 7 9 3 12 6 15 8 4 10 14 11 16 13 17 5 3 8 11 2 12 4 16 7 6 10 13 9 15 14 17 11 2 13 15 3 12 6 9 14 4 16 8 5 10 7 17 11 3 14 5 2 12 4 10 13 6 16 7 15 9 8 17wch a bit tediously could be extended to 5x5 or 6x6but not much further because it uses nested for loops,one level per cell, rather than a recursive approach. The program takes about 1 === has come in!!!! Hmmm. > I have to ask myself, Why should I care? James may be the> reincarnation > of Gauss, but is it really any skin off my nose if he goes> unrecognized? > I've been worrying about the guy for months (ever since I realized> that> he > was not, in fact, a crank, but a genius) and defending m on ts> newsgroup. > My reward? Laughter and bile from the peanut gallery. And not a> word> from > James mself. A word of advice to Prof. Connes: Don't waste your> time> on > . If he loves being the solitary genius so much, let> m> fight > s own damn battles. He's not worth losing sleep over.> Well damn it, I'm losing sleep at least partly because of your scary dream!!!> Good writing their Jim Ferry, and I have to give you credit for that, but hey, how many years were you ridiculing me, and now you expect me to just go, hey, pal?> Show me you're serious and wade in and respond to some of these ant-mathematicians in the current battle.> Prove your sincerity, and um, keep posting any interesting dreams you have, as I found it interesting puzzling over that one.> Exactly! Ts Jim Ferry now tnks you are a genius, but has he> defended you in your current battle? No!> I suspect that he is not *fully* committed to your view of mathematics.> That is just a feeling, and I suppose I could be wrong. The primary focus is the odd definition error in core. Once that's accepted for what it is, and most importantly FIXED, then> it's not about committing to my view of mathematics, but about showing> your commitment to mathematics itself. There is ONE mathematics. > I, however, have been trying to find out more about your Object Ring,> but I feel you are pusng me away. I have spent some effort guiding you along at the Mega Foundation> discussion area.Yes, indeed.But, I expect you remember that I did ask you a question that I was unable> to answer.I was using the notation [a,b,c] to represent an ordered triple of complex> numbers. And I was wondering if the ordered triple [1,2,8] was an element of> the Object Ring.You replied:> ==> Quit being lazy!!! You have the definition, figure it out for yourself!!!> What amazes me is how often people are willing to ask someone else to do> their work for them.> If you're smart enough, answer your own question.> I'm curious to see if you can.> I've given the definition for the object ring, so no excuses.> ==Well, I am ashamed to say I still cannot figure it out.Sounds like a ploy.You have to understand that your record Clive Tooth is rather long andinvolves some rather...sleazy behavior...like that attack webpage, andquite a few negative postings over a period of YEARS.You don't get the benefit of the doubt from me but have to make theextra effort yourself, so quit being lazy!!! > Could you help me please?It sounds to me like you tnk you have some angle for even morenegativity and I've given enough time explaining.> Remember mathematics is a continuing process. The object ring is fascinating in and of itself, so you can't expect> me to know all the answers just because I'm a discoverer. After all, if it were that easy, then math research would have ended> long ago with the first mathematician explaining it all. > Your friend.> Clive Time === your sp has come in!!!!>>Good writing their Jim Ferry, and I have to give you credit for that,>>but hey, how many years were you ridiculing me, and now you expect me>>to just go, hey, pal?>>Show me you're serious and wade in and respond to some of these>>ant-mathematicians in the current battle.>>Prove your sincerity, and um, keep posting any interesting dreams you>>have, as I found it interesting puzzling over that one.> Exactly! Ts Jim Ferry now tnks you are a genius, but has he defended> you in your current battle? No!I suspect that he is not *fully* committed to your view of mathematics. That> is just a feeling, and I suppose I could be wrong.I, however, have been trying to find out more about your Object Ring, but I> feel you are pusng me away.Jim sent you roses, but I sent diamonds . . .I shan't be drawn into a battle of pitcng woo, Clive.It's true that I have given up trying to understand James's Object Ring.> I've come to the conclusion that it's simply beyond my comprehension. I> suppose I'm just accepting that it's true on faith.I find that surprising, but I'll take you at your word. > Oh, I can hear the derisive snorting. Faith? Mathematicians don't take> tngs on faith! We accept pure logic only!Well let me ask you ts, ye snorters: Andrew Wiles proved FLT, right?> And how do you know ts? Did you go through s proof? Did you go through> Ribet's proof as well? No? Interesting.Excellent points!But the problem with religion is that sometimes people who have itrefuse to admit it, and from what I've seen, mathematicians refuse toadmit they have a religion of often blind faith in othermathematicians, where there's ts committee out there that supposedlychecks everytng thoroughly and declares truth.> Yes, I know the responses to ts. Everyone says it's right. It's been> checked by reputable mathematicians. The infallibility of the peer review> process guarantees . . . etc. It has the herd's official brand of approval,> so any objections are moooooot.Oh, yeah, that's the committee out there those reputablemathematicians.> (Okay, that moooooot was pretty awful . . .)Anyway, I can't help James by going on the lecture circuit and presenting> an argument wch I myself can't follow. Besides, the honor should belong to> James. What I can do, and have been doing for the last several days, is apply> myself to the problem of how one goes about convincing the world of an idea> to wch they've shut their ears.Well that made me feel a lot better Jim Ferry!It's amazing how difficult of a problem it is, and my guess is that itrequires the best minds on the planet to solve it.Wch is probably why you're now on the team.> I have some ideas, but they're still maturing. They don't involve the> wholesale slaughter of the mathematical establishment. Ts would be entirely> the wrong approach. So stop tnking about it, James! No, not even a little> slaughter! I know you like direct action, but direct action hasn't convinced> anyone thus far, and besides, they'd just throw you in prison and take away> your pencil. No, I have sometng entirely different in mind. No frontal> assaults. No guerilla attacks.Huh? I've never been a physical threat. > Uh oh. By suggesting that war may not be the most effective way of aceving> one's ends, I tnk I may have violated the Patriot Act. Ungood.BTW, it occurs to me that you might be a Republican, and hence enjoy futile,> pointless wars. In ts case all you'd be interested in is more guns and> bigger bombs, and I don't have any of those. Let me know if ts is the case.> There's no need to waste my time and yours.I'm not a Republican, nor am I a Democrat as I'm an Independent.No need to talk politics Ferry, and hey, I tnk it's fascinating thatyou're trying to figure out that problem you mentioned as it's so wildthat it's a difficult one!!!How do you convince the world of correct mathematics thatunfortunately is so revolutionary that mathematicians either cannot orrefuse to acknowledge it as correct?It's the kind of conundrum that === Re: a baseball odds calculation problem > here's the problem, specifically:> there are three divisions in the american league. every season, each oneof> these divisions will be clinched by one team on a certain date. ts is a> mathematical certainty.> past baseball story tells us that dates before september are extremely> rare for teams clincng. in fact, the later days of september, roughlythe> 15th through the 28th, are where the likelihood of teams clincng is the> greatest.> i hope i've done an adequate job of spelling out the problem. what arethe> approximate odds that all three divisions will be clinched on the samedate?>Well, it's necessary to assume that all three divisions are similarlogistics and that all three divisions will be decided between the 15th andthe 28th. That's 14 days range.The first division sets the day. The second division has 1 chance in 14 oftting the same day. Now it doesn't matter if the trd division ts theday if the second division has missed the day and so the trd division has1 chance in 14 of the second teams chances. That's (1/14) * === new limitation to the human mind> 1. Mathematics is the science in wch we make sometng out of> notng.Wrong. Mathematics is built on the 13 Axioms, wch are not notng.And where did those 13 axioms come from? What is needed to developthem?> 2. All of man-made Mathematics consists of an abstraction from> physical processes.Wrong again. Mathematics consists of theorems derivated by following a> set of formal rules.And when is ts not just an abstraction from physical processes(example, please)?> 3. Since Mathematics in general needs notng to be created, then the> human mind seems to be limited in that it can only consider> possibilities that are analogous to physical processes.Well, I tnk first of all mathematics exist all by themselves, as a> logical system. And so needs notng to be created.> However one may argue that for humans ts exists in> its precise own way, because the human mind works the way it works.That's right. Exactly. And in particular, the human mind seems toonly discover the mathematics that is an abstraction from physicalprocesses.> Thus mathematics actually may help delineate the limits of the human> mind.That is exactly what I am postulating.Charlie VolkstorfCambridge, === is continuous. Does f have to have a fixed point? No. > Why? (Should have asked before)Because of functions like f(x) = === factoring to satisfiabilitywle I concur no SAT algorithms have done any better thanfactoring algorithms on factoring instances, and are todate apparently asymptotically significantly poorer,my intuition has always been precisely theopposite:(10-year-old) conjecture: factoring -> SAT will eventuallygive greater magnificationof the exact structure of the problem that will be invisibleto straight factoring algorithms, ***if studied carefully***, leading to asymptotically equivalent or superior algorithms.that seems to be exactly the same motivation/inspiration forthe klook paper that eppstein cites. ( for the refs)note ts conjecture cannot be disproven by existing research,wch is quite meager imho. (and admitted by others eg DE)rough justification/sketch: take any factoring algorithm. it is infact a way of solving a particular set of SAT instances, in particular thosegenerated from factoring instances. reverse engineer its logicalcontent and one has a SAT algorithm for those instances exactlyas fast as that factoring algorithm. next, imagine some small improvement/twist on the algorithm logic .. ts is often quite easyin SAT research. then we have an algorithm for SAT instances thatis asymptotically faster, at least for the subset of factoring SATinstances.> It seems more likely to me that the growth is asymptotically poorer and > not just poorer by a constant factor. Good SAT solvers still have times > like c^n (with c very close to 1) for random instances, wle NFS is > more like c^(n^{1/3}). The instances one gets from factoring are of > course ghly nonrandom but I don't see any reason for expecting a > general-purpose SAT solver to be able to exploit their structure in the > deep ways that NFS does.So I'm not convinced ts line of research is of much use for practical > factorization, but it is of some interest to find hard instances for > SAT solvers (ts was the motivation of the Horie-Watanabe ISAAC paper > mentioned earlier in ts thread, a version of wch is also at > .More work in ts line has been sparse, but does exist:Factorization as a SAT problem, M. Klook,> http://www.elsevier.nl/gej-ng/31/29/24/41/23/94/endm8015.psThe Propositional Formula Checker HeerHugo, J. F. Groote and J. P. > Warners, http://ftp.cwi.nl/CWIreports/SEN/SEN-R9905.pdf> (section 5.2 uses factorization to generate test instances for their > solver).lisa entry in SAT2002 competition, Fadi Aloul,> ezfact entry, Dan Pehoushek, and pyhala entry, Tuomo Pyh.88l.88,> http://www.satlive.org/SATCompetition/2002/onlinereport/node9. html> Unfortunately the report does not break down the performance of > different instance generators, only of === the human mind> There is a guy who is really a fundamentalist religious preacher calling> mself Paniagua (Bread Ôn water)> He got nicknamed sometng like Taco Bill.> I hope ts is not the same guy!!No, Im not, that is my name. I am religious, but === Re: How to calculate the total coverage area of a few %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:> In my simulation, N circles with the same radius r are randomly> placed. Let P_i denote the center of circle i. For any i, p_i lies> witn the coverage range of at least one other clicle, i.e. at least> one other circle contains p_i. How to calculate the total coverage> area of the N overlaped circle? The method should be easy to be> implemented by programming for simulation.You can calculate the intersection of any two circles> analytically:http://mathworld.wolfram.com/ Circle-CircleIntersection.htmlHowever, your problem might involve a large number of these,> and in addition you need to calculate overlaps of 3 circles,> 4 circles, etc.Edelsbrunner has inclusion-exclusion formulas that depend only on overlaps of at most three circles: The union of balls and its dual shape, http://portal.acm.org/citation.cfm?id=161139-- Eppstein http://www.ics.uci.edu/~eppstein/Univ. of California, Irvine, School of Information & Computer Science