mm-1319 === Subject: Re: Modern censorship You are a vile human being! === Subject: Re: Modern censorship === >Subject: Modern censorship >You read in history books about individuals harried by mobs, >continually barraged with various attacks, who face outrageous >behavior at the hands of some group, and then we come to a supposedly >enlightened age with extraordinary tools for sharing information--and >the same damn thing happens again. Oh, you're back again. I thought you said you didn't need us anymore. >Over the years that I've posted on PUBLIC forums the thing that has >stood out is how out there, without shame, and without even hesitating >to hide what they're doing--groups of people have made it their >business to try to censor what I write. Yeah, so what? It's a free country. There's no law against censoring you. >Time after time they'd post to tell me to go away, or to tell others >to ignore me, as these people made it their business to try to >control. Go away. >When harassing me on Usenet wasn't enough, they took to putting up >webpage--and if another one of you tries to claim that I harassed >David Ullrich, a math professor who after harassing me for years >dragged race into the picture talking about racial slurs, because I >complained about him bringing race into the picture to his university, >then you just make my point that much more forcefully. Race is not a >tool to be used by some unethical math professor who thinks it's a >neat way to insult someone on Usenet. David Ullrich was wrong to try >to attack me with race, and I was right to call him on it, and >complain to his school. You were wrong. You were an asshole then and you're still an asshole. >You people cheat. Tough titty for you, isn't it? >to a journal, so some of you emailed that journal to get the paper >censored. So what are you going to do about it? >You are disgusting cretins who follow no rules, no moral obligations, >and you are irrational. >I can argue point for point, point by point, explain over and over >again, as I've done for years, and one of you will just disagree to be >disagreeable! You can argue the points 'til you're blue in the face, you're still wrong. >I offer compromise and get spat upon. P-tui! >History shows that there are always those of the mob, who take it upon >themselves to try and control the few, or especially, the one. >I'm thankful that this plays out over the Internet. We're not. What's the matter, not getting any hits on your blog? >In the past you are the people who would be tying someone to a stake >to burn them and then, blaming the victim, shout your morality to the >heavens, as if God listens to loudness above reason. >You are the reason that we have a society of today where so many >problems will not get addressed because a few people fight for control >they do not have. Most people just blame the Republicans. >You wish to control others, to dominate the conversation, to force >your will upon people who might want to say something you don't want >to hear. So? >So still the webpages are there--some of you illegally using my >copyrighted material to insult me--and you people refuse to give up >your attempts at control despite the years, despite the stupidity of >it all, despite the immorality. Boo-hoo. >But you do not control me. I post as I will despite your webpages, We've already noticed that. >despite emails you might send, despite the dedication with which you >try to push me this way or that, or to push others, though, yes, most >posters bend to your will. >They are cows, and cowards. I watch them come and go over the years >terrified to ever say anything at all objective about my work for fear >that they'll be mobbed, as they will. As they should. Anyone who supports you deserves to have a new asshole reamed. >But I will not be ruled. I will not be controlled by you. I will not >be conquered. And you won't get published, either. Guess what matters. >And I will win. I have a paper at a major math journal. If they try >to slide out of publication like others before them, it will go to >another, and another, and another, as I adjust, shift the wording, >learn the game, play the politics necessary to get published. Sure, if one journal got suckered by your fraud, there's probably another. Not that it matters. It won't survive being made public. >And if it takes a decade I WILL GET PUBLISHED. Care to make a wager on that? >And then you will be part of history, part of the sad story of >mobbings, and angry people willing to do so much wrong for the sake of >their own sense of control. Yet another sad sorry tale among so many >in a world of people who never learn their history. You mean there was another math crank who turned out to be right all along? >In a world where people refuse to learn from the mistakes of the past, >not only to repeat them, but to wallow in the misery they create for >others, to celebrate the destruction they wreak, and to pride >themselves until the day they finally fall. Please, please tell us what happened to The Hammer. My theory is that you accidentally deleted it off your computer. Am I right? >And humanity is so much the worse for all of the stink of it. Gee, we'll just have to muddle through, deprived of your insights. >James Harris -- Mensanator Ace of Clubs === Subject: Re: Modern censorship > You read in history books about individuals harried by mobs, > continually barraged with various attacks, who face outrageous > behavior at the hands of some group, and then we come to a supposedly > enlightened age with extraordinary tools for sharing information--and > the same damn thing happens again. And the one thing they all have in common: They are dead. Perhaps that is your problem, you are still alive. Maybe we will see your brilliance only in your death. - Tim -- Timothy M. Brauch NSF Fellow Department of Mathematics University of Louisville email is: news (dot) post (at) tbrauch (dot) com === Subject: Re: Modern censorship > I can argue point for point, point by point, explain over and over > again, as I've done for years, But you don't. You just start a new thread and repeat yourself. === Subject: Re: Modern censorship > You read in history books about individuals harried by mobs, I think there's a movie called Married to the Mob. Your movie could be Harried by the Mob! === Subject: Re: Modern censorship >Time after time they'd post to tell me to go away, or to tell others >to ignore me, as these people made it their business to try to >control. When people brush away annoying flies, they're not trying to control anything; they just want to be left in peace. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if you're afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Help on a probabilty problem, HELP Consider a particular Ph.D. student; we know that after she begins her Ph.D. program, the number of years it takes her to complete her studies is a random variable with distribution exp(1/4) (independent of when she started). Suppose you know that the student completed her Ph.D. today. We wish to estimate how long ago she started. Assume that the a priori distribution of X is uniform on [3, 6]. a. Let X represent the number of years ago the student started. Let Y represent the observation of when she completes, relative to today (so the given observation is Y = 0). Find fY | X(y | x) for x >= 0. can not understand the meaning very precisely ,Can not understand the === Subject: Re: infinitesimal calculation ? >I am trying to get addtionnal data on infinitesimal numbers dx. A Mathematical perspective or a historical perspective? Historically, you are talking about what Newton called fluxions, and the whole concept is inconsistent. However, there are two modern[1] concepts using similar nomenclature. 1. Differential forms, which are defined in terms of germs of functions, that is, dF_X is the set of all continuous[2] functions that agree with F in a neighborhood of F. 2. Infinitesimals in nonstandard analysis (NSA), and axiom systems that produce the same types of number systems with different machinery. >I think (memory) that the definition may be something like whatever >e>o, 0<|dx|o, 0<|dx|Therefore, I lately (a shame for me) discover that we can extend >consistently (and simply) the axiomatic set theory >(zermelo-fraenkel) with a new property standard and just 3 axioms That isn't strictly necessary; it's just a convenience to allow you to avoid dealing with ultrfailters. If you're willing to do the work, you can define everything that you need within ZFC. [1] But not particularly new. [2] Generally with additional differentiability conditions imposed. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: infinitesimal calculation ? >I am trying to get addtionnal data on infinitesimal numbers dx. > A Mathematical perspective or a historical perspective? mathematics (with possible application to physics). I am almost sure that this topic has been already handled since 1960, but I must admit that I have missed this aspect. >Therefore, I lately (a shame for me) discover that we can extend >consistently (and simply) the axiomatic set theory >(zermelo-fraenkel) with a new property standard and just 3 axioms > That isn't strictly necessary; it's just a convenience to allow you to > avoid dealing with ultrfailters. If you're willing to do the work, you > can define everything that you need within ZFC. This is one of the points I want to quickly know/understand. I do not know the ultra filters, but I think, it may be a way to construct the non standard objects based on the ZFC axioms. I also understand that the 3 added axioms (Idealisation, transfer and standardisation: it is a translation, so I do not know if it is the correct terms in English ;) of the non standard analysis are ~logically independent (they can be added or not to ZFC). Therefore, the introduction of these objects should require the addition of new definitions covering at least a part of these axioms. Does the ultra filters theory add new axioms/definitions equivalent to a part of the non standard analysis axioms? An almost identical point concerns the possible construction of the non standard (and also standard) real numbers through infinite countable real number sequences (ZFC). Now, If I assume that the collection of all infinite countable real sequence is a ZFC set (I haven't tried to demonstrate it, so may be It is a wrong hypothesis ;), I do not see why we need the idealisation axiom to introduce the non standard reals (we just need the ZFC without any new axioms/objects). Can anyone help me? Seratend. === Subject: Re: infinitesimal calculation ? >> Hi everybody, >> I am trying to get addtionnal data on infinitesimal numbers dx. >> I am not sure about the terminology, I have heard it a long ... long >> time ago in a mathematic lecture (my memory may be wrong, may be it >> was during a dream? |-). >> I think (memory) that the definition may be something like whatever >> e>o, 0<|dx|> For example, the infinitesimal number definition may be considered >> as the smallest open set containing x in the usual |R topology?). >> Does anyone can provide some information about this? >> Seratend. >> http://www.math.wisc.edu/~keisler/calc.html >> The trouble with Keisler's book as an intro to calculus using >> infinitesimals is that he spends entirely too much time worrying about >> their existence. Since no one worries about the existence or ordinary >> reals (well hardly anyone; Errett Bishop certainly did and some of his >> followers do too), why worry about whther finitesimals exist. I mean >> no elementary calc book starts by talking about Dedekind cuts, >Maybe that's because, by the time one reads an elementary calculus book, >one is already familiar with real numbers. Also, elementary analysis >books _do_ start with Dedekind cuts. Unfortunately, by the time most students see calculus, they have been imbued with the idea that all of mathematics is calculation. They not only are not familiar with real numbers, but they are not familiar with integers except in knowing how to carry out the arithmetic operations. I would not start with Dedekind cuts, but with nested intervals, as they are easier to understand. Elementary analysis should precede calculus, or the same thing will happen in calculus as in arithmetic; it becomes only memorizing formulas and calculation, with no understanding of what anything means. It is easy to go from concepts and structure to computing, and the computational processes are understood. The other way is difficult, and most never tie calculus to analysis. The nitpicking in Keisler's book is needed to introduce the students to mathematics, and get them away from only thinking about how to compute the answers. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: do I understand co-variant vs. contra-variant? <4190db6a$12$fuzhry+tra$mr2ice@news.patriot.net> >Some people use different notations, I hope >these are clear. Not really, because you seem to be mixing some very different things. > S^u = (&S^u / &X^v) X^v (contravariant) That's fine as long as you're dealing with affine coordinates. It breaks down when you have curvilinear coordinates. > S_u = (&X^v / &S^u) X_v (covariant) You haven't defined S_u. Presumably you're assuming a metrix and setting S_u to g_uv S^v. >Do a bit of algebra on the contravariant, Things get trick once you drag in differential forms or operators. The form dX^v is covariant, but the system of forms {dX^v} is contravariant. > S^u dX^v = (&S^u / &X^v) X^v dX^v > = dS^u X^v >Multiply by g_uv and get > S dX = X dS (contravariant). ITYM S.dX = X.dS (scalar) -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: do I understand co-variant vs. contra-variant? >Some people use different notations, I hope >these are clear. > Not really, because you seem to be mixing some very different things. > S^u = (&S^u / &X^v) X^v (contravariant) > That's fine as long as you're dealing with affine coordinates. It > breaks down when you have curvilinear coordinates. That's why it was presented as basic example. > S_u = (&X^v / &S^u) X_v (covariant) > You haven't defined S_u. Presumably you're assuming a metrix and > setting S_u to g_uv S^v. Yes, if the g_uv cannot be transformed to unity then the covariant-contravariant difference cannot be transformed away. I initially solved S dS = X dX as S^2 = X^2 + k^2 (covariant) and the k being invariant cannot be transformed away. In GR this is sometimes refered to as a Generally Covariant relation because SdS = XdX is obviously invariant. >Do a bit of algebra on the contravariant, > Things get trick once you drag in differential forms or operators. The > form dX^v is covariant, but the system of forms {dX^v} is > contravariant. Well, that's a terminology I won't use. > S^u dX^v = (&S^u / &X^v) X^v dX^v > > = dS^u X^v >Multiply by g_uv and get > S dX = X dS (contravariant). > ITYM S.dX = X.dS (scalar) What happened, I don't understand that??? Ken S. Tucker === Subject: Re: do I understand co-variant vs. contra-variant? > Edward, I think I can speak for the mathematical > community this way, (by definition) > S*dS = X*dX == covariant > S*dX = X*dS == contravariant > Starting with that would any SOB like to argue > mathematics with me? Fellow traveller Ken: It would seem scurilous of me not to reply to your painstakingly contructed follow up posts here, even if I can't understand just what you are talking about. So let me say that I find the above mathematical haiku very beautiful, and I am quite sure it applies to _something_! However, let me take this opportunity to gratuitously synopsize the results of my own heady researches into covariant and contravariant vectors. Consider X',Y', the column vector representations of a vector X and a covector Y in a first coordinate system, X,Y, the representations in a second, and a matrix A such that X = AX'. Further consider possible distinct applications of the matrix operators t and i (transpose and inverse) to A. Then: X = AX' ; Y = A^it Y' ; X' = A^i X ; Y' = A^t Y So (A,A^it) forms a pair giving the forward transformation of vectors and covectors, (A^i,A^t) a second pair giving the reverse transformations. Which matrix we write unadorned is completely arbitrary, as none of the four matrices is more direct inverse or transpose than any of the others; a similar remark applies to vectors vs. covectors. === Subject: Re: do I understand co-variant vs. contra-variant? > Edward, I think I can speak for the mathematical > community this way, (by definition) > > S*dS = X*dX == covariant > > S*dX = X*dS == contravariant > > Starting with that would any SOB like to argue > mathematics with me? > Fellow traveller Ken: > It would seem scurilous of me not to reply to your painstakingly > contructed follow up posts here, even if I can't understand just what > you are talking about. So let me say that I find the above > mathematical haiku very beautiful, and I am quite sure it applies to > _something_! Ha, if math is beautiful, make it look pretty. > However, let me take this opportunity to gratuitously synopsize the > results of my own heady researches into covariant and > contravariant vectors. > Consider X',Y', the column vector representations of a vector X and a > covector Y in a first coordinate system, X,Y, the representations in > a second, and a matrix A such that X = AX'. Further consider > possible distinct applications of the matrix operators t and i > (transpose and inverse) to A. > Then: > X = AX' ; Y = A^it Y' ; X' = A^i X ; Y' = A^t Y > So (A,A^it) forms a pair giving the forward transformation of vectors > and covectors, (A^i,A^t) a second pair giving the reverse > transformations. Which matrix we write unadorned is completely > arbitrary, as none of the four matrices is more direct inverse or > transpose than any of the others; a similar remark applies to > vectors vs. covectors. Looks good. (I usually use component notation as that is more often used in physics but this is cross-posted to sci.math, the bright guys). What you've written is true in algebra, but is it true in geometry. The basis of covariant and contravariant objects is there relation. May I ask you to relate 1 cm to 1 inch? Ken S. Tucker === Subject: Re: do I understand co-variant vs. contra-variant? > Consider X',Y', the column vector representations of a vector X and a > covector Y in a first coordinate system, X,Y, the representations in > a second, and a matrix A such that X = AX'. Further consider > possible distinct applications of the matrix operators t and i > (transpose and inverse) to A. > Then: > X = AX' ; Y = A^it Y' ; X' = A^i X ; Y' = A^t Y > So (A,A^it) forms a pair giving the forward transformation of vectors > and covectors, (A^i,A^t) a second pair giving the reverse > transformations. Which matrix we write unadorned is completely > arbitrary, as none of the four matrices is more direct inverse or > transpose than any of the others; a similar remark applies to > vectors vs. covectors. Sure, as far as that goes. But vectors are fundamentally different from covectors, because a vector points and a covector copoints. That is, vectors are the base objects, and covectors are defined as real functions on vectors. And while there is an isomorphism between them that can be used to interchange vectors with covectors, that is not natural (this is one type of duality transform). A vector can be visualized as a little arrow with an obvious interpretation of points. A covector can be visualized as a set of nested surfaces which copoint in the direction of a vector normal to the surfaces. The covector is a function of a vector, and that can be visualized as a count of the number of surfaces pierced by the arrow. Vectors and covectors behave in a fundamentally different manner when one applies a mapping to the underlying manifold: a vector behaves naturally under the push-forward of the mapping, and a covector behaves naturally under the pull-back, so vectors are naturally covariant and covectors are naturally contravariant. This is intrinsic to their definitions, and the only ambiguity is the historical confusion in the definitions of covariant and contravariant. Beware: don't confuse mappings of the manifold with transforms among coordinate systems applied to the manifold! The latter affect your representations of vectors and covectors, but don't affect the vectors or covectors themselves. Given suitable conditions, the mappings of the manifold don't affect your representations but do affect the vectors and covectors themselves. The existence of a commuting category diagram here permits (sloppy) physicists to ignore the distinctions, and historically most have done so (as have many mathematicians). The search for a theory of Quantum Gravity has required much more care in this area.... Tom Roberts tjroberts@lucent.com === Subject: Re: do I understand co-variant vs. contra-variant? > X = AX' ; Y = A^it Y' ; X' = A^i X ; Y' = A^t Y > > So (A,A^it) forms a pair giving the forward transformation of vectors > and covectors, (A^i,A^t) a second pair giving the reverse > transformations. Which matrix we write unadorned is completely > arbitrary, as none of the four matrices is more direct inverse or > transpose than any of the others; a similar remark applies to > vectors vs. covectors. > Sure, as far as that goes. > But vectors are fundamentally different from covectors, because a vector > points and a covector copoints. That is, vectors are the base objects, > and covectors are defined as real functions on vectors. And while there > is an isomorphism between them that can be used to interchange vectors > with covectors, that is not natural (this is one type of duality > transform). I'm not sure if the last comment refers to raising and lowering of indices, or to the symmetry between vectors and co-vectors each being linear maps on the other. > A vector can be visualized as a little arrow with an obvious > interpretation of points. A covector can be visualized as > a set of nested surfaces which copoint in the direction of > a vector normal to the surfaces. The covector is a function of > a vector, and that can be visualized as a count of the number > of surfaces pierced by the arrow. I've been musing on the possible distinctions between vectors and co-vectors, and (harp arpegia): The covectors can be defined as linear maps on the vectors. However, the vectors form a space of linear maps on the covectors -- or are isomorphic to such a space, for purists. This doesn't mean they are the same class of objects, but at least suggests a kind of mirror or rotational symmetry. In given physical situations there is certainly a distinction between instances of vectors and covectors: the gradient is distinct from infinitesimal displacments, and the reciprocal lattice vectors are different from the lattice vectors. However, given a metric -- and the common vector spaces of physics always come equipped with one via an inner product -- we can express any vector in the form of an associated covector, and any covector in the form of a vector. This challenges the idea that there is anything fundamentally vector or covector like about a given object, and suggests it's more a matter of conventional representation. How about the bongs of a bell image which the authors of that fat book on gravity are so fond of? This is a colorful way of visualizing the inner product : we can understand such a product as taking the projection of Y on the direction of the X, multiplied by a constant to adjust for non-unit norm of X. The surfaces are a way of picturing this projection, their spacing includes the constant. But this understanding is reflexive: if we were to regard X as the starting point rather than Y, we can interpret the product as recording the number of surfaces pierced by X orthogonal to Y -- their spacing now including |Y|. Conclusion? I don't have a clean one. Starting with a given object regarded as a vector field there is at least sometimes a strong distinction between objects expressed as covector fields relative to that initial assignement. But there is also significant symmetry, which blurs the distinction between these instances as examples of fundamentally different types of object. In QM the physical distinction vectors and linear maps on them seems to vanish entirely, and we are left with a formal one: the space of linear maps on the wave functions is occupied solely by ... the wave functions! By limiting ourselves to nice coordinate systems, the operation of finding the associated covector is transparent: we merely write , and throw in a conjugation if performing an explicit calculation. I'm not sure why it is that we only encounter one physical species of object here. > Vectors and covectors behave in a fundamentally different manner when > one applies a mapping to the underlying manifold: a vector behaves > naturally under the push-forward of the mapping, and a covector behaves > naturally under the pull-back, so vectors are naturally covariant and > covectors are naturally contravariant. This is intrinsic to their > definitions, and the only ambiguity is the historical confusion in the > definitions of covariant and contravariant. I like the terminology push-forward vs. pull-back (even if I'm not sure how to apply it) and I claim to understand perfectly well the differing transformation properties. However, one might wonder, given that we can express any covector as a vector, which therefore transforms as a vector! Maybe we can say this is an unnatural representation. It all amounts to the same damn thing, over and over, though: coordinate systems change, some things remain the same, and this requirement gives us our transformation laws. > Beware: don't confuse mappings of the manifold with > transforms among coordinate systems applied to the manifold! > The latter affect your representations of vectors and > covectors, but don't affect the vectors or covectors > themselves. Given suitable conditions, the mappings of > the manifold don't affect your representations but do > affect the vectors and covectors themselves. Under what circumstances would one consider different mappings of the manifold as distinguished from different coordinate systems? Please explicate. > The existence > of a commuting category diagram here permits (sloppy) > physicists to ignore the distinctions, and historically > most have done so (as have many mathematicians). The search > for a theory of Quantum Gravity has required much more care > in this area.... === Subject: Re: do I understand co-variant vs. contra-variant? <4190db6a$12$fuzhry+tra$mr2ice@news.patriot.net> Tensor Analysis by Ed Nelson (Princeton U press) has a nice little, though old book on this. There are so many modern treatments I won't try to pick any out. (I learned from Spivak--I don't even know if his Intro to Diff. Geo. series ever got published in decent form-I mean in decent text and hardback, or how many of the 5 vols got out). Anyway, Nelson defines vector fields as derivations of scalars, and forms as the dual space. Van === Subject: Re: do I understand co-variant vs. contra-variant? Great post Tom, and Edward, I'll keep that as a ref. [snip good stuff] > A vector can be visualized as a little arrow with an obvious > interpretation of points. A covector can be visualized as > a set of nested surfaces which copoint in the direction of > a vector normal to the surfaces. The covector is a function of > a vector, and that can be visualized as a count of the number > of surfaces pierced by the arrow. The word surface, while conventional may exclude covariant and contravariant definitions in 1 dimension. For example, in GR we are entitled to analyse unit lengths in the direction of radius by employing only ds^2 = g_11 dx^1 dx^1 = g^11 dx_1 dx_1 , (dt=0). and we know g_11 ~ 1/g^11, hence g_11 =/= g^11, so the covariant and contravariant measurements differ in 1D. Along that 1D line, a small unit length x is transformed two ways, x' = (&x'/&x)*x == contravariant x' = (&x/&x')*x == covariant and provides, x' dx = x dx' == contravariant x' dx' = x dx == covariant (I previously expressed that using S and dS for x' and dx', sorry for the confusion). [...] > The search > for a theory of Quantum Gravity has required much more care > in this area.... You've mentioned that before, may we ask why you think that. Ken S. Tucker > Tom Roberts tjroberts@lucent.com === Subject: Re: do I understand co-variant vs. contra-variant? > But vectors and tensors are the natural language of physics. One key > observation is that physical phenomena are utterly independent of > humans, or of their concepts; so physical phenomena must be independent > of coordinate system, and tensors are the natural mathematical objects > to model this. Yahbut. Tensors are multi-linear mappings from a cartesian product of n copies of a vector space and m copies of the dual vector space into the real numbers. All of which are abstractions cooked up by humans to -model- some aspects of nature. Tensors and vectors live up in our heads, not Out There. The co-ordinate free formulation of tensors is a matter of simplifying and cleaning up the math. Bob Kolker === Subject: Re: do I understand co-variant vs. contra-variant? >> But vectors and tensors are the natural language of physics. One key >> observation is that physical phenomena are utterly independent of >> humans, or of their concepts; so physical phenomena must be >> independent of coordinate system, and tensors are the natural >> mathematical objects to model this. > Yahbut. Tensors are multi-linear mappings from a cartesian product of n > copies of a vector space and m copies of the dual vector space into the > real numbers. All of which are abstractions cooked up by humans to > -model- some aspects of nature. Tensors and vectors live up in our > heads, not Out There. Yes, of course. That's what I said (...to model this). That's what physics IS. > The co-ordinate free formulation of tensors is a matter of simplifying > and cleaning up the math. Based on observations of myself over several decades, and dozens of people in this newsgroup, the simplifying involved gives a qualitatively different feel for and understanding of the mathematical and physical concepts involved. I simply did not understand GR from component-based books like Weinberg; but when I started studying MTW a lot of things fell into place quite quickly. And most of the people around here who are wedded to component notation simply don't have a clue.... Tom Roberts tjroberts@lucent.com === Subject: Re: Sum of an infinite series .... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAI29pE24373; >Could someone let me know how to calculate the sum of the following series: >1 + 4/7 + 9/49 + 16/343 + 25/2401 + . >> I assume you mean the infinite series >> 1+ (2^2)/7 + (3^2)/7^2 + (4^2)/7^3 +..., >> in other words, >> sum_1^infty (n^2)/7^(n-1). >> You could start with >> 1/(1-x)=sum_0^infty x^n >> Differentiate with respect to x, multiply by x, then differentiate >> again. Finally, put x=1/7. >> --Dan Grubb >Wow. That's amazing! That is sooo indirect but elegant. How did you >ever thought of that approach? >Kira It's a pretty well-known technique. Get hold of Concrete Mathematics by Graham, Knuth, & Patashnik and read about generating functions, and have fun. Todd Trimble === Subject: Conversion of (x,z) 2D-space coordinates to (u,c) 2D-space coordinates by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAI29ox24319; Hello there, I'm a Computing Science student from the Eindhoven University of Technology in the Netherlands and I'm currently writing my master's thesis on texture synthesis for data visualization. I've created a texture synthesis model that uses u, v and c (spatial frequency, regularity and physical contrast) as its input parameters. The output is the texture itself and the (x,y,z)-coordinates of the texture in a perceptually uniform texture space. X, y, and z are functions of u, v, and c and my UVC-To-XYZ routines are working perfectly. I am now trying to solve my XYZ-To-UVC routines based on the functions that I use to convert from u, v, and c to x, y, and z. If v and y are left out and I try to solve for u and c, the problem boils down to the following equations: x = 1.11 + (-1.11 + 2.22 * u) * c^(0.5) z = 2.13 * c^(0.87) - 1.14 * c * u^(2.46) I simply want to solve for u and c (i.e. u = ... and c = ... in terms of only x and y), so I tried to use Mathematica (5.0.1.0) on this: NSolve[ { x == 1.11 + (-1.11 + 2.22 * u) * c^(0.5), z == 2.13 * c^(0.87) - 1.14 * c * u^(2.46) }, {u,c}] Strangely enough, Mathematica simply goes on and on (even on powerful PC's) but doesn't come up with a solution. Danny Holten. === Subject: Simply connected, analytic The following question occurred to me : Is it possible to map a non-simply connected domain in C to a simply connected domain in C via an analytic function? I cant think of any examples. Isaac === Subject: Re: Simply connected, analytic >The following question occurred to me : Is it possible to map a non-simply >connected domain in C to a simply connected domain in C via an analytic >function? I cant think of any examples. Simnple examples have been _given_ in posts right here, in replies to _your_ posts. C is simply connected. What is exp(C)? >Isaac ************************ David C. Ullrich === Subject: Re: Simply connected, analytic > The following question occurred to me : Is it possible to map a non-simply > connected domain in C to a simply connected domain in C via an analytic > function? I cant think of any examples. > Isaac What do you mean by to a simply connected domain? One can simply include a non-simply connected domain to a simply connected one, so I presume you at least mean surjection, but you should be more precise what you mean. === Subject: Re: Simply connected, analytic >> The following question occurred to me : Is it possible to map a >> non-simply connected domain in C to a simply connected domain in C via an >> analytic function? I cant think of any examples. >> Isaac > What do you mean by to a simply connected domain? One can simply include > a non-simply connected domain to a simply connected one, so I presume you > at least mean surjection, but you should be more precise what you mean. Yes I'm sorry, I mean surjection. Is this possible? === Subject: Re: Simply connected, analytic >> The following question occurred to me : Is it possible to map a >> non-simply connected domain in C to a simply connected domain in C via an >> analytic function? I cant think of any examples. >> Let's see... C - {1}, the plane minus a point, is not simply connected. The image under f(z) = z^2 is C, which is simply connected. Is that what you wanted? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Simply connected, analytic >> >>> The following question occurred to me : Is it possible to map a >>> non-simply connected domain in C to a simply connected domain in C via >>> an >>> analytic function? I cant think of any examples. >>> > Let's see... C - {1}, the plane minus a point, is not simply connected. > The image under f(z) = z^2 is C, which is simply connected. > Is that what you wanted? > -- > G. A. Edgar > http://www.math.ohio-state.edu/~edgar/ Yes thank you. === Subject: Number theory reference In some number theory book the following assertion is proved. Consider eqn.(1) under the given conditions. Aa^(1/2) + Bb^(1/2) = Cc^(1/2) + Dd^(1/2) (1) Condition: A, B, C, D are integers each > 1 and a, b, c, d are square free positive integers. Assertion: If (1) is satisfied then either a = c and b = d or a = d and b = c. I would greatly appreciate if someone can kindly provide me with a reference. === Subject: Re: Number theory reference > In some number theory book the following assertion is proved. > Consider eqn.(1) under the given conditions. > Aa^(1/2) + Bb^(1/2) = Cc^(1/2) + Dd^(1/2) (1) > Condition: A, B, C, D are integers each > 1 > and a, b, c, d are square free positive integers. > Assertion: > If (1) is satisfied then either a = c and b = d or a = d and b = c. > I would greatly appreciate if someone can kindly provide me with a reference. This is a *special case* of the general fact that the set off all square-free positive integers is linearly independent over the rational numbers. Proof by induction over the number of distinct prime factors of the square-free numbers. The proof is similar to, and only slightly more involved than the proof that sqrt(2) is irrational. Thomas === Subject: Question on e^(1/z)... e^(1/z) is analytic on C {0}. However, on any circle around 0 of arbitrary radius, the integral over that circle of e^(1/z) is 2*pi*i since the residue is 1. Does this mean that e^(1/z) does not have a primitive on C {0} ? For if it did, then the integral around the closed curve would be zero. Is my thinking logically correct? Is this a good way to check that a general complex valued function does not have a primitive somewhere? Isaac === Subject: Re: Question on e^(1/z)... >e^(1/z) is analytic on C {0}. However, on any circle around 0 of >arbitrary radius, the integral over that circle of e^(1/z) is 2*pi*i since >the residue is 1. Does this mean that e^(1/z) does not have a primitive on >C {0} ? For if it did, then the integral around the closed curve would be >zero. Is my thinking logically correct? >Is this a good way to check that a general complex valued function does >not have a primitive somewhere? Yes, this is the best way to give a clear and concise proof of things that would otherwise be proved by a somewhat roundabout method. >Isaac ************************ David C. Ullrich === Subject: Re: Question on e^(1/z)... > e^(1/z) is analytic on C {0}. However, on any circle around 0 of > arbitrary radius, the integral over that circle of e^(1/z) is 2*pi*i since > the residue is 1. Does this mean that e^(1/z) does not have a primitive on > C {0} ? For if it did, then the integral around the closed curve would be > zero. Is my thinking logically correct? Yes, an analytic function f on an open set V has a primitive in V iff int_gamma f(z) dz = 0 for every closed contour gamma in V. === Subject: Re: Question on e^(1/z)... >e^(1/z) is analytic on C {0}. However, on any circle around 0 of >arbitrary radius, the integral over that circle of e^(1/z) is 2*pi*i since >the residue is 1. Does this mean that e^(1/z) does not have a primitive on >C {0} ? For if it did, then the integral around the closed curve would be >zero. Is my thinking logically correct? Yes, of course. >Is this a good way to check that a general complex valued function does >not have a primitive somewhere? Yes. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Question on e^(1/z)... the path integral of an analytic function defined on a set D over a closed curve is zero only if the WHOLE interior of the curve lies in D. This is not the case here. Karl > e^(1/z) is analytic on C {0}. However, on any circle around 0 of > arbitrary radius, the integral over that circle of e^(1/z) is 2*pi*i since > the residue is 1. Does this mean that e^(1/z) does not have a primitive on > C {0} ? For if it did, then the integral around the closed curve would be > zero. Is my thinking logically correct? > Is this a good way to check that a general complex valued function does > not have a primitive somewhere? > Isaac === Subject: Re: Question on e^(1/z)... >the path integral of an analytic function defined on a set D over a >closed curve is zero only if the WHOLE interior of the curve lies in D. >This is not the case here. >Karl This is not true; it is true with if, rather than only if. The function e^(1/x^2)) is analytic on C {0}, and has residue 0 at 0. So its integral over a closed curve in the domain is 0; however, if we multiply it by x, this is no longer the case. >> e^(1/z) is analytic on C {0}. However, on any circle around 0 of >> arbitrary radius, the integral over that circle of e^(1/z) is 2*pi*i since >> the residue is 1. Does this mean that e^(1/z) does not have a primitive on >> C {0} ? For if it did, then the integral around the closed curve would be >> zero. Is my thinking logically correct? >> Is this a good way to check that a general complex valued function does >> not have a primitive somewhere? >> Isaac -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Question on e^(1/z)... > the path integral of an analytic function defined on a set D over a > closed curve is zero only if the WHOLE interior of the curve lies in D. False: The path integral of 1/z^2 over the unit circle is 0. === Subject: Re: Question on e^(1/z)... > the path integral of an analytic function defined on a set D over a closed > curve is zero only if the WHOLE interior of the curve lies in D. > This is not the case here. > Karl I understand that. What I was saying, though is that it is zero as well IF it has a primitive defined and analytic on a domain containing the curve. That is why I was trying to conclude that e^(1/z) doesn't have a derivative. Am I right? >> e^(1/z) is analytic on C {0}. However, on any circle around 0 of >> arbitrary radius, the integral over that circle of e^(1/z) is 2*pi*i >> since the residue is 1. Does this mean that e^(1/z) does not have a >> primitive on C {0} ? For if it did, then the integral around the >> closed curve would be zero. Is my thinking logically correct? >> Is this a good way to check that a general complex valued function does >> not have a primitive somewhere? >> Isaac === Subject: Set theory I have been asked to solve the following problem. Suppose X is an infinite set. Show that the cardinal number of X is less than the cardinal number of its power set. Now I have a problem with this. Suppose the cardinality of X is c. Now I thought that the only infinite cardinalities are c and aleph null which would mean that the cardinality of X is not less than the power set of X. What is going on here????? === Subject: Re: Set theory > I have been asked to solve the following problem. Suppose X is an infinite > set. Show that the cardinal number of X is less than the cardinal number of > its power set. Now I have a problem with this. Suppose the cardinality of X > is c. Now I thought that the only infinite cardinalities are c and aleph > null which would mean that the cardinality of X is not less than the power > set of X. What is going on here????? There are infinitely many infinite cardinals. If PX is the power set of X and #X is the cardinality of X then #X < #PX = 2^{#X}. === Subject: Re: Set theory > I have been asked to solve the following problem. Suppose X is an infinite > set. Show that the cardinal number of X is less than the cardinal number of > its power set. You don't need the hypothesis that X is an infinite set. The conclusion holds for every X, whether finite or infinite. > Now I have a problem with this. Suppose the cardinality of X > is c. Now I thought that the only infinite cardinalities are c and aleph > null which would mean that the cardinality of X is not less than the power > set of X. What is going on here????? You were misinformed. There are infinitely many different infinite cardinalities. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Set theory > I have been asked to solve the following problem. Suppose X is an infinite > set. Show that the cardinal number of X is less than the cardinal number of > its power set. > You don't need the hypothesis that X is an infinite set. The conclusion > holds for every X, whether finite or infinite. > Now I have a problem with this. Suppose the cardinality of X > is c. Now I thought that the only infinite cardinalities are c and aleph > null which would mean that the cardinality of X is not less than the power > set of X. What is going on here????? > You were misinformed. There are infinitely many different infinite > cardinalities. > -- > Dave Seaman > Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. > Dr. Seaman, Goodnight, Steven === Subject: Re: Set theory >I have been asked to solve the following problem. Suppose X is an infinite >set. Show that the cardinal number of X is less than the cardinal number of >its power set. Classic diagonalization. The subsets of X correspond to functions from X to {0,1}, depending on whether or not a given member is included or excluded in the subset. Assume that there is a surjection f:X->P(X), and construct a subset of X that is not the image of any member of X; this is done is analogous manner as the diagonalization proof of the uncountability of reals, except that instead of a sequence of 'digits', one has the 'digits' (0 and 1, or exclusion/inclusion) indexed by the set X. Requiring X to be infinite is actually irrelevant. >Now I have a problem with this. Suppose the cardinality of X >is c. Now I thought that the only infinite cardinalities are c and aleph >null which would mean that the cardinality of X is not less than the power >set of X. What is going on here????? No, the power set operation always builds a larger cardinality. The cardinality of the reals is c (by definition), and the cardinality of the power set of the realsis 2^c > c. --- Stan Liou === Subject: Re: Set theory > I have been asked to solve the following problem. Suppose X is an infinite > set. Show that the cardinal number of X is less than the cardinal number of > its power set. Now I have a problem with this. Suppose the cardinality of X > is c. Now I thought that the only infinite cardinalities are c and aleph > null which would mean that the cardinality of X is not less than the power > set of X. What is going on here????? Not so. You can get larger and larger cardinalities by using the power set operator. Let P{A) be the power set of A. The |A| < |P(A)| < |P(P(A))| ... etc Bob Kolker === Subject: Re: Set theory > I have been asked to solve the following problem. Suppose X is an infinite > set. Show that the cardinal number of X is less than the cardinal number of > its power set. Now I have a problem with this. Suppose the cardinality of X > is c. Now I thought that the only infinite cardinalities are c and aleph > null which would mean that the cardinality of X is not less than the power > set of X. What is going on here????? > > Not so. You can get larger and larger cardinalities by using the power > set operator. Let P{A) be the power set of A. The |A| < |P(A)| < > |P(P(A))| ... etc True, but that is the thing to be proved! I wonder who asked a student to come up with this proof on their own. It's not the kind of thing one would think of ... in fact most people need to go through it many times before they really start to understand and believe it. === Subject: Re: Set theory >> I have been asked to solve the following problem. Suppose X is an infinite >> set. Show that the cardinal number of X is less than the cardinal number of >> its power set. Now I have a problem with this. Suppose the cardinality of X >> is c. Now I thought that the only infinite cardinalities are c and aleph >> null which would mean that the cardinality of X is not less than the power >> set of X. What is going on here????? >> >> Not so. You can get larger and larger cardinalities by using the power >> set operator. Let P{A) be the power set of A. The |A| < |P(A)| < >> |P(P(A))| ... etc >True, but that is the thing to be proved! >I wonder who asked a student to come up with this proof on their own. >It's not the kind of thing one would think of ... Depends on what was covered recently. >in fact most people >need to go through it many times before they really start to understand >and believe it. Really? I don't see why that would be. ************************ David C. Ullrich === Subject: Re: Set theory > I have been asked to solve the following problem. Suppose X is an infinite > set. Show that the cardinal number of X is less than the cardinal number of > its power set. Now I have a problem with this. Suppose the cardinality of X > is c. Now I thought that the only infinite cardinalities are c and aleph > null which would mean that the cardinality of X is not less than the power > set of X. What is going on here????? > Not so. You can get larger and larger cardinalities by using the power > set operator. Let P{A) be the power set of A. The |A| < |P(A)| < > |P(P(A))| ... etc > Bob Kolker Bob, So are you saying that there are more than 2 infinite cardinalities? If so (and it seems that you are saying just that) are there a countable or uncountable number of infinite cardinalities? === Subject: Re: Set theory >> Not so. You can get larger and larger cardinalities by using the power >> set operator. Let P{A) be the power set of A. The |A| < |P(A)| < >> |P(P(A))| ... etc >> Bob Kolker >Bob, >So are you saying that there are more than 2 infinite cardinalities? If so >(and it seems that you are saying just that) are there a countable or >uncountable number of infinite cardinalities? Mr. Kolker is correct. There are definitely more than countably many infinite cardinalities, but as for the 'number' of infinite cardinalities, there is no such thing, in the sense that the set of all infinite cardinal numbers does not exist, so we cannot take its cardinality. In some sense, there are too many cardinals to form a set (analogous to the situations like set of all sets, etc.) --- Stan Liou === Subject: Re: irreducible components > f(x,y,z) is defined by y^2=xz & z^2=y^3 > does this give me only these two irreducible components? > 1) z=0, y=0 & x=x > 2) z=1, y^2=x, 1=xy > I think I goofed bigtime y=0 <=> z=0 so your first component is correct. Otherwise y and z are both nonzero hence yy = xz, zz = yyy <=> yy = xz, zz = yxz <=> yy = xxy, z = xy <=> y = xx, z = xxx Therefore (x, xx, xxx) is the second component. The above deduction holds in any integral domain, --Bill Dubuque === Subject: Re: irreducible components >> f(x,y,z) is defined by y^2=xz & z^2=y^3 >> does this give me only these two irreducible components? >> 1) z=0, y=0 & x=x >> 2) z=1, y^2=x, 1=xy >> I think I goofed bigtime > y=0 <=> z=0 so your first component is correct. > Otherwise y and z are both nonzero hence > yy = xz, zz = yyy > <=> yy = xz, zz = yxz > <=> yy = xxy, z = xy > <=> y = xx, z = xxx > Therefore (x, xx, xxx) is the second component. > The above deduction holds in any integral domain, Above should be commutative semigroup. The curve (t,t^2,t^3) is called the TWISTED CUBIC. A web search will turn up much more about it, e.g. here is a 3-d graph of the twisted cubic which you can interactively manipulate http://www.math.rutgers.edu/courses/535/535-f02/pictures/twistedcubic.html --Bill Dubuque === Subject: Post-sci-math Popular software at low low prices boundary=--1951814562350648 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with SMTP id iAI3MOF30190; --------------------------------------------------------------------- TOP quality software:

Special Offer #1:
Windows XP Professional+Microsoft Office XP Professional = only $80
Special Offer #2:
Adobe - Photoshop 7, Premiere 7, Illustrator 10 = only $120
Special Offer #3:
Also:
Windows 2000 Workstation
Windows 2000 Server
Windows 2000 Advanced Server
Windows 2000 Datacenter
Windows NT 4.0
Windows Millenium
Windows 98 Second Edition
Windows 95
Office XP Professional
Office 2000
Office 97
MS Plus
MS SQL Server 2000 Enterprise Edition
MS Visual Studio .NET Architect Edition
MS Works 7
MS Picture It Premium 9
Adobe Photoshop
Adobe PageMaker
Adobe Illustrator
Adobe Acrobat 6 Professional
Adobe Premiere
Macromedia Freehand MX 11
Corel Draw Graphics Suite 12
Corel Draw Graphics Suite 11
Corel Photo Painter 8
Corel Word Perfect Office 2002
Borland Delphi 7 Enterprise Edition
Quark Xpress 6 Passport Multilanguage

Enter Here











?2N47A3zVjC9Xk2yaba|post-sci-m ath@mathforum.org>or un*su*bs*cr*ibe
ambulant advice testicle virginian illustrate lob melinda bohemia corkscrew idiot lumbar opinionate beside bertram demography gunnery risen binuclear commandant elide furious gloucester beast septa fraught laundry dirge guardhouse lanky confucius christlike kyoto circumsphere chassis exhibition pedantic inconspicuous americana compress dowager polygon gauleiter hippocrates floor darius atalanta influx brewster astm saud berglund peek usgs tony dragonfly shoestring === Subject: ANNO: new yahoo forum for curves and surfaces I've created a yahoo mailing list for discussing curves and surfaces. Please see: http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html http://xahlee.org/surface/gallery.html http://groups.yahoo.com/group/curves_surfaces/ Xah xah@xahlee.org http://xahlee.org/PageTwo_dir/more.html === Subject: suggest an easier text ?? The following book seems useful for what I'm doing currently. Author Freidlin, M. I. (Mark Iosifovich) Title Random perturbations of dynamical systems / M.I. Freidlin, A.D. Wentzell ; translated by Joseph Sz.9fcs Published New York : Springer-Verlag, c1984 However the level seems to be over my head. can someone suggest a text that deals with similar topics that is geared towards undergraduate/ lower graduate levels? === Subject: Re: suggest an easier text ?? > The following book seems useful for what I'm doing currently. > Author Freidlin, M. I. (Mark Iosifovich) > Title Random perturbations of dynamical systems / M.I. Freidlin, A.D. > Wentzell ; translated by Joseph Sz.9fcs > Published New York : Springer-Verlag, c1984 > However the level seems to be over my head. > can someone suggest a text that deals with similar topics that is > geared towards undergraduate/ lower graduate levels? I am not familiar with this text. Can you elaborate a little bit on what you are trying to do/learn? Are you trying to get over a specific hump, or are you trying to learn a broad section of stochastic processes? -- === Subject: Re: The construction of a tree saver from rabbits relating to tokamaks (most snipped) > So what I do is get 6 diameter black plastic drain piping and use a > very sharp high quality knife to cut it. I use the same piping year > after year. But when I first began to use it I cut a line down the I am not very good at estimating a distance length without measuring it with a meter stick. There are some that can look at a piece of board or pipe and tell instantly whether it is 4 or 6. I am not. I made a mistake in that the plastic pipe I am using is 4 diameter. When I make a mistake like this I go back and change it in my other posts with the symbol of (sic) after making the change. So it would look like this in my old post-- 4 (sic). I used to use brackets instead of paranthesis but found out that some computer protocol looks at brackets as a command rather than plain text and so have gone from brackets to paranthesis. I used to have trouble with the reverse symbol of > in that the protocol on websites does not treat that symbol as plain text. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: The decimal system is outlawed now ! ! ! We have to use the hexadecimal system now or the math-cops will arrest us! So start using it now , because in $07D5 , all offenders still using the decimal system will be executed ! === Subject: Re: The decimal system is outlawed now ! ! ! Hans-Marc Olsen scribbled the following: > We have to use the hexadecimal system now or the math-cops will arrest > us! > So start using it now , because in $07D5 , all offenders still using > the decimal system will be executed ! If the decimal system is so evil, why are you writing the year as $07D5? That shows an underlying decimal bias. Why not simply write it as 7D5? Actually even the name hexadecimal itself has a decimal bias. Hexa is Greek for six and deca is Latin for ten, yet when you add those two numbers together, the hexadecimal number you get is 10, not 16. -- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland -------- -------------------------------------------------------- rules! --------/ Holy Banana of this, Sacred Coconut of that, Magic Axolotl of the other. - Guardian in Jinxter === Subject: Re: The decimal system is outlawed now ! ! ! > Hans-Marc Olsen scribbled the following: >>We have to use the hexadecimal system now or the math-cops will arrest >>us! >>So start using it now , because in $07D5 , all offenders still using >>the decimal system will be executed ! > If the decimal system is so evil, why are you writing the year as $07D5? > That shows an underlying decimal bias. Why not simply write it as 7D5? > Actually even the name hexadecimal itself has a decimal bias. Hexa > is Greek for six and deca is Latin for ten, yet when you add those two > numbers together, the hexadecimal number you get is 10, not 16. You are understanding! A+6=10. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: The decimal system is outlawed now ! ! ! - Etymology The Germans have the correct word: hexadekadisches System. BTW, the Latin word is decem = ten; the Greek is .83å.83Ì.83é.83À = deka, in English transliterated as deca. In Cymru: ten = deg, in French: dix; a heavily worn-out form of decem; etc. etc. It is all Indo-European. Johan E. Mebius >> Hans-Marc Olsen scribbled the following: > We have to use the hexadecimal system now or the math-cops will arrest > us! > So start using it now , because in $07D5 , all offenders still using > the decimal system will be executed ! >> If the decimal system is so evil, why are you writing the year as $07D5? >> That shows an underlying decimal bias. Why not simply write it as 7D5? >> Actually even the name hexadecimal itself has a decimal bias. Hexa >> is Greek for six and deca is Latin for ten, yet when you add those two >> numbers together, the hexadecimal number you get is 10, not 16. > You are understanding! A+6=10. === Subject: Re: The decimal system is outlawed now ! ! ! - Etymology > The Germans have the correct word: hexadekadisches System. > BTW, the Latin word is decem = ten; the Greek is í¡.8còí[Hyphen][A Ring]Çí¡.8cò[Capit alIGrave][Hyphen].8c[Micro]í¡.8c[CapitalU Acute]í[Hyphen].8c.bcí¡.8c òí[Hyphen].8c± = deka, in English transliterated as deca. > In Cymru: ten = deg, in French: dix; a heavily worn-out form of decem; > etc. etc. It is all Indo-European. or dhasa=10 in Sanskrit(vowel a short and long). But base 10 may remain even if or after 10 fingers of the hands are genetically modified. BTW,it is a surprise(to me) Greek letters could show correctly in Compose or Preview modes here. === Subject: Re: The decimal system(...) - Etymology - Greek alphabet About Greek and other non-Latin alphabets in webpages and in Email: (1) under MS Windows you can select several different input locales via >>Start >Settings >Control Panel >Regional Options >Tab Input Locales. Once you are back in text typing mode you obtain the desired keyboard setting among the input locales you selected previously by keying LeftAlt-LeftShift as often as needed. Watch the blue square in the system tray. The Netscape Email and HTML composer programs generate the official W3C tokens, for instance α for lowercase letter alpha and ω for lowercase letter omega. If in Email both the sending and the receiving parties have the correct character encoding settings, then EMail letters will be shown as intended by the sender. With HTML the browser will do its work. No special settings needed there. BTW, when viewing the raw EMail text in hexadecimal you will find the W3C tokens transformed into Unicode tokens. For instance, lowercase alpha is CEB1h. You really need a basic viewer program that does not perform any interpretation; just shows the characters in (extended) ASCII and in hexadecimal. BTW, about modification, genetic or otherwise: you may know that the Simpson family uses the octal number system. >>The Germans have the correct word: hexadekadisches System. >>BTW, the Latin word is decem = ten; the Greek is í.8e.8cÇí.8e.8c[Mic ro]í.8e.8c.bcí.8e.8c± = deka, in English transliterated as deca. >>In Cymru: ten = deg, in French: dix; a heavily worn-out form of decem; >>etc. etc. It is all Indo-European. >> >or dhasa=10 in Sanskrit(vowel a short and long). But base 10 may >remain even if or after 10 fingers of the hands are genetically >modified. >BTW,it is a surprise(to me) Greek letters could show correctly in >Compose or Preview modes here. === Subject: Re: The decimal system is outlawed now ! ! ! - Etymology JEMebius scribbled the following: > The Germans have the correct word: hexadekadisches System. > BTW, the Latin word is decem = ten; the Greek is .91Ç.91[Micro].91.bc[EDoubleD ot]± = deka, in English > transliterated as deca. > In Cymru: ten = deg, in French: dix; a heavily worn-out form of decem; > etc. etc. It is all Indo-European. I don't think you understand. If the hexadecimal system were free of decimal bias, it wouldn't have a two-part name, one part meaning six, the other ten, at all, no matter what language those words are in. In contrast, its name would simply be based on a word meaning ten or something. Not of course ten as we know it - the number of fingers on our hands - but a similar name. -- /-- Joona Palaste (palaste@cc.helsinki.fi) ------------- Finland -------- -------------------------------------------------------- rules! --------/ To err is human. To really louse things up takes a computer. - Anon === Subject: Re: The decimal system is outlawed now ! ! ! - Etymology the chicken-egg format can be utilized: base of four to the second-power? > I don't think you understand. If the hexadecimal system were free of > decimal bias, it wouldn't have a two-part name, one part meaning six, > the other ten, at all, no matter what language those words are in. --Give Earth a Trickier Dick Cheeny -- out of office, after gigayears! http://tarpley.net/bush12.htm http://www.benfranklinbooks.com/ http://members.tripod.com/~american_almanac http://www.wlym.com/pdf/iclc/howthenation.pdf http://www.rand.org/publications/randreview/issues/rr.12.00/ http://www.rwgrayprojects.com/synergetics/plates/figs/plate02.html === Subject: Re: The decimal system is outlawed now ! ! ! >We have to use the hexadecimal system now or the math-cops will arrest >us! >So start using it now , because in $07D5 , all offenders still using >the decimal system will be executed ! Please let this threat become a thread in a different newsgroup. This is against the current trend of using more and more paper year by year! Paper manufacturers have been advocating the binary system since times immemorable. To put an end to joking for now: In the 1950s chess champion Max Euwe made an issue of teaching binary arithmetic in primary schools. Johan E. Mebius === Subject: Re: The decimal system is outlawed now ! ! ! In sci.math, JEMebius <419C955C.2080004@xs4all.nl>: >>We have to use the hexadecimal system now or the math-cops will arrest >>us! >>So start using it now , because in $07D5 , all offenders still using >>the decimal system will be executed ! Note: $07D5 = next year. >> > Please let this threat become a thread in a different newsgroup. Followups redirected to alt.politics.usa.misc. (It's about the only country that might seriously give any thought at all to this proposal outside of engineering circles... :-) ) > This is against the current trend of using more and more paper year by year! > Paper manufacturers have been advocating the binary system since times > immemorable. > To put an end to joking for now: > In the 1950s chess champion Max Euwe made an issue of teaching binary > arithmetic in primary schools. Well, one has to admire its simplicity. :-) + 0 1 x 0 1 0 0 1 0 0 0 1 1 0 1 0 1 Makes memorization of the multiplication tables a *lot* easier. Of course one drawback is that the national debt becomes harder US national debt is estimated to be $1101100011010111000101010000000101110011100.1010000 base 2. (from http://www.brillig.com/debt_clock/) > Johan E. Mebius -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: what is the method for disproving limit in analysis? Hi all, I am wondering what is the method for disproving limits in analysis. I mean: for proving limits, I just need to state the following: Given any epsilon > 0, I can find a N0(epsilon), such that for all n>N0, |X_n - X_limit| 0, I can not find a N0(epsilon), such that for all n>N0, |X_n - X_limit| epsilon? === Subject: Re: what is the method for disproving limit in analysis? >Hi all, >I am wondering what is the method for disproving limits in analysis. Which method is easiest (or even applicable) depends on the sequence. >I mean: for proving limits, I just need to state the following: >Given any epsilon > 0, I can find a N0(epsilon), such that for all n>N0, >|X_n - X_limit| 0...]. If I use L for X_limit, A and E for universal and existential quantifiers, then this is [EL][Ae>0][EN][An>N][|X_n - L|But now I want to disprove: that's to say, prove that the series does not >converge to that limit. >Do I say: >Given any epsilon > 0, I can not find a N0(epsilon), such that for all n>N0, >|X_n - X_limit|0 ... . >Sometimes the above is difficult to formulate, can I say the following >instead? >There exists an epsilon, such that |X_n - X_limit| > epsilon? Well, as long as you intend to mean [AL][Ee>0][AN][En>N][ |X_n-L| >= e ], then yes. Another way of proving a sequence does (or doesn't) have a limit is to prove that it is (or isn't) Cachy: (Def.) A sequence X_n is Cauchy iff for all eps>0, there exists N such that for all n,m>N, |x_n-x_m|>Hi all, >>I am wondering what is the method for disproving limits in analysis. > Which method is easiest (or even applicable) depends on the > sequence. >>I mean: for proving limits, I just need to state the following: >>Given any epsilon > 0, I can find a N0(epsilon), such that for all n>N0, >>|X_n - X_limit| Alright, as long as you prepend that with there exists an > X_limit such that [given any epsilon > 0...]. > If I use L for X_limit, A and E for universal and existential > quantifiers, then this is [EL][Ae>0][EN][An>N][|X_n - L|>But now I want to disprove: that's to say, prove that the series does not >>converge to that limit. >>Do I say: >>Given any epsilon > 0, I can not find a N0(epsilon), such that for all >>n>N0, >>|X_n - X_limit| There does not exist an X_limit such that for all eps>0 ... . >>Sometimes the above is difficult to formulate, can I say the following >>instead? >>There exists an epsilon, such that |X_n - X_limit| > epsilon? > Well, as long as you intend to mean > [AL][Ee>0][AN][En>N][ |X_n-L| >= e ], > then yes. > Another way of proving a sequence does (or doesn't) have a limit > is to prove that it is (or isn't) Cachy: > (Def.) A sequence X_n is Cauchy iff for all eps>0, there exists > N such that for all n,m>N, |x_n-x_m| Intuitively, the difference between a term and subsequent terms > becomes arbitrarily small. The definitions of `converging > sequence' and `Cauchy sequence' are equivalent in every complete > metric space (e.g., real numbers), and the Cauchy criterion is > easier to work with in many situations. > Sequences with specific properties are easier to deal with. For > example, if the sequence is monotone, it has a limit iff it is > bounded. If one can isolate a _subsequence_ that does not > converge, or two subsequences that converge to a different > value, then this is also a proof of non-convergence. A good > example of that is X_n = (-1)^n[1 - 1/n]. > --- > Stan Liou could you please write the statement of disproving the existence of limits in plain English... I am even more confused by the [AL], [EL], etc... === Subject: Re: what is the method for disproving limit in analysis? >> >>There exists an epsilon, such that |X_n - X_limit| > epsilon? > Well, as long as you intend to mean > [AL][Ee>0][AN][En>N][ |X_n-L| >= e ], > then yes. > could you please write the statement of disproving the existence of limits > in plain English... Did you not see my post where I did exactly that? > I am even more confused by the [AL], [EL], etc... Yes, it's hard to read. === Subject: Re: what is the method for disproving limit in analysis? > I am wondering what is the method for disproving limits in analysis. > I mean: for proving limits, I just need to state the following: > Given any epsilon > 0, I can find a N0(epsilon), such that for all n>N0, > |X_n - X_limit| But now I want to disprove: that's to say, prove that the series does not > converge to that limit. > Do I say: > Given any epsilon > 0, I can not find a N0(epsilon), such that for all n>N0, > |X_n - X_limit|oo) x_n = a when for all e > 0, some n in N with for all j > n, |x_j - a| < e The negation is (by DeMorgan rules for quantifiers and connectives) is some e > 0 with for all n in N, some j > n with |x_j - a| >= e In otherwords find an e and a subsequence (x_nj)_j of (x_n)_n with for all j, |x_nj - a| >= e > Sometimes the above is difficult to formulate, can I say the following > instead? > There exists an epsilon, such that |X_n - X_limit| > epsilon? No. It's a toughie that's best slogged out using symbolic logic approach. === Subject: A Geometry question NEED HELP ABCD is a paralleloram. E is a point on BC and F a point on CD. AE cuts BF at G. AF cuts DE at H and BF cuts DE at K. Prove that [AGKH]=[BEG]+[CEKF]+[DFH] === Subject: Re: A Geometry question NEED HELP Kessie escribi.97: > ABCD is a paralleloram. E is a point on BC and F a point on CD. AE > cuts BF at G. AF cuts DE at H and BF cuts DE at K. Prove that > [AGKH]=[BEG]+[CEKF]+[DFH] [ABF] = [AED] = (1/2)[ABCD] Then [BEG] + [CEKF] + [DFH] = [ABCD] - ([ABF / AED]) = [ABCD] - ([ABF] + [AED] - ([ABF / AED])) = [ABCD] - ([ABF] + [AED]) + ([ABF] / [AED])) = [ABF] / [AED] = [AGKH] -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: A Geometry question NEED HELP > ABCD is a paralleloram. E is a point on BC and F a point on CD. AE cuts BF at G. > AF cuts DE at H and BF cuts DE at K. Prove that > [AGKH]=[BEG]+[CEKF]+[DFH] You'd better choose unit such that [ABCD] = 1. Compute [AED], [ABF] Compute [ADF] + [FBC] Write DAE as disjoint union of polygons. Same with ADF U FBC. Conclude. === Subject: Re: A Geometry question NEED HELP > ABCD is a paralleloram. E is a point on BC and F a point on CD. AE cuts BF at G. > AF cuts DE at H and BF cuts DE at K. Prove that > [AGKH]=[BEG]+[CEKF]+[DFH] i can prove it wrong: take a square as ABCD and set E and F equal to C. points G, H and K will be on C as well. [AGKH] is therefor sqrt(2), while [BEG] is 1, [CEKF] is 0 and [DFH] is 1. you get sqrt(2)=2 and that's wrong. === Subject: Re: sci.math.moderated? >> Quite likely. What is practically a certainty is that unless moderated >> these narrower groups would soon contain all sorts of stuff having >> nothing to do with the narrow focus, and there would be pointless > Imagine a mathematics without points! It's called point-free. http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=45 >> complaints about this. And there would of course still be a need for >> a group without any narrow focus. -- http://hertzlinger.blogspot.com === Subject: Algebra counterexample Is there a simple example of a ring homomorphism f: A -> B where A and B are both associative K-algebras with 1, such that f is not an algebra homomorphism, i.e., f is not K-linear, i.e., f(ka) <> k*f(a) for some k in K and a in A? -- Jim Heckman === Subject: Re: Algebra counterexample days. My association with the Department is that of an alumnus. >Is there a simple example of a ring homomorphism f: A -> B where >A and B are both associative K-algebras with 1, such that f is >not an algebra homomorphism, i.e., f is not K-linear, i.e., >f(ka) <> k*f(a) for some k in K and a in A? Complex conjugation for C as a C-algebra? -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Algebra counterexample > Is there a simple example of a ring homomorphism f: A -> B where > A and B are both associative K-algebras with 1, such that f is > not an algebra homomorphism, i.e., f is not K-linear, i.e., > f(ka) <> k*f(a) for some k in K and a in A? With K=A=B= the complex numbers, take f to be complex conjugation. === Subject: Re: Algebra counterexample >> Is there a simple example of a ring homomorphism f: A -> B where >> A and B are both associative K-algebras with 1, such that f is >> not an algebra homomorphism, i.e., f is not K-linear, i.e., >> f(ka) <> k*f(a) for some k in K and a in A? > With K=A=B= the complex numbers, > take f to be complex conjugation. In fact, any automorphism of a field extension. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Algebra counterexample >> Is there a simple example of a ring homomorphism f: A -> B where >> A and B are both associative K-algebras with 1, such that f is >> not an algebra homomorphism, i.e., f is not K-linear, i.e., >> f(ka) <> k*f(a) for some k in K and a in A? > With K=A=B= the complex numbers, > take f to be complex conjugation. > In fact, any automorphism of a field extension. -- Jim Heckman === Subject: Re: Banach limits and ultrafilters > > Now let F be any free ultrafilter. If we put > x_1+x_2+...+x_n > (1) LIM (x_n) = F-lim ----------------- > n > then LIM is a Banach limit. > > Question: Is each Banach limit of the form (1)? > > ...or maybe the answer is at the opposite extreme... the set of > Banach limits is a convex set, and those of the form (1) > are the extrememe points. What do you think? > To disprove the original conjecture, you would only have > to show the set of Banach limits of the form (1) is > not convex. Take two different free ultrafilters, then > use the midpoint of their corresponding Banach limits. > Show that there is no third ultrafilter that achieves that limit. I should apologize for 2 things. The first one is late response. The second one is that I forgot to put references in my posting. [1] and [2] are books I used for Banach limit (but they can be found elsewhere). [3] is a paper I found after sending the question to Google Groups. [1] B. Balcar, P. Stepanek: Teorie Mnozin (Set Thoery, in Czech) [2] K.P.S. Bhaskara Rao, M. Bhaskara Rao: Theory of Charges [3] J. Connor: Almost none of the sequences of 0's and 1's are almost convergent, Internal. J. of Math. and Math. Sci. 13(4) (1990), 775-778. I tried first to follow your hint and to prove that the set of linear functionals of the form (1) is not convex. I tried choosing two appropriate sequences and use them to show this. But this method was rather cumbersome and led to lot of computation. I wasn't able to think up a simpler proof. Your note that each linear combination of functionals of the form (1) is Banach limit saggest to use infinite linear combinations. I.e. if L_1, L_2,... are of the form (1) and Sum (c_i) =1, then L := Sum (c_i L_i) (2) is also Banach limit. Afterwards I came across the paper [3] of J. Connor (from Ohio University at that time). I learned from it that the set of all Cesaro convergent sequences has Lebesgue measure 1. (A sequence is Cesaro convergent to L if the arithmetic means of first n temrs conveges to L as n tends to infinity.) On the other hand, the set of all almost convergent sequences has Lebesgue measure 0. (A sequence is almost convergent if each Banach limit of this sequence has the same value.) Each almost convergent sequence is Cesaro convergent. So there exists a sequence of 0's and 1's which is Cesaro convergent but not almost convergent. For such a sequence each functional of the form (2) has the same value, but the value of Banach limit is not determined uniquely. Hence there are Banach limits which aren't of the form (2). So at last my question is: Is some characterization of all Banach limits known? (My feeling is that this question isn't easy.) TIA Martin === Subject: New Web Site Dealing with Theoretical Physics(now is under construction) This is the scientific place for discussion about some Quantum theories like Quantum Electrodynamics(QED), Quantum Chromodynamics (QCD), Quantum Field Theories(QFT), Quantum Gravity Theories(QGT) and some other new topics in theoretical high energy physics(hep) like String and Superstring theories, M-Theory (11 dimensional supergravity at low energies), F-Theory (12 spacetime dimensions) and Grand Unified Theories (GUTs),.... Hope to see you as an active and useful member in this club. K.Niknejad --------------------------------------------------------------------- For finding this site, you can see below URL: http://www.kiarash-niknejad.com/ --------------------------------------------------------------------- === Subject: Diophantian equations by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3wk11785; Let p be a prime, p>3. Is it true, that the equation x_1+2x_2...+(p-1)x_{p-1} +px_0(x_0-1)/2+px_1(x_1-1)/2+...+px_{p-1}(x_{p-1}-1)/2 -x(x-1)/2=n, where x=x_0 +x_1+ ... +x_{p-1}, has a solution for every positive integer n with integer x_0, x_1, ... ..., x_{p-1}? It is true if p=5. === Subject: Re: Algebra counterexample by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3sX11607; >Is there a simple example of a ring homomorphism f: A -> B where >A and B are both associative K-algebras with 1, such that f is >not an algebra homomorphism, i.e., f is not K-linear, i.e., >f(ka) <> k*f(a) for some k in K and a in A? >-- >Jim Heckman How about A = B = C, the complex numbers, viewed as C-algebras, and f the conjugation map? Todd Trimble === Subject: Re: Math is a SIN ! ! ! ! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3sn11615; >> Jesus Christ does not want you to waste your time with silly numbers >> or you won't go to heaven ! >Wasn't it Weierstrass who said, God created integers, all the rest is >man's handiwork. >Anyway, in my bible, God tells Noah to build an ark and gives dimensions. >I think we have proof God created a coordinate grid system. Oh, I almost >forgot, Jesus taught us to multiple using manipulatives, loaves and fishes. Didn't Jesus use loaves and fishes to teach us the Banach-Tarski paradox? Todd Trimble === Subject: Re: Math is a SIN ! ! ! ! > Anyway, in my bible, ... Anyway,now maths is IN !! ..that we all are in.. that's in how you see. === Subject: Re: Math is a SIN ! ! ! ! Yup. Especially trigonometry. Thomas === Subject: Re: Algebra counterexample by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3sr11621; >Is there a simple example of a ring homomorphism f: A -> B where >A and B are both associative K-algebras with 1, such that f is >not an algebra homomorphism, i.e., f is not K-linear, i.e., >f(ka) <> k*f(a) for some k in K and a in A? >-- >Jim Heckman A=B=complex numbers as an algebra over itself f=complex conjugation H === Subject: Re: Set theory by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3ts11637; >I have been asked to solve the following problem. Suppose X is an infinite >set. Show that the cardinal number of X is less than the cardinal number of >its power set. Now I have a problem with this. Suppose the cardinality of X >is c. Now I thought that the only infinite cardinalities are c and aleph >null which would mean that the cardinality of X is not less than the power >set of X. What is going on here????? There are infinitely many infinite cardinalities. Aleph_0 and c are two of them, and P(c), PP(c), ... give more. I don't know where you got the thought from. Todd Trimble === Subject: Re: NEAT PRODUCTS OF TRIG FUNCTIONS by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3ts11628; >I put problems like this (fun with rational values of trigonometric >expressions) on our local mathematical competitions: >Prove (and where appropriate, evaluate): >sec(4*pi/9) + sec(2*pi/9) - sec(pi/9) is an integer. >(The angles in Fahrenheit being 80, 40, 20) >cos(pi/7) + cos(2*pi/7) - (3*pi/7) is rational. >8*cos(arccos(7/128)/3) is an integer. >sin(pi/18) * sin(5*pi/18) * sin(7*pi/18) is rational. >cos(pi/5) - cos(2*pi/5) is rational. >Let A=arctan(2); show that for all positive integer numbers n, >the numbers 5^(n/2)*sin(n*A) are even integers. >Finally: >For fun, plot the curve and find the area inside it: >x^2 + (y-(x^2)^(1/3))^2=1 I have found these fascinating, but have a question about your last problem. Is the correct relation equivalent to x^2 + [y - x^(2/3)]^2 = 1? === Subject: Diophantian equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3tD11656; Let p be a prime, p>3. Is it true, that the equation 0x_0+1x_1+...+(p-1)x_{p-1} +px_0(x_0-1)/2+px_1(x_1-1)/2+...+px_{p-1}(x_{p-1}-1)/2 -x(x-1)/2=n, where x is x_0+x_1+...+x_{p-1}, has a solution (x_0,x_1,...,x_{p-1}) with integer x_0,x_1,...,x_{p-1}? It is true if p=5. === Subject: Re: Multidimensional Abel's/Schroder's functional equations by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAID3ta11661; Peter, thank you very much for the assisstance! === Subject: Re: Mahler's equation cordialy, lolo === Subject: Applications of proper classes? Has the notion of a proper class been necessary to solve any problems in number theory or real analysis? (See Wikipedia entry on Classes at http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) Dan Download DC Proof 1.0 at http://www.dcproof.com === Subject: Re: Applications of proper classes? > Has the notion of a proper class been necessary to solve any problems in > number theory or real analysis? > (See Wikipedia entry on Classes at > http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) > Dan > Download DC Proof 1.0 at http://www.dcproof.com No. ZFC has no proper classes, and (so far) all advances in number theory and real analysis can be formalized in ZFC (plus, perhaps, some large cardinal axioms). === Subject: Re: Applications of proper classes? >>Has the notion of a proper class been necessary to solve any problems in >>number theory or real analysis? >>(See Wikipedia entry on Classes at >>http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) > No. ZFC has no proper classes, and (so far) all advances in > number theory and real analysis can be formalized in ZFC > (plus, perhaps, some large cardinal axioms). Why are large cardinal axioms better than classes? I presume the reference to large cardinal axioms is to cover the uses of Grothendieck universes. But then, there are theories of sets+classes that are conservative extensions of ZFC (for example Ackermann's, or ZFC+reflection principle a la Feferman's proposal with terminology change) that can do all the things that Grothendieck universes are called upon to do. And is the elimination of classes by using Grothendieck universes really work as advertised? Even those who use Grothendieck universes talk of proper classes when nobody is looking: They refer to >the< category of sets, meaning all sets, rather than the category of sets in a fixed universe V; the French usage of 'small' means that the category of V-small sets is still a proper class, even if mathematicians in the trenches are completely unaware of it. [I am reminded of one who thought that the set of all sets of rank < 2omega, a pretty small topos, is same as the 'set' of all sets of cardinality < 2omega, whatever the latter means!] The only way out is to define V-small to mean 'is an element of V'. It seems to be the case that all uses of universes need only one (in any case, a finite number): 'small' means is in V, large means in not in V, and 'huge sets' needed are only class abstracts of the kind usable even in ZFC. Now replace 'small' by 'set' and 'large' by class, and it looks a lot like Ackermann's theory of sets and classes, except that we have a strong version of replacement whose necessity for theorems actually being proven is doubtful. === Subject: Re: Applications of proper classes? >>Has the notion of a proper class been necessary to solve any problems in >>number theory or real analysis? >>(See Wikipedia entry on Classes at >>http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) >>Dan >>Download DC Proof 1.0 at http://www.dcproof.com > No. ZFC has no proper classes, and (so far) all advances in > number theory and real analysis can be formalized in ZFC > (plus, perhaps, some large cardinal axioms). The class of all sets would be a proper class. It's just that sets are more interesting. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Applications of proper classes? >>Has the notion of a proper class been necessary to solve any problems in >>number theory or real analysis? > No. ZFC has no proper classes, and (so far) all advances in > number theory and real analysis can be formalized in ZFC > (plus, perhaps, some large cardinal axioms). > The class of all sets would be a proper class. It's just that sets are > more interesting. Furthermore, it can be shown that anything you can do with proper classes (including super-classes of THESE, etc), can in a suitable sense already be done with sets plus large cardinals. ---------------------------------------------------------------------------- -- Bill Taylor W.Taylor@math.canterbury.ac.nz ---------------------------------------------------------------------------- -- Mathematics runs under set theory just as Freecell runs under windows. ---------------------------------------------------------------------------- -- === Subject: Re: Applications of proper classes? >>>Has the notion of a proper class been necessary to solve any problems in >>>number theory or real analysis? >> No. ZFC has no proper classes, and (so far) all advances in >> number theory and real analysis can be formalized in ZFC >> (plus, perhaps, some large cardinal axioms). >> The class of all sets would be a proper class. It's just that sets are >> more interesting. >Furthermore, it can be shown that anything you can do with proper classes >(including super-classes of THESE, etc), can in a suitable sense already >be done with sets plus large cardinals. NBG is a conservative extension of ZF with proper classes. Any theorem of NBG not explicitly involving proper classes is also a theorem of ZF. Large cardinals give MORE than proper classes. Also, it cannot be proved that large cardinals are consistent, with or without choice. The main advantages of proper classes are those of convenience. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Applications of proper classes? > > >>Has the notion of a proper class been necessary to solve any problems in >>number theory or real analysis? >> >>(See Wikipedia entry on Classes at >>http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) >> >>Dan >>Download DC Proof 1.0 at http://www.dcproof.com >> > > No. ZFC has no proper classes, and (so far) all advances in > number theory and real analysis can be formalized in ZFC > (plus, perhaps, some large cardinal axioms). > The class of all sets would be a proper class. It's just that sets are > more interesting. I was under the impression that you can't build the class of all sets using ZFC, even in principal. 'cid 'ooh === Subject: Re: Applications of proper classes? >Has the notion of a proper class been necessary to solve any problems in >number theory or real analysis? >(See Wikipedia entry on Classes at >http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) >Dan >Download DC Proof 1.0 at http://www.dcproof.com >> No. ZFC has no proper classes, and (so far) all advances in >> number theory and real analysis can be formalized in ZFC >> (plus, perhaps, some large cardinal axioms). > The class of all sets would be a proper class. It's just that sets are > more interesting. The class of all sets would be a proper class if ZFC had proper classes, but it doesn't. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Applications of proper classes? >> >> >Has the notion of a proper class been necessary to solve any problems in >number theory or real analysis? > >(See Wikipedia entry on Classes at >http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) > >Dan >Download DC Proof 1.0 at http://www.dcproof.com > >> >> No. ZFC has no proper classes, and (so far) all advances in >> number theory and real analysis can be formalized in ZFC >> (plus, perhaps, some large cardinal axioms). > The class of all sets would be a proper class. It's just that sets are > more interesting. > The class of all sets would be a proper class if ZFC had proper classes, > but it doesn't. How do we exclude the existence of a class whose properness (as I understand it, representability as a set) is undecidable? I'm speaking from ignorance, but can't a class C have the form if (undecidable proposition) then x in C iff (definition yielding proper class); otherwise x in C iff (definition yielding a set))? Maybe I just misunderstand what you mean by ZFC doesn't have proper classes (or maybe I don't understand what a class is!). === Subject: Re: Applications of proper classes? >> >>> >>> >>Has the notion of a proper class been necessary to solve any problems in >>number theory or real analysis? >> >>(See Wikipedia entry on Classes at >>http://en.wikipedia.org/wiki/Class_%28set_theory%29 ) >> >>Dan >>Download DC Proof 1.0 at http://www.dcproof.com >> >>> >>> No. ZFC has no proper classes, and (so far) all advances in >>> number theory and real analysis can be formalized in ZFC >>> (plus, perhaps, some large cardinal axioms). >> >> The class of all sets would be a proper class. It's just that sets are >> more interesting. >> The class of all sets would be a proper class if ZFC had proper classes, >> but it doesn't. > How do we exclude the existence of a class whose properness (as I > understand it, representability as a set) is undecidable? We don't have to. There is no axiom of ZFC that talks about anything being excluded. The axioms give certain conditions for concluding that sets exist with certain properties. Note that the word set appears nowhere in the axioms of ZFC. That's because it's implicit that everything that exists in ZFC is what we commonly call a set. That is, for all X in an axiom is to be understood as for every set X, and there exists an X means there exists a set X. There is no room for anything that doesn't exist. > I'm speaking from ignorance, but can't a class C have the form if > (undecidable proposition) then x in C iff (definition yielding proper > class); otherwise x in C iff (definition yielding a set))? You are no longer talking about ZFC, but about some different set theory (perhaps NBG, which does have proper classes). > Maybe I just misunderstand what you mean by ZFC doesn't have proper > classes (or maybe I don't understand what a class is!). ZFC does not have proper classes, because everything in ZFC is a set. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Algebra counterexample by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAIE6ZN17641; Mathematicians seem to be a bit lazy ... === Subject: Best method of combining estimates I have inherited a processing scheme that looks for features of a certain size in an image. This is done by applying a series of filters, each of which has a maximum output at a certain feature size and whose output drops off at bigger and smaller sizes. At present, size detection is carried out by simply selecting the filter with the greatest output. The outputs of the filters are fairly smooth, and overlap significantly. The outputs are proportional to image contrast, which can vary significantly from image to image. I would like to try to get a bit more accuracy from the existing processing scheme. I thought of some method of looking at the ratios of the output of the 'maximum' filter to those of the filters on either side. This gives me two numbers that should allow some form of 'interpolation' for a more accurate result. I could run the filters with features of different lengths and get enough information for a 2-D lookup table for each filter (1-D for the ones at the ends), but I can't help thinking that first, there should be a more elegant (and storage-efficient) way of doing this; second, it might be useful to be able to extend the scheme to more than just the ones either side, and that would mean a many-D lookup table; and third, someone must have done this before. Does anyone have any suggestions, references, links, hints, etc? Jon === Subject: Help with analysis of complex function Math folks, I am trying to show that the magnitude of the following function achieves its maximum at z = R + Pi I. E^(3 z)/(1 + E^z) where z ranges from z = R (>0) to z = R + 2 Pi I. When z = R, the value of the function is E^(3 R)/(1 + E^R) When z = R + Pi I, the value of the function is E^(3 R)/(-1 + E^R) When z = R + 2 Pi I, the value of the function is E^(3 R)/(1 + E^R) again. Help would be appreciated. Diana -- God made the integers, all else is the work of man. L. Kronecker, Jahresber. DMV 2, S. 19. === Subject: Re: Help with analysis of complex function > I am trying to show that the magnitude of the following function achieves > its maximum at z = R + Pi I. > E^(3 z)/(1 + E^z) where z ranges from z = R (>0) to z = R + 2 Pi I. |e^(3z)/(1 + e^z)| = |e^(3z)|/|1 + e^z| = e^(3R)|/|1 + e^z|. You maximize the last expression by minimizing the denominator. But 1 + e^(R+it) describes a circle of radius e^R, centered at 1, as t goes from 0 to 2Pi. Where does that cirlce have minimum modulus? === Subject: Re: Help with analysis of complex function > I am trying to show that the magnitude of the following function achieves > its maximum at z = R + Pi I. > > E^(3 z)/(1 + E^z) where z ranges from z = R (>0) to z = R + 2 Pi I. > |e^(3z)/(1 + e^z)| = |e^(3z)|/|1 + e^z| = e^(3R)|/|1 + e^z|. You maximize > the last expression by minimizing the denominator. But 1 + e^(R+it) > describes a circle of radius e^R, centered at 1, as t goes from 0 to 2Pi. > Where does that cirlce have minimum modulus? === Subject: Need Letters Journal for Chaos Theory Does anybody know of a mathematical letters journal that would be appropriate for publishing a short latter on chaos in planetary orbits? TIA. === Subject: Re: Cantor's diagonal proof wrong? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAIFPLW24454; >> I don't think this is true at all. In my experience, if you ask a >> roomful of non-mathematicians whether i, the square root of -1, >> exists they will mostly claim it doesn't. >Nonsense. The imaginary number i is 1 turned 90 degrees >counterclockwise on the complex plane. It has as much existence as any >integer or real number. >Bob Kolker He said *non-mathematicians*. I'm guessing he himself is a mathematician who already knew that. (Although Galois theory teaches us that we could just as well say i is the 90 degree clockwise turn!) Todd Trimble === Subject: Computer language and category theory Is there any work done one computer languages and category theory? To clarify, what I am looking into is something like the following: Given the language S ::= A | B A ::= aa | ba | A-aa | B-ba B ::= ab | bb | A-ab | B-bb Thus, the set of sentences in S is sequences {a,b}{a,b}-{a,b}{a,b}-... such that the letter in front of the dash (-) is the same as the letter behind it. I think this language can be described as a category as follows: C= Obj={A,B} Morph=Hom(A,A) U Hom(A,B) U Hom(B,A) U Hom(B,B) Hom(x,y) = any sentence s in S such that it starts with an x and ends with an y. E.g. bb-ba, ba, ba-aa-ab-ba in Hom(B,A) The operator * is defined as follows: Given x in Hom(X,Y) and y in Hom(Y,Z) such that neither x nor y is a unit morphism. Then x*y = x-y The unit elements of Hom(A,A) and Hom(B,B) is just an empty sentence handled specially: x * e = e * x = x (a) Does this look sensible? However, I don't have such a simple language, but rather a slightly more complicated one: T ::= A | B A ::= aa | ba | A-aa | B-ba B ::= ab | bb | A-ab | B-bb | A-K K ::= ( A )-B | ( B )-B The K, introduced in order to be able to create a tree-like structure, makes the elements of T unsuited to form a set of morphisms, as each morphism in a category have excactly one source object and one destination object. (b) What kind of mathematical tools should I study to handle language constructs like the language T? In reality, our objects are a little different, but not in essence. My collegue wants to define something like a partial parametriced monoid, an animal I never before have encountered. Partiality follows from the fact that you cannot take two arbitrary elements of S (or T) and form a new element of S (or T). Parametrization does occure in some sense in the language T, where we must choose whether to insert an A sentence into the ( A )-B or a B into the same meta sentence. (Was this understandable?) However, I don't like such animals. So I wonder if there is some use of category theory or something else that have been used to model language like constructs. -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: 3D Model for allocation of total capacity by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAIFVxL25040; Can anyone help? I have an interesting issue. I've chosen to study maths and computer science and one of the problems I've been asked to solve, potentially using a 3D Modeling formula is: In a Shared Server environment, where total capacity of a server is measured by total CPU capacity, total disk capacity and total memory capacity, I need a formula to help me understand how much each user is requesting of the total capacity. By example: On a single server total CPU is 2.88 Ghz, 16Gb Memory and 146Gb of Disk Space. I have a user who only wants to use 1Ghz CPU, 2Gb Mem and 2Gb of Disk Space. What formula can I use that tells me what percentage of the total capacity of the machine this person is requiring? Nick === Subject: Re: 3D Model for allocation of total capacity Possible the maximum of the 3 percentages. If you need more than 100% of the available disk space you are done for. But then again the same may not true of the CPU. If a user is doing something real-time they may need x Ghz but if they are not they may get by with less. Something similar could be said of memory as this can be paged, but with consequences for CPU and disk. > Can anyone help? I have an interesting issue. I've chosen to study maths and computer science and one of the problems I've been asked to solve, potentially using a 3D Modeling formula is: > In a Shared Server environment, where total capacity of a server is measured by total CPU capacity, total disk capacity and total memory capacity, I need a formula to help me understand how much each user is requesting of the total capacity. > By example: > On a single server total CPU is 2.88 Ghz, 16Gb Memory and 146Gb of Disk Space. > I have a user who only wants to use 1Ghz CPU, 2Gb Mem and 2Gb of Disk Space. What formula can I use that tells me what percentage of the total capacity of the machine this person is requiring? > Nick === Subject: Re: Russia Developing New Nuclear Missile > >>Russia Developing New Nuclear Missile >> >>Russia's new nuclear missile can carry up to 10 nuclear warheads >>weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >>combat payload. > >Is this an offensive or a defensive weapon? >> >>Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >>anti-conventional defense, hence anti-ballistic defense = defense^2, >>ergo this new Russian guy = anti-defensive^2 defense = defense^3.) >Seems quite logical. What comes after defense^3? >> No, Topolski (rex, etc) missed a solution. He is saying this is designed >> to penetrate Star Wars Shield, with which I absolutely agree, but his >> maths is pure rubbish. Defensive and offensive are related by defensive >> = - offensive. Is not this the logic ?! Then surely >> defensive^3=(-offensive)^3=-offensive^3! and the missile is both >> affensive and defensive in equal amounts!!! > Blasphemy aside, how come that the alleged new Russian missile is > a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see > above). Now by your algebra the guy is -offensive^3, ergo still defensive. > Q.E.D. Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to put it in plain kindergarten maths that even Brookski can understand, then you do not know whether y is positive or negative until you use a particular x. Ergo, logic sez that if you use missile offensively, then it will be offensive (negative x, thus negative y), and vice versa. This is logic. === Subject: Re: Russia Developing New Nuclear Missile >> >Russia Developing New Nuclear Missile > >> >Russia's new nuclear missile can carry up to 10 nuclear warheads >weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >combat payload. >> >>Is this an offensive or a defensive weapon? > >Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >anti-conventional defense, hence anti-ballistic defense = defense^2, >ergo this new Russian guy = anti-defensive^2 defense = defense^3.) >> >>Seems quite logical. What comes after defense^3? > No, Topolski (rex, etc) missed a solution. He is saying this is designed > to penetrate Star Wars Shield, with which I absolutely agree, but his > maths is pure rubbish. Defensive and offensive are related by defensive > = - offensive. Is not this the logic ?! Then surely > defensive^3=(-offensive)^3=-offensive^3! and the missile is both > affensive and defensive in equal amounts!!! >> Blasphemy aside, how come that the alleged new Russian missile is >> a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see >> above). Now by your algebra the guy is -offensive^3, ergo still >> defensive. Q.E.D. > Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to > put it in plain kindergarten maths that even Brookski can understand, then > you do not know whether y is positive or negative until you use a > particular x. Ergo, logic sez that if you use missile offensively, then it > will be offensive (negative x, thus negative y), and vice versa. This is > logic. Hey Homo, Use Russian math...first I smack you up side your head. If you get up, I smack you a second time. 1+1=2 This is what MTRP understands. It is a more elegant solution anyway. No decimal points are required. We only smack you up side the head in whole numbers. === Subject: Re: Russia Developing New Nuclear Missile > >>Russia Developing New Nuclear Missile >> > >>Russia's new nuclear missile can carry up to 10 nuclear warheads >>weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >>combat payload. > >Is this an offensive or a defensive weapon? >> >>Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >>anti-conventional defense, hence anti-ballistic defense = defense^2, >>ergo this new Russian guy = anti-defensive^2 defense = defense^3.) > >Seems quite logical. What comes after defense^3? >> >> No, Topolski (rex, etc) missed a solution. He is saying this is >> designed to penetrate Star Wars Shield, with which I absolutely agree, >> but his maths is pure rubbish. Defensive and offensive are related by >> defensive = - offensive. Is not this the logic ?! Then surely >> defensive^3=(-offensive)^3=-offensive^3! and the missile is both >> affensive and defensive in equal amounts!!! > Blasphemy aside, how come that the alleged new Russian missile is > a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see > above). Now by your algebra the guy is -offensive^3, ergo still > defensive. Q.E.D. >> Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to >> put it in plain kindergarten maths that even Brookski can understand, >> then you do not know whether y is positive or negative until you use a >> particular x. Ergo, logic sez that if you use missile offensively, then >> it will be offensive (negative x, thus negative y), and vice versa. This >> is logic. > Hey Homo, > Use Russian math...first I smack you up side your head. If you get up, I > smack you a second time. > 1+1=2 > This is what MTRP understands. It is a more elegant solution anyway. No > decimal points are required. We only smack you up side the head in whole > numbers. Hey SCRMC, I did not know Brookski maths was still alive in Russia, I though it disappeared in stone age when they figured out how to count on fingers ;) === Subject: Re: Russia Developing New Nuclear Missile >Russia Developing New Nuclear Missile >> >> >Russia's new nuclear missile can carry up to 10 nuclear warheads >weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >combat payload. >> >>Is this an offensive or a defensive weapon? > >Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >anti-conventional defense, hence anti-ballistic defense = defense^2, >ergo this new Russian guy = anti-defensive^2 defense = defense^3.) >> >>Seems quite logical. What comes after defense^3? >No, Topolski (rex, etc) missed a solution. He is saying this is designed >to penetrate Star Wars Shield, with which I absolutely agree, but his >maths is pure rubbish. Defensive and offensive are related by defensive >= - offensive. Is not this the logic ?! Then surely >defensive^3=(-offensive)^3=-offensive^3! and the missile is both >affensive and defensive in equal amounts!!! >>Blasphemy aside, how come that the alleged new Russian missile is >>a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see >>above). Now by your algebra the guy is -offensive^3, ergo still defensive. >>Q.E.D. > Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to > put it in plain kindergarten maths that even Brookski can understand, then > you do not know whether y is positive or negative until you use a particular > x. Ergo, logic sez that if you use missile offensively, then it will be > offensive (negative x, thus negative y), and vice versa. This is logic. No. Cuz U add one wrong assumption that plain nukes are offensive, whereas even Brookski knows they are not. Thus the sign of our x, and hence y, is already determined - ponimayesh? === Subject: Re: Russia Developing New Nuclear Missile >>Russia Developing New Nuclear Missile > > >>Russia's new nuclear missile can carry up to 10 nuclear warheads >>weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >>combat payload. > >Is this an offensive or a defensive weapon? >> >>Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >>anti-conventional defense, hence anti-ballistic defense = defense^2, >>ergo this new Russian guy = anti-defensive^2 defense = defense^3.) > >Seems quite logical. What comes after defense^3? >> >>No, Topolski (rex, etc) missed a solution. He is saying this is designed >>to penetrate Star Wars Shield, with which I absolutely agree, but his >>maths is pure rubbish. Defensive and offensive are related by defensive >>= - offensive. Is not this the logic ?! Then surely >>defensive^3=(-offensive)^3=-offensive^3! and the missile is both >>affensive and defensive in equal amounts!!! >Blasphemy aside, how come that the alleged new Russian missile is >a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see >above). Now by your algebra the guy is -offensive^3, ergo still >defensive. Q.E.D. >> Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to >> put it in plain kindergarten maths that even Brookski can understand, >> then you do not know whether y is positive or negative until you use a >> particular x. Ergo, logic sez that if you use missile offensively, then >> it will be offensive (negative x, thus negative y), and vice versa. This >> is logic. > No. Cuz U add one wrong assumption that plain nukes are offensive, whereas > even Brookski knows they are not. Thus the sign of our x, and hence y, is > already determined - ponimayesh? No, even Brookski knows that though offensive/defensive split of nuclear weapons' use is about 1/99, this does not in any way limit the offensive effectiveness of these weapons. Kapish? === Subject: Re: Russia Developing New Nuclear Missile >> >Russia Developing New Nuclear Missile >> >> >Russia's new nuclear missile can carry up to 10 nuclear warheads >weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >combat payload. >> >>Is this an offensive or a defensive weapon? > >Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >anti-conventional defense, hence anti-ballistic defense = defense^2, >ergo this new Russian guy = anti-defensive^2 defense = defense^3.) >> >>Seems quite logical. What comes after defense^3? > >No, Topolski (rex, etc) missed a solution. He is saying this is designed >to penetrate Star Wars Shield, with which I absolutely agree, but his >maths is pure rubbish. Defensive and offensive are related by defensive >= - offensive. Is not this the logic ?! Then surely >defensive^3=(-offensive)^3=-offensive^3! and the missile is both >affensive and defensive in equal amounts!!! >> >>Blasphemy aside, how come that the alleged new Russian missile is >>a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see >>above). Now by your algebra the guy is -offensive^3, ergo still >>defensive. Q.E.D. >Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to >put it in plain kindergarten maths that even Brookski can understand, >then you do not know whether y is positive or negative until you use a >particular x. Ergo, logic sez that if you use missile offensively, then >it will be offensive (negative x, thus negative y), and vice versa. This >is logic. >>No. Cuz U add one wrong assumption that plain nukes are offensive, whereas >>even Brookski knows they are not. Thus the sign of our x, and hence y, is >>already determined - ponimayesh? > No, even Brookski knows that though offensive/defensive split of nuclear > weapons' use is about 1/99, this does not in any way limit the offensive > effectiveness of these weapons. Kapish? Of course it does - even by your Brookski logic the probability would be merely 0.01010101010. But let us stop here. First I was kidding and second nobody can match Our Majesty in a pointless discussion anyway. So sez sci. === Subject: Re: Russia Developing New Nuclear Missile >>Russia Developing New Nuclear Missile > > >>Russia's new nuclear missile can carry up to 10 nuclear warheads >>weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >>combat payload. > >Is this an offensive or a defensive weapon? >> >>Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >>anti-conventional defense, hence anti-ballistic defense = defense^2, >>ergo this new Russian guy = anti-defensive^2 defense = defense^3.) > >Seems quite logical. What comes after defense^3? >> >>No, Topolski (rex, etc) missed a solution. He is saying this is designed >>to penetrate Star Wars Shield, with which I absolutely agree, but his >>maths is pure rubbish. Defensive and offensive are related by defensive >>= - offensive. Is not this the logic ?! Then surely >>defensive^3=(-offensive)^3=-offensive^3! and the missile is both >>affensive and defensive in equal amounts!!! >Blasphemy aside, how come that the alleged new Russian missile is >a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see >above). Now by your algebra the guy is -offensive^3, ergo still >defensive. Q.E.D. >> Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to >> put it in plain kindergarten maths that even Brookski can understand, >> then you do not know whether y is positive or negative until you use a >> particular x. Ergo, logic sez that if you use missile offensively, then >> it will be offensive (negative x, thus negative y), and vice versa. This >> is logic. > No. Cuz U add one wrong assumption that plain nukes are offensive, whereas > even Brookski knows they are not. They are neither offensive or defensive but they are both. There is nothing to say that nuclear weapons could not be used offensively. It is only the naive assumption that they will not be used offensively (first strike) that defines them as defensive weapons. The US anti-missile star wars program is a true defensive system though. It is only designed to shoot down missiles from an agressor nation. Thus the sign of our x, and > hence y, is already determined - ponimayesh? === Subject: Re: Russia Developing New Nuclear Missile >> >Russia Developing New Nuclear Missile >> >> >> >Russia's new nuclear missile can carry up to 10 nuclear warheads >weighing a total of 4.4 tons,compared with the Topol-M's 1.32-ton >combat payload. >> >>Is this an offensive or a defensive weapon? > >Super-defensive ... actually defensive^3. (Easy math: Plain nukes = >anti-conventional defense, hence anti-ballistic defense = defense^2, >ergo this new Russian guy = anti-defensive^2 defense = defense^3.) >> >>Seems quite logical. What comes after defense^3? > >No, Topolski (rex, etc) missed a solution. He is saying this is designed >to penetrate Star Wars Shield, with which I absolutely agree, but his >maths is pure rubbish. Defensive and offensive are related by defensive >= - offensive. Is not this the logic ?! Then surely >defensive^3=(-offensive)^3=-offensive^3! and the missile is both >affensive and defensive in equal amounts!!! >> >>Blasphemy aside, how come that the alleged new Russian missile is >>a[o]ffensive? In my deduction it is defensive^3, ergo defensive (see >>above). Now by your algebra the guy is -offensive^3, ergo still >>defensive. Q.E.D. >Q.E.D. aside, if the solution for offensiveness-defensiveness is y=x^3 to >put it in plain kindergarten maths that even Brookski can understand, >then you do not know whether y is positive or negative until you use a >particular x. Ergo, logic sez that if you use missile offensively, then >it will be offensive (negative x, thus negative y), and vice versa. This >is logic. >>No. Cuz U add one wrong assumption that plain nukes are offensive, whereas >>even Brookski knows they are not. > They are neither offensive or defensive but they are both. Hi Brookski. Welcome back - your new ID is approved. > There is nothing to say that nuclear weapons could not be used offensively. > It is only the naive assumption that they will not be used offensively > (first strike) that defines them as defensive weapons. > The US anti-missile star wars program is a true defensive system though. > It is only designed to shoot down missiles from an agressor nation. >> Thus the sign of our x, and hence y, is already determined - ponimayesh? === Subject: Re: Russia Developing New Nuclear Missile >The US anti-missile star wars program is a true defensive system though. >It is only designed to shoot down missiles from an agressor nation Well Putin couldn't have given GW a better Christmas present. Now GW can push hard for his Star Wars anti missile shield program. If I were a conspiracy theorist I would have said GW got Putin to make the announcement to....................... === Subject: Re: Russia Developing New Nuclear Missile >>The US anti-missile star wars program is a true defensive system though. >>It is only designed to shoot down missiles from an agressor nation > Well Putin couldn't have given GW a better Christmas present. Now GW > can push hard for his Star Wars anti missile shield program. > If I were a conspiracy theorist I would have said GW got Putin to make > the announcement to....................... you may be onto something there :) === Subject: Re: Russia Developing New Nuclear Missile >>The US anti-missile star wars program is a true defensive system though. >>It is only designed to shoot down missiles from an agressor nation > Well Putin couldn't have given GW a better Christmas present. Now GW > can push hard for his Star Wars anti missile shield program. > If I were a conspiracy theorist I would have said GW got Putin to make > the announcement to....................... Not to worry! Putin will only nuke European rogue nations. Russia just signed a major agreement with Boeing on 7E7 production. Russia needs a new civilian airliner fleet. Now they get latest US technology, part of the pie and new fleet of aircraft. You don't nuke somebody who's giving you all this stuff. Airbus sucks! === Subject: Coset codes of a Reed-Solomon code? I need some help on a problem regarding coset codes of Reed-Solomon codes (using GF(2^m)). I will quickly describe the problem. Define the same_symbol_distance of a codeword c as the maximum number of same symbols contained in c. Define the same_symbol_distance of a codebook C as the number of same symbols of c_s, where c_s is a codeword in C containing the maximum number of same symbols. As an example, the same_symbol_distance of the all-zero codeword of an (n,k) RS code is equal to n. Also, the same_symbol_distance of the all e codeword (e in GF(2^m)) is equal to n. Thus the same_symbol_distance of an (n,k) RS codebook is equal to n. Given an (n,k) Reed-Solomon code C_rs, what is the minimum same_symbol_distance that can be obtained for C = C_rs + h, using the appropriate coset h? How can I determine the cosets that will give a minimum same_symbol_distance? I have done a simulation using a (7,4) RS code, with g(x) = (x-alpha^1)(x - alpha^2)(x-alpha^3). I used the following polynomials as cosets: a_0 + a_1*x + a_2*x^2, where a_i in GF(2^m). (Thus, all polynomials with degree < degree(g(x)) was used). This is from the fact that any polynomial p(x) with degree >= degree(g(x)) will be in C_rs + h_1 where h_1 = p(x) mod g(x). I have also found that for the above code, the same_symbol_distance is 4. Will the same_symbol_distance always equal k? Another result is that the cosets generating a minimum same_symbol_distance have weight equal to (n-k). However, this is a necessary condition but not a sufficient condition. To see that this is necessary, consider the same_symbol_distance of c_0 + h, where c_0 is the all zero codeword and h is an arbitrary coset. Clearly, the same_symbol_distance of (c_0 + h) = n - weight(h). Then same_symbol_distance(C_rs + h) <= n-weight(h). (The above is only true if it can be shown that the minimum same_symbol_distance of (C_rs + h) is equal to k, for some h.) Also, from the simulation, I found that there are more than one coset h_min generating the minimum same_symbol_distance. In C_rs + h_min, more than one codeword have a same_symbol_distance = same_symbol_distance(C_rs+h_min). Is there maybe a mathematical way to prove the above? Any help and/or suggestions will be greatly appreciated Jaco Versfeld === Subject: Re: Mahler's equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAIGbnm31173; >cordialy, lolo I would like your looking at some recent threads on functional equation solving: 7 nov int f(x)*g(x,y) , 15 nov solving f(h(x),y,z-x)=.... 15 nov multidimensional Abel's Moroever, studying 'forms' is very interesting : phi(x) | phi(h(x))=phi(x)+1 properties of f(x,y,z)=m^[y.phi(x)+z+c](L) ? Bracket is invariant for (x,y,z) ->(h(x),y,z-x) f->f for z -> z+1 f-> m(f) Friendly,Alain. === Subject: Re: Element of? >As a layman I wonder if the basic meanings of all symbols used in axioms >of set theory have already been completely and unambiguously clarified. >>No. The axioms frequently serve to *help* clarify the meanings of the >>symbols. Also, any grammar rules *help* clarify the meanings. >>Ultimately, however, the meanings only exist unambiguously for a model >>of the axioms/symbols. > A mathematics that is not just a self-satisfying game has to be > carefully fitted into the whole knowledge of menkind. This requires at > first properly chosen most elementary basics. The Atoms of mathematics > are not the axioms but what one intends to express for instance with the > symbol for element of. That is, as I understand it, how math is actually developed. The people who built ZF and ZFC versions of set theory *did* have an idea of what that symbol means. However, that doesn't mean they started by defining it. The purpose of the axioms is to describe what it means without referring to the model they had in mind. >>I have no idea what this is supposed to mean. Real and rational numbers >>are no more uncertain than integers. > If physicists and engineers prefer integers, they often do so because > these genuine numbers are absolutely precise. Calculation with reals or > more precisely rationals always depends on the chosen accuracy. To work with integers is to impose a level of accuracy as well. Rarely do objects travel at integer velocities or cover integer distances. >Given, zero does not exist as a rational number. >>Zero *is* a rational number. What made you think it is not? > I gave several reasons in de.sci.mathematics > Let me add a further one: > Zero is supposed to be a neutral number without any sign. Can you > imagine to divide a by b and yield a result without sign? Perhaps you are not aware of either of the common definitions of a rational number. Q = {m/n | m and n are integers, n =/= 0} Choosing m=0, n=1 gives 0 as a rational number. As for your issue with signs, if moving right is positive, and moving left is negative, does it make sense to refer to not moving as being right or left motion? >Given, the concept of real numbers covers zero just in case of the >potential infinity. >>Huh? What do you mean by covers zero and potential infinity? > As did Weyl, I consider a continuum a sauce. That makes no sense to me. Granted, I haven't looked up Weyl's comment. > The term potential infinity was introduced by Aristoteles and means > infinity is a fiction outside the wealth of numbers. There is actually > no infinite number. Thus the fact that infinity is not a number. Why obfuscate things with the term potential? >Is there any justification for including or excepting a rational or real >zero in physics? >>I thought we were talking about math, which is a tool used in physics. > I respect mathematics, but I am an engineer who demands flawless tools. As an engineer you should know that we have no flawless tools. More importantly, as an engineer it is very likely that you have little or no understanding of the mentality behind most of mathematics. >Couldn't reals be interpreted as integers divided by an denominator of >infinite size? I would conclude from that: Reals are of quite different >quality. >>You can't have a denominator of infinite size, under any normal >>interpretation. You think the reals of quite different quality from >>*what*? > The entity of reals as a sauce is quite different from the notion of a > number. What do you mean by as a sauce? > What are you referring to with IZ, IQ, >>IR, and IC? > Sorry for my awkward letters. I meant integer, rational, real, and > complex numbers. Those are usually just referred to as Z, Q, R, C. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Element of? >> As a layman I wonder if the basic meanings of all symbols used in axioms >> of set theory have already been completely and unambiguously clarified. > > No. The axioms frequently serve to *help* clarify the meanings of the > symbols. Also, any grammar rules *help* clarify the meanings. > Ultimately, however, the meanings only exist unambiguously for a model > of the axioms/symbols. > A mathematics that is not just a self-satisfying game has to be > carefully fitted into the whole knowledge of menkind. This requires at > first properly chosen most elementary basics. The Atoms of mathematics > are not the axioms but what one intends to express for instance with the > symbol for element of. This sounds reasonable, but I'm not sure it's true. Generations of mathematicians have engaged quite deliberately in self-satisfying games - read Hardy's A mathematician's apology. Then decades later things (like abstract algebra, in particular finite fields) turn out to have direct application to the real world, (quite possibly getting classified by the US gumint as munitions [sic]). It's all part of the unreasonable effectiveness of mathematics [or words very Brian Chandler http://imaginatorium.org === Subject: Re: Element of? >>I thought we were talking about math, which is a tool used in physics. > Of course, mathematics is much, much more than a tool used in physics. Absolutely. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Cantor's diagonal proof wrong? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAIHHaI02526; >> But now I'm starting to wonder if there may be value in looking at this >> differently. Might it be valid to think of division of 1 by 9 as an >> infinite processes (algorithm) that produces closer and closer >> approximations, yet is always unable to produce the actual point on >> the line? >> Nonesense. One can always divide a line segment into N equal parts by a >> well known geometric construction. So getting the point on the segment >> corresponding to k/N for k = 0,1,...N is trivial. Or equivalently choose >> a unit length and lay out multiples of this length on an infinite ray. >> Again one easily constructs points corresponding to k*N. Why do you >> complicate a very straightforward matter? >Because most things in life that seem straightforward turn out later to be >anything but that. I enjoy finding the exceptions, so I search. But like >fishing, many times, there is just nothing to be caught. >-- >Curt Welch http://CurtWelch.Com/ >curt@kcwc.com http://NewsReader.Com/ Would you argue that 1/5 is also defined by an algorithm? Probably not, because 1/5 = 0.2, a nice number. But why should the method of obtaining 1/5 differ from 1/9? It shouldn't. This occurs because of your choice of the decimal system. If you were to choose the system of 9 digits, then 1/9 would be a nice 0.1 . Nut then 1/5, in this system, wouldn't be so nice anymore. Certainly, a theory about infinity, cannot depend on how many digits or symbols you use in your system. facedancer === Subject: Re: Cantor's diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > Would you argue that 1/5 is also defined by an algorithm? Yes. All mathematical operations are procedures or algorithms. Some make reference to procedures that can never produce an answer, like 1/0, and some make reference to procedures that never terminate, like .111.... -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? > 1/0, and some make reference to procedures that never terminate, like > .111.... 1/9 terminates. Using a well known construction one can divide any length by an integer ( > 0 of course). It terminates very nicely think you. Once again you confuse the mathematics with the method of representing quantities. Don't quit your day job to take up a career in mathematics. You have not got the Right Stuff. Bob Kolker === Subject: Re: Cantor's diagonal proof wrong? >> Would you argue that 1/5 is also defined by an algorithm? > Yes. All mathematical operations are procedures or algorithms. > Some make reference to procedures that can never produce an answer, like > 1/0, and some make reference to procedures that never terminate, like > .111.... Infinites can have the start and the end. Consider the reals in the closed area [0,1]. Zero is the start and 1 is the end. There are still infinite many reals. How are you so sure that 0.111... is NOT this type of infinite string? Tapio > -- > Curt Welch > http://CurtWelch.Com/ > curt@kcwc.com > http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > Infinites can have the start and the end. Consider the reals in the > closed area [0,1]. Zero is the start and 1 is the end. There are still > infinite many reals. How are you so sure that 0.111... is NOT this type > of infinite string? > Tapio This is a pefect example of how people let language confuse them. If you take a finite sized line, and keep cuting it in half, you have the description of an infinite procedure which (in the world of math and geometry at least) has no end. But the line does have two ends. So then the guy above points to the end of the line and says, look, there's an infinity with two ends!. The ends he was pointing to has nothing to do with the end of the infinite procedure under discussion. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? > So then the guy above points to the end of the line and says, look, > there's an infinity with two ends!. Yes, but somebody else have done that as they defined [0,1]. > The ends he was pointing to has nothing to do with the end of the > infinite procedure under discussion. How you can be so sure about that? :-) Tapio > -- > Curt Welch > http://CurtWelch.Com/ > curt@kcwc.com > http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > So then the guy above points to the end of the line and says, look, > there's an infinity with two ends!. > Yes, but somebody else have done that as they defined [0,1]. > The ends he was pointing to has nothing to do with the end of the > infinite procedure under discussion. > How you can be so sure about that? :-) I can't. But I can pretend to be sure on Usenet. :) -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? > I can't. But I can pretend to be sure on Usenet. :) Has anyone told you that you are a horse's arse. No? O.K. You are a horse's arse. Bob Kolker === Subject: Re: Cantor's diagonal proof wrong? > I can't. But I can pretend to be sure on Usenet. :) So you admit you're a troll? === Subject: Re: Cantor's diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > I can't. But I can pretend to be sure on Usenet. :) > So you admit you're a troll? No, I'm not trolling. But I'm sure some people see my posts like that. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? >> Would you argue that 1/5 is also defined by an algorithm? >Yes. All mathematical operations are procedures or algorithms. >Some make reference to procedures that can never produce an answer, like >1/0, and some make reference to procedures that never terminate, like >.111.... Does 0.20000000.... terminate or not? Does it make a difference if I tell you that it was written in base 3? Alan -- Defendit numerus === Subject: Re: Cantor's diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w >> Would you argue that 1/5 is also defined by an algorithm? >Yes. All mathematical operations are procedures or algorithms. >Some make reference to procedures that can never produce an answer, like >1/0, and some make reference to procedures that never terminate, like >.111.... > Does 0.20000000.... terminate or not? No. But 0.2 does. :) > Does it make a difference if I > tell you that it was written in base 3? No. The ... langauge you used means does not terminate. > Alan -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? :> :>> Would you argue that 1/5 is also defined by an algorithm? :> :>Yes. All mathematical operations are procedures or algorithms. :> :>Some make reference to procedures that can never produce an answer, like :>1/0, and some make reference to procedures that never terminate, like :>.111.... :> Does 0.20000000.... terminate or not? : No. But 0.2 does. :) So does 1/5 terminate or not? Does 1/5 = 0.2, or does it equal 0.20000000.....? I would say that it equals both and the idea of 1/5 'terminating' is not well defined. Stephen === Subject: Re: Cantor's diagonal proof wrong? >:> >:>> Would you argue that 1/5 is also defined by an algorithm? >:> >:>Yes. All mathematical operations are procedures or algorithms. >:> >:>Some make reference to procedures that can never produce an answer, like >:>1/0, and some make reference to procedures that never terminate, like >:>.111.... >:> Does 0.20000000.... terminate or not? >: No. But 0.2 does. :) >So does 1/5 terminate or not? Does 1/5 = 0.2, or does it equal >0.20000000.....? I would say that it equals both and the >idea of 1/5 'terminating' is not well defined. And after you are done answering that, does 2/3 terminate or not? Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3). Alan -- Defendit numerus === Subject: Re: Cantor's diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > And after you are done answering that, does 2/3 terminate or not? > Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3). the silly questions (and answers) back to the big picture of why on earth I was talking about procedures that do or do not terminate. You should read that if you want to understand why I was talking about these things. I believe math was created as a langauge for talking about what can, and can not do, and know, about procedures. But somewhere, it feels to me, like it went off track and made up some rules of logic which violate the laws of nature (the laws of procedures). Did it go off track or not? Is it a bad thing even if it did? I don't know for sure. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? :> And after you are done answering that, does 2/3 terminate or not? :> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3). : the silly questions (and answers) back to the big picture of why on earth I : was talking about procedures that do or do not terminate. You should read : that if you want to understand why I was talking about these things. : I believe math was created as a langauge for talking about what can, and : can not do, and know, about procedures. But somewhere, it feels to me, : like it went off track and made up some rules of logic which violate the : laws of nature (the laws of procedures). Did it go off track or not? Is : it a bad thing even if it did? I don't know for sure. Why do you believe this, and why do you think if went off track? You have not actually given any evidence for your laws of nature or that something has gone off track? Despite your belief math is clearly capable of discussing things that are not physically possible. Even something as simple as 358203583*2347922423 = 841034224524641609 is not a procedure that a human can complete. Do you think that you enough objects that you could arrange in a grid 358 million by 2.3 billion, then count up all the objects needed to fill in the grid and verify that it is indeed 841 quintillion? In any case, I thought you were interested in AI, and that you were going to solve the AI problem? I would expect a successful AI to understand Cantor's proof. I would also expect a successful AI to understand why the diagonal argument does not apply to the natural numbers or the rational numbers. The fact that you do not understand Cantor's proof does not mean that your AI should not. Stephen === Subject: Re: Cantor's diagonal proof wrong? %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > :> And after you are done answering that, does 2/3 terminate or not? > :> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3). > : get the silly questions (and answers) back to the big picture of why on > : earth I was talking about procedures that do or do not terminate. You > : should read that if you want to understand why I was talking about > : these things. > : I believe math was created as a language for talking about what can, > : and can not do, and know, about procedures. But somewhere, it feels to > : me, like it went off track and made up some rules of logic which > : violate the laws of nature (the laws of procedures). Did it go off > : track or not? Is it a bad thing even if it did? I don't know for > : sure. > Why do you believe this, and why do you think if went off track? > You have not actually given any evidence for your laws of nature > or that something has gone off track? Yeah, that's a good question to ask. It's the question I've hard a hard time finding a way to answer. I've put forth a lot of evidence, but I've not put forth a convincing argument for anyone yet. I present an argument that I hope some people will see shows how the logic of the diagonal argument has problems. This counter argument doesn't depend on the definition of numbers, or reals, or anything said in set theory. It simply shows that any argument that takes the form of the diagonal argument to prove that a value can not be in an infinite table is invalid. Maybe this will help some people open their minds to the possibility that what they are looking at is just an illusion. That there might be more hidden behind the language than they every suspected. The trick to this problem is that it's a matter of faith. Like believing in God, once you accept it as fact, it's hard to see anything else. You build a huge set of defensive arguments to support the initial belief, and any single piece of evidence presented is easily ignored against the fortress of defense built to support the first belief. You can't even see the other position until you are first willing to, if only for a second, open a crack in your defenses and look with new eyes at the evidence - even though it seems to contradict everything you believe in. > Despite your belief math > is clearly capable of discussing things that are not physically > possible. Even something as simple as > 358203583*2347922423 = 841034224524641609 > is not a procedure that a human can complete. Do you think > that you enough objects that you could arrange in a grid 358 > million by 2.3 billion, then count up all the objects needed > to fill in the grid and verify that it is indeed 841 quintillion? The larger the problem, to more expensive in time and energy it becomes to solve. If it's a finite problem, then the only question is do you have enough time and energy and matter to do it. At some upper end of that problem, there may not be enough matter in the universe to build a machine that could solve the problem - but we don't really know the full nature of the universe yet. What we do know however, is that any infinite processes you specify will require an infinite amount of time and energy, and will never complete. > In any case, I thought you were interested in AI, and that you were > going to solve the AI problem? Yeah, I am. And BTW, I don't need to shed light on this area of math in order to solve AI. This is not something I must resolve to finish my AI work. It's just something that because of my AI work, I was able to spot a problem with. So I became curious to further understand the nature of this problem I spotted. > I would expect a successful AI > to understand Cantor's proof. I would also expect a successful > AI to understand why the diagonal argument does not apply to > the natural numbers or the rational numbers. The fact that > you do not understand Cantor's proof does not mean that your > AI should not. I understand Cantor's proof. It's trivial to understand. What's much harder to understand is why it's an invalid proof. You might remember I started this thread by explaining I used to believe the proof. I was taught the proof some 25 years ago in school and thought it was a very cool proof and understood the logic instantly even though the results were surprising. I spent the next 25 years believing it was an obviously valid proof and when I've seen the same proof used in other fields like computer science, I instantly believed the results of those proofs. I no longer believe that. I know things know that I did not know 25 years ago. I now know what we are and why we do the things we do. I know what language is now. My AI would have no problem believing that Cantor's proof is valid just like I did and just like you do. It would have no problem learning to talk just like you do. If it were educated the same way you were, it too would be posting the type of messages you post in order to understand what this fool named Curt thought he was talking about. But with enough of the proper education, it would also be able to understand what I was talking about. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Cantor's diagonal proof wrong? >> :> And after you are done answering that, does 2/3 terminate or not? >> :> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3). >> : get the silly questions (and answers) back to the big picture of why on >> : earth I was talking about procedures that do or do not terminate. You >> : should read that if you want to understand why I was talking about >> : these things. >> : I believe math was created as a language for talking about what can, >> : and can not do, and know, about procedures. But somewhere, it feels to >> : me, like it went off track and made up some rules of logic which >> : violate the laws of nature (the laws of procedures). Did it go off >> : track or not? Is it a bad thing even if it did? I don't know for >> : sure. >> Why do you believe this, and why do you think if went off track? >> You have not actually given any evidence for your laws of nature >> or that something has gone off track? >Yeah, that's a good question to ask. It's the question I've hard a hard >time finding a way to answer. I've put forth a lot of evidence, but I've >not put forth a convincing argument for anyone yet. >I present an argument that I hope some people will see shows how the logic >of the diagonal argument has problems. This counter argument doesn't >depend on the definition of numbers, or reals, or anything said in set >theory. It simply shows that any argument that takes the form of the >diagonal argument to prove that a value can not be in an infinite table is >invalid. No, it does not show any such thing. At _most_ it shows something of the form if you look at it that way then the proof is invalid, which is simply irrelevant, because the way you suggest people look at it is simply not consistent with what the words in the statement of the theorem _mean_. _If_ one interprets the reals are uncountable as meaning the moon is made of green cheese then the reals are uncountable does indeed become false, but this is simply silly because of that if - the reals are uncountable does _not_ mean the moon is made of green cheese. >Maybe this will help some people open their minds to the >possibility that what they are looking at is just an illusion. That there >might be more hidden behind the language than they every suspected. >The trick to this problem is that it's a matter of faith. Like believing >in God, once you accept it as fact, it's hard to see anything else. You >build a huge set of defensive arguments to support the initial belief, and >any single piece of evidence presented is easily ignored against the >fortress of defense built to support the first belief. You can't even see >the other position until you are first willing to, if only for a second, >open a crack in your defenses and look with new eyes at the evidence - even >though it seems to contradict everything you believe in. Find a mirror somewhere. This is a precise description of what the rest of us see you doing: You started with an explanation why the proof was wrong. You eventually agreed that your initial explanation was totally bogus. But that doesn't seem to have had any effect on your conviction that the proof is wrong - you just continue to invent new explanations, gradually becoming more and more vague and less relevant to what the theorem actually _says_. >> [...] >What we do know however, is that any infinite processes you specify will >require an infinite amount of time and energy, and will never complete. And this has no relevance whatever, because the statement of the theorem has nothing to do with processes. >> In any case, I thought you were interested in AI, and that you were >> going to solve the AI problem? >Yeah, I am. And BTW, I don't need to shed light on this area of math in >order to solve AI. This is not something I must resolve to finish my AI >work. It's just something that because of my AI work, I was able to spot a >problem with. So I became curious to further understand the nature of this >problem I spotted. >> I would expect a successful AI >> to understand Cantor's proof. I would also expect a successful >> AI to understand why the diagonal argument does not apply to >> the natural numbers or the rational numbers. The fact that >> you do not understand Cantor's proof does not mean that your >> AI should not. >I understand Cantor's proof. If you think that the statement above about completing infinite processes has some relevance then you don't even understand the _statement_ of the theorem, much less the proof. >It's trivial to understand. What's much >harder to understand is why it's an invalid proof. You might remember I >started this thread by explaining I used to believe the proof. I was >taught the proof some 25 years ago in school and thought it was a very cool >proof and understood the logic instantly even though the results were >surprising. I spent the next 25 years believing it was an obviously valid >proof and when I've seen the same proof used in other fields like computer >science, I instantly believed the results of those proofs. I no longer >believe that. I know things know that I did not know 25 years ago. I now >know what we are and why we do the things we do. I know what language is >now. >My AI would have no problem believing that Cantor's proof is valid just >like I did and just like you do. It would have no problem learning to talk >just like you do. If it were educated the same way you were, it too would >be posting the type of messages you post in order to understand what this >fool named Curt thought he was talking about. But with enough of the >proper education, it would also be able to understand what I was talking >about. ************************ David C. Ullrich === Subject: Re: Cantor's diagonal proof wrong? > I suspect what this adds up to is that I just have to finish my AI work, > build machines that are smarter than humans, and then once people become > interested in how these machines work, and learn for themselves how they > work, they will begin to understand the problem with the foundation of > mathematics. If you can't handle mathematics you will not succed with your AI work. Yoda says: Doomed are you Young Curt. Hold not your breath until AI have