mm-552 === Subject: Re: Running wide >Let A be a simple closed convex curve in R^2 with [CapitalThorn]nite length. >Let B be the set of points outside A distant 1 from A. >Is (length of B) - (length of A) = 2 Pi ? >Yes. Hard to believe but it's so. Luckily the delay between my saying that and the email asking for a proof gave me a little time to think about how to prove it... Calling the following a proof might be a slight exaggeration, but it's pretty convincing, I think: Suppose the curves are smooth. Say c_2(t) (0 <= t <= 1) is a parametrization of B. Now for every t let c_1(t) be the point of A closest to c_2(t). The convexity shows that c_1 is a parametrization of A. Now, using complex notation instead of vectors in R^2: The fact that |c_2(t) - c_1(t)| = 1 shows that there is a continuous function a(t) such that c_2(t) = c_1(t) + e^{ia(t)}, so c_2'(t) = c_1'(t) + i a'(t) e^{ia(t)}. The fact that c_1(t) is the point of A closest to c_2(t) shows that c_2(t) - c_1(t) is perpendicular to c_1'(t); if you draw a little picture you convince yourself that hence c_1'(t) and i a'(t) e^{ia(t)} are parallel and in opposite directions. So |c_2'(t)| = |c_1'(t) + i a'(t) e^{ia(t)}| = |c_1'(t)| + |i a'(t) e^{ia(t)}| = |c_2'(t)| = |c_1'(t)| + |a'(t)| So length(B) = int|c_2'| = length(A) + int|a'| = length(A) + 2Pi. ************************ David C. Ullrich === Subject: Re: Four people crossing a bridge >That's a good point. If the bridge could not in fact hold more than two >people at a time .. well, with four people, you should be able to build >a better bridge! > >John > Maybe so, but it'll be a real bitch building the bridge in the middle > of the night if you only have one ßashlight. > Wait until morning. Then they will be able to cross the bridge in 22 minutes. === Subject: Re: The Meaning of Abstract >I'm surprised that some people take abstract as a vague >common-sense concept. To me, it has a precise technical meaning: >lossy compression. >That's a very narrow meaning that has little currency in everyday >speech. >Your meaning fails to accommodate almost any of the uses of abstract >in mathematics, for instance. Algebra is not a lossy compression of >number theory or whatever else you might want to plug in. >It's no wonder that you're a fan of Chaitin. You're more than willing >to spread horribly sloppy attempts at thought. > lossy compression is far from horribly sloppy. It's obviously correct. > Anyone that doesn't understand the reference doesn't understand > abstraction. > You can also think of abstraction as pattern matching or signal [CapitalThorn]ltering. > It's just different ways to talk about the same thing. Well its not so very totally obviously correct to me; although i must admit that i ßashed as such when i [CapitalThorn]rst read it. Thing is that an abstraction is not located in time and space which is the most usual criteria of something that is not physical. However, a compressed set of marks is de[CapitalThorn]nitely located in time and space. patty === Subject: Re: The Meaning of Abstract >I'm surprised that some people take abstract as a vague common-sense >concept. To me, it has a precise technical meaning: lossy compression. >A program is abstract, because it *loses* the architectural details of >a computation, and it is concise. A blueprint of a house is abstract >because it *loses* the architectural and material properties of an >actual house, and it is concise. >Instantiation is most certainly a Platonist word which includes >counter-factuals in its meaning. An abstract entity represents >another entity in a purposeful way, it is a sign that points to >another object or sign. >Maybe we should all study semiotics instead of Platonist computer >science and mathematics! Maybe that is how one truly becomes a >hard-core materialist! Hi Eray - I didn't know that instantiation comes from Plato. The word comes from Latin and means to provide a concrete instance of. But I still don't like it because people seem to use it to indicate the the agency of physical realization of an abstract rather than a mere concrete instance or example described by an abstract. I'm not sure your telling us a lot either by de[CapitalThorn]ning abstract as lossy compression. These aspects of an abstract may be true without being de[CapitalThorn]nitive. When I inhale air there is lossy compression too. I think you'd be better off just analyzing predicates in relation to one another and recognizing that predicates are abstractions because they're objective and they're objective for reasons that aren't plain but that relate to the mechanics of differences between differences. In point of fact the abstraction comes from differential mechanics in which the lossy part of the abstraction refers to the loss of identity as the result of taking differences between concrete things out there. === Subject: Re: The Meaning of Abstract Discussion, linux) > > I'm surprised that some people take abstract as a vague > common-sense concept. To me, it has a precise technical meaning: > lossy compression. > That's a very narrow meaning that has little currency in everyday > speech. > Your meaning fails to accommodate almost any of the uses of abstract > in mathematics, for instance. Algebra is not a lossy compression of > number theory or whatever else you might want to plug in. > It's no wonder that you're a fan of Chaitin. You're more than willing > to spread horribly sloppy attempts at thought. > lossy compression is far from horribly sloppy. It's obviously > correct. Anyone that doesn't understand the reference doesn't > understand abstraction. I guess I don't understand abstraction. Is abstract supposed to be a synonym for lossy compression? If so, do you regard an mp3 ripped from a CD as an abstraction of the original wave [CapitalThorn]le? > You can also think of abstraction as pattern matching or signal > [CapitalThorn]ltering. It's just different ways to talk about the same thing. These uses don't capture the full meaning of abstraction in either mathematics or philosophy, near as I can [CapitalThorn]gure. I don't see that, for instance, algebra has arisen via any of the notions of lossy compression, pattern matching or signal [CapitalThorn]ltering from the various concrete mathematical structures which preceded it. The act of [CapitalThorn]ltering a signal or lossily compressing a source is very different to my mind than the mathematical (or philosophical) act of abstraction. Evidently, what is obviously correct in your view is horribly sloppy in mine. -- Not all features that are found on the Security tab are designed to help make your documents and [CapitalThorn]les more secure. --Microsoft on Of[CapitalThorn]ce security features (after it was pointed out by a third party that a certain password setting is easily bypassed.) === Subject: Re: George Green, portrait? > George Green 1793--1841 > ... the general mathematical theory of potential developed by an > obscure, self-taught miller's son would lead to the mathematical > theories of electricity underlying twentieth-century industry. > quoted at > > Most of the biographies at The MacTutor History of Mathematics archive > have portraits of the subject. (Even Euclid and Archimedes...) But > not Green. Are there no known portraits of this man? > -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ Interesting reading (George Green 1793-1841) Ref: http://www.nottingham.ac.uk/physics/gg/ http://www.stpetersnottingham.org/sermon/green.htm http://www-groups.dcs.st-andrews.ac.uk/~history/ Mathematicians/Green.html === Subject: Re: The Meaning of Abstract > I guess I don't understand abstraction. > Is abstract supposed to be a synonym for lossy compression? If > so, do you regard an mp3 ripped from a CD as an abstraction of the > original wave [CapitalThorn]le? I would de[CapitalThorn]nitely understand an mp3 as an abstract representation of the raw waveform data contained on a CD. Do you have a better example for the point you are trying to make? > The act of [CapitalThorn]ltering a signal or lossily compressing a source is very > different to my mind than the mathematical (or philosophical) act of > abstraction. Evidently, what is obviously correct in your view is > horribly sloppy in mine. I wouldn't go as far as to say it's obviously correct, however it lossy compression is vaguely how I understand abstraction as well. A few different questions: Does it have to be lossy? Can't abstraction be lossless as well? If it is lossless, is it abstraction or merely a different form representation? How can we universally quantify the level of abstraction? Nathan === Subject: Re: Running wide I'm looking for a proof of the problem as stated (or counterexample), not one that assumes the existence of derivatives. Please say if you think that is not realistic. === Subject: Re: The Meaning of Abstract %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT'{w X-Url: http://CurtWelch.Com/ > lossy compression is far from horribly sloppy. It's obviously > correct. Anyone that doesn't understand the reference doesn't > understand abstraction. > You can also think of abstraction as pattern matching or signal > [CapitalThorn]ltering. It's just different ways to talk about the same thing. > Well its not so very totally obviously correct to me; although i must > admit that i ßashed as such when i [CapitalThorn]rst read it. Thing is that an > abstraction is not located in time and space which is the most usual > criteria of something that is not physical. However, a compressed set > of marks is de[CapitalThorn]nitely located in time and space. The belief that an abstraction is not located in time and space is just a false, and useless belief. We have the power to make stuff up that has never existed by combining features from real things together in ways that we have never seen before. Such as a ßying elephant. Our power to generate these ideas are limitless. But, does the idea of a ßying elephant help you understand yourself or the universe? Is it real? Well, the concept is real beacuse I just created it for this post. And we all understand the concept so it's real in that sense even if there are no ßying elephants in the world. Is the belief that an abstraction is not located in time and space a ßying elephant or is it real? It is in fact just a ßying elephant. Show me an abstraction that doesn't exist in time and space. You can't do it because anything you can show me, by de[CapitalThorn]ntion has to exist in time and space. All you can do is talk to me about ßying elephants and say they are real and tell me that you believe they are real. To say that you believe something is real is to believe that you will one day see one, but that you just haven't seen one yet. To say that something is real but can never been seen, is your perosonal choice to believe that things which can never be senseed, are in fact real. And if you belive that, then I assume you believe not only that abstractions don't exist in time and space, but that ßying elephants are also just as real. Our culture accepts fairly easilly the idea that things we can't sense are real. This happens for many reasons. Mostly it happens because we put a large amount of faith in second-hand knowledge. Bob said he saw a ßying elephant, and I trust Bob, so even though I've never seen one, I too believe they are real. Other people said they have seen time distort according to Einstein's predictions so I believe it is real even though I have never seen it happen. Everyone knows that concepts exist in a dimension outside of time and space so everyone believes it. Everyone is wrong. Or, more accurately, they are just talking about ßying elephants because it's doesn't hurt to do so - until you try to [CapitalThorn]gure out how the brain works and what consciousness is. The other reason we accept the idea so easily is the fact that these things happen inside our head - where we can sense them in our thoughts - but where we can not tie them to our other sensory inputs from the physcial world. When we see an action, and can tie that to a sound, and a smell, and a taste, and touch, then we know the thing we sensed exists in the world of all those senses. But when we sense our thoughts, we see no connections to our other senses - we can not hear, see, taste, or feel, our own thoughts. That makes us believe our thoughts exist in a separte world from the world of the other 5 senses. But, we have learned enough about the brain to know this is not true. Eerything we think about is linked to brain activity, which we can sense in physical world - even if most of us have never gotten the chance to do that. What we in fact have is a brain full of pattern recognition hardware. We have pattern recognition hardware that can detect elephants, and hardware which can detect ßying things. And if we ever saw a ßying elephant, we would be able to detect it in a heartbeat. We would know what we were looking at. And we have trained our pattern recognition hardware for detecting elephant to also respond to the word elephant. But we do not confuse the real thing for the word because we know that elepahant in the context of the hearing the word elephant is different than elepahnt in the context of seeing a large animal. It's the functioning of this pattern matching hardware which is creating abstractions. It's a lossy compression system for responding to some aspects of the data, and ignoring other aspects. Our pattern matching hardware is the de[CapitalThorn]ntion of all abstractions we know about. The abstraction exists in the form of the hardware which produces it, which is very real and physical and exists in time and space, and in the outputs it generates each time it is used, which is also something that is very real in time and space. The belief that an abstraction could be timeless is the belief that these pattern matching machines could exist forever, and could exist without a physcial form - both which are impossible - i.e., ßying elephants. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com Webmaster for http://NewsReader.Com/ === Subject: Re: The Meaning of Abstract %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT'{w X-Url: http://CurtWelch.Com/ > > > I'm surprised that some people take abstract as a vague > common-sense concept. To me, it has a precise technical meaning: > lossy compression. > > That's a very narrow meaning that has little currency in everyday > speech. > > Your meaning fails to accommodate almost any of the uses of abstract > in mathematics, for instance. Algebra is not a lossy compression of > number theory or whatever else you might want to plug in. > > It's no wonder that you're a fan of Chaitin. You're more than willing > to spread horribly sloppy attempts at thought. > lossy compression is far from horribly sloppy. It's obviously > correct. Anyone that doesn't understand the reference doesn't > understand abstraction. > I guess I don't understand abstraction. > Is abstract supposed to be a synonym for lossy compression? If > so, do you regard an mp3 ripped from a CD as an abstraction of the > original wave [CapitalThorn]le? Of course it is. Is there not a large set of CDs that all have different data values stored on them, yet they would all be abstractly represented by the same mp3 [CapitalThorn]le? To make it easier to see, think of a lossy compression algorithm which is easy to reverse. Just sum the values of all the bytes in a [CapitalThorn]le together using 8 bit math so that you end up with a single 8 bit check sum value to represent the [CapitalThorn]le with. Are there not an in[CapitalThorn]nite numbner of [CapitalThorn]les which are abstractly represented with the same 8 bit sum? Am I not able to talk about all [CapitalThorn]les with an 8 bit check sum value of 107 as an abstract set of [CapitalThorn]les? > You can also think of abstraction as pattern matching or signal > [CapitalThorn]ltering. It's just different ways to talk about the same thing. > These uses don't capture the full meaning of abstraction in either > mathematics or philosophy, near as I can [CapitalThorn]gure. They do, but you just don't see it yet. > I don't see that, for instance, algebra has arisen via any of the > notions of lossy compression, pattern matching or signal > [CapitalThorn]ltering from the various concrete mathematical structures which > preceded it. Why it arose of course is a harder idea to get a handle on but the fact that it is created by combining abstractions together is not so hard to grasp once you learn to look at it like that. Do you not see that numbers are just abstactions created by combining different properties of real things together? 3 sheep and 3 oranges share a common abstract property which we call 3. Do you not see that all properties are abstractions of real things? > The act of [CapitalThorn]ltering a signal or lossily compressing a source is very > different to my mind than the mathematical (or philosophical) act of > abstraction. Evidently, what is obviously correct in your view is > horribly sloppy in mine. Sure, I know we look at it very differently. I'm just trying to show you how it's possible to adjust your set of abstractions about the world to make it [CapitalThorn]t my set of abstractions and about how the world becomes much simpler when you do that (i.e., many things which seemed unrelated are actually very much related). -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com Webmaster for http://NewsReader.Com/ === Subject: Re: The Meaning of Abstract charset=utf-8 Nathan Funk [CapitalEth][EDouble Dot][Micro] .b3 .b9[EDo ubleDot].b9 > A few different questions: Does it have to be lossy? Can't abstraction > be lossless as well? If it is lossless, is it abstraction or merely a > different form representation? How can we universally quantify the level > of abstraction? One possible way to visualize whatever the original poster means by abstraction, would be to look at some of the evolved computer languages that were (and are) used to represent various objects abstractly. In most (if not all) of these languages, the notion of pointer is of fundamental importance. Imo, the most natural way to de[CapitalThorn]ne abstraction for a certain set of data, is to provide a pointer to the data. In such models, abstraction doesn't have to be lossy and it appears to be a different form of representation of the same object. In away, the Index of an encyclopedia (or book) seems to me like an abstraction of the actual contents. > Nathan -- Ioannis Galidakis http://users.forthnet.gr/ath/jgal/ ------------------------------------------ Eventually, _everything_ is understandable === Subject: Re: The Meaning of Abstract Discussion, linux) > I guess I don't understand abstraction. > Is abstract supposed to be a synonym for lossy compression? If > so, do you regard an mp3 ripped from a CD as an abstraction of the > original wave [CapitalThorn]le? > I would de[CapitalThorn]nitely understand an mp3 as an abstract representation of > the raw waveform data contained on a CD. Do you have a better example > for the point you are trying to make? Most of us don't regard the relation between mp3 and wave [CapitalThorn]le as the same as the relation between the concept of triangle and *all* of the particular three-sided [CapitalThorn]gures with which we are familiar[1]. That x is created by lossy compression from a source y puts x in a relationship with a particular y. When we abstract, we forget certain details of each in a collection of instances, not the details of just one particular instance. But even this forgetting is not really the de[CapitalThorn]ning feature of abstraction. Consider the operation that takes algebras to their underlying sets by forgetting the algebraic structure. This operation loses some data, just like lossy compression, but it is not an act that most of us would call abstraction. One doesn't abstract algebras to reach a notion of set[2]. The fact that data is lost or forgotten doesn't capture the notion of abstraction, although it may be a feature of abstraction (similarly, there is nothing about compression as far as the forgetting is concerned). Anyway, if you want to say that an mp3 is an abstraction of a particular wave [CapitalThorn]le, well, you can have that reductio ad absurdum. Footnotes: [1] I'm not taking the position that the notion of triangle is really somehow abstracted from our experience with particular three-sided-[CapitalThorn]gures here. [2] I can't say why this *isn't* a good example of abstraction, but I think one would [CapitalThorn]nd that most philosophers of mathematics wouldn't call this forgetful functor an instance of abstraction. -- So, at this time, I'd like to assure you that I am not interested in making sure mathematicians worldwide get [CapitalThorn]red. I've rethought my desire to go to Congress and try to get funding for mathematicians cut. -- James Harris is a reasonable man. Whew! === Subject: Re: The Meaning of Abstract Discussion, linux) > > > I'm surprised that some people take abstract as a vague > common-sense concept. To me, it has a precise technical meaning: > lossy compression. > > That's a very narrow meaning that has little currency in everyday > speech. > > Your meaning fails to accommodate almost any of the uses of abstract > in mathematics, for instance. Algebra is not a lossy compression of > number theory or whatever else you might want to plug in. > > It's no wonder that you're a fan of Chaitin. You're more than willing > to spread horribly sloppy attempts at thought. > > lossy compression is far from horribly sloppy. It's obviously > correct. Anyone that doesn't understand the reference doesn't > understand abstraction. > I guess I don't understand abstraction. > Is abstract supposed to be a synonym for lossy compression? If > so, do you regard an mp3 ripped from a CD as an abstraction of the > original wave [CapitalThorn]le? > Of course it is. Is there not a large set of CDs that all have > different data values stored on them, yet they would all be > abstractly represented by the same mp3 [CapitalThorn]le? Most of us distinguish between quotienting by an equivalence relation and abstracting. To repeat the example I posted a moment ago: [CapitalThorn]x a signature S and consider the functor which takes S-algebras to their underlying set. Just like your example, for each set, there are a slew of S-algebras which are mapped to that set. I would not think that this forgetful functor is an example of abstracting. I think it's fairly evident that you and I are using the word abstraction in different senses. [...] > You can also think of abstraction as pattern matching or signal > [CapitalThorn]ltering. It's just different ways to talk about the same thing. > These uses don't capture the full meaning of abstraction in either > mathematics or philosophy, near as I can [CapitalThorn]gure. > They do, but you just don't see it yet. Don't patronize. > I don't see that, for instance, algebra has arisen via any of the > notions of lossy compression, pattern matching or signal > [CapitalThorn]ltering from the various concrete mathematical structures which > preceded it. > Why it arose of course is a harder idea to get a handle on but the fact > that it is created by combining abstractions together is not so hard to > grasp once you learn to look at it like that. > Do you not see that numbers are just abstactions created by combining > different properties of real things together? 3 sheep and 3 oranges share > a common abstract property which we call 3. Do you not see that all > properties are abstractions of real things? Oh, no, no, no. I am not walking down this path with you. I can already predict that a discussion of the origins of the concept of number will be somewhat more painful than licking razor blades. > The act of [CapitalThorn]ltering a signal or lossily compressing a source is very > different to my mind than the mathematical (or philosophical) act of > abstraction. Evidently, what is obviously correct in your view is > horribly sloppy in mine. > Sure, I know we look at it very differently. I'm just trying to show you > how it's possible to adjust your set of abstractions about the world to > make it [CapitalThorn]t my set of abstractions and about how the world becomes much > simpler when you do that (i.e., many things which seemed unrelated are > actually very much related). The only thing common to all of these notions is that in each, certain information is lost. Big deal. That's not all that deep and it doesn't seem to capture the essence of abstraction. In particular, the actual acts of compression or signal [CapitalThorn]ltering take a single input to produce a new, simpler output. That's different than the abstraction from certain particulars of individuals in a set to arrive at a new concept in which each individual is an instance. These are just different conceptual acts. I believe that this is also different than pattern matching, in which one [CapitalThorn]nds common features rather than [CapitalThorn]nding a broader notion the includes each of the particulars. -- Jesse F. Hughes History will hate you and love me. I'm the misunderstood and persecuted genius. You're the assholes. -- James Harris === Subject: Foundations textbooks My name is Andrew Cousino and I am very interested in studying logic. In particular, I am interested in studying recursion theory. A professor at my undergraduate school turned me on to the subject. Unfortuneately, the graduate school that I am going to does not offer any extensive study in logic. I would appreciate it if I got some suggestions for books on logic and eventually recursion theory out of which I could teach myself. Assume that my knowledge of logic and set Andrew Cousino acousino@knox.edu === Subject: laser cooling Varney does not have the right to talk to me because of his iq. Varney is a loser. It is probably true most 12 year old kids understand physics better than him. Varney is from the physics groups. It is slime like him that destroyed the physics groups. Now only slime, ideits, fools, nuts and people who are more with just helping to destroy the physics groups driving away people who know a lot about physics. The slime needs to slime to the math groups to make them worse. I guess slime needs to spread like aids. Most dogs would not read the physics groups and that kind of tells what the drop food from dogs on those people. Varney helped to make the physics groups as bad as his lousey life. === Subject: Re: The Meaning of Abstract Discussion, linux) > Of course it is. Is there not a large set of CDs that all have > different data values stored on them, yet they would all be > abstractly represented by the same mp3 [CapitalThorn]le? > Most of us distinguish between quotienting by an equivalence relation > and abstracting. > To repeat the example I posted a moment ago: [CapitalThorn]x a signature S and > consider the functor which takes S-algebras to their underlying set. > Just like your example, for each set, there are a slew of S-algebras > which are mapped to that set. I would not think that this forgetful > functor is an example of abstracting. Stupid! A simpler example. Take the map N -> N taking 0 to 0 and n+1 to n. This is a lossy compression scheme. Would you *really* want to call it an abstraction? -- Jesse F. Hughes Even I, who know beyond doubt that my death will be caused by a silly girl, will not hesitate when that girl passes by. -- Merlin, as reported by John Steinbeck. === Subject: Re: Foundations textbooks Discussion, linux) > My name is Andrew Cousino and I am very interested in studying logic. > In particular, I am interested in studying recursion theory. A > professor at my undergraduate school turned me on to the subject. > Unfortuneately, the graduate school that I am going to does not offer > any extensive study in logic. I would appreciate it if I got some > suggestions for books on logic and eventually recursion theory out of > which I could teach myself. Assume that my knowledge of logic and set I recommend Boolos, Burgess and Jeffrey's Computability and Logic. It is a nice introduction to logic with a special emphasis on its relationship to computability. -- Quincy, would you rather do epistemology or conceptual analysis? You know what? I'd rather [CapitalThorn]ght on an aircraft carrier.... And Mama and Baba (Papa) would [CapitalThorn]ght on an aircraft carrier, too. -- Quincy P. Hughes, age 3 1/2 === Subject: Re: The Meaning of Abstract > I'm surprised that some people take abstract as a vague > common-sense concept. To me, it has a precise technical > meaning: lossy compression. I have to agree with Mr. Hughes. If you [CapitalThorn]nd that to be the essential feature of abstraction, then I fear you haven't abstracted the notion properly. ;) I think a better meaning is: to identify the necessary features of a concept. Lossy compression is insuf[CapitalThorn]cient to describe this version because it often includes features that are *not* essential to the abstraction of an instance. If I show you a picture of a girl, and you produce a JPEG image of it and present it to me as an abstraction, I'll say you're full of it. Now, if I ask a 5 year old to draw what he sees in the picture, he will most likely draw a stick [CapitalThorn]gure of a girl, identifying what he believes to be the essential features of the image. And I'm fairly certain that a large survey with our two abstractions will show that the stick [CapitalThorn]gure is a much better abstraction than the JPEG. Why? Because the JPEG contains too much information. It takes out redundant information, but only on a local level. The JPEG compression algorithm necessarily cannot identify the information contained in the picture that depends on having human knowledge and experience. And that is why it does not produce a useful abstraction. The same can be said of the MP3 [CapitalThorn]le. An MP3 is not a very good abstraction of the original. But a piano rendition with a monophonic melody and simple chord accompaniment *would* be what many musicians regard as an abstraction of the original song. > A program is abstract, because it *loses* the architectural details > of a computation, and it is concise. Actually, programs aren't very abstract at all. They tend to be rather concrete engineering solutions to practical problems. And a review of www.dictionary.com shows that concrete and practical are practically antonyms of abstract. > A blueprint of a house is abstract because it *loses* the > architectural and material properties of an actual house, and it is > concise. > [...] A blueprint is certainly more abstract than an actual house, but not because of what it is missing. It is abstraction because of what it has, which is the essential features of the house. Whether it speci[CapitalThorn]es the placement of all the electrical outlets or not is of no concern. Whether it conveys the proper layout of rooms and ßoors is. If I created a blueprint that located all the electrical outlets in the house but not any of the walls, according to your de[CapitalThorn]nition, it is an abstraction of the house, because I have *lost* detail, and created a concise version of the house. But merely losing detail is not the key feature. The key feature is identifying the details that we use to classify the concrete instances. Abstraction is a notion concerned with classes and types, not details and data representation. Such a de[CapitalThorn]nition is not suf[CapitalThorn]ciently abstract. Dave --- Outgoing mail is certi[CapitalThorn]ed Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). === Subject: Re: Four people crossing a bridge billwilkinson@mindspring.com mumbles... >That's a good point. If the bridge could not in fact hold more than two >people at a time .. well, with four people, you should be able to build >a better bridge! > >John >Maybe so, but it'll be a real bitch building the bridge in the middle >of the night if you only have one ßashlight. > Wait until morning. >Won't the batteries be dead by then? Batteries?? http://www.peak-berlin.com/artbild/11485.jpg -- *MM Replace Ôdot' with a dot to reply === Subject: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax X-Meow: Approved X-Tra: time X-Files: off Kirk posted: > BTW: explain me how a how a gradient can be negative. > [Hammond] > Perfect example of your scienti[CapitalThorn]c naivete..... Ellipses are not necessary in an debate, unless you want to attract the attention of an imaginary public, but maybe you're just a clown all the time? > for chrissakes up hill > means a positive gradient down hill means a negative gradient. > increading gravity means positive gradient decreasing gravity means > negative gradient My dear, totally retarded and illiterate ignoramus: http://mathworld.wolfram.com/Gradient.html A gradient is a vector [CapitalThorn]eld (and please don't come and asc me what that means), how on earth can a vector be negative. Can you tell me if the vector (0.1271,-1.3357,0.4478) is positive or negative. Crossposted to sci.math because on topic there. -- mhm 27x12 smeeter #28 Usenet Valhalla Circle #19 & #21 Bartlo's hate lits #1: <40376AD8.C83FBF5A@enter.net> CEO Alcatroll Labs Inc. Alexa knows everything about PSI, electromagnetic radiation and quantum mechanics: They are beaming earth with EMR which is interfering with natural human PSI abilities, which is likely quantum. in alt.alien.visitors === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax >posted: > BTW: explain me how a how a gradient can be negative. > [Hammond] > Perfect example of your scienti[CapitalThorn]c naivete..... >Ellipses are not necessary in an debate, unless you want to attract the >attention of an imaginary public, but maybe you're just a clown all the >time? > for chrissakes up hill > means a positive gradient down hill means a negative gradient. > increading gravity means positive gradient decreasing gravity means > negative gradient >My dear, totally retarded and illiterate ignoramus: >http://mathworld.wolfram.com/Gradient.html >A gradient is a vector [CapitalThorn]eld (and please don't come and asc me what that >means), how on earth can a vector be negative. >Can you tell me if the vector (0.1271,-1.3357,0.4478) is positive or >negative. >Crossposted to sci.math because on topic there. === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax > A gradient is a vector [CapitalThorn]eld (and please don't come and asc me what that > means), how on earth can a vector be negative. That isn't the only meaning of gradient. In the case of a scalar function of a single variable, f(x), the gradient is the [CapitalThorn]rst derivative df/dx which is a scalar. Gib === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax > A gradient is a vector [CapitalThorn]eld (and please don't come and asc me what that > means), how on earth can a vector be negative. >That isn't the only meaning of gradient. In the case of a scalar >function of a single variable, f(x), the gradient is the [CapitalThorn]rst >derivative df/dx which is a scalar. >Gib === Subject: Re: The physics of practical daily life > (For the record, the seven SI units are: meter, kilogram, > second, Ampere, Kelvin, candela, and mole. AFAIK, most light > bulbs, however, use lumens (candela sterradian). Hint hint.) I did say that energy can be derived from Length; Force, and Time: Ever see heat energy generated by persistent pounding or bending; to the point of glowing: Ever see sparks emanating from an emery wheel? Ever generate electricity by patting a cat or sliding accross a seat or a rug: Ouch(;^) Ain't friction grand(:-? It can even make beautiful music; when a rozened bow is properly rubbed over a [CapitalThorn]ddle's strings. Probably has a lot to do with Descartes spinning vortices; don't ask me exactly how; I can only speculate on that. === Subject: Re: Periodic function > f is a real analytic function such that: > f(x, y) = f(x + y, y) > Is it possible f to be expressed in closed form without the use of > trigonometric functions? If g is any periodic function with period 1, then g(x/y) will do. An example would be g(x) = fractional part of x (e.g. g(2.334) = .334) Actually, it's not entirely clear to me what you're asking for. === Subject: Re: The physics of practical daily life > >I've challenged him to explain a light bulb using his three >fundamental measures. > Could you instead suggest an experiment that would distroy his computer? > Maybe he IS a computer? That just occurred to me. His inability to grasp > simple concepts and trying to make them into something that needs gigabytes > of data necessary to explain them all makes sense now. All we need to do > is to get him to calculate pi to the last degree and he'll implode! > DaveL :) Flattery will get you nowhere, and who needs the ratio of a plane circle's circumference divided by it diameter to more than 360 degrees of arc with three or four decimal places per degree? === Subject: Re: Help with testing for nonlinearity by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i51NHoA24613; I am not an expert in stats, but had to solve (test for nonlinearity of time trends) a similar problem a few months ago. I used the higher order statistics based method described in a paper of a colleague of mine: Diagnosis of poor Control loop Performance using Higher Order Statistics M.A.A.S. Choudhary, S.L. Shah and N. Thornhill You can download the preprints of the paper from: http://www.ualberta.ca/dept/chemeng/control/reports.html Hope this helps Vinay >Its been awhile since I've done stats work and I am very rusty. Any >help would be greatly appreciated. >I'm testing for linearity of some datasets against a time trend with a >comparison of the correlation ratio and the correlation coef[CapitalThorn]cient. >I'm running into a few cases where my derived Eta-squared is less than >my R-squared, which, by de[CapitalThorn]nition, cannot be true, right? I've >doubled over my [CapitalThorn]gures and can't [CapitalThorn]nd where I'm going astray. My >correlation coef[CapitalThorn]cients look sound, and I'm deriving my Eta-squared >as the SS(between groups)/SS(total) >Any help/suggestions/observations would be fantastic! >-FX === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax X-Meow: Approved X-Tra: time X-Files: off Kirk posted: > A gradient is a vector [CapitalThorn]eld (and please don't come and asc me what that > means), how on earth can a vector be negative. > That isn't the only meaning of gradient. In the case of a scalar > function of a single variable, f(x), the gradient is the [CapitalThorn]rst derivative > df/dx which is a scalar. > Gib let f: -> f(x) be the function that attaches to each 3d coordinate the degree of humidity (some insects like that function). it maps a three dimensional vector space into the scalars (1 dim reals). The gradient of that function is a 3D vector [CapitalThorn]eld, in each point indicating the direction (and intensity) to [CapitalThorn]nd a local maximum of the function. REMARK: there can be many local maxima and some singulatities of f can guide you the wrong way :-) -- mhm 27x12 smeeter #28 Usenet Valhalla Circle #19 & #21 Bartlo's hate lits #1: <40376AD8.C83FBF5A@enter.net> CEO Alcatroll Labs Inc. Alexa knows everything about PSI, electromagnetic radiation and quantum mechanics: They are beaming earth with EMR which is interfering with natural human PSI abilities, which is likely quantum. in alt.alien.visitors === Subject: Re: looking for old HP 41cv or cx calculator >I am lookning for a HP 41CV or HP 41CX calculator, mine is on it's >last leg and I need a replacement. If you have one you would like to There are currently about 5 such calculators for sale on E-Bay. Just go to ebay.com and type in hp 41 calculator in the the search box to see descriptions. --Lynn === Subject: Re: The physics of practical daily life > In sci.math, Donald G. Shead > > Most physists know more theoretical physics than is good for them. It > is suf[CapitalThorn]cient to know that physics is the science of physically > existent object's; bodies and massive accumulations of descrete bits > of matter moving and pushing each other around in the vast emptyness > of space. > precisely how? > BTW: objects, discrete, emptiness. already run out of viable names for them. Quit wasting the governors money. > About three or four hundred years ago Rene' Descartes, a philosopher > of physics visualized the universe as consisting of countless > vortexes of these masses of various densities; each centered and > whirling around a center of mass, that contained a nucleus consisting > of one or more of these descrete bits of matter. > Rene died in 1650. cut< He envisioned the whole universe as consising of whirling vortices: > At one exteme the whole universe, and each star is a vortex. The sun, > and the planets orbiting around it make up a whole system, with each > body and atom making up smaller vortexes. > Well, that's somewhat accurate; the spinning accretion disk, > however, isn't quite a vortex. Then again, I'm not sure > whether a vortex requires a sink or not. The sink for each vortex is the nucleus around which it spirally whirls; which gradually grows as it absorbs the material which spirals into it. > Such a system of vortexes within vortexes requires only three basic > concepts: _Matter_ of which the physically existent object's; bodies > and massive accumulations of descrete bits consist; _Space_ the vast > emptiness, in which they have room to exist and move around, and > _Time_ for all of these vortexes to move and interact sequentially. > This requires only three measures: Length, to measure the sizes, > distances and displacements of the bodies in space; Force, to measure > the magnitude of the pushes exerted on and/or by the bodies, and Time, > to measure the durations of these displacements and pushes. > All systems of weights and measures need these three fundamental > measures; from which they may then go on to develop whatever other > measures they may need: Such as mass, momentum, energy, density and so > forth. > Hm. And a reference force is generated precisely how again? Depends; what do you need it for? > Through the years each country or locality has developed its own units > of measure, and there has become a glut of them to the point where > confusion reins supreme. > So far, the foot-pound-second system developed by the U.S. appears to > me to be the best. Counting by tens has only aggravated the situation; > with its seven fundamental units and hundreds of derived units the > metric SI system is fraught with its own errors and omissions. Avoid > it at all costs. It's a strawman and has already cost physics its > sanity, not to mention that of otherwise intellegent scientists. > OK. Describe an operating light bulb using only force, > length, and time. You are allowed to derive units of > course such as pressure (force / length^2) and velocity > (length / time), of course. > If a light bulb's too complicated, describe a ßaming brand. > Good luck. Rub two sticks together; when you get one ßamming good, you can toss it. === Subject: Re: The physics of practical daily life > it at all costs. It's a strawman and has already cost physics its > sanity, not to mention that of otherwise intellegent scientists. > Provide a listing of these scientists with names, testimonials, and dates they > were comitted to asylums. First you've got to prove their lack of sanity; then catch them, and convict them before committing them: So far they've remained slippery as elm, and insistently proclaim they're sane and I'm not. As a general rule I'd say anyone who believes real physics consists of examined. === Subject: Re: The physics of practical daily life > > So far, the foot-pound-second system developed by the U.S. appears to > me to be the best. Counting by tens has only aggravated the situation; > with its seven fundamental units and hundreds of derived units the > metric SI system is fraught with its own errors and omissions. Avoid > it at all costs. It's a strawman and has already cost physics its > sanity, not to mention that of otherwise intellegent scientists. > SI is easier to use and less prone to error, Shead. In a pig's eye; the errors are multiples of ten too, Sam. === Subject: Re: George Green, portrait? > George Green 1793--1841 > ... the general mathematical theory of potential developed by an > obscure, self-taught miller's son would lead to the mathematical > theories of electricity underlying twentieth-century industry. > quoted at > > Most of the biographies at The MacTutor History of Mathematics archive > have portraits of the subject. (Even Euclid and Archimedes...) But > not Green. Are there no known portraits of this man? I've found a website, that claims that Ôthe circumstances of his life were such that no portrait was ever made'. Apparently he is Ôcommemorated by a plaque in the ßoor of Westminster Abbey', which is Ôadorned with a representation of a windmill, presumedly green's mill (once home to george green according to this website about it:http://www.greensmill.org.uk/) The website I got the information from is here: http://www.ee.umd.edu/~taylor/frame2.htm I'm a Student (graduating this year) at nottingham university, which has a george green library. I don't know if they have a similar plaque, but I think they have some display, that shows the windmill. I don't remember there being a portrait of him. It's too later to look now, but if I remember next time I pass I'll have a look. === Subject: Re: SOLVING for a variable? I found Goal Seek, thank you. Where is Solver? === Subject: Re: SOLVING for a variable? On the tools menu. If it is not there then check add-ins on the tools menu and tick it. If it is not on that list then get out your install discs and install it. > I found Goal Seek, thank you. Where is Solver? === Subject: Re: SOLVING for a variable? > How many patches are in your walls from hitting at a ßy with a > sledgehammer? > Use the quadratic formula as others have suggested. All the SOLVE > subroutine does in a TI83+ calculator is look at the equation and > determine which of several available subroutines to use on it; itthis > case teh TI83+ would pick the quadratic formula. > --OL Well, as it turns out, I solved the equation for X in terms of R by myself using the quadratic formula, as people suggested. But that was a few minutes after I posted the message. I still needed to know how to use the solve function on Excel because many of the equations I work with (generally chemical rate laws and equilibriums) are NOT quadratics. They include X^3 and X^4 terms. The equation I posted was not a very good example because it was so simple that I could solve it as it was. Sorry. And thank you all for the help. B.W. === Subject: Re: resolving Will's misunderstanding >Let's try a different approach, starting with de[CapitalThorn]nitions of everything >under consideration: > >We are talking about functions, but the real concern is what algorithms >are used to compute those functions. For simplicity's sake, we can >restrict our discussion to functions mapping from the whole numbers to >the whole numbers f:W->W. For an algorithm to correctly represent >f(x)=y, it should start with input x and produce output y after [CapitalThorn]nitely >many steps, where each step is computed from the input and the previous >steps. For a given function f, f may or may not be de[CapitalThorn]ned for all/any >numbers in the whole numbers. As a result, the algorithm describing f >is only required to be [CapitalThorn]nite for those values of x for which f is >de[CapitalThorn]ned. A function f with such an algorithm is in general called a >partial function. A partial function f which is de[CapitalThorn]ned on all elements >of the whole numbers is called a total function and is considered to be >computable (or recursive). > >At this point, I believe we can start making some clear observations. >First: the halting problem is only available for discussion if we a >dealing with partial functions (which have algorithms that do not halt >in [CapitalThorn]nite time for some input values). >Second: it appears to be your desire to discuss only total functions. >Third: if we restrict ourselves to the total functions, discussing the >halting problem is meaningless. > >A question that is relevent to the above: if we are restricting >ourselves to total functions, is there a way to specify the algorithms >of all total functions in such a way that there is no possibility of >dealing with potentially partial functions? > >Before moving forward, do you have any comments on the above? Right now >I'm just trying to set a framework for a detailed analysis, but if the >basic framework is in dispute, I may need to adjust for that. > >The term to be de[CapitalThorn]ned is valid construction. Facts & FALSE -> anything. > >Agreed. I just wanted to make sure there were no objections before >diving into the details. If the basic concepts to be de[CapitalThorn]ned in detail >can't be agreed on, then there isn't much point in going further. > >For the moment, let's allow functions to be n-ary for ease of notation. >(It can be shown that we can restrict ourselves to unary functions, but >they get messier.) > >Let's start by de[CapitalThorn]ning primitive recursive functions These functions >are guaranteed to be total functions and can be constructed by a [CapitalThorn]nite >number of applications of the following rules: > >1) f(x) = x+1 is primitive recursive >2) f(x1,x2,...,xn) = m is primitive recursive where n and m >= 0 >3) f(x1,x2,...,xn) = xi is primitive recursive where 0 < i <= n >4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is >primitive recursive and m-ary then so is n-ary f de[CapitalThorn]ned as >f(x1,x2,...,xn) = h(g1(x1,x2,...,xn),...,gm(x1,x2,...,xn)) >5) For n>=1, if f is n-ary, g is (n-1)-ary, h is (n+1)-ary, and g and h >are primitive recursive, then so is f de[CapitalThorn]ned by the following two rules: >a) f(0,x2,x3,...,xn) = g(x2,x3,...,xn) >b) f(x1+1,x2,x3,...,xn) = h(x1,f(x1,x2,...,xn),x2,...,xn) > >These rules appear (to my eye) to correspond with the de[CapitalThorn]nition of a >function guaranteed to halt you offered in another thread. > >Using these it is possible (though messy) to construct virtually all >normal mathematical functions on the whole numbers. In particular, >prime decomposition that will allow a number to represent a sequence of >numbers by its prime decomposition. This enables us to formally get >back to unary functions, if need be. > >Any comments at this point? The next question will be, do primitive >recursive functions describe ALL total computable functions? > >nope > >There's two questions, which one are you saying nope to? > >the 1st one, you're supposed to be establishing the *next* question. > >Since there are only 5 rules for deriving primitive recursive functions, >each one can be indexed based on what rules/functions are used to >construct it. Consider one such indexing f0, f1, f2, f3, ... and let us >construct a new function g0(x) = fx(x) + 1. Since fx can be formally >considered unary, regarless of what n is for n-ary fx, and x determines >the construction of f, g0(x) is de[CapitalThorn]ned on all x in the Whole Numbers. >Moreover, we have included all partial recursive functions on our list >f0, f1, f2, .... Therefore, g0(x) is a total function, it is >computable, but it is not a primitive recursive function. By prepending >g0(x) to the listing of primitive recursive functions, we can construct >g1(x), g2(x), etc. Since there are multiple possible indexing schemes >available for the primitive recursive functions, there are also multiple >sequences of gi(x) available. > >Any issues with the above? > > >I got no idea what rule 5 does. It looks like a step -1 for loop that pushes the >parameters into the 2nd parameter of the 2nd parameter recursively, making >a longer parameter list every iteration. > >It's a construction similar to f(0) = c, f(x+1) = g(f(x)). The idea is >to de[CapitalThorn]ne f(0) as something known, and de[CapitalThorn]ne f(x+1) in terms of f(x). >The parameter list doesn't actually get longer. > > > > >construct a new function g0(x) = fx(x) + 1 > >this is just diagonalisation all over again, f with godel x applied to its own godel number >with a modi[CapitalThorn]cation. > >You are quite right. Now, does that invalidate the conclusion? > > > > >what's to stop me de[CapitalThorn]ning consistent p.r.f. where variable references are contextually free. >fx(x) = fx(y) for any x, y > >Those functions are described in construction 2, they are the constant >functions. This has nothing to do with the validity or invalidity of >the argument. > > > > >this stops dummy functions like the above that double usage the x in g0(x) >I've shown that getting a parameter x and using it for 2 arguments is >enough to change a total system to non halting system. > >We have not gotten to anything that does not halt. All that is being >said is: a suf[CapitalThorn]cient restriction to prevent functions that don't halt >also excludes some that do always halt. > >ok, but how hard is it to exclude functions of type fx(x), they're the ones going >into in[CapitalThorn]nite loops. what computation are they performing that can't be >done some other way? > >The functions fx do not contain in[CapitalThorn]nite loops. Nor do the functions gi >built from them. The point was that the primitive recursive functions, >while guaranteed to halt, are not *all* of the functions that are >guaranteed to halt. The various functions gi are also guaranteed to >halt, but they were found using a diagonal-style argument, and are >therefore not primitive recursive. > >Having said all that, MOST functions and operations that we would want >to perform would be primitive recursive functions, including things like >the addition function, subtraction function (which produces 0 for >negative results), multiplication, integer division, modular division, >exponentiation, etc. As long as the computations can be broken down >into the 5 rules applied to various primitive recursive functions, we >are good to go. > >So far, there are no in[CapitalThorn]nite loops, and everything is guaranteed to >halt. It's just that not everything that halts is primitive recursive. > The trick is to see if we can [CapitalThorn]nd something that will include ALL >functions that halt, and still exclude those that don't. > > > >If fx(x) is a construct that halts, then UTMs must be impossible. Because >mUTM(mUTM) doesn't halt, as we've been through. > >If UTMs are not part of primitive recursive functions, then they're not complete >anyway. > >You are putting the cart before the horse. UTM has not been de[CapitalThorn]ned >yet. All I have said is there are functions that are guaranteed to halt >that are not primitive recursive. > >Even so, what does g0(x) do? I still say you are entrenched in *names* of >functions and not *what* they actually compute. Two algorithms can perform >the same computation can't they? > >g0(x) computes something different from ALL primitive recursive >functions. It proves that there is a function that is guaranteed to >halt on all inputs, yet is not primitive recursive. > true, I'm not familiar with them. How you know they are de[CapitalThorn]ned properly? > Herc > As references: > http://en.wikipedia.org/wiki/Primitive_recursive > http://mathworld.wolfram.com/PrimitiveRecursiveFunction.html > or, for a less readable one: > Robert Soare's Recursively Enumerable Sets and Degrees published by > Springer-Verlag. > All three of them are saying essentially the same thing. What I posted > above is a translation of Soare's work into normal language. > As far as being de[CapitalThorn]ned properly, if the people who are working with > this branch of mathematics agree to the same, or equivalent, de[CapitalThorn]nitions > of primitive recursive, then that is the proper de[CapitalThorn]nition. The issue > is, does it have the properties that the people who created it hoped it > would have? They hoped that all computable functions would be primitive > recursive. It turns out that this is not the case. Did you perform an analysis of their mind when they stated their goal? Aren't they just trying to close an avenue to support bogus mathematics texts by working backwards from a diag proof into enumerated functions? What is the value for x when Fx(y) = y^2? Herc === Subject: Re: resolving Will's misunderstanding Note: I snipped out a few tangents that appear to have died. I am also checking the general ßow of this conversation, it seems to be getting away from the topic at hand (at least as I perceive it). >For the moment, let's allow functions to be n-ary for ease of notation. >(It can be shown that we can restrict ourselves to unary functions, but >they get messier.) > >Let's start by de[CapitalThorn]ning primitive recursive functions These functions >are guaranteed to be total functions and can be constructed by a [CapitalThorn]nite >number of applications of the following rules: > >1) f(x) = x+1 is primitive recursive >2) f(x1,x2,...,xn) = m is primitive recursive where n and m >= 0 >3) f(x1,x2,...,xn) = xi is primitive recursive where 0 < i <= n >4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is >primitive recursive and m-ary then so is n-ary f de[CapitalThorn]ned as >f(x1,x2,...,xn) = h(g1(x1,x2,...,xn),...,gm(x1,x2,...,xn)) >5) For n>=1, if f is n-ary, g is (n-1)-ary, h is (n+1)-ary, and g and h >are primitive recursive, then so is f de[CapitalThorn]ned by the following two rules: >a) f(0,x2,x3,...,xn) = g(x2,x3,...,xn) >b) f(x1+1,x2,x3,...,xn) = h(x1,f(x1,x2,...,xn),x2,...,xn) > >These rules appear (to my eye) to correspond with the de[CapitalThorn]nition of a >function guaranteed to halt you offered in another thread. > >Using these it is possible (though messy) to construct virtually all >normal mathematical functions on the whole numbers. In particular, >prime decomposition that will allow a number to represent a sequence of >numbers by its prime decomposition. This enables us to formally get >back to unary functions, if need be. > >Any comments at this point? The next question will be, do primitive >recursive functions describe ALL total computable functions? > >nope >Since there are only 5 rules for deriving primitive recursive functions, >each one can be indexed based on what rules/functions are used to >construct it. Consider one such indexing f0, f1, f2, f3, ... and let us >construct a new function g0(x) = fx(x) + 1. Since fx can be formally >considered unary, regarless of what n is for n-ary fx, and x determines >the construction of f, g0(x) is de[CapitalThorn]ned on all x in the Whole Numbers. >Moreover, we have included all partial recursive functions on our list >f0, f1, f2, .... Therefore, g0(x) is a total function, it is >computable, but it is not a primitive recursive function. By prepending >g0(x) to the listing of primitive recursive functions, we can construct >g1(x), g2(x), etc. Since there are multiple possible indexing schemes >available for the primitive recursive functions, there are also multiple >sequences of gi(x) available. > >Any issues with the above? > >construct a new function g0(x) = fx(x) + 1 > >this is just diagonalisation all over again, f with godel x applied to its own godel number >with a modi[CapitalThorn]cation. > >You are quite right. Now, does that invalidate the conclusion? I notice that you did not answer this question. Stated another way, do you believe that diagonalization is an invalid technique at all times, or do you believe that it can work in certain circumstances? Rereading the thread, it appears to me that everything following is either a sign of not accepting some aspect of the preceding material, or of confusion between what came before and what will inevitably come, if the presentation continues. If everything below is actually a tangent, I would just as soon stop that line of discussion and return to [CapitalThorn]nding out which rule(s) must be added to the [CapitalThorn]rst [CapitalThorn]ve to be able to describe every total function, and what possible consequences exist that could prove inconvenient. Note: I have responded below in case the following material is not a tangent. >what's to stop me de[CapitalThorn]ning consistent p.r.f. where variable references are contextually free. >fx(x) = fx(y) for any x, y > >Those functions are described in construction 2, they are the constant >functions. This has nothing to do with the validity or invalidity of >the argument. > >this stops dummy functions like the above that double usage the x in g0(x) >I've shown that getting a parameter x and using it for 2 arguments is >enough to change a total system to non halting system. > >We have not gotten to anything that does not halt. All that is being >said is: a suf[CapitalThorn]cient restriction to prevent functions that don't halt >also excludes some that do always halt. > >ok, but how hard is it to exclude functions of type fx(x), they're the ones going >into in[CapitalThorn]nite loops. what computation are they performing that can't be >done some other way? > >The functions fx do not contain in[CapitalThorn]nite loops. Nor do the functions gi >built from them. The point was that the primitive recursive functions, >while guaranteed to halt, are not *all* of the functions that are >guaranteed to halt. The various functions gi are also guaranteed to >halt, but they were found using a diagonal-style argument, and are >therefore not primitive recursive. > >Having said all that, MOST functions and operations that we would want >to perform would be primitive recursive functions, including things like >the addition function, subtraction function (which produces 0 for >negative results), multiplication, integer division, modular division, >exponentiation, etc. As long as the computations can be broken down >into the 5 rules applied to various primitive recursive functions, we >are good to go. > >So far, there are no in[CapitalThorn]nite loops, and everything is guaranteed to >halt. It's just that not everything that halts is primitive recursive. >The trick is to see if we can [CapitalThorn]nd something that will include ALL >functions that halt, and still exclude those that don't. > > > >If fx(x) is a construct that halts, then UTMs must be impossible. Because >mUTM(mUTM) doesn't halt, as we've been through. > >If UTMs are not part of primitive recursive functions, then they're not complete >anyway. > >You are putting the cart before the horse. UTM has not been de[CapitalThorn]ned >yet. All I have said is there are functions that are guaranteed to halt >that are not primitive recursive. > > >Even so, what does g0(x) do? I still say you are entrenched in *names* of >functions and not *what* they actually compute. Two algorithms can perform >the same computation can't they? > >g0(x) computes something different from ALL primitive recursive >functions. It proves that there is a function that is guaranteed to >halt on all inputs, yet is not primitive recursive. >true, I'm not familiar with them. How you know they are de[CapitalThorn]ned properly? >Herc >As references: >http://en.wikipedia.org/wiki/Primitive_recursive >http://mathworld.wolfram.com/PrimitiveRecursiveFunction.html >or, for a less readable one: >Robert Soare's Recursively Enumerable Sets and Degrees published by >Springer-Verlag. >All three of them are saying essentially the same thing. What I posted >above is a translation of Soare's work into normal language. >As far as being de[CapitalThorn]ned properly, if the people who are working with >this branch of mathematics agree to the same, or equivalent, de[CapitalThorn]nitions >of primitive recursive, then that is the proper de[CapitalThorn]nition. The issue >is, does it have the properties that the people who created it hoped it >would have? They hoped that all computable functions would be primitive >recursive. It turns out that this is not the case. > Did you perform an analysis of their mind when they stated their goal? I give them the bene[CapitalThorn]t of the doubt, in that they have a stated goal, present their solution, and someone else punches a hole in the solution. If the same person found the hole, it would be more reasonable to question their true motives. > Aren't > they just trying to close an avenue to support bogus mathematics texts by > working backwards from a diag proof into enumerated functions? What does this question have to do with the material presented so far? The question remains, do you accept the construction of the g's from enumerations of the primitive recursive functions as a method for generating non-primitive recursive functions that are guaranteed to halt? > What is the value for x when Fx(y) = y^2? It depends on the enumeration speci[CapitalThorn]ed. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: resolving Will's misunderstanding > The question remains, do you accept the construction of the g's from > enumerations of the primitive recursive functions as a method for > generating non-primitive recursive functions that are guaranteed to halt? no. they're not primitive to start, they're speci[CapitalThorn]cally designed for self reference. what is so different about this to the ordinary function diag proof? > What is the value for x when Fx(y) = y^2? > It depends on the enumeration speci[CapitalThorn]ed. such as? if the UTM is disallowed to parse constructions that copy a parameter, like parsing 0 0 0 x 0 0 0 and writing 0 0 0 x 0 0 0 x 0 0 then diag contradictions are no longer possible. if an analagous rule stopping function constructions of the form g(x) = f(x,x) or g(x) = fx(x) or g(x) = x * x can be embedded into PRF then it should be a suf[CapitalThorn]cient patch for a halting enumeration. but I know nothing about them unless you want to demo how x*x would work. Herc === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax [CapitalThorn]eld (and please don't come and asc me what that >means), how on earth can a vector be negative. >That isn't the only meaning of gradient. In the case of a scalar >function of a single variable, f(x), the gradient is the [CapitalThorn]rst derivative >df/dx which is a scalar. >Gib > let f: -> f(x) be the function that attaches to each 3d coordinate the > degree of humidity (some insects like that function). it maps a three > dimensional vector space into the scalars (1 dim reals). > The gradient of that function is a 3D vector [CapitalThorn]eld, in each point > indicating the direction (and intensity) to [CapitalThorn]nd a local maximum of the > function. > So what? That doesn't alter the fact that the gradient of a scalar > function of a scalar variable is a scalar. In this case (which is a > common one) the gradient has a sign (if non-zero). > Gib Your case is a special case of the regular gradient: you're considering the case of a function on the 1 dimensional vector space R over R. So you're right that the gradient of a scalar function is a scalar, but it's also a vector in an important, unifying way. Ôcid Ôooh === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax > [CapitalThorn]eld (and please don't come and asc me what that > >means), how on > earth can a vector be negative. >That isn't the only meaning > of gradient. In the case of a scalar >function of a single > variable, f(x), the gradient is the [CapitalThorn]rst derivative > >df/dx which is > a scalar. >Gib > let f: -> f(x) be the function > that attaches to each 3d coordinate the > degree of humidity (some > insects like that function). it maps a three > dimensional vector > space into the scalars (1 dim reals). > The gradient of that > function is a 3D vector [CapitalThorn]eld, in each point > indicating the > direction (and intensity) to [CapitalThorn]nd a local maximum of the > function. > So what? That doesn't alter the fact that the gradient of a > scalar > function of a scalar variable is a scalar. In this case > (which is a > common one) the gradient has a sign (if non-zero). > > Gib > Your case is a special case of the regular gradient: you're > considering the case of a function on the 1 dimensional vector space R > over R. So you're right that the gradient of a scalar function is a > scalar, but it's also a vector in an important, unifying way. OK, if you want to consider it a vector in this case, but then you also have to accept that a vector can have a sign - in this special case. It is equivalent to considering a scalar to be a vector in an important, unifying way. This is getting a bit away from the point I was addressing, someone's contention that a gradient cannot have a sign. I think we all [CapitalThorn]rst learned about gradients for the simple (special, if you like) case in which a gradient is a scalar. Gib Gib === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax > > > [CapitalThorn]eld (and please don't come and asc me what that > >means), how on > earth can a vector be negative. >That isn't the only meaning > of gradient. In the case of a scalar >function of a single > variable, f(x), the gradient is the [CapitalThorn]rst derivative > >df/dx which is > a scalar. >Gib > let f: -> f(x) be the function > that attaches to each 3d coordinate the > degree of humidity (some > insects like that function). it maps a three > dimensional vector > space into the scalars (1 dim reals). > The gradient of that > function is a 3D vector [CapitalThorn]eld, in each point > indicating the > direction (and intensity) to [CapitalThorn]nd a local maximum of the > > function. > > So what? That doesn't alter the fact that the gradient of a > > scalar > function of a scalar variable is a scalar. In this case > (which is a > common one) the gradient has a sign (if non-zero). > > Gib > > Your case is a special case of the regular gradient: you're > considering the case of a function on the 1 dimensional vector space R > over R. So you're right that the gradient of a scalar function is a > scalar, but it's also a vector in an important, unifying way. > OK, if you want to consider it a vector in this case, but then you also > have to accept that a vector can have a sign - in this special case. I never said otherwise, though I would like to say that the unifying way of talking about things leaves room for this vector to not have a sign. (What does have a sign is its single real component). Anyway, this is all just conventional anyways--there's no math involved here. > It > is equivalent to considering a scalar to be a vector in an important, > unifying way. Ôcid Ôooh === Subject: Re: Sequence for experts, this was never solved before Any integer in the set {0, ... ,9} since every sequence can be continued in[CapitalThorn]nitely many ways. That is, there are in[CapitalThorn]nitely many sequences that start with 2 4 4 6 5 4 4 4 4. Ôcid Ôooh === Subject: Re: Turing Machine with Tape === Subject: Re: Turing Machine with Tape >As a result, after the annual audit you will [CapitalThorn]nd that you have >unallocated funds suf[CapitalThorn]cient for the purchase of a new tape. > Except for handling, shipping, deduction for portion sent, > inßation, devaluation of currency and misc. charges such as > locating old (3-month old) model tapes. Likely I'd have suf[CapitalThorn]cient > funds for just half a tape, but a tape 1/2, or 1/3 the width > because of cost of new case, would perform unreliably. >Fortunately, for a minimal cost we can also sell you a new tape >head, which will only require 1/4 width tape. Unfortunately your 1/2 >and 1/3 tapes will not [CapitalThorn]t the new tape head. Alas, during downtime, because of the problem being solved, our machine became water logged and sunk to the bottom of the data pool. In desperate relief we've decided to outsource the problem. What's the buoyancy of a ßoating-point? ---- === Subject: Re: =?ISO-8859-1?Q?Revista/magazine_Investigaci=F3n_ciencia?= > Simplemente para dar respuesta al ingl.8es que buscaba una revista > cienti[CapitalThorn]ca en esp. > Con temas muy actuales, bien explicados, perfectamente explicados( > f.92sica, computaci.97n, qu.92mica, algo de teor.92a de n.9cmeros, criptolog.92a, > seguridad inform.87tica), la revista Investigaci.97n ciencia es una > maravilla. Y algo que es de admirar de esta revista es que sus > trabajos son de una calidad pasmosa, explican cosas que hace falta > saber algo sino te pierdes. No vale para pasar el rato sin tener nada > en la cabeza. Y s.97lo leer las notas a pie de p.87gina. > Las secciones que incluye a parte de los art.92culos son variadas, a mi > me gusta la de Nexos, mediante nexos entre personas, hechos, etc... te > relacionan 2 cosas que en principio eran muy inconexas. Otra secci.97n > muy buena es la de construyelo tu mismo, siempre proponiendo inventos > caseros muy ingeniosos. > Y luego la secci.97n: Hace 50, 100, 150 a.96os. Que est.87 bien para ver > c.97mo hemos evolucionado. Yo sol.92a comprar la revista Investigaci.97n ciencia en una libreria propiedad de chilenos en el centro-oeste de Estocolmo, alguien conoce como se llama para que pueda contactar el due.96o?? I used to buy a magazine called Investigaction ciencia in a chiliean owned bookstore in centre west Stockholm, Anyone know what the store name is- I want to get in touch with the owner! === Subject: Re: Foundations textbooks > I would appreciate it if I got some suggestions for books on logic and > eventually recursion theory out of which I could teach myself. For recursion theory I can highly recommend: P. Odifreddi, Classical Recursion Theory, North Holland, 1999. Achim -- _____________________________________________________________ ___________ | _____/ | Achim Blumensath O/ ___/ | LaBRI / Bordeaux =o= / | www-mgi.informatik.rwth-aachen.de/~blume / o----| _____________________________________________________________ __________| === Subject: Re: Foundations textbooks I'm reading Yu. I. Manin's A Course in Mathematical Logic. It covers everything I'm interested in (it does Goedel's theorem *after* recursive function theory!), but I'm not so keen on the presentation. You may want to check it out. There's also Mendelson's Introduction to Mathematical Logic. It's good, but incredibly dense. Ôcid Ôooh === Subject: Any quick way to [CapitalThorn]nd a remainder ? I have a problem from a friend, i want to know the solution. What is the remainder if the following numver x is divided by 13. x = 1^2001 + 2^2001 + 3^2001 + .... + 2001^2001 Any help would be appreciated. === Subject: Re: Any quick way to [CapitalThorn]nd a remainder ? >What is the remainder if the following numver x is divided by 13. >x = 1^2001 + 2^2001 + 3^2001 + .... + 2001^2001 Since 2001 is odd, (-k)^2001 = -k^2001. Thus, 1^2001 + 12^2001 = 0 mod 13, 2^2001 + 11^2001 = 0 mod 13, 3^2001 + 10^2001 = 0 mod 13, 4^2001 + 9^2001 = 0 mod 13, 5^2001 + 8^2001 = 0 mod 13, 6^2001 + 7^2001 = 0 mod 13. Therefore, 12 --- 2001 > k = 0 mod 13 --- k=0 Since 2001 = 154*13-1, you should be able to use the summation above to get the answer you want. Rob Johnson take out the trash before replying === Subject: Re: Any quick way to [CapitalThorn]nd a remainder ? 2002=13[Times]154 1^2001+...+12^2001+13^2001 same remainder mod 13 as 1^2001+...12^2001 so answer is same remainder as 154[Times](1^2001+....+12^2001) mod 13 154 = 11 mod 13 so = 11[Times](1^2001+...+12^2001) mod 13 a^(p-1) = 1 mod p for p prime which 13 is (this is fermats theorem) 2001=(12[Times]12[Times]12)+(22[Times]12)+9 so a^2001=(((a^12)^12)^12)[Times](a^12)^22[Times]a^9 = a^9 mod 13 that is a^n mod p = a^(n mod (p-1)) mod p I think. so 11[Times](1^9+...12^9) mod 13 that is [CapitalThorn]nd the remainder of 11[Times](1^9+...12^9) excel spread sheet should handle it from here use the =mod(x,13) function. I came up with 0 as remainder which suggests maybe something trickier than my simpli[CapitalThorn]cations OH I see 1^n+...k^n is always divisible by k[Times](k+1) so 1^2001+....+12^2001 is divisible by 12[Times]13 > I have a problem from a friend, i want to know the solution. > What is the remainder if the following numver x is divided by 13. > x = 1^2001 + 2^2001 + 3^2001 + .... + 2001^2001 > Any help would be appreciated. === Subject: Re: Any quick way to [CapitalThorn]nd a remainder ? > I have a problem from a friend, i want to know the solution. > What is the remainder if the following numver x is divided by 13. > x = 1^2001 + 2^2001 + 3^2001 + .... + 2001^2001 > Any help would be appreciated. Take for example the remainder of (1307^2001) divided by 13. You get the same result if you take [CapitalThorn]rst the remainder of 1307 / 13, that is 7. So you calculate 7^2001 divided by 13. Make a table: 7^1 divided by 13, remainder is 7. 7^2 divided by 13 = 49/13 = 3 remainder 10. 7^3 divided by 13 = (10 * 7) / 13 = 70/13 = 5 remainder 5. 7^4 divided by 13 = (5 * 7) / 13 = 35/13 = 2 remainder 9 and so on. You will see a pattern. Once you spot the pattern, the remainder of 7^2001 divided by 13 is easy. Do the same for 0, 1, 2 to 12. Of the numbers 1 to 2001, [CapitalThorn]nd how many produced the same result as (0^2001), how many produced the same result as (1^2001) and so on. Five to ten minutes work, that is all. === Subject: Re: Any quick way to [CapitalThorn]nd a remainder ? > I have a problem from a friend, i want to know the solution. Not from me, but you'll get some hints. > What is the remainder if the following numver x is divided by 13. > x = 1^2001 + 2^2001 + 3^2001 + .... + 2001^2001 By Fermat's little theorem, x^12 = 1 (mod 13) when x is not a multiple of 13. So this is congruent to > x = 1^9 + 2^9 + 3^9 + .... + 2001^9. Now, x^9 + (x+1)^9 + ... + (x+12)^9 = 0 (mod 13) for any positive integer x. Apply this and get ..... -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Boolean Algebra with non-binary numbers > I've heard of boolean algebras without atoms but can't give an example. > The clopen subsets of Q. of Zen awe. === Subject: Re: Boolean Algebra with non-binary numbers This post not CC'd by email >Except for the talk about other bases, the question points to probability >theory as the answer your instructor is looking for. G'day G'day Keith, Delighted you think of me as a student. It's quite ßattering at my age. As it happens I have been teaching for 35 years. While waiting for a new teaching contract I am currently exploring some favourite haunts to keep my mind active. >Probabilities are real numbers in the range [0,1], they have commutative, >distributive OR (+) and AND (.) operations just like booleans. The >probability of any event OR its inverse is 1, and the probability of an event >AND its inverse is 0. >--Keith Lewis klewis {at} mitre.org >The above may not (yet) represent the opinions of my employer. -- Quentin Grady ^ ^ / New Zealand, >#,#< [ / / ... and the blind dog was leading. http://homepages.paradise.net.nz/quentin === Subject: Re: Boolean Algebra with non-binary numbers >Except for the talk about other bases, the question points to probability >theory as the answer your instructor is looking for. >G'day G'day Keith, > Delighted you think of me as a student. It's quite ßattering at my >age. As it happens I have been teaching for 35 years. While waiting >for a new teaching contract I am currently exploring some favourite >haunts to keep my mind active. Ha! I *knew* the question was written by a teacher. Of course I also thought said teacher already had my answer in mind... We're all here to learn, eh? :^) --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Boolean Algebra with non-binary numbers > Computing uses binary number systems so when Boolean algebra is >used in computing courses one becomes accustomed to seeing Boolean >algebra applied to binary numbers and only to binary numbers. It >comes as a bit of a shock to consider that Boolean algebra can be >applied to class of elements provides certain postulates are met. >Here is one description of Boolean algebra. >A class of elements B together with two binary operations (+) and (.) >where a.b may be written as ab is a Boolean algebra, if and only if >the following postulates hold. >P1 The operations (+) and (.) are commutative. >P2 There exist in B distinct identity elements 0 and 1 relative to >operations (+) and (.) respectively. >P3 Each operation is distributive over the other. >P4 For every a in B there exists an element a' in B such that > a + a' = 1 and aa' = 0 >Can someone please provide a non-binary number system example of >Boolean algebra that [CapitalThorn]ts this description? I'm not looking for >Venn diagrams but rather an example in another number base. >How about a ternary example? Except for the talk about other bases, the question points to probability theory as the answer your instructor is looking for. Probabilities are real numbers in the range [0,1], they have commutative, distributive OR (+) and AND (.) operations just like booleans. The probability of any event OR its inverse is 1, and the probability of an event AND its inverse is 0. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: laser cooling > Varney does not have the right to talk to me because of his iq. > Varney is a loser. It is probably true most 12 year old kids understand > physics better than him. > Varney is from the physics groups. It is slime like him that destroyed > the > physics groups. Why are you telling sci.math this? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: laser cooling > Varney does not have the right to talk to me because of his iq. 182 on the STANFORD-BINET L-M (pre 1986) You should feel lucky that I would take the time to insult you. > Varney is a loser. It is probably true most 12 year old kids understand > physics better than him. LOL! You are an idiot Kurt, as always. > Varney is from the physics groups. It is slime like him that destroyed the > physics groups. The famous Kurt slime! Will he misspell idiots next? > Now only slime, ideits, fools, nuts and people who are more happy > with just helping to destroy the physics groups driving away people who know a > lot about physics. You do not know anything about physics, Kurt... and yet you were driven away. > The slime needs to slime to the math groups to make them > worse. > I guess slime needs to spread like aids. > Most dogs would not read the physics groups and that kind of tells what the to > drop > food from dogs on those people. > Varney helped to make the physics groups as bad as his lousey life. Quit whining Kurt, and die already. === Subject: Re: My paper passed peer review > You sure waste a lot of time writing math-less rebuttals, don't you? And how do you waste your time? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: My paper passed peer review > You sure waste a lot of time writing math-less rebuttals, don't you? You sure waste a lot of time writing math-less and physics-less posts, don't you Schoenfeld? === Subject: Re: A Repunit Observation >Let Rn denote the repunit of n digits, De[CapitalThorn]ne s(n) as the sum of the >exponents of the prime factors of Rn. Consider the following REPUNIT >OBSERVATION: If s(n)is equal to less than 3, then n is prime. > >This observation is based on existing factorization tables as reviewed >on Internet. Is the REPUNIT OBSERVATION or Conjecture embedded in an >exiting theorem or has it been previously announced? I would >appreciate any help on this line of research. > > Research? This is nothing to research. Trivially if n = pq > then 10^n-1 is divisible by 10^p-1 and 10^q-1. It is also divisible > by 9. Thus, s(n) >= 4 if n is composite. > True, but OP isn't talking about 10^n - 1, OP is talking about > the repUNIT numbers, which are (10^n - 1)/9. There's an extra factor that's been forgotten about, which makes it non-trivial (i.e. it's not always true/false). If n=pq 10^n-1 = Phi(n,10) * Phi(p,10) * Phi(q,10) * Phi(1,10) R(pq) = Phi(pq,10) * Phi(p,10) * Phi(q,10) So we're looking for n=pq s.t. those three factors are prime. Phi(x,10) is prime for 2, 19, and 38 11111111111111111111111111111111111111 So the OP was, erm, not very good, shall we say, at investigating his conjecture. Clue to OP - 1) If making conjectures about factors of cyclotomic numbers, get hold of either the cunningham tables, or the extended cunningham tables, and use that previously-discovered corpus. 2) Try your conjectures in an arbitrary-precision algebra package such as GP/Pari,. Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: A Repunit Observation > >Let Rn denote the repunit of n digits, De[CapitalThorn]ne s(n) as the sum of the >exponents of the prime factors of Rn. Consider the following REPUNIT >OBSERVATION: If s(n)is equal to less than 3, then n is prime. > >This observation is based on existing factorization tables as reviewed >on Internet. Is the REPUNIT OBSERVATION or Conjecture embedded in an >exiting theorem or has it been previously announced? I would >appreciate any help on this line of research. > > Research? This is nothing to research. Trivially if n = pq > then 10^n-1 is divisible by 10^p-1 and 10^q-1. It is also divisible > by 9. Thus, s(n) >= 4 if n is composite. > > True, but OP isn't talking about 10^n - 1, OP is talking about > the repUNIT numbers, which are (10^n - 1)/9. > There's an extra factor that's been forgotten about, which makes > it non-trivial (i.e. it's not always true/false). > If n=pq > 10^n-1 = Phi(n,10) * Phi(p,10) * Phi(q,10) * Phi(1,10) > R(pq) = Phi(pq,10) * Phi(p,10) * Phi(q,10) > So we're looking for n=pq s.t. those three factors are prime. I don't think so. OP didn't write, If s(n) is less than or equal to 3, then n is prime. 3 is not equal to less than 3. To refute OP you have to [CapitalThorn]nd a composite n such that s(n) is 2. > Phi(x,10) is prime for 2, 19, and 38 > 11111111111111111111111111111111111111 > So the OP was, erm, not very good, shall we say, at investigating his > conjecture. > Clue to OP - > 1) If making conjectures about factors of cyclotomic numbers, get hold > of either the cunningham tables, or the extended cunningham tables, > and use that previously-discovered corpus. > 2) Try your conjectures in an arbitrary-precision algebra package > such as GP/Pari,. Clue to sci.math generally; make sure you understand what it is you're responding to before you respond to it. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: my work on numbers > me. > I am Abhishek Bansal (Dob: 6 Feb,1983). > Here I am submitting my work on numbers as jpeg images which I have got > scanned: > http://mathforum.org/web.comments/abhishek.html > Although my work could be very simple,making me a matter of laugh. But > that doesn't matter to me,as I am Contented. This would be my last > contribution. As I had been a victim of unfortunates and many many other > things. > If you consider me as a talent and think it can be ben[CapitalThorn]cial to you Or > getting wasted I can offer myself and can come to america or australia .But > I need 100% scholarship for that , if you consider me for that. > Else if you know someone in delhi. > Please Note here I nowhere is saying iam brilliant. What I said I am a > thinker, had made sincere attempt. > Please comment on my work .I would check my e-mail on june,2 (ISTmorning > ,from cyber cafe) after that any of your reply will probably of less use or > no use. > With reference to (6): > 4^(4^3)+1 = 59649589127497217*5704689200685129054721 The OP can search for Generalised Fermat Numbers (or Ôgeneralized') for more information. Number 3 is also false, as it's possbile to construct covering sets of prime factors such that for every a, N+/-2^a has a small factor. This is intimately related to Reisel Numbers and Sierpinski Numbers. e.g.: if N+2^1 is divisible by 3, then N+2^3, N+2^5, N+2^7,... N+2^(2x+1) are all divisible by 3. Similarly, if N+2^2 is divisible by 5, then N+2^6, N+2^10 ... N+2^(4x+2) are all divisible by 3. It's possible to place several other small factors, which are factors of 2^x-1 for small x, such as 3, 5, 241, so that every power of 2 is covered. ``3 5 241'', for more info. (For the latter, you can't go wrong with the Kurt Foster and Douglas Zare posts.) Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: my work on numbers me. > I am Abhishek Bansal (Dob: 6 Feb,1983). > Here I am submitting my work on numbers as jpeg images which I have got scanned: > http://mathforum.org/web.comments/abhishek.html > Although my work could be very simple,making me a matter of laugh. But that doesn't matter to me,as I am Contented. This would be my last contribution. As I had been a victim of unfortunates and many many other things. > If you consider me as a talent and think it can be ben[CapitalThorn]cial to you Or getting wasted I can offer myself and can come to america or australia .But I need 100% scholarship for that , if you consider me for that. > Else if you know someone in delhi. > Please Note here I nowhere is saying iam brilliant. What I said I am a thinker, had made sincere attempt. > Please comment on my work .I would check my e-mail on june,2 (ISTmorning ,from cyber cafe) after that any of your reply will probably of less use or no use. With reference to (6): 4^(4^3)+1 = 59649589127497217*5704689200685129054721 -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Finding Natural f: Doubly asymptotic to constant, 1/x > I am looking for a continuous function f(x) for x>0 which satis[CapitalThorn]es: > (1) is decreasing > (2) As x-->0, f(x)-->k (constant); stays near this constant value for > small x, decreasing slowly as x gets large. > (3) As x-->in[CapitalThorn]nity, f(x)-->1/x > f could be recursive or a closed form. Does any such function occur in > nature (biology, physics etc)? Appreciate any help. to occur in nature in one place and I was wondering where else in nature this sequence arises. Also, is there a collection of sequences available on the net? === Subject: Asymptotic expansion... function: $$ D(y) = sum_{n=0}^{infty} (-1)^n frac{y^{n+1}}{(n+1)!(n+1)^frac{1}{2}}$$ The author deduces from this expansion the following asymptotic expansion for large y: $$ D(y) = 2pi^{-frac{1}{2}}(ln y)^{frac{1}{2}}left[ 1+0.2886(ln y)^{-1} - 0.2473 (ln y)^{-2} + 0.3403(ln y)^{-3} - ldotsright]$$ How he obtains the second expansion from the [CapitalThorn]rst is not explained. more details. Could anyone help ? === Subject: Re: Asymptotic expansion... > function: > $$ D(y) = sum_{n=0}^{infty} (-1)^n > frac{y^{n+1}}{(n+1)!(n+1)^frac{1}{2}}$$ > The author deduces from this expansion the following asymptotic > expansion for large y: > $$ D(y) = 2pi^{-frac{1}{2}}(ln y)^{frac{1}{2}}left[ 1+0.2886(ln > y)^{-1} - 0.2473 (ln y)^{-2} + 0.3403(ln y)^{-3} - ldotsright]$$ > How he obtains the second expansion from the [CapitalThorn]rst is not explained. > more details. Enough with the TEX already. === Subject: Re: Asymptotic expansion... Just a guess but from the coef[CapitalThorn]cients and the computing ability available in the late 60's I would say he plotted the function and did a nonlinear regression analysis against log(y). > function: > $$ D(y) = sum_{n=0}^{infty} (-1)^n > frac{y^{n+1}}{(n+1)!(n+1)^frac{1}{2}}$$ > The author deduces from this expansion the following asymptotic > expansion for large y: > $$ D(y) = 2pi^{-frac{1}{2}}(ln y)^{frac{1}{2}}left[ 1+0.2886(ln > y)^{-1} - 0.2473 (ln y)^{-2} + 0.3403(ln y)^{-3} - ldotsright]$$ > How he obtains the second expansion from the [CapitalThorn]rst is not explained. > more details. > Could anyone help ? === Subject: Re: P vs. NP Problem; 1WayFx Model a possible solution. > 1WayFx Model demonstrates the existence of a class of problems in > NP?P, and therefore P is strictly included in NP. I don't understand the following de[CapitalThorn]nition from your paper http://1wayfx.com/21012.html: ? ? S : algorithm ? tests all the elements in the set S This looks incomplete to me. You don't say what you mean by algorithm (a multi-tape Turing machine?) and what it means for an algorithm to test an element. -- Gareth Rees === Subject: Re: Linear ODEs - Elimination method > Solving say the following system: > Dx - 3y = 0 .... (1) > 2x - Dy = 0 .... (2) > Ô = d/dt > x' = 3y; y' = 2x > y = 2x' = 6y > y = ae^kt + be^-kt; k^2 = 6 > y' = ake^kt - bke^-kt > x = y'/2 = (ake^kt - bke^-kt)/2 Great, but you slighted my question(s). The above is obvious except for the fact that you used equation (2) for the last substitution to [CapitalThorn]nd x. Why could you haved used (1) as well? [That's what I asked, cf. QUESTIONS end of my post]. > by elimination is straighforward - e.g. operate on (1) by D, and > multiply (2) by -3, and then adding the new (1) and (2) eliminates y, > and gives D^2(x) - 6x = 0, solving, we get x(t) in 2 arb constants. A > similar procedure gives us y(t) in 2 arbitrary constants. > Why do twice the work? Once x(t) or y(t) is solved, then > substitue in appropriate equation to get resp. y(t) or x(t). For the same reason one learns Gauss reduction on simple linear systems [CapitalThorn]rst - to get the gist of it so we can generalize to more complex ones, and through this systemization of the procedure speed up an otherwise dull process. R === Subject: Performing an update operation in O(lg n) time Hi . Given a red-black tree with n nodes (its height is O(lg n)) , how can I update values (decreasing a speci[CapitalThorn]c value from all the nodes , that their key is less than / equal to a given key) in time of O(lg n) ? In the worst case , we need to update all the (speci[CapitalThorn]c) values in the red-black tree (worst case is when the given key is the maximum key value in the tree , then we need to update all the nodes in the tree) . The tree has n nodes , so how the update can be done in O(lg n) time ? Someone told that an additional value is needed to be saved , in order to update each node (which is needed to be updated) in O(1) time (constant time) . What is the additional value which is needed to be saved here , for each node ? Lior . === Subject: Re: proof of the Riemann hypothesis ? But isn't de Branges (sp?) also supposed to have a proof? That would be much more plausible ... I looked at one document on the subject by him, but it seemed more a historial survey than a proof. -- 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: proof of the Riemann hypothesis ? > I stumbled upon this paper yesterday : > http://arxiv.org/abs/math.GM/0405531 || ^^ > Anyone heard of it ? > On page 2 he says: > (c_1 - r_1) * ... * (c_n - r_n) = (c_1 - 1) * ... * (c_n -1) > implies > r_1 = ... = r_n = 1 > for complex c_i != 0 and real r_i != 0. > Therefore I conclude: > (2-4)*(3-4) = (2-1)*(3-1) > implies > 1=4 Obviously your numbers weren't complex enough. > In an other paper, Ferreira has proven the denumerability of the real > numbers. That might indeed be correct if 4 = 1 in his theory. Who was it who came up with a witty meaning for GM? Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: proof of the Riemann hypothesis ? > I stumbled upon this paper yesterday : > http://arxiv.org/abs/math.GM/0405531 > Anyone heard of it ? On page 2 he says: (c_1 - r_1) * ... * (c_n - r_n) = (c_1 - 1) * ... * (c_n -1) implies r_1 = ... = r_n = 1 for complex c_i != 0 and real r_i != 0. Therefore I conclude: (2-4)*(3-4) = (2-1)*(3-1) implies 1=4 In an other paper, Ferreira has proven the denumerability of the real numbers. That might indeed be correct if 4 = 1 in his theory. Christian === Subject: Re: The physics of practical daily life > This post is off-topic in sci.math. Please honor, or at least consider, > the purpose of the newsgroups to which you post. OH C'mon Charly; mathematics isn't off any topic; especially everyday physics. > So far, the foot-pound-second system developed by the U.S. appears to > me to be the best. Counting by tens has only aggravated the situation; > with its seven fundamental units and hundreds of derived units the > metric SI system is fraught with its own errors and omissions. Avoid > it at all costs. It's a strawman and has already cost physics its > sanity, not to mention that of otherwise intellegent scientists. Anybody who believes in two dimensional worms existing; or doesn't know that there's always something beneath a two dimensional surface, a dolt: There's gold in them thar hills, but sometimes it takes a little digging to get it out. > -- > There are two things you must never attempt to prove: the unprovable -- > and the obvious. Sounds like you might be talking about two gallons comming out of a 1 gallon klein jug. > -- > Democracy: The triumph of popularity over principle. Proving that humans are gullible. === Subject: Re: The physics of practical daily life > This post is off-topic in sci.math. Please honor, or at least consider, > the purpose of the newsgroups to which you post. > OH C'mon Charly; mathematics isn't off any topic; especially everyday > physics. Strange... I posted a similar request to an earlier off-topic post of yours, but I only sent it to sci.math. You didn't respond. Apparently you are cross-posting to newsgroups you don't even visit! According to your position stated above, *every* newsgroup post should be cross-posted to sci.math. Ridiculous! -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: The physics of practical daily life > This post is off-topic in sci.math. Please honor, or at least consider, > the purpose of the newsgroups to which you post. > > OH C'mon Charly; mathematics isn't off any topic; especially everyday > physics. >Strange... I posted a similar request to an earlier off-topic post of yours, but I only sent >it to sci.math. You didn't respond. Apparently you are cross-posting to newsgroups you don't >even visit! He doesn't visit sci.physics either. All he does in hang out in the bogus alt.sci.physics, and clutter up a couple of other groups he never reads. Gene Nygaard === Subject: Re: The physics of practical daily life > Rene died in 1650. > cut< > He envisioned the whole universe as consising of whirling vortices: Really? I didn't know they had LSD way back then! Double-A === Subject: Re: The physics of practical daily life > > So far, the foot-pound-second system developed by the U.S. appears to > me to be the best. Counting by tens has only aggravated the situation; > with its seven fundamental units and hundreds of derived units the > metric SI system is fraught with its own errors and omissions. Avoid > it at all costs. It's a strawman and has already cost physics its > sanity, not to mention that of otherwise intellegent scientists. > SI is easier to use and less prone to error, Shead. In a pig's eye; the errors are multiples of ten too, Sam. === Subject: Where did the Koszul complex come from? Let A be a noetherian local ring, k be its residue [CapitalThorn]eld. It is well-known that the Koszul complex over A gives a [CapitalThorn]nite free resolution of k(see, for example, Serre's Local Algebra). So the global homological dimension of A is [CapitalThorn]nite. My question is where on earth did it come from? === Subject: Re: What is a universal mapping problem? > |According to a book, it is like this; > |Let C, D be categories, X be an object of D, > |Z be an object of C, T: C -> D be a functor, > |g: X -> T(Z) be a morphism. > |For each morphism f: X -> T(Y), there exists a unique > |morphism h: Z -> Y such that f = T(h)g. > |Then we call g a universal morphism. > |Given this de[CapitalThorn]nition, let's consider the following example. > |Let A be a commutative ring, > |S be a multiplicatively closed subset of A. > |Let A[1/S] be the localization of A by S. > |Let g: A -> A[1/S] be the canonical homomorphism. > |Let B be a commutative ring. Let f: A -> B be a ring homomorphism > |such that f(s) is invertible for every s in S. > |Then there exists a unique homomorphism h : A[1/S] -> B > |such that f = hg. > |I would like to call g a universal homomorphism, > |but how do we [CapitalThorn]t this example to the above de[CapitalThorn]nition > |of a universal morphism or how do we modify it? > the morphism that you think should be playing the role of g lives in > the category that should be playing the role of c, whereas the > book's de[CapitalThorn]nition says that g should live in d, and that's > probably one of the main reasons that you're confused. > take: > c = the category of commutative rings. > d = the category where an object is a commutative ring equipped with a > multiplicatively closed subset. > t = the functor assigning to a commutative ring r the pair > (r,{invertible elements in r}). > g = the obvious morphism (a,s)->t(a[1/s]) > then your candidate for g, namely the canonical homomorphism > a->a[1/s], is not the true correct g, but rather u(g) where u:d->c is > a certain functor which seems more or less completely extraneous to > the problem. > if the category d seems like a somewhat arti[CapitalThorn]cial category created > only for the purpose of shoehorning your example to [CapitalThorn]t the book's > de[CapitalThorn]nition, then: > 1. to some extent yes, but auxiliary categories of more or less > this sort can be useful at times, and > 2. it's dif[CapitalThorn]cult to be sure whether the book that you got the > de[CapitalThorn]nition from is doing a good job of explaining the surprisingly > simple and intuitive ideas that are lurking around here. your example > would require a bit less shoehorning if certain simpler and slightly > more general concepts were used, and i don't know whether the book in > question adequately covers these simpler concepts. Mathematician. I just couldn't come up with the functor T. === Subject: Re: Running wide >I'm looking for a proof of the problem as stated (or counterexample), not >one that assumes the existence of derivatives. >Please say if you think that is not realistic. It's certainly realistic - I didn't realize that that's what you wanted (I [CapitalThorn]nd the fact that it's true for smooth curves suf[CapitalThorn]ciently amazing.) First I should say that this is a standard result - it's Someone's Theorem (probably you need a slightly better reference than that to search for it, sorry.) Anyway, I can think of at least two ways to take what I posted and make it into a proof that works in general: (i) show that any convex curve can be approximated in an appropriate sense by a smooth convex curve, (ii) use a little real analysis: although a convex curve need not be smooth, it does have to have a tangent almost everywhere, etc. (For example, the problem as stated talked about a convex curve with [CapitalThorn]nite length; you evidently didn't realize that _any_ convex curve has [CapitalThorn]nite length.) I suspect that either of (i) and (ii) works. I also suspect that (ii) gives a _simpler_ proof, although it's less elementary. Here's a proof along the lines of (ii) - by proof I mean proof, except for just assuming various things that it seems to me are clear from the convexity, although they have to be actually proved to give an actual _proof_: Say K is a compact convex set in the complex plane C, with nonempty interior. For convenience assume that 0 is in the interior of K; let A be the boundary of K and let B be the boundary of the set of points at distance at most 1 from K. Note that the fact that 0 is in the interior of K and K is convex shows that a ray starting at the origin intersects A in exactly one point, and similarly for B. So if c_2 : [a,b] -> B is a parametrization of B then arg(c_2(t)) either increases by 2Pi or decreases by 2Pi as t runs from a to b. (Here arg is as in complex variables: arg(re^{it}) = t.) Let c_2 be a parametrization of B by arclength; c_2 : [0,L] -> B, where L is the length of B, and |c_2'(t)| = 1 for almost all t (in particular, for all t such that c_2'(t) exists.) Suppose that c_2 runs around K clockwise, ie in the direction of increasing argument. Then |c_2(t) - c_2(s)| <= |t - s|, so in particular c_2 is absolutely continuous. Let c_1(t) be the point of A closest to c_2(t) as before. Now c_1 need not be what one would usually call a parametrization of A, since c_1 need not be 1-1 except for the endpoints (for example if A has a corner then c_1 will be constant on an interval.) But c_1 is nonetheless close enough to a paramterization of A, because arg(c_1(t)) is a _nondecreasing_ function of t (so although c_1 may be constant on intervals it never doubles back over parts of A it's already traced.) So the length of A is the integral of |c_1'|. De[CapitalThorn]ne a(t) as before; since we speci[CapitalThorn]ed this time that our curves are clockwise it follows that a(t) is nondecreasing, in particular |a'| = a' almost everywhere. The same argument as before shows that |c_2'| = |c_1'| + |a| = |c_1'| + a almost everywhere, and hence as before int|c_2'| = int|c_1'| + 2Pi, QED. There's a lot of stuff there that must be true because of the convexity, that I didn't prove. It seems like the subtlest gap is showing that c_1 is absolutely continuous; we need the absolute continuity of c_1 to conclude that the length of A is int|c_1'|. It's clear that c_1 must be absolutely continuous because (*) |c_1(t) - c_1(s)| <= |c_2(t) - c_2(s)| (at least for s close to t), so c_2 ac implies c_1 ac. (Why is (*) true? Well, if you look at a picture it seems clear... Now, could be we're not looking at the right picture. But if we assume that c_1 is absolutely continuous then the fact that |c_1| <= |c_2| implies (*). Of course that's not a proof, because we're using (*) to show that c_1 is absolutely continuous. But it convinces me that (*) must in fact be true - so one proves (*) directly from the convexity somehow and we're set...) ************************ David C. Ullrich === Subject: Re: Running wide >I'm looking for a proof of the problem as stated (or counterexample), not >one that assumes the existence of derivatives. >Please say if you think that is not realistic. >It's certainly realistic - I didn't realize that >that's what you wanted (I [CapitalThorn]nd the fact that it's >true for smooth curves suf[CapitalThorn]ciently amazing.) >First I should say that this is a standard >result - it's Someone's Theorem (probably you >need a slightly better reference than that to >search for it, sorry.) >Anyway, I can think of at least two ways to take what >I posted and make it into a proof that works in >general: (i) show that any convex curve can be >approximated in an appropriate sense by a smooth >convex curve, (ii) use a little real analysis: >although a convex curve need not be smooth, it >does have to have a tangent almost everywhere, >etc. (For example, the problem as stated >talked about a convex curve with [CapitalThorn]nite length; >you evidently didn't realize that _any_ >convex curve has [CapitalThorn]nite length.) >I suspect that either of (i) and (ii) works. >I also suspect that (ii) gives a _simpler_ >proof, although it's less elementary. Here's >a proof along the lines of (ii) - by >proof I mean proof, except for just assuming >various things that it seems to me are clear >from the convexity, although they have to be >actually proved to give an actual _proof_: >Say K is a compact convex set in the complex >plane C, with nonempty interior. For convenience >assume that 0 is in the interior of K; let A >be the boundary of K and let B be the boundary >of the set of points at distance at most 1 from >Note that the fact that 0 is in the interior >of K and K is convex shows that a ray starting >at the origin intersects A in exactly one point, >and similarly for B. So if c_2 : [a,b] -> B is >a parametrization of B then arg(c_2(t)) either >increases by 2Pi or decreases by 2Pi as t runs >from a to b. (Here arg is as in complex >variables: arg(re^{it}) = t.) >Let c_2 be a parametrization of B by arclength; >c_2 : [0,L] -> B, where L is the length of B, >and |c_2'(t)| = 1 for almost all t (in particular, >for all t such that c_2'(t) exists.) Suppose that >c_2 runs around K clockwise, ie in the direction >of increasing argument. Then |c_2(t) - c_2(s)| ><= |t - s|, so in particular c_2 is absolutely >continuous. >Let c_1(t) be the point of A closest to c_2(t) >as before. Now c_1 need not be what one would >usually call a parametrization of A, since c_1 >need not be 1-1 except for the endpoints (for >example if A has a corner then c_1 will be >constant on an interval.) But c_1 is >nonetheless close enough to a paramterization >of A, because arg(c_1(t)) is a _nondecreasing_ >function of t (so although c_1 may be constant >on intervals it never doubles back over >parts of A it's already traced.) So the length >of A is the integral of |c_1'|. De[CapitalThorn]ne a(t) >as before; since we speci[CapitalThorn]ed this time that >our curves are clockwise it follows that a(t) >is nondecreasing, in particular |a'| = a' almost >everywhere. The same argument as before shows >that |c_2'| = |c_1'| + |a| = |c_1'| + a almost >everywhere, and hence as before int|c_2'| >= int|c_1'| + 2Pi, QED. >There's a lot of stuff there that must >be true because of the convexity, that I >didn't prove. It seems like the subtlest >gap is showing that c_1 is absolutely >continuous; we need the absolute continuity >of c_1 to conclude that the length of A is >int|c_1'|. It's clear that c_1 must be >absolutely continuous because >(*) |c_1(t) - c_1(s)| <= |c_2(t) - c_2(s)| >(at least for s close to t), so c_2 ac >implies c_1 ac. (Why is (*) true? Well, >if you look at a picture it seems clear... >Now, could be we're not looking at the >right picture. But if we assume that c_1 >is absolutely continuous then the fact >that |c_1| <= |c_2| implies (*). Of course >that's not a proof, because we're using >(*) to show that c_1 is absolutely continuous. >But it convinces me that (*) must in fact >be true - so one proves (*) directly >from the convexity somehow and we're set...) Here's a proof of (*). Choose coordinates so that c_2(t) = (0,1) and c_1(t) = (0,0). Then all of K must lie below or on the x-axis, because if (x,y) is a point of K with y > 0 then some point on the segment joining (0,0) to (x,y) is at distance less than 1 from (0,1). Say c_2(s) = (x,y) = p and c_1(s) = (X,Y) = P. Let N be the line through P perpendicular to the segment pP. The same argument shows that all of K must lie on or below N. Suppose that X > 0. Then the fact that (0,0) lies on or below N shows that N has negative slope; hence the segment pP has positive slope. (Or it could happen that N is horizontal and then pP is vertical, but pP cannot be a line with negative slope.) This show that x >= X and 1 - y >= -Y, which implies (*). ************************ David C. Ullrich === Subject: Re: Running wide > although a convex curve need not be smooth, it does have to have a tangent almost everywhere Ah, I hadn't thought of that. > you evidently didn't realize that _any_ convex curve has [CapitalThorn]nite length. Right. As you can see, I don't know much (maybe I should have mentioned that earlier). I was interested in what sort of arguments would be needed. === Subject: Re: Running wide >Let A be a simple closed convex curve in R^2 with [CapitalThorn]nite length. >Let B be the set of points outside A distant 1 from A. >Is (length of B) - (length of A) = 2 Pi ? First some differential geometry background. Let f(t) be a parameterization of the curve where f'(t) does not vanish. Then, the unit tangent to the curve at f(t) is f'(t) T(t) = ------- [1] |f'(t)| Let the matrix R rotate right by pi/2, that is, [ 0 1 ] R = [ ] [2] [ -1 0 ] Note that R^2 = -I. Since T(t) is a unit vector, T'(t) is perpindicular to T(t), therefore parallel to the unit vector RT(t). If k(t) is the right curvature of the curve at f(t), then T'(t) ------- = k(t) RT(t) [3] |f'(t)| Without loss of generality, assume that the curve circles its interior once clockwise. Then, the integral of its right curvature with respect to arclength is 2 pi. That is, | | k(t) |f'(t)| dt = 2 pi [4] | Using [1], [3], and the fact that R^2 = -I, we get that RT'(t) = -k(t) T(t) |f'(t)| = -k(t) f'(t) [5] The set of points you describe above can be parameterized by f(t) - RT(t) [6] Note that the derivative of [6] is f'(t) - RT'(t) = f'(t) + k(t) f'(t) = (1 + k(t)) f'(t) [7] If k(t) >= -1, the length of the points outside the curve is given by | | | f'(t) - RT'(t) | dt | | = | |1 + k(t)| |f'(t)| dt | | = | (1 + k(t)) |f'(t)| dt | | = | |f'(t)| dt + 2 pi [8] | Thus, [8] says that the length of the points you describe above is the length of the original curve plus 2 pi as long as k(t) >= -1 so that |1 + k(t)| = 1 + k(t). Thus, your statement is true even if the curve has some concavity, as long as the right curvature is no less than -1. Rob Johnson take out the trash before replying === Subject: Re: Running wide >[...] >Thus, [8] says that the length of the points you describe above is the >length of the original curve plus 2 pi as long as k(t) >= -1 so that >|1 + k(t)| = 1 + k(t). >Thus, your statement is true even if the curve has some concavity, as >long as the right curvature is no less than -1. Is the fact that The set of points you describe above can be parameterized by f(t) - RT(t) still true in this case? >Rob Johnson >take out the trash before replying ************************ David C. Ullrich === Subject: Re: Running wide >[...] >Thus, [8] says that the length of the points you describe above is the >length of the original curve plus 2 pi as long as k(t) >= -1 so that >|1 + k(t)| = 1 + k(t). >Thus, your statement is true even if the curve has some concavity, as >long as the right curvature is no less than -1. >Is the fact that The set of points you describe above can >be parameterized by f(t) - RT(t) still true in this case? Yes it is. As the right curvature gets closer to -1, the points that are 1 unit outside the curve come closer to a singularity, where the right curvature becomes in[CapitalThorn]nite. In fact, the right curvature of f(t) - RT(t) is k(t) ------ k(t)+1 which tends to in[CapitalThorn]nity as k(t) tends to -1. If k(t) < -1, then the curve f(t) - RT(t) intersects itself and we need to deal with ugly interior loops. Rob Johnson take out the trash before replying === Subject: Re: Running wide > Let f(t) be a parameterization of the curve where f'(t) does not vanish. Why is this possible? Is there a theorem that tells us that such a parametrization is always possible? The curve is continuous ( the image of a continuous function on [0,1] ), but that doesn't imply a tangent exists anywhere at all, does it? Or does convexity + [CapitalThorn]nite length somehow imply the existence of a tangent? If so, I don't see how - please explain! === Subject: Re: Running wide It would be nice if the relevant parts of previous posts to which you refer are quoted so that each post makes more sense, both to [CapitalThorn]rst time readers as well as to refresh the memories of those who are following the thread. For example, in this thread, we are talking about simple, closed, convex curves in R^2. > Let f(t) be a parameterization of the curve where f'(t) does not vanish. >Why is this possible? >Is there a theorem that tells us that such a parametrization is always >possible? >The curve is continuous ( the image of a continuous function on [0,1] ), but >that doesn't imply a tangent exists anywhere at all, does it? >Or does convexity + [CapitalThorn]nite length somehow imply the existence of a tangent? >If so, I don't see how - please explain! It is a theorem that any convex function has monotonically increasing left and right derivatives, and at each point, the right derivative is greater than or equal to the left derivative. It is a simple corollary that the left and right derivatives are equal except at a countable set of points and that the sum of the differences of the derivatives at the countable set of points is less than or equal to the difference of the maximum of the left derivative and the minimum of the right derivative. A simple adaptation of this proof gives us that a simple, closed, convex curve in R^2 has left and right tangents at each point. If the curve circles its interior clockwise, then the arguments of these tangents are monotonically decreasing, and the argument of the right tangent is less than or equal to the argument of the left tangent at each point. We get the same corollary that the left and right derivatives are equal except at a countable set of points and that the sum of the differences of the arguments at the countable set of points is less than or equal to 2 pi. Since the arguments of the tangents are monotonically decreasing, they are differentiable almost everywhere, so the curvature exists almost everywhere. Furthermore, the length integral of the curvature plus the sum of the differences of the arguments equals 2 pi. -------------------------------- I had intended the differential geometry proof to be used on smooth approximations to the curve and then limits taken. The comments I made above regarding tangents and curvature, allow this to be done. Rob Johnson take out the trash before replying === Subject: Re: Running wide >[...] >-------------------------------- >I had intended the differential geometry proof to be used on smooth >approximations to the curve and then limits taken. The comments I made >above regarding tangents and curvature, allow this to be done. Well, that's not so clear. You say that one can get the curve consisting of points at unit distance from the original curve by adding the unit normal - that's not quite right if the curve has corners. (This is why I started with the second curve; if c_2(t) is a parametrization of the second curve and c_1(t) is the point of A closest to c_2(t) then c_1 _is_ a parametrization (sort of) of A, regardless. >Rob Johnson >take out the trash before replying ************************ David C. Ullrich === Subject: Re: Running wide >It would be nice if the relevant parts of previous posts to which you >refer are quoted so that each post makes more sense, both to [CapitalThorn]rst time >readers as well as to refresh the memories of those who are following >the thread. >For example, in this thread, we are talking about simple, closed, convex >curves in R^2. ... >I had intended the differential geometry proof to be used on smooth >approximations to the curve and then limits taken. The comments I made >above regarding tangents and curvature, allow this to be done. For actually *seeing* why the result is true, however, I think that starting from Lynn Kurtz's observation that it's obviously true for a convex polygon, then doing the limit details starting from there (instead of from smooth curves), is a better approach, and involves even less analysis in those details. Lee Rudolph === Subject: Re: Running wide >It would be nice if the relevant parts of previous posts to which you >refer are quoted so that each post makes more sense, both to [CapitalThorn]rst time >readers as well as to refresh the memories of those who are following >the thread. >For example, in this thread, we are talking about simple, closed, convex >curves in R^2. >... >I had intended the differential geometry proof to be used on smooth >approximations to the curve and then limits taken. The comments I made >above regarding tangents and curvature, allow this to be done. >For actually *seeing* why the result is true, however, I think that >starting from Lynn Kurtz's observation that it's obviously true for >a convex polygon, then doing the limit details starting from there >(instead of from smooth curves), is a better approach, and involves >even less analysis in those details. You could be right, but I don't see it. Honest, my intention was to do it this way in my subthread, but I did it the way I did instead because I didn't see how to work out those details (didn't see how to convince myself that the necessary limits come out right, which is a much weaker requirement than actually giving a rigorous proof...) Let's say that if C is a convex closed curve, the boundary of the compact convex set K, then Out(C) is the set of points at distance 1 from K. Now given A it's clear that there's an inscribed polygon P such that length(P) is close to length(A); this is just the de[CapitalThorn]nition of arclength. What I don't see how to show, even sloppily (other than by just saying it's obvious) is why length(Out(P)) is close to length(Out(A)). ??? >Lee Rudolph ************************ David C. Ullrich === Subject: Re: Running wide > Yes. Hard to believe but it's so. >Proof ? For a heuristic argument notice that it is trivially true for convex polygons. Now all you have to show is that for a sequence of such polygons approximating C1 their one unit away neighbors approach C2. :-) --Lynn === Subject: Re: Latex tabular? Huh?!? and yes, it's accessible through Google, (ii) it is quite active, (iii) I can say for sure that *that* post appeared in comp.text.tex since I'm actually following it regularly, (iv) I may have given you a wrong link for the FAQ, but then a search string like uk tug tex faq in Google *should* give you the right one as (one of) the [CapitalThorn]rst hit(s). Also, your TeX distro may come with a static copy of it. HTH, Michele -- > Comments should say _why_ something is being done. Oh? My comments always say what _really_ should have happened. :) - Tore Aursand on comp.lang.perl.misc === Subject: Re: irreducible polynomails in a [CapitalThorn]eld of polynomials === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax > A gradient is a vector [CapitalThorn]eld (and please don't come and asc me what that > means), how on earth can a vector be negative. >That isn't the only meaning of gradient. In the case of a scalar >function of a single variable, f(x), the gradient is the [CapitalThorn]rst >derivative df/dx which is a scalar. No, the gradient is the vector (df/dx). Not that there would ever be any reason to make the distinction except in a silly context like this. But R and R^1 are not exactly the same thing... >Gib ************************ David C. Ullrich === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax > posted: >A gradient is a vector [CapitalThorn]eld (and please don't come and asc me what that >means), how on earth can a vector be negative. >That isn't the only meaning of gradient. In the case of a scalar >function of a single variable, f(x), the gradient is the [CapitalThorn]rst derivative >df/dx which is a scalar. >Gib > let f: -> f(x) be the function that attaches to each 3d coordinate the > degree of humidity (some insects like that function). it maps a three > dimensional vector space into the scalars (1 dim reals). > The gradient of that function is a 3D vector [CapitalThorn]eld, in each point > indicating the direction (and intensity) to [CapitalThorn]nd a local maximum of the > function. So what? That doesn't alter the fact that the gradient of a scalar function of a scalar variable is a scalar. In this case (which is a common one) the gradient has a sign (if non-zero). Gib === Subject: Re: Can a vector be positive or negative? ->Re: How to write a pseudo scienti[CapitalThorn]cal hoax >let f: -> f(x) be the function that attaches to each 3d coordinate the >degree of humidity (some insects like that function). it maps a three >dimensional vector space into the scalars (1 dim reals). >The gradient of that function is a 3D vector [CapitalThorn]eld, in each point >indicating the direction (and intensity) to [CapitalThorn]nd a local maximum of the >function. >REMARK: there can be many local maxima and some singulatities of f can >guide you the wrong way :-) === Subject: order of elements, I need help with problem this should probably be easy, but I am stuck: Suppose b has order 1072 (mod m), m >= 2, what is the order of b^1508? anyone smart enough? === Subject: Re: order of elements, I need help with problem > this should probably be easy, but I am stuck: > Suppose b has order 1072 (mod m), m >= 2, what is the order of b^1508? If a has order k, then a^r has order k/gcd(k,r). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: order of elements, I need help with problem > this should probably be easy, but I am stuck: > Suppose b has order 1072 (mod m), m >= 2, what is the order of b^1508? >If a has order k, then a^r has order k/gcd(k,r). === Subject: Re: order of elements, I need help with problem Sorry, the order of b should be 1073, NOT 1072. === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) Well, metre is the UK spelling (like colour or gaol), while meter is the US spelling (like color or jail). -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) >Well, metre is the UK spelling (like colour or gaol), while >meter is the US spelling (like color or jail). That's what Webster's New World Dictionary says: metre n. chießy Brit. sp. of METER Thomas === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) > Well, metre is the UK spelling (like colour or gaol), while > meter is the US spelling (like color or jail). Where on earth (America, probably) does this bizarre opinion, that Ôjail' is US English and not English, come from? Formal discussions of the choice between jayl and gaol spellings in English date to at least as far back as 1668. John bleedin' Milton (Londoner, if you hadn't guessed) spelt it Ôjail', and he was pushing up daisies in 1674. There was no such thing as the years later, for example. Before that, jaiole, jayle, jaile, and jayll were all attested Middle English spellings. Bring back the ÔGayhole' spelling (C13), that's what I say. It appears we've been practising irony for 8 centuries! Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many >However, 1 meter minus 1 meter >does de[CapitalThorn]nitely not equal 0 meter before rounding off because any >measurement is uncertain to some extent. >measurement is a physical, not mathematical concept. >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) > Keep in mind that Eckard is German. And in German meter means > metre. It's even pronounced the same as metre. Well, meter was perhaps not the best choice since Englishmen write metre while Americans write meter. I could substitute one meter alias metre by one mile if you like. I would rather prefer to close my door than exchanging useless words with a Chapman. He will never give up lecturing and even insulting me in a penetrant manner. Someone responded privately to this idea of mine: > I cannot imagine any justi[CapitalThorn]cation for the distinction between: > uncountable and countable in[CapitalThorn]nite. To my understanding the latter > does not really make sense. One can never reach the in[CapitalThorn]nite by > counting. > There is a very precise de[CapitalThorn]nition of the difference and one can > prove (if you accept bi-valued logic) that the reals are NOT > countably in[CapitalThorn]nite. I never doubted that reals are not countable. On the contrary, I merely objected against Weierstrass's notion of the in[CapitalThorn]nity which I consider biased in favor of numbers for historical reasons. In other words, I do not see any justi[CapitalThorn]cation for ascribing countability to anything without any limit. I guess, the difference between countable in[CapitalThorn]nite and simply in[CapitalThorn]nite is perhaps a useless arbitrary construct. At least, I did not [CapitalThorn]nd any genuine argument why and how to distinguish between 1000.. (in[CapitalThorn]nitely many zeros) and 5000.. (in[CapitalThorn]nitely many zeros). For my feeling of English language, in[CapitalThorn]nitely many is a misnomer. Eckard Blumschein === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many >However, 1 meter minus 1 meter >does de[CapitalThorn]nitely not equal 0 meter before rounding off because any >measurement is uncertain to some extent. >measurement is a physical, not mathematical concept. >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) > Keep in mind that Eckard is German. And in German meter means > metre. It's even pronounced the same as metre. What mistake? I see no mistake of mine. > Well, meter was perhaps not the best choice since Englishmen write metre > while Americans write meter. My dictionary gives meter for metre as speci[CapitalThorn]cally a US usage, and Blumschein posts with a .de address. > I would rather prefer to close my door than exchanging useless words > with a Chapman. He will never give up lecturing and even insulting me in > a penetrant manner. Aaaaaaaaah! > I never doubted that reals are not countable. On the contrary, I merely > objected against Weierstrass's notion of the in[CapitalThorn]nity which I consider > biased in favor of numbers for historical reasons. favour. > In other words, I do > not see any justi[CapitalThorn]cation for ascribing countability to anything without > any limit. I guess, the difference between countable in[CapitalThorn]nite and > simply in[CapitalThorn]nite is perhaps a useless arbitrary construct. At least, I > did not [CapitalThorn]nd any genuine argument why and how to distinguish between > 1000.. (in[CapitalThorn]nitely many zeros) and > 5000.. (in[CapitalThorn]nitely many zeros). Crass formalism. Neither of these are representations of numbers (for similar blunders see is 0.99999 = 1 threads ad nauseam). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many > (for similar blunders see is 0.99999 = 1 threads ad nauseam). Point of ambiguity - is that see (is 0.99999 = 1 threads ad nauseam) or see (is 0.99999 = 1 threads) ad nauseam ? :-) Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many >measurement is a physical, not mathematical concept. > What mistake? I see no mistake of mine. Really? >Well, meter was perhaps not the best choice since Englishmen write metre >while Americans write meter. > My dictionary gives meter for metre as speci[CapitalThorn]cally a US > usage, and Blumschein posts with a .de address. Do you infer from that I have to use English English? > favour. Of course. >In other words, I do >not see any justi[CapitalThorn]cation for ascribing countability to anything without >any limit. I guess, the difference between countable in[CapitalThorn]nite and >simply in[CapitalThorn]nite is perhaps a useless arbitrary construct. At least, I >did not [CapitalThorn]nd any genuine argument why and how to distinguish between >1000.. (in[CapitalThorn]nitely many zeros) and >5000.. (in[CapitalThorn]nitely many zeros). > Crass formalism. Neither of these are representations of numbers I agree: The essence of in[CapitalThorn]nity evades traditional mathematical thinking. It simply means not countable, no matter with which cardinality. However, I wonder if a mathematican objects against formalism. > (for similar blunders see is 0.99999 = 1 threads ad nauseam). The word blunder might rather be justi[CapitalThorn]ed as to judge any attempt of theoretically reducing mathematics just to numbers. I would like to express my higher estimation to numerical mathematics as compared to a questionable because arbitrary Weierstrassian theory of numbers. Ad nauseam? Yes, youngsters with a sound feeling should keep clear from those who are forcing Buridan's donkey to suffer starvation over the centuries and often demand to exclude excactly zero by all means but do not care for values within its close proximity. === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many >measurement is a physical, not mathematical concept. > What mistake? I see no mistake of mine. > Really? Really! >Well, meter was perhaps not the best choice since Englishmen write metre >while Americans write meter. > My dictionary gives meter for metre as speci[CapitalThorn]cally a US > usage, and Blumschein posts with a .de address. > Do you infer from that I have to use English English? English English? Better than double Dutch I suppose! > Ad nauseam? Yes, youngsters with a sound feeling should keep clear from > those who are forcing Buridan's donkey to suffer starvation over the > centuries and often demand to exclude excactly zero by all means but do > not care for values within its close proximity. You have a bit of a donkey fetish it seems :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many <40BD83EE.8070305@et.uni-magdeburg.de> Discussion, linux) > I never doubted that reals are not countable. On the contrary, I merely > objected against Weierstrass's notion of the in[CapitalThorn]nity which I consider > biased in favor of numbers for historical reasons. > favour. Surely at this point, one would guess that Blumschein prefers American English spelling. Regardless of his .de address. (Curiously I prefer American English spelling too, despite my work-related .nl email address.) -- It seems to me that in wartime Americans shouldn't be attacking each other in this way on a *worldwide* forum. Then again, I know I'm an American, but I have no way of knowing that you are, which would explain a lot. --James Harris, on why Yanks should accept his proof === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many > I never doubted that reals are not countable. On the contrary, I merely > objected against Weierstrass's notion of the in[CapitalThorn]nity which I consider > biased in favor of numbers for historical reasons. > favour. > Surely at this point, one would guess that Blumschein prefers American > English spelling. Regardless of his .de address. Why, I wonder? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many > I never doubted that reals are not countable. On the contrary, I merely > objected against Weierstrass's notion of the in[CapitalThorn]nity which I consider > biased in favor of numbers for historical reasons. > favour. > Surely at this point, one would guess that Blumschein prefers American > English spelling. Regardless of his .de address. >Why, I wonder? Regardless, surely spelling is the least of his problems... Seems silly to waste time addressing the sort of issues raised here. I know that _I_ would never waste time on something like that. ************************ David C. Ullrich === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many <40BD83EE.8070305@et.uni-magdeburg.de> <87llj6axf0.fsf@phiwumbda.org> Discussion, linux) > I never doubted that reals are not countable. On the contrary, I merely > objected against Weierstrass's notion of the in[CapitalThorn]nity which I consider > biased in favor of numbers for historical reasons. > favour. > Surely at this point, one would guess that Blumschein prefers American > English spelling. Regardless of his .de address. > Why, I wonder? Maybe he just reads more American English writing than British English? Anyway, who knows? For whatever reason, it's apparent that his spelling tends to the American English conventions. -- Jesse F. Hughes Leaving things always seems to [CapitalThorn]x me, Running seems to ease my worried mind. -- Bad Livers, Honey, I've Found a Brand New Way === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many > However, 1 meter minus 1 meter > does de[CapitalThorn]nitely not equal 0 meter before rounding off because any > measurement is uncertain to some extent. >measurement is a physical, not mathematical concept. >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) > Keep in mind that Eckhard is German. And in German meter means > metre. It's even pronounced the same as metre. A meter in German is > a Messer, which is also a knive... (OK, I better stop now) But he was writing in English :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many Discussion, linux) > However, 1 meter minus 1 meter > does de[CapitalThorn]nitely not equal 0 meter before rounding off because any > measurement is uncertain to some extent. >measurement is a physical, not mathematical concept. >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) > Keep in mind that Eckhard is German. And in German meter means > metre. It's even pronounced the same as metre. A meter in German is > a Messer, which is also a knive... (OK, I better stop now) > But he was writing in English :-( But surely meter is the American English spelling of the unit of length. Or was he writing in British English? I couldn't tell. -- Well *supposedly* a correct and profound math paper can get published in a Ôreputable journal' which means that the journals I've faced so far may lose a lot of their luster once the full story comes out. --- James Harris, on the quality of math journals rejecting his paper === Subject: Re: in[CapitalThorn]nitely much - not in[CapitalThorn]nitely many > > However, 1 meter minus 1 meter > does de[CapitalThorn]nitely not equal 0 meter before rounding off because any > measurement is uncertain to some extent. > >measurement is a physical, not mathematical concept. > >(Incidentally, the SI unit of length is the metre. A meter >is a measuring device, e.g., an odometer.) > > Keep in mind that Eckhard is German. And in German meter means > metre. It's even pronounced the same as metre. A meter in German is > a Messer, which is also a knive... (OK, I better stop now) > But he was writing in English :-( > But surely meter is the American English spelling of the unit of > length. But he appears to neither American by nationality or residence. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Speci[CapitalThorn]c Gravity or Relative Density > Copied from > : > ÔThe density of water is often taken as a standard and the density of > other materials compared relative to it. The relative density of water > is the density of water over the density of water, 62.4 pounds per > cubic foot over 62.4 pounds per cubic foot, so that it is 1.0. The > relative density of a rock that is 2 times as dense as water is the > density of the rock (124.8 pounds per cubic foot) divided by the > density of water (62.4 pounds per cubic foot) to give 2.0. The > relative density of steel is 7.7 and that of mercury is 13.6.' > Speci[CapitalThorn]c gravity is the same in either the customary, or the SI system > of weights and measures. >Donald, >SG is de[CapitalThorn]nately NOT a fundamental quantity these days. Useful sometimes, >but not fundamental. >What point are you trying to make? >Owen >P.S. keep to SI units and you avoid confusion when mixing standards. for >example AFAIK the Ôpound' is NOT a unit of mass, although the Ôounce' may be >such. What a weird idea!! Does that mean that if I buy 8 oz of cheese, it is a measurement of mass? But if I buy 2 lb of cheese, it is a measurement of force? Get real. Since you mentioned standards, what exactly do you imagine to be the standard for your pound? What is the nature of the standard? Something electrical, something mechanical, or what? Who, exactly, made it the standard? For whom does that standard apply--IOW, for whom does the de[CapitalThorn]ning agency have the authority to set the standards? When, exactly, was this standard established? Just the year will do, or a range of years if that's the best you can come up with. If you can't do that, Og, you might as well crawl back into your cartoon cave. >Since density is always mass/volume, we shouldn't mix Ôpounds' and >'cubit feet', coz that's just confusing to normal folks. We express density in pounds per cubic foot, or pounds per cubic yard, or pounds per cubic inch, or pounds per U.S. gallon, and the like in the real world. When physics textbooks used to use English units, which they don't even in the U.S. any more, in the period from around 1940 to 1970 and occasionally later (never earlier) express those densities in slugs per cubic foot, and that usage found limited application in the real world in the U.S. and Canada, and hardly any use anywhere else. But the thing is, if you do use slugs, then the only thing you can mix them with is those cubic feet. In the only system which contains slugs, there are no gallons or pints of any sort, there are no cubic inches, there are no cubic yards. Mixing slugs with any of those units is more confusing than mixing pounds with any of those units in measurements of density. Now, let's get back to the standards. Since you appear to be from the U.K., I'd suggest you look up the Weights and Measures Act of 1963, and see how pounds are de[CapitalThorn]ned there. Or just look at the U.S. law, which describes the 1959 agreement among the national standards laboratories of six countries on which that U.K. de[CapitalThorn]nition is based, the Notice in the Federal Register of 1 July 1959, http://www.ngs.noaa.gov/PUBS_LIB/FedRegister/FRdoc59-5442.pdf http://gssp.wva.net/html.common/re[CapitalThorn]ne.pdf Announcement. Effective July 1, 1959, all calibrations in the U.S. customary system of weights and measures carried out by the National Bureau of Standards will continue to be based upon metric measurement standards and except for the U.S. Coast and Geodetic Survey as noted below, will be made in terms of the following exact equivalences and appropriate multiples and submultiples: 1 yard = 0.9144 meter 1 pound (avoirdupois) = 0.453 592 37 kilogram Currently, the units de[CapitalThorn]ned by these same equivalences, which have been designated as the International Yard and the International Pound, respectively, will be used by the National Standards Laboratories of Australia, Canada, New Zealand, South Africa, and United Kingdom; thus there will be brought about international accord on the yard and pound by the English-speaking nations of the world, in precise measurements involving these basic units. This document also discusses the earlier U.S. de[CapitalThorn]nition of the pound as a slightly different exact fraction of a kilogram, from the late 19th Century. Gene Nygaard === Subject: re:a new way round diagonalisation-one in[CapitalThorn]nity?meta set theory Hi Ross F., > I think if you have the set of the zero and each of its successors > that that is the set of natural numbers. Many set zero to be the > empty set. I think you have the set of natural numbers from de[CapitalThorn]ning > successor and zero. I think, that it is more dif[CapitalThorn]cult to get the set of natural numbers in meta set theory - and that is no surprise! As I hope, that in meta set theory Goedel's incompleteness theorem does not hold, it should not be easy to get natural numbers and arithmetics. I look for a de[CapitalThorn]nition as near as possible to axiom A3. For this and your other questions I need further time to think - but I will publish the results here. Oskar Trestone (more used to German ...) ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: Re: a new way round diagonalisation-one in[CapitalThorn]nity?meta set theory Hi Oskar, When you start with the empty set, and generate sets from it, inductively any set thus generated is [CapitalThorn]nite, yet you can generate in[CapitalThorn]nitely many sets. To get in[CapitalThorn]nite sets requires in[CapitalThorn]nite induction. The series 1/2^x sums over the naturals to two. It is said that the summation converges to two. Its value is equal to two. That's a point about in[CapitalThorn]nite induction, that the value of the expression is the same whether it is considered inductively or synthetically. Researching in[CapitalThorn]nite induction leads to some consideration of why there are in[CapitalThorn]nite ordinals. My naive question about it was if there are in[CapitalThorn]nite integers are there in[CapitalThorn]nite integers. With only the empty set and successor there is basically induction that there are not [CapitalThorn]nitely many sets, the amount of sets is in-[CapitalThorn]nite not [CapitalThorn]nite. If the set theory model is restricted to contain only the empty set and its successors, then the set of all sets, including irregular sets but not Russell's non-set, the set of all sets or even integers/ordinals is not constructible by any [CapitalThorn]nite process, but only by consideration of all possible elements synthetically. Abour Russell's non-set, its de[CapitalThorn]nition exists without it having to be a set, yet I think the set of all sets is a set. It's kind of similar to the de[CapitalThorn]nition of point vis-a-vis line, except the [CapitalThorn]nite ordinals can be de[CapitalThorn]ned without de[CapitalThorn]ning the set of all sets, except they form that collection. http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va= synthetic In a model with in[CapitalThorn]nite ordinals, the set of all sets is both similar and different. I was thinking about it the other day, and thought of another way to equate null and all. Along with that, I was more heavily thinking about how to get in[CapitalThorn]nity from all this [CapitalThorn]niteness, [CapitalThorn]nity. I ßeetingly think I thought of a way to get in[CapitalThorn]nity from [CapitalThorn]nity, or rather think I ßeetingly thought. That is so, although it may have been along the lines of the empty set being a proper class of sorts. I'll have to try and remember it, it was quite good. Determining some rationalization of the in[CapitalThorn]nite from the in[CapitalThorn]nite would indeed be interesting. What tools can you use to integrate a collection of points? (That's to deter Ullrich.) And so on: how do you explain the Banach-Tarski result? Ross F. === Subject: Re: mass-density or weight-density > > The weight density of a material is the weight of a given volume of > the material divided by that volume. An example is that a 1 cubic foot > volume of water weighs 62.4 pounds. The density of water is then 62.4 > pounds per cubic foot. > > The weight density of an electron is immeasurablely small; as must be > its mass > The mass has been measured very precisely, Sheas > Mass of an electron is 9.1093897 x 10^-31 kg (accuracy 0.59 ppm) > : Which is the product of its density and its volume: For any > measurable quantity, inertia is the measure of the quantity of matter, > or mass that it contains! > > No Shead, inertian is a property of mass--it is not a measure of anything. > You are confusing inertia with intertial mass. No Sam, inertia is the measure of inertial mass; the ratio of how much (net) force [f] is required, divided by the acceleration [a] of the mass that is caused by that net force acting upon it: For any given mass of matter: Inertial Mass = f/a :: which is equal to its Gravitational Mass = w/g. Forget about mass varying with velocity Sam: A body's mass only changes when material is added or taken away from it. Theoretical physics is trying to get at the truth about them; so far it's all speculation; mostly erroneous AFAIK. === Subject: Re: mass-density or weight-density > No Sam, inertia is the measure of inertial mass Nope--inertia is NOT a measure. That's one of the reasons you are confused. === Subject: Re: mass-density or weight-density > Please re-read and please pay attention this time Missing period.^ === Subject: Re: exp(C) charset=iso-8859-7 The World Wide Wade > Is the image of the Complex circle, C={z: |z|=1/e}: exp(C), a circle? > No, in fact e^z takes no circle in the plane to a circle. We need only show > the curvature of the image curve is not constant. For simplicity consider > a circle centered at 0 of radius r. The image curve is parameterized by > p(t) = e^[re^(it)], t in R. The velocity vector is v(t) = > e^[re^(it)]*ire^(it). The speed is |v(t)| = e^(rcos(t))*r. So the unit > tangent vector is T(t) = v(t)/|v(t)| = ie^[i(rsin(t)+t)]. Finally > kappa = |T'(t)|/|v(t)| = |rcos(t) + 1|/r*e^(rcos(t)), > which is easily seen to be nonconstant. -- Ioannis Galidakis http://users.forthnet.gr/ath/jgal/ ------------------------------------------ Eventually, _everything_ is understandable === Subject: Re: exp(C) > Is the image of the Complex circle, C={z: |z|=1/e}: exp(C), a circle? No, in fact e^z takes no circle in the plane to a circle. We need only show the curvature of the image curve is not constant. For simplicity consider a circle centered at 0 of radius r. The image curve is parameterized by p(t) = e^[re^(it)], t in R. The velocity vector is v(t) = e^[re^(it)]*ire^(it). The speed is |v(t)| = e^(rcos(t))*r. So the unit tangent vector is T(t) = v(t)/|v(t)| = ie^[i(rsin(t)+t)]. Finally kappa = |T'(t)|/|v(t)| = |rcos(t) + 1|/r*e^(rcos(t)), which is easily seen to be nonconstant. === Subject: Re: Sober Space === Subject: Sober Space > Or for that matter, what motivated the de[CapitalThorn]nition? >First observe that the topology of a space is a complete >lattice in which [CapitalThorn]nite infs distribute over arbitrary sups. >Such lattices are called complete Heyting algebras (presumably >owing to the connection between them and intuitionistic set theory) >or CHAs. Is every CHA the open set lattice of some space? No. >They must be atomic in a certain sense. For all x, x = sup{ a atom | a <= x } a atom when a minimally nonzero ie, a /= bottom = 0, for all x < a, x = 0 ? No, isn't atomic in sense of Boolean algebra naive for Heyting algebra? >Can a map between spaces induce an isomorphism between open >set lattices without being a homeomorphism? Yes, unfortunately. For example? >But there is a way to go back, more precisely a (contravariant) >functor from the category of CHAs and homomorphisms that preserve >[CapitalThorn]nite infs and arbitrary sups, to the category of topological spaces >that is adjoint to the functor that takes a space to its open set >lattice. (The two functors are adjoint on the right.) Latin, Greek, Hebrew or Sanskrit? >This sets up an equivalence between a certain subcategory of Top >--the sober spaces--and a subcategory of CHAs--the atomic ones. Does that mean anything more than a bijection between atomic CHA and sober spaces? Yes, each pair are morphs, open sets bijectively mapped to elements of CHA with preservation of operations via intersection <-> glb and Big Union <-> sup Hence I surmise a space produces an atomic CHA and an atomic CHA will produce a sober space, which is the original space providing it's sober and additionally a CHA can produce a topology only if it's atomic. So where did the points go, or rather, when do they return? Pointless topology, Heyting algebra, Complete Heyting algebra, Stone duality. That's too much, and now I should wiki Category Theory also? Not one of my favorites, Category Theory, is that like pro[CapitalThorn]ling? ---- === Subject: Re: Sober Space === Subject: Re: Sober Space > What a hassle, getting the hang of Wiki-editing. How you > make comments, like notice exactly one in the history listing? >You put the comment in the Edit summary box. Found it. [3]Edit summary: ____________________________________________________________ -- What are the radial buttons (*) in the history page for? Compare selected versions -- >registering is just a matter of choosing a username and a password. Fairly painless as compared to others. >It certainly helps with collaboration I'll keep it in mind for possible future occasions. ---- === Subject: OT Wikipedia (was Re: Sober Space) > What are the radial buttons (*) in the history page for? Click one in the left column, then click a higher one in the right column, then click the Compare selected versions button, and you get a diff of the two versions. If you have JavaScript enabled, some of the buttons are hidden to avoid inconsistent choices (right choice lower than left choice). === Subject: Re: Sober Space >Suppose you have a lattice which is isomorphic to the lattice of >open sets of some topological space. There may be many spaces with >the right lattice, but only one (upto homeomorphism) is sober. > I suspect that's related to: > Sobriety of X is precisely the condition that forces the ring > C^0(X, R) of continuous real-valued functions on X to > determine X up to homeomorphism. Maybe; that's something I haven't looked at. >The sober one is in a sense the nicest one. > Nice because they have inverses, namely their locales > that generate back the original topology? Well, I only really meant it in a vague sense. Sobriety is a sort of separation axiom, so sober spaces are nicer than non-sober spaces in a similar sense that T_1 spaces are nicer than non-T_1 spaces. > The Wikipedia entry misses on two points: >Now [CapitalThorn]xed. > make comments, like notice exactly one in the history listing? You put the comment in the Edit summary box. > Does that come with registration? No, it should work whether or not you're registered. In any case, registering is just a matter of choosing a username and a password. > One bene[CapitalThorn]t I see from registering is ability to collaborate with other > editors and contributors. That could make for fewer, yet better revisions. It certainly helps with collaboration, since it's much easier for people to remember a username than an IP address. Also you get a user page, watchlist, etc. > Am I correct, irreducible is equivalent to hyperconnected? Yes. > Then why two terms for the same thing? Probably they originated in different contexts. === Subject: Debits and Credits by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52CmO232465; === Subject: Re: Debits and Credits Debits are assets and expenses; Credits are liabilities and income. Debits are considered positive; credits negative. Every transaction must have an algebraic sum of zero. Hope this helps. Retired CPA -- To e-mail me get rid of the cats and dogs. === Subject: Re: Debits and Credits charset=Windows-1252 > Need some good ideas on how to explain debits verses credits to adults. > Debits are assets and expenses; Credits are liabilities and income. > Debits are considered positive; credits negative. > Every transaction must have an algebraic sum of zero. Although that's mathematically correct, I've never met an accountant or bookkeeper who actually thinks in terms of maintaining a zero balance by adding negative and positive amounts -- instead they think of maintaining Debits = Credits, where all amounts are positive. In a sense the difference is trivial, but in practice it often leads to great confusion. The following version is, I think, closer to how the terms are most commonly used in practice ... If A,L,Q,I,E denote Assets, Liabilities, eQuity, Income and Expenses, respectively (all positive), the fundamental equation of accounting is A = L + Q + (I - E) or A + E = L + Q + I [*] Debit means an increase on the LHS or a decrease on the RHS of [*]; Credit means an increase on the RHS or a decrease on the LHS of [*]. The accounts of type A,E are called debit accounts because they are increased by debits, while accounts of type L,Q,I the RHS of [*]) are called credit accounts because they are increased by credits. (Some textbooks actually encourage thinking of debits & credits as synonyms for LHS & RHS, respectively, since the standard form in which to write transactions is with debits in a left-hand column and credits in a right- hand column. Again, this seems like a trivial thing, but such standards help tremendously to prevent bookkeeping confusion.) Every transaction maintains Debits = Credits by debiting some account(s) and crediting some other account(s). Note that both debits & credits can be applied to *any* account, increasing one kind and decreasing another. This hardly registers as mathematics, but the subject has a long and (some think) interesting history. I understand that this accounting method was [CapitalThorn]rst described by Pacioli in the same work that contained his failed attempts to solve the problem of points. --r.e.s. === Subject: Re: Proof of twin-prime conjecture? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52CmPE32472; Reading this proof (or attempting to) reminds me that I know little or no analytic number theory. Can anyone recommend a good book? >Arenstorf has uploaded an MS purporting to prove the twin >prime conjecture to the arXiv: >http:// uk.arxiv.org/abs/math .NT/0405509 >This does not look completely stupid, in fact >(whether or not it is all correct) it appears to be >a substantial piece of analytic number theory. >-- >Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html >Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 >Francis Wheen, _How Mumbo-Jumbo Conquered the World_ >Finally, a discussion about some real and signi[CapitalThorn]cant developments in math on sci.math. >Anthony J. Natoli === Subject: Re: Proof of twin-prime conjecture? Stephen Miller TOP-POSTED: > Reading this proof (or attempting to) reminds me that I know little or no > analytic number theory. Can anyone recommend a good book? M. Ram Murty, Problems in Analytic Number Theory: http://www.amazon.co.uk/exec/obidos/ASIN/0387951431/ -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: George Green, portrait? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52CmPp32479; > George Green 1793--1841 > ... the general mathematical theory of potential developed by an > obscure, self-taught miller's son would lead to the mathematical > theories of electricity underlying twentieth-century industry. > quoted at > <http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/ Green.html > Most of the biographies at The MacTutor History of Mathematics archive have portraits of the subject. (Even Euclid and Archimedes...) But not Green. Are there no known portraits of this man? >The website I got the information from is here: href=http://www.ee.umd.edu/~taylor/frame2.htm>http:// www.ee.umd.edu/~tay lor/frame2.htm Quoted from the above site Photography has not appeared yet...The circumstances of his life were such that no portraits of him were ever made. Thus no likeness of Green is available. If he had met a helpful friend or relative earlier in life, he may have prospered and lived longer with his outstanding contribution in mathematics. === Subject: Is this sequence uniforemly convergence ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52CmPL32515; I would appreciate for solving this problem? suppose that A={r_{1},r_{2},....,r_{n},...} is the set of all rational numbers and the sequence of function f_{n} is de[CapitalThorn]ned as below : f_{n}=1 if x is in A and f_{n}=0 if x is not in A in this case discuss about uniformly convergence and pointwise convergence. === Subject: Re: Is this sequence uniforemly convergence ? >I would appreciate for solving this problem? > suppose that A={r_{1},r_{2},....,r_{n},...} is the set of > all rational numbers and the sequence of function f_{n} > is de[CapitalThorn]ned as below : > f_{n}=1 if x is in A and f_{n}=0 if x is not in A > in this case discuss about uniformly convergence > and pointwise convergence. It seems very likely that you have stated the problem incorrectly. All the f_n are the same! (So of course they do converge uniformly...) ************************ David C. Ullrich === Subject: Re: proof of the Riemann hypothesis ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52CmOd32417; >I stumbled upon this paper yesterday : >http://arxiv.org/ abs/math.GM/04 05531 >Anyone heard of it ? >Michael Throughout the paper he uses the representation zeta(s)=product(1/(1-1/p^s))) where p runs through all primes, which is only valid for Re(s)>1. Johann === Subject: Re: Journal editors and reviewers, speak up > Perhaps I wasn't clear enough in my writing. Arturo and David were > mentioned just as examples of mainstream mathematicians in general. > When I said *they* were the ones who encouraged you I meant some of the > mathematicians who post here, not those two in particular. Actually, > I don't remember who in particular made those suggestions to James, > but I do recall that several of the regulars here did so. > That's a nice way to re-tract a lie. Here's my original paragraph: > You don't seem to have noticed that the mainstream mathematicians you've > been attacking here (like Professors Magidin and Ullrich) also make use > of those established and reputable outlets for their mathematical work. > It's not dif[CapitalThorn]cult to [CapitalThorn]nd published papers for the professionals who post > here. Have you forgotten that *they* were the ones who encouraged you > to try to get your work published? USENET is a hobby and a diversion > for them, not the professional outlet *you've* tried to make of it. The phrase *they* were the ones refers to unspeci[CapitalThorn]ed members of the Ômainstream mathematicians' you've been attacking here and the professionals who post here. The phrase (like Professors Magidin and Ullrich) presents them merely as *examples* of those groups, as indicated by the word like. Now, do you care to re-tract your accusation, or is your reading comprehension as lacking as your knowledge of hyphenation? -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: 1st World Congress and School on Universal Logic (UNILOG 2005) ************************************************************* *************** ************************************************************* **************** ********** 1st World Congress and School on Universal Logic (UNILOG 2005) http://www.uni-log.org Montreux Switzerland, School : March 26-30 ; Congress : March 31 - April 3, 2005 This event will focus on: 1) Techniques that can be used for a general theory of logics (Labelled deductive systems, Kripke structures, Logical matrices, etc.) ; 2) Studies of classes of logics (Substructural logics, Non monotonic logics, Paraconsistent logics, etc.) 3) Scope of validity and domain of application of fundamental theorems of logic (Completeness, Deduction, Cut-elimination, etc.) 4) Philosophical considerations about the nature of logic and the universality of some logical laws or axioms The school is intended for advanced students and young researchers. There will be about 20 tutorials on many subjetcs: combination of logics, multiple conclusion logic, combinatory logic, logics and games, abstract model theory, logic as language vs. logic as calculus, category theory for logics, etc. Invited speakers of the congress will include A.Avron, D.Batens, J.Corcoran, M.Dunn, D.Gabbay, R.Jansana, A.Koslow, V.de Paiva, K.Segerberg. Contributed papers for the congress can be submitted before October 30, More information on the website: http://www.uni-log.org ************************************************************* *************** ************************************************************* *** === Subject: Re: Calculate probability of lung cancer death? > >Of course the analysis is slightly incorrect because it implicitly >assumes > that >smokers can only die from lung cancer. > It also implicitly assumes that those with lung cancer died from smoking > cigarettes. The correlation was looked for and, because it was looked > for, there was bias interpreting data. >I don't see much support for your position. I simply assumed that the >calculated [CapitalThorn]gure of 0.00223 deaths by lung cancer per year was correct. I >did not make any causality assumptions. I also assumed that smokers only >die from lung cancer which is clearly a false assumption, but I don't have >the smoker mortality rate. If I had the smoker mortality rate, I could >produce better estimates. Basicly: >Survivors of Year X = ( 1 - MortalityRate ) * (Survivors of Year X-1) >Deaths by Lung Cancer in Year X = 0.00223 * (Survivors of Year X-1) >Clearly the mortality rate is at least 0.00223. A large mortality rate >means that lots of smokers are dying from other causes (heart disease, >mouth cancer, etc.). >So a better statement would have been that out of a sample of 100,000 >smokers, we should expect more than 1% of them to die within 5 years. >1% is an overestimate of the number that will die from lung cancer, but an >underestimate of the number that will die from all causes. Sigh! Given the age group, your numbers are low no matter what habits those people had. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Calculate probability of lung cancer death? > Did I make a mistake? > Yes, you became and remain a smoker. > I was watching Tom Selleck portray Eisenhower on TV tonight. I wonder if it > will take him a while to kick the cigarette habit now that the role is > [CapitalThorn]nished. Just what do you think he was smoking? Tobacco? === Subject: Re: a+b+c= Pi and cos(a)+cos(b)+cos(c) < = 3/2 > Ken Pledger a .8ecrit : > > >.... >I am trying to show that if a,b and c are the angles of a >triangle, then cos(a)+cos(b)+cos(c) < = 3/2 just using >trigonometry (with calculus is easy).... > > > > Here's another proof to add to your collection. > Your proof is essentially the same as Robin Chapman's proof, > (there's no disgrace, quite the reverse). There is a quite different > proof I saw posted to fr.sci.maths. This proof uses the fact that cos is > concave over [0,Pi/2]. > Pascal I just woud like tu see the proof of Pascal. Do you have the link? And someone talked about a geometric proof do you know it? just by couriosity. === Subject: Re: a+b+c= Pi and cos(a)+cos(b)+cos(c) < = 3/2 Jesus Rogelio Perez Buendia a .8ecrit : > I just woud like tu see the proof of Pascal. Do you have the link? The proof is not mine. Here's the link http://minilien.com/?FQLUmz0NZM Pascal === Subject: Help with an integral equation I would like to solve the next eigenvalue/eigenfunction problem: Int_{-oo}^{+oo} K(x,x')f(x')dx' = e f(x), where f(x) is odd, and the symmetric kernel K(x,x') = xx'd(x-x') + A ln |(x-x')/(x+x')|, with some real constant A. d is the Dirac delta-function. I appreciate any help or hint. PS. In the case A=0, f_a(x)=d(x-a)-d(x+a) with e_a=a^2. === Subject: Re: Consecutive ingegers: Coprime > I'd like to have te bibliography of the books you are talking. Can > you send me that please? > Somehow I have this feeling that if I send anything > to yoyontzin@yahoo-dot-com.no-spam.invalid > it's going to bounce right back to me. > Anyway, if you have internet access, I've given you > the author & title of three books, how hard can it be > for you to track down whatever additional bibliographical details > you might want? I am also interested about the bibliography. Pleas give me the names and the autors and I look for the rest. === Subject: Is this series uniforemly convergence ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52DYkB05862; I appreciate for solving this problem. Is this series convergent? If yes , is it uniforemly or pointwise? 00 ------------ 1 --------------------------------- / 3/2 / 1+ n [1 - f ( x )] ------------ n n=1 wher A={r ,r ,r ,...,r ,...} is the set of all rational numbers 1 2 n and f (x)= 1 if x is in A and is 0 if it is not in A . === Subject: Re: Is this series uniforemly convergence ? > I appreciate for solving this problem. > Is this series convergent? If yes , is it uniforemly or pointwise? > 00 > ------------ 1 > > --------------------------------- > / 3/2 > / 1+ n [1 - f ( x )] > ------------ n > n=1 > wher A={r ,r ,r ,...,r ,...} is the set of all rational numbers > 1 2 n > and f (x)= 1 if x is in A and is 0 if it is not in A . What did you really mean. Did you lose a subscript on the last f? And even so, why write it with a subscript if it does not depend on it? Perhaps you should try again to write the question. === Subject: Eisenstein and pythagorean triples It's widely known that if (a,b,c) is a positive integer solution to the equation a^2+b^2=c^2 then 3, 4,and 5 divide abc. But did you know that if (a,b,c) is a positive integer solution to the equation a^2+b^2-ab=c^2 then 3, 5 and 7 divide ab(a-b)c; and if (a,b,c) is a positive integer solution to a^2+b^2+ab then 3, 5, and 7 divide ab(a+b)c. To see these, and other interesting(?) little facts about Eisenstein and Pythagorean triples, see http://www.geocities.com/fredlb37/index.html === Subject: Is this sequence uniforemly convergence ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52Ds7707858; I appreciate for solving this problem. Is this sequence convergent? If yes , Is it uniforemly orpointwise? 2 |n x 0 < x < 1/n | - - f (x)=| 2 |-n x + 2n 1/n < x < 2/n | - - | |0 2/n < x < 1 - - === Subject: RFC: odd n=a^2+b^2+c^2+d^2 n odd == 1 [mod 2] n = ((n+1)/2)^2 - ((n-1)/2)^2 = ((n+1)/2)^2 - (n-1)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 == ((n+1)/2)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 [mod 2] Can induction now complete the proof? [not sure, but I suspect very possibly] Each square strictly positive, so maybe - but haven't quite seen it yet. Can anyone help, please? This is in relation to: http://www.bearnol.pwp.blueyonder.co.uk/Math/4sq.htm James === Subject: Re: RFC: odd n=a^2+b^2+c^2+d^2 >n odd == 1 [mod 2] >n = ((n+1)/2)^2 - ((n-1)/2)^2 > = ((n+1)/2)^2 - (n-1)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 > == ((n+1)/2)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 + ((n-1)/2)^2 [mod 2] >Can induction now complete the proof? [not sure, but I suspect very >possibly] >Each square strictly positive, so maybe - but haven't quite seen it yet. >Can anyone help, please? >This is in relation to: >http://www.bearnol.pwp.blueyonder.co.uk/Math/4sq.htm >James Why not: ((n+1)/2)^2 - ((n-1)/2)^2 = =(n^2 + 2n + 1)/4 - (n^2-2n+1)/4 =(n^2 + 2n + 1 -n^2 +2n -1)/4 = 4n/4 = n q.e.d Hans v. D. Randers, Danmark. === Subject: approach to solving problems when approaching dif[CapitalThorn]culties of intellectual nature, modeling the problem into approximate abstaction provides a mechanism of analysis. however, it may not ALWAYS be neccessary to reduce variables to subnuclear axioms. one should aim to construct a scale of attack. this should eliminate unnecessary energy expenditure. don't you think? === Subject: Re: approach to solving problems > when approaching dif[CapitalThorn]culties of intellectual nature, modeling the > problem into approximate abstaction provides a mechanism of analysis. > however, it may not ALWAYS be neccessary to reduce variables to > subnuclear axioms. one should aim to construct a scale of attack. this > should eliminate unnecessary energy expenditure. > don't you think? 1) Examine problem. 2) Think. 3) Write down correct answer. How do you do it? -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Accidental limit >Hi! >Ive just found this limit accidentally and empirically: >Lim (n->oo) ( [ SUM ( i = (n^2) -> (n+1)^2 ) SQRT(i) ] ) = 1/6 >... where [x] means Ôdecimal part of x'. i.e. [3.1234]= 0.1234 >Is this true? > Yes, it's true. My [CapitalThorn]rst guess was that something like this > couldn't possibly be true, but it is: > First note that if N < M and N, M are integers then > |sum_N^M f(i) - int_{N-1/2}^{M-1/2} f(x) dx| > <= c int_{N-1/2}^{M+1/2} |f''(x)| dx. > Using the notation o(1) for something that tends to zero > it follows that > sum_{n^2}^{(n+1)^2} sqrt(i) > = o(1) + int_{n^2-1/2}^{(n+1)^2+1/2} sqrt(x) dx. > Now since the derivative of sqrt(x) tends to zero we can > use trivial approximations for the parts of the integral > at the ends (approximating int_a^b f by (b-a) f(x) for > some x in [a,b]) and we get > sum_{n^2}^{(n+1)^2} sqrt(i) > = o(1) + (n + (n+1))/2 + int_{n^2}^{(n+1)^2} sqrt(x) dx > = o(1) + n + 1/2 + (2/3)((n+1)^3 - n^3) > = o(1) + n + 1/2 + (2/3)(3n^2 + 3n +1) > = o(1) + integer + 1/2 + 2/3 > = o(1) + integer + 1/6. > So the fractional part of the sum tends to 1/6. >Usefull? > I can't imagine why. It does seem very curious, though. So, if k+1 is prime, we get... lim_{n->inf} FractionalPart(sum_{i=n^k}^{(n+1)^k} i^(1/k)) = 1/2 - 1/(k+1) -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Accidental limit >So, if k+1 is prime, we get... >lim_{n->inf} FractionalPart(sum_{i=n^k}^{(n+1)^k} i^(1/k)) > = 1/2 - 1/(k+1) Yes, that is true. We can use the Euler-Maclaurin Sum Formula correctly to verify this. Asymptotically, n --- 1/k k 1/k+1 1 1/k 1 1/k-1 1/k-3 > j = --- n + - n + --- n + O(n ) --- k+1 2 12k j=1 n^k --- 1/k k 1+k 1 1 1-k 1-3k > j = --- n + - n + --- n + O(n ) --- k+1 2 12k j=1 (n+1)^k --- 1/k k (n+1)^{k+1} - n^{k+1} - 1 k 1 > j = n + k ------------------------- + --- + - --- k+1 k+1 2 j=n^k 1 (n+1)^{k-1} - n^{k-1} -3k - --- --------------------- + O(n ) 12k ((n+1)n)^{k-1} Thus, asymptotically the fractional part is 1 1 k-1 -k -k-1 - - --- - --- n + O(n ) 2 k+1 12k which tends to 1/2 - 1/(k+1) from below as n tends to in[CapitalThorn]nity. Rob Johnson take out the trash before replying === Subject: Re: Accidental limit >Hi! > >Ive just found this limit accidentally and empirically: > >Lim (n->oo) ( [ SUM ( i = (n^2) -> (n+1)^2 ) SQRT(i) ] ) = 1/6 > >... where [x] means Ôdecimal part of x'. i.e. [3.1234]= 0.1234 > >Is this true? > Yes, it's true. My [CapitalThorn]rst guess was that something like this > couldn't possibly be true, but it is: > First note that if N < M and N, M are integers then > |sum_N^M f(i) - int_{N-1/2}^{M-1/2} f(x) dx| > <= c int_{N-1/2}^{M+1/2} |f''(x)| dx. > Using the notation o(1) for something that tends to zero > it follows that > sum_{n^2}^{(n+1)^2} sqrt(i) > = o(1) + int_{n^2-1/2}^{(n+1)^2+1/2} sqrt(x) dx. > Now since the derivative of sqrt(x) tends to zero we can > use trivial approximations for the parts of the integral > at the ends (approximating int_a^b f by (b-a) f(x) for > some x in [a,b]) and we get > sum_{n^2}^{(n+1)^2} sqrt(i) > = o(1) + (n + (n+1))/2 + int_{n^2}^{(n+1)^2} sqrt(x) dx > = o(1) + n + 1/2 + (2/3)((n+1)^3 - n^3) > = o(1) + n + 1/2 + (2/3)(3n^2 + 3n +1) > = o(1) + integer + 1/2 + 2/3 > = o(1) + integer + 1/6. > So the fractional part of the sum tends to 1/6. >Usefull? > I can't imagine why. It does seem very curious, though. >So, if k+1 is prime, we get... >lim_{n->inf} FractionalPart(sum_{i=n^k}^{(n+1)^k} i^(1/k)) > = 1/2 - 1/(k+1) Hmm, looks like we'd get Frac(1/2 + k/(k+1)((n+1)^(k+1)-n^(k+1))), and if it's true that the binomial coef[CapitalThorn]cient C(k+1,j) is divisible by k+1 except for j = 0 and j = k+1, which I imagine is obvious to everyone but me, then we have Frac(1/2 + k/(k+1)) = 1/2 = 1/(k+1), yes. Now why is... oops, it's not so. C(4,2) is not divisible by 4, for example. So for k = 3 we get Frac(1/2 + 3/4((n+1)^4-n^4)) = Frac(1/2 + (9/2)n^2 +3/4), which doesn't _have_ a limit as n -> in[CapitalThorn]nity. Oh wait, 4 is not prime. Yes, if k+1 is prime we get what you said. ************************ David C. Ullrich === Subject: Re: Accidental limit >I've just found this limit accidentally and empirically: > >Lim (n->oo) ( [ SUM ( i = (n^2) -> (n+1)^2 ) SQRT(i) ] ) = 1/6 > >... where [x] means Ôdecimal part of x'. i.e. [3.1234]= 0.1234 > >Is this true? Usefull? >It is true. Use the Euler-Maclaurin Sum Formula >Upon reading G. A. Edgar's response to this question, I realized that >I put the wrong order of derivative in the last term. The series above >should be > n > --- 1/2 2 3/2 1 1/2 1 -1/2 1 -5/2 > k = - n + - n + -- n - ---- n + ... > --- 3 2 24 1920 > k=1 > (n+1)^2 > --- 1/2 2 7 1 1 1 5n^4+10n^3+10n^2+5n+1 > k = 2n + 3n + - - -- ------ - ---- ---------------------- + ... > --- 6 24 n(n+1) 1920 (n(n+1))^5 > k=n^2 > 2 7 1 1 1 1 1 1 > = 2n + 3n + - - -- ( --- - --- + --- - --- ) + O( --- ) > 6 24 n^2 n^3 n^4 n^5 n^6 >The [CapitalThorn]nal big-O answer remains the same. I missed a sign change; -1/1920 should be +1/1920. As before, this does not affect the big-O estimate. Rob Johnson take out the trash before replying === Subject: Re: determining contributions to a function > However I don't think this would be correct for any other form of f (but I > may be wrong). In such a case is there a general way to determine > contributions of a speci[CapitalThorn]c variable to the value of a function of that > variable (and other variables)? > > I would guess > x.f_x(x,y,z)/f(x,y,z) > What does x.f_x(x,y,z) mean? > . is same as *; f_x is the partial derivated with respect to x. I've been thinking about the above de[CapitalThorn]nition of contribution in the context of the post by Lance Lamboy. I'm not sure I understand why (or how) the above de[CapitalThorn]nition indicates the contribution of x to the value of the function. Could you explain the reasoning that leads to the above de[CapitalThorn]nition? Rajarshi === Subject: Re: f(x+1)=f(x)+g(x)+y, where f(g(x))=x > f(x) is a monotonically increasing continuous real->real function > where > f(x+1) = f(x) + g(x) + y for all x >= 0, > where g(x) is the inverse of f(x), ie. f(g(x)) = x for all x >= 0, > and f(0) = g(0) = 0. > y is a positive real constant. > What can be said about f(x)? > Does it have a closed form? You mean something like this? y = 1.414, f(x)=0.846 x - 0.082 x^2 + 1.256 x^3 - 0.605 x^4 +..., g(x)=1.182 x + 0.136 x^2 - 2.421 x^3 + ... And then we can ask whether y=sqrt(2), and whether that is the only value of y for which there is a solution analytic at 0... -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: f(x+1)=f(x)+g(x)+y, where f(g(x))=x === Subject: f(x+1)=f(x)+g(x)+y, where f(g(x))=x >f(x) is a monotonically increasing continuous real->real function >f(x+1) = f(x) + g(x) + c for all x >= 0, >g(x) is the inverse of f(x), ie. f(g(x)) = x for all x >= 0, >f(0) = g(0) = 0. >c is a positive real constant. This is redundant as, f(1) = c f(2) = g(1) ff(2) = fg(1) = 1 >What can be said about f(x)? f(1) < f(2) ff(1) < ff(2) = 1 < 2 fff(1) < f(1) < f(2) f^4(1) < ff(1) < ff(2) = 1 < 2 f^5(1) < f^3(1) < f(1) < f(2) f^6(1) < f^4(1) < ff(1) < ff(2) = 1 < 2 etc. ---- === Subject: mapping of a function (need help) d >= 2, m >= 2, d divides m Let f be a map, f: Z/mZ -> Z/dZ de[CapitalThorn]ned by: f([a]_m) = [a]_d a element of Z How do I show that f is well de[CapitalThorn]ned? Is is suf[CapitalThorn]cient to say that the number of elements of Z/mZ is greater or equal to the number of elements of Z/dZ ? How do I explain that f induces a map f* : U_m -> U_d ? (U_n is the group of units of Z/nZ) === Subject: Re: mapping of a function (need help) days. My association with the Department is that of an alumnus. > d >= 2, m >= 2, d divides m >Let f be a map, f: Z/mZ -> Z/dZ de[CapitalThorn]ned by: >f([a]_m) = [a]_d a element of Z >How do I show that f is well de[CapitalThorn]ned? Formally, you would need to show that if [a]_m = [b]_m, then [a]_d = [b]_d; that is, you must prove that if a = b (mod m), then a=b (mod d). The fact that d divides m should prove... crucial... in showing this is true. > Is is suf[CapitalThorn]cient to say that the >number of elements of Z/mZ is greater or equal to the number of >elements of Z/dZ ? Certainly not. That would be true if d=2, m=3 (Z/mZ has more elements than Z/2Z). But then the map that sends [a]_3 to [a]_2 is not well de[CapitalThorn]ned: [1]_3 = [4]_3, since 1=4 (mod 3), but [1]_1 is not equal to [4]_2. >How do I explain that f induces a map f* : U_m -> U_d ? >(U_n is the group of units of Z/nZ) You show that f respects multiplication and maps the multiplicative identity of Z/mZ to the multiplicative identity of Z/dZ. Therefore, since f(x*y) = f(x)*f(y), if xy=1, then f(x)*f(y)=1, so f(x) is also a unit. -- 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: mapping of a function (need help) > d >= 2, m >= 2, d divides m > Let f be a map, f: Z/mZ -> Z/dZ de[CapitalThorn]ned by: > f([a]_m) = [a]_d a element of Z > How do I show that f is well de[CapitalThorn]ned? The crucial fact is that if a = b (mod m) then a = b (mod d). > Is is suf[CapitalThorn]cient to say that the > number of elements of Z/mZ is greater or equal to the number of > elements of Z/dZ ? No. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: mapping of a function (need help) f([a]_m) = f([am]_m) = [am]_d = [a]_d would this proove that f is well de[CapitalThorn]ned? === Subject: Re: wishart matrix >Does anybody know whether wishart distribution is necessarily >associated with Normal distribution only? If X is a normally >distrbuted matrix with 0 mean, X'X is proportional to covariance >matrix, and follows wishart distribution >[ check the link for reference : >http://www.quantlet.com/mdstat/scripts/mva/htmlbook/ mvahtmlframe67.html >However, can I apply properties of wishart matrix to X'X if X is NOT >normally distributed? (may be zero mean or non-zero mean) X'X has a Wishart distribution with covariance matrix Sigma, but only asymptotically is X'X proportional to Sigma. Assuming the rows of X are independent and identically distributed, one can give examples of non-normality, but not of anything which can be considered particularly reasonable. Any linear combination of the elements of a row has to have the property that its absolute value is distributed as the absolute value of a normal random variable with mean 0. -- 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: A Repunit Observation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52FKdc17443; >Let Rn denote the repunit of n digits, De[CapitalThorn]ne s(n) as the sum of the >exponents of the prime factors of Rn. Consider the following REPUNIT >OBSERVATION: If s(n)is equal to less than 3, then n is prime. > >This observation is based on existing factorization tables as reviewed >on Internet. Is the REPUNIT OBSERVATION or Conjecture embedded in an >exiting theorem or has it been previously announced? I would >appreciate any help on this line of research. > Research? This is nothing to research. Trivially if n = pq > then 10^n-1 is divisible by 10^p-1 and 10^q-1. It is also divisible > by 9. Thus, s(n) >= 4 if n is composite. >True, but OP isn't talking about 10^n - 1, OP is talking about >the repUNIT numbers, which are (10^n - 1)/9. even without the extra factors of 3. Please note: (Yes, I know you know this) that if n has at least 3 distinct prime factors, then the conjecture is obviously false. If n = pq, p,q, prime, then N = (10^n-1)/9 is divisible by R = (10^p-1)/9 and S = (10^q-1)/9 and (10^n-1)/(9R*S) which is > 1. Thus, the conjecture [s(n) < 3] is true only when n is prime. i.e. after dividing out (x^p-1) and (x^q-1) from (x-1)(x^n-1) one is still left with a non-trivial cyclotomic polynomial. Just let x=10. It might be possible that s(n) = 3,if all of R, S, and R_n/(RS) are prime. I do not know of any examples. n = 19*23 does not work, nor does n = 19*317; I have not checked n = 23*317. Examples would be VERY VERY rare. In fact, there should be at most [CapitalThorn]nitely many; the expected number is O(sum(1/log(N) * 1/log(R) * 1/log(S))) by PNT even assuming in[CapitalThorn]nitely many prime R_p. This sum converges quite rapidly. === Subject: Re: A Repunit Observation >Let Rn denote the repunit of n digits, De[CapitalThorn]ne s(n) as the sum of the >exponents of the prime factors of Rn. Consider the following REPUNIT >OBSERVATION: If s(n)is equal to less than 3, then n is prime. > >This observation is based on existing factorization tables as reviewed >on Internet. Is the REPUNIT OBSERVATION or Conjecture embedded in an >exiting theorem or has it been previously announced? I would >appreciate any help on this line of research. > > Research? This is nothing to research. Trivially if n = pq > then 10^n-1 is divisible by 10^p-1 and 10^q-1. It is also divisible > by 9. Thus, s(n) >= 4 if n is composite. >True, but OP isn't talking about 10^n - 1, OP is talking about >the repUNIT numbers, which are (10^n - 1)/9. > even without the extra factors of 3. False? The conjecture is that s(n) < 3 implies n is prime. It seems that below you are trying to prove the conjecture true, not false. Perhaps false was a typo for trivial? > Please note: (Yes, I know you know this) > that if n has at least 3 distinct prime factors, then the conjecture > is obviously false. If n = pq, p,q, prime, then N = (10^n-1)/9 > is divisible by R = (10^p-1)/9 and S = (10^q-1)/9 and > (10^n-1)/(9R*S) which is > 1. Thus, the conjecture [s(n) < 3] is > true only when n is prime. i.e. after dividing out (x^p-1) and (x^q-1) from > (x-1)(x^n-1) one is still left with a non-trivial cyclotomic > polynomial. Just let x=10. You've left out the case where n is the square of a prime. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Is this sequence uniforemly convergence ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52FKdU17447; I appreciate you for solving this problem. suppose that f is a bounded variation function and lim f(x)=0 + x-->0 if {a } is a strictly decreasing function and is convergent to n 0 ({a } is in [0,1]). prove that n ------------ n ( - 1) f(a ) is convergent. / n / ------------ === Subject: Re: Is this sequence uniforemly convergence ? >I appreciate you for solving this problem. >suppose that f is a bounded variation function and lim f(x)=0 > + > x-->0 >if {a } is a strictly decreasing function and is convergent to > n >0 ({a } is in [0,1]). prove that > n > ------------ > n > ( - 1) f(a ) is convergent. > / n > / > ------------ Sum by parts. (Or use Dirichlet's test...) ************************ David C. Ullrich === Subject: Re: Integration Problems... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52FYtf18797; >Hallo to everyone! >I'm trying to integrate the following function: >F(z)=cos(m*pi*z/L)*T(z), >where m=1,2,3,4,... and T(z) an arbitrary function of z. >The integration limits are from z=0...L, that is: >integrate(F(z),z=0..L). I have tried the simpson and boole rules but >they give enormous errors. Can someone point me to an another >direction? Is it possible to take advantage of the cos(m*pi*z/L) and >replace it with something better in order to minimize the errors?? >Yannis Hi Yannis, try integrating dI/dz = cos(m*pi*z/L)*T(z) , I(0) = 0, in the range [0,L] with an integrator for ordinary differential equations with adaptive step size control. The stepsize in such codes is chosen according to the (local) error of integration that you are willing to tolerate. Best wishes Torsten. === Subject: Autocovariance and autocorrelation : performances of estimators ? I have to estimate locally (i.e. on small size areas) the 2D autocorrelation/autocovariance of random [CapitalThorn]elds in order to characterize texture. Usually performances of sample correlation/covariance estimates are given for iid random pairs. Since I have only one realisation of small size to perform the estimation, I have to take into account the correlation between sample pairs in order to compute the bias and the variance of these estimators. Do anyone knows books or publications dealing with performances of such estimates ? O.D. === Subject: .999... ?= 1 Ok, ok, calm down, folks. Yes, I know .999... = 1. But certain friends of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that waste anyone's time, but I've about given up proving this to them, and one of them commented that he won't believe it until a PhD says it's true. Here's the best proof I've seen or used: If x = 0.999... 10x = 9.999... - x = 0.999... ---------------- = 9x = 9 Or, x = 1 As a side request, does anyone know a proof that might be more understandable to an idiot? -- Mekkala, Atheist #2148 Atheism is ... the bed-rock of sanity in a world of madness. --Emmett F. Fields === Subject: Re: .999... ?= 1 > .... Yes, I know .999... = 1. But certain friends of mine don't .... > and one of them commented that he won't believe it until a PhD says > it's true. That shows a view of mathematics which saddens me. The subject is self-authenticating, and doesn't depend on appeals to the authority of a Ph.D. or anyone else. Reasoning with your friends is intellectually the right thing to do, although it's never guaranteed to work. ;-( > Here's the best proof I've seen or used: > If x = 0.999... > 10x = 9.999... > - x = 0.999... > ---------------- > = 9x = 9 > Or, x = 1 > .... Other people's technical criticisms of that argument are correct but perhaps not helpful to you in convincing your friends. Many people worry about 0.999... and some of the experience of dealing with their worries is distilled into the alt.algebra.help FAQ at http://aah.ryan-usa.com/node30.html which may give you some helpful ideas. Ken Pledger. === Subject: Re: .999... ?= 1 >Ok, ok, calm down, folks. Yes, I know .999... = 1. But certain friends >of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that >waste anyone's time, but I've about given up proving this to them, and >one of them commented that he won't believe it until a PhD says it's >true. >Here's the best proof I've seen or used: > If x = 0.999... > 10x = 9.999... >- x = 0.999... >---------------- >= 9x = 9 >Or, x = 1 That's not a proof; at least it's not a _complete_ proof. Proof that there's something missing: Let x = ...999. Then 10x = ...9990, so -9x = 9, x = -1. So: the same proof shows that ...999 = -1, which is nonsense. (What's missing in your non-proof is a proof that there is such a thing as 0.999... . Which of course we can't prove until we [CapitalThorn]rst de[CapitalThorn]ne what it _means_...) >As a side request, does anyone know a proof that might be more >understandable to an idiot? ************************ David C. Ullrich === Subject: Re: .999... ?= 1 >Ok, ok, calm down, folks. Yes, I know .999... = 1. But certain friends >of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that >waste anyone's time, but I've about given up proving this to them, and >one of them commented that he won't believe it until a PhD says it's >true. >Here's the best proof I've seen or used: > If x = 0.999... > 10x = 9.999... >- x = 0.999... >---------------- >= 9x = 9 >Or, x = 1 > That's not a proof; at least it's not a _complete_ > proof. > Proof that there's something missing: > Let x = ...999. Then 10x = ...9990, so -9x = 9, x = -1. I don't follow that last step. I have x - 10x = -8991. > So: the same proof shows that ...999 = -1, which is > nonsense. Yes, but it seems to be YOUR nonsense. === Subject: Re: .999... ?= 1 >Ok, ok, calm down, folks. Yes, I know .999... = 1. But certain friends >of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that >waste anyone's time, but I've about given up proving this to them, and >one of them commented that he won't believe it until a PhD says it's >true. > >Here's the best proof I've seen or used: > > If x = 0.999... > > 10x = 9.999... >- x = 0.999... >---------------- >= 9x = 9 > >Or, x = 1 > That's not a proof; at least it's not a _complete_ > proof. > Proof that there's something missing: > Let x = ...999. Then 10x = ...9990, so -9x = 9, x = -1. >I don't follow that last step. >I have x - 10x = -8991. ??? Oh - you left out the ... . The number ...999 was supposed to consist of a string of in[CapitalThorn]nitely many 9's extending to the _left_. When you do the subtraction they all cancel except for one of them. (No, of course there's no such number as ...999. Which is exactly my point - before we can prove anything about 0.999... we need to de[CapitalThorn]ne exactly what we mean by the notation, and then show that there _is_ such a number.) > So: the same proof shows that ...999 = -1, which is > nonsense. >Yes, but it seems to be YOUR nonsense. Uh, right. The obviously wrong proof I gave that ...999 = -1 uses exactly the same sort of reasoning as the purported proof that 0.999... = 1. So it shows that there's something important missing from that proof. ************************ David C. Ullrich === Subject: Re: .999... ?= 1 Your proof is just multiplying by 10, does nothing at all for Mathematicians. If you take an error bounds of the difference between the two numbers and show that it becomes arbitrarily small with increasing digits, then you have something. > Ok, ok, calm down, folks. Yes, I know .999... = 1. But certain friends > of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that > waste anyone's time, but I've about given up proving this to them, and > one of them commented that he won't believe it until a PhD says it's > true. > Here's the best proof I've seen or used: > If x = 0.999... > 10x = 9.999... > - x = 0.999... > ---------------- > = 9x = 9 > Or, x = 1 > As a side request, does anyone know a proof that might be more > understandable to an idiot? > -- > Mekkala, Atheist #2148 > Atheism is ... the bed-rock of sanity in a world of madness. > --Emmett F. Fields === Subject: Re: .999... ?= 1 > Ok, ok, calm down, folks. Yes, I know .999... = 1. > But certain friends > of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that > waste anyone's time, but I've about given up proving this to them, and > one of them commented that he won't believe it until a PhD says it's > true. > Here's the best proof I've seen or used: > If x = 0.999... > 10x = 9.999... > - x = 0.999... > ---------------- > = 9x = 9 > Or, x = 1 You're just following rules for manipulating decimals you learned somewhere. The precise meaning of .9999... appears nowhere. It's confusion about this meaning that causes all the arguments. === Subject: Re: .999... ?= 1 > As a side request, does anyone know a proof that might be more > understandable to an idiot? Strictly between any two distinct real numbers there exists another real number. Ask them to name a real number strictly between .999... and 1. === Subject: Re: .999... ?= 1 back with a beer, ruminated at length, fell asleep, woke up, lit up a joint, then fell asleep again after thoughtfully blurting out: > As a side request, does anyone know a proof that might be more > understandable to an idiot? > Strictly between any two distinct real numbers there exists another real > number. Ask them to name a real number strictly between .999... and 1. Asked that. They said the number in between is 2.999... / 3. *rolls eyes* -- Mekkala, Atheist #2148 Atheism is ... the bed-rock of sanity in a world of madness. --Emmett F. Fields === Subject: Re: .999... ?= 1 > 2.999... / 3 Which equals ...? -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: .999... ?= 1 kicked back with a beer, ruminated at length, fell asleep, woke up, lit up a joint, then fell asleep again after thoughtfully blurting out: > 2.999... / 3 > Which equals ...? 1 -- Mekkala, Atheist #2148 Atheism is ... the bed-rock of sanity in a world of madness. --Emmett F. Fields === Subject: Re: .999... ?= 1 > 2.999... / 3 > Which equals ...? A real number strictly between 0.999... and 1 which, sadly, is not expressible in decimal notation. This clearly shows an inadequacy inherent to decimal and similar notations. DWC (totally in jest, of course) === Subject: Re: .999... ?= 1 >Here's the best proof I've seen or used: > If x = 0.999... > 10x = 9.999... >- x = 0.999... >---------------- >= 9x = 9 >Or, x = 1 That's solid. Here are some others: 1/3 = 0.3333... therefore 3/3 = 0.9999... 1.00000000... - 0.99999999... --------------- 0.00000000... --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: .999... ?= 1 Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >Here's the best proof I've seen or used: > 10x = 9.999... >- x = 0.999... >---------------- >= 9x = 9 >That's solid. No it isn't. It's just a plausibility argument, not a proof. There's no reason to believe that the multiplication by ten and the subtraction are right unless you prove them from the de[CapitalThorn]nition of ... decimals. And once you have that de[CapitalThorn]nition, you can prove that 0.999... = 1 directly. > Here are some others: Same problem. -- Richard === Subject: Re: .999... ?= 1 kicked back with a beer, ruminated at length, fell asleep, woke up, lit up a joint, then fell asleep again after thoughtfully blurting out: >Here's the best proof I've seen or used: > 10x = 9.999... >- x = 0.999... >---------------- >= 9x = 9 >That's solid. > No it isn't. It's just a plausibility argument, not a proof. > There's no reason to believe that the multiplication by ten and the > subtraction are right unless you prove them from the de[CapitalThorn]nition of > ... decimals. And once you have that de[CapitalThorn]nition, you can prove > that 0.999... = 1 directly. There's no need to do anything with the de[CapitalThorn]nition of a repeating decimal. When you move a decimal point one digit to the right, you are multiplying by 10. Moving the decimal point one digit to the right in .999... (repeating 9) yields 9.999... Since 9.999... is the equivalent of 9 + 0.999... (by the de[CapitalThorn]nitions of addition and decimals), 9.999... - 9 = 0.999... Again, the proof I posted has nothing to do with the de[CapitalThorn]nition of a repeating decimal, and would work just as well with a non-repeating decimal: If x = 0.9 10x = 9 - x = 0.9 ---------- = 9x = 8.1 Or x = 0.9 (which we already know is true) Your objection would be correct if I had tried to prove it by saying that: 0.999... - 1 ---------- = 0 The above is true, of course, but you can't actually perform that subtraction on paper, so it would be, as you said, a plausibility argument. -- Mekkala, Atheist #2148 Atheism is ... the bed-rock of sanity in a world of madness. --Emmett F. Fields === Subject: Re: .999... ?= 1 > There's no need to do anything with the de[CapitalThorn]nition of a repeating > decimal. You're sadly mistaken. > When you move a decimal point one digit to the right, you are > multiplying by 10. Moving the decimal point one digit to the right in > .999... (repeating 9) yields 9.999... Yada yada yada ... these are just rules you learned in 6th grade or whatever. You don't know why they are valid for nonterminating decimal expansions - because you don't know what the latter are. And until you do, your idiot friends are really no more lacking in understanding than you are. === Subject: Re: .999... ?= 1 kicked back with a beer, ruminated at length, fell asleep, woke up, lit up a joint, then fell asleep again after thoughtfully blurting out: > There's no need to do anything with the de[CapitalThorn]nition of a repeating > decimal. > You're sadly mistaken. > When you move a decimal point one digit to the right, you are > multiplying by 10. Moving the decimal point one digit to the right > in .999... (repeating 9) yields 9.999... > Yada yada yada ... these are just rules you learned in 6th grade or > whatever. You don't know why they are valid for nonterminating decimal > expansions - because you don't know what the latter are. And until you > do, your idiot friends are really no more lacking in understanding > than you are. Then please, enlighten me. -- Mekkala, Atheist #2148 Atheism is ... the bed-rock of sanity in a world of madness. --Emmett F. Fields === Subject: Re: .999... ?= 1 Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >When you move a decimal point one digit to the right, you are >multiplying by 10. Proof? >Since 9.999... is the equivalent of 9 + 0.999... Proof? >(by the de[CapitalThorn]nitions of addition and decimals) Only if your de[CapitalThorn]nitions of addition and decimals include numbers like 9.999... The proof relies on the plausibility of multiplication by ten and subtraction working for repeating decimals in an analogous way to how it works for [CapitalThorn]nite ones, but unless you prove that, you haven't proved the result. >Again, the proof I posted has nothing to do with the de[CapitalThorn]nition of a >repeating decimal, and would work just as well with a non-repeating >decimal: Yes, it would work for non-repeating decimals, because you presumably agree on the meaning of non-repeating decimals. -- Richard === Subject: Re: .999... ?= 1 > Ok, ok, calm down, folks. Yes, I know .999... = 1. But certain friends > of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that > waste anyone's time, but I've about given up proving this to them, and > one of them commented that he won't believe it until a PhD says it's > true. > Here's the best proof I've seen or used: > If x = 0.999... > 10x = 9.999... > - x = 0.999... > ---------------- > = 9x = 9 > Or, x = 1 > As a side request, does anyone know a proof that might be more > understandable to an idiot? If your friends are not convinced by this proof, ask them what they mean by 0.999...; for me it's the limit of .9, .99, .999, .9999, ..., so it's obviously 1. -- Maxi === Subject: Re: .999... ?= 1 > Ok, ok, calm down, folks. Yes, I know .999... = 1. But certain friends > of mine don't, so if a few mathematicians here could just con[CapitalThorn]rm that > waste anyone's time, but I've about given up proving this to them, and > one of them commented that he won't believe it until a PhD says it's > true. How odd! We get an unusually long 0.999...-free period, and then suddenly two references are posted within hours of each other. If you had been keeping up with that other thread, then you would know that there is a proof (or maybe it was a disproof) involving Buridan's donkey, but it only works if you use American-style spellings. Derek Holt. === Subject: Approximating the surface area of n-ellipsoids, a la Muir A simple and effective method, possibly new, for approximating the surface area of ellipsoids in n dimensions is presented. In 1883, T. Muir presented an approximation for the perimeter of an ellipse. With its semiaxes of lengths a and b, the approximation has the form 2 pi (1/2 (b^p + a^p))^(1/p) . Muir's value of p = 3/2 optimizes an approximation in this form for nearly 3-dimensional ellipsoid with semiaxes of lengths a, b, and c, Knud Thomsen proposed an approximation which may be regarded as a generalization of the form used by Muir 4 pi (1/3 ((bc)^p + (ac)^p + (ab)^p))^(1/p) together with the value p = 1.6075, which he chose in order to minimize worst |relative error|. I then suggested that p = 8/5 exactly may be used in order, among other things, to optimize the approximation for nearly spherical ellipsoids. (It is fortuitous that p = 8/5 is quite close to Thomsen's value, for it means that, in optimizing for nearly spherical ellipsoids, we do not then substantially increase worst |relative error|.) This generalization can be extended nicely to provide approximations for surface areas of ellipsoids in n dimensions. Let S_n denote the surface area of the unit n-sphere: If n is even, S_n = 2 pi^(n/2)/(n/2 - 1)! If n is odd, S_n = 2^((n+1)/2) pi^((n-1)/2)/(n-2)!! [Note the double factorial. Example: 7!! = 7*5*3*1, rather than (7!)!] With semiaxes of lengths a_1, a_2, a_3,..., and a_n, the form of the approximation is (to be viewed in a [CapitalThorn]xed-width font, please) ( 1 n p ) 1/p S_n ( - sum (product a_j) ) ( n i=1 j<>i ) Thus, for example, for a four-dimensional ellipsoid with semiaxes of lengths a, b, c, and d, the surface area would be approximated by 2 pi^2 (1/4 ((bcd)^p + (acd)^p +(abd)^p +(abc)^p))^(1/p) . The question remaining is: In order to optimize such approximations for nearly spherical ellipsoids, what values of p must be used? Answer: p = 2(n + 1)/(n + 2) [That answer is based on calculations of the optimal values of p up to n = 7. The pattern seems quite obvious, but I have not done a proof.] How well do these approximations work for ellipsoids which are not almost spherical? I have not had the opportunity to examine worst relative error for n>3. However, we may use the data presented by Garry Tee in Table 2 on page 17 of his Surface area and capacity of ellipsoids in n dimensions at to make a few comparisons. I also give the relative error provided by the YNOT-style approximations proposed recently by Knud Thomsen (see Gerard Michon's ) and, independently, by me at . [Areas computed by Tee are for ellipsoids having semiaxes lengths in arithmetic progression from a_1 = 1 to the stated a_n. I give the computed areas truncated to only 5 signi[CapitalThorn]cant digits, although Tee gave them to 12 signi[CapitalThorn]cant digits.] Computed Relative errors n a_n area YNOT-style Muir-style ---------------------------------------------- 4 2 6.3922*10^1 -0.14% 0.025% 10 2 8.2918*10^2 -0.31% 0.015% 20 13 2.5891*10^14 -3.11% 1.773% 256 3 1.6264*10^-78 -0.70% 0.005% 256 36 8.1862*10^147 -5.80% 0.749% 256 100 7.5617*10^254 -7.65% 2.629% [Looking at this data, readers may be tempted to surmise that the surface area is always underestimated by the YNOT-style approximations (except in the degenerate and spherical cases) and always overestimated by the Muir-style approximations (except in the spherical case). Such a surmise would be incorrect. (For example: Muir's own approximation _under_estimates the perimeter of noncircular ellipses. And for n=3, both YNOT- and Muir-style approximations both over- and underestimate surface area at various places.)] We might also consider the singly degenerate case, in which exactly one axis is of length zero. The relative error, pi (n-1)!!/(2^(n/2) n^((n+2)/(2(n+1))) (n/2 - 1)!) - 1 if n is even and 2^((n-1)/2) ((n-1)/2)!/(n^((n+2)/(2(n+1))) (n - 2)!!) - 1 if n is odd, is an increasing function, negative for n=2 and positive for n>=3. As n increases, relative error seems to approach a value slightly larger than 1/4. [Is this limit perhaps Sqrt(pi/2) - 1 exactly?] This is not good; but of course, one may then use the YNOT-style approximations instead, which work perfectly in this particular case. Thoughtful comments are always invited. David W. Cantrell === Subject: Re: Fourier transform of non-negative function > I have a problem in which I am performing a discrete inverse Fourier > transform to get a function which I know must be non-negative. All of > the odd Fourier coef[CapitalThorn]cients except the 1st coef[CapitalThorn]cient are unknown. > Since there is a constraint on the function (f(x) >= 0), there should > also be a constaint on these unknown coef[CapitalThorn]cients. What does this > constraint on the coef[CapitalThorn]cients look like? > I don't have a clue where to start. The interplay of the basis functions > to achieve a non-negative function appears complex... > Nathan I think that the notion you are after is positive de[CapitalThorn]nite. For the Fourier series this is something like sum_{m,n} a_{m-n} x_m x_n >= 0 for all sequences x_m, (i.e. the matrix whose (m,n)th element is a_{m-n} is positive de[CapitalThorn]nite) and the same is true for the discrete Fourier transform. (So you could look at the matrix whose entries are a_{m-n} and then compute its eigenvalues to see if they are all positive, but actually the eigenvalues will simply be the coef[CapitalThorn]cients of the original function, so the easiest way to compute them is by computing the Fourier transform.) === Subject: Some easy problems This isn't homework. 1) Let G be a transitive permutation group on the [CapitalThorn]nite set A with |A| > 1. Show that there is some s in G such that s(a) =! a for all a in A. I'm completely stumped on this one. 2) Let g_1, g_2, ..., g_r be representatives of the conjugacy classes of the [CapitalThorn]nite group G and assume these elements pairwise commute. Prove that G is abelian. This one looks like it should be easy, but I can't do it. The problem is equivalent to proving G = Z(G), i.e., each conjugacy class has only one element. I tried proof by contradiction by assuming xgx^-1 = g' =! g for some x, g in G, but that didn't work too well. === Subject: Re: Some easy problems >This isn't homework. >1) Let G be a transitive permutation group on the [CapitalThorn]nite set A with >|A| > 1. Show that there is some s in G such that s(a) =! a for all a >in A. Hint: for any a in A, what fraction of the members of G have s(a) = a? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: 640,000,000,000 TO 1 > Paul Nutteing wonders: > Is it blind faith, arrogant indifference or conspiracy of > silence over uninvestigated unrelated matches in databases. > Not sure in the UK. The FSS remains a bit of a mystery to me. > But in Aus (& NZ) its a combination of the with a large dollop > of bullying from lab managers and forensic scientists with > an agenda. > The worst of the latter is Bruce Budowle of corrupt FBI > DNA lab fame (and who also has a major [CapitalThorn]nancial interest > in ABI). There are highly respected forensic scientists who > can't get their work questioning multi-billion to one match > odds published in the forensic journals because of the > inßuence wielded by Budowle. > The Ômere' billion to one odds usually used by FSS experts > represents a small victory of sorts by some of the Ôdissenters' > over Budowle's insistence on using odds that suggest > absolute uniqueness in the history of the human race. > Its not Budowle all by himself of course - there is big (public) > money sunk into the databases which has been justi[CapitalThorn]ed with > hyperbolic claims as to their infallibility. > michael >the FBI now wants to be able to state to a scienti[CapitalThorn]c certainty that two > DNA samples match. This is an assertion no other forensic DNA laboratory > would dare make -- because there are no certainties in science, only > probabilities< I would rather those forensic scientists quoted the baseline chance of any match in a population a la Birthday problem. 1 in 400,000 for 10 loci, 1 in 100,000 for 9 loci or whatever for the modal group in a community. After that again working from data in the real world and analysing locus by locus and rarity of allele frequencies they can then state whether the given individual is more or less likely to have a match in the total population. There is one [CapitalThorn]rm rule from my simulations - anyone with a rare allele is the least likely to be falsely matched. Billions , even heptillions I saw a reference to this last week in an old report is just cloud-cookoo land. *********** Latest simulation (will become dnas8.htm [CapitalThorn]le eventually joining the rest ) FSI, 115 (2001) 111-112 FSI, 119 (2001) 113-115 Basque data: all samples were from people who had gparents with Basque birth-surnames and born in the Basque country. The Basque data compared to South Spain data for the 3 principal alleles at each locus , summing allele frequencies showed following percentages Locus Spain Basque D3 67 73 FGA 50 50 D5 89 85 D13 73 79 D7 68 70 D21 54 63 D18 44 51 D16 79 81 Results for simulating 9 loci,18 datapoint for the Basques data giving 1 million pro[CapitalThorn]les Pairs of matches in [CapitalThorn]rst 12 digit - 11,100 14 digit - 1,004 16 digit - 76 18 digit - 9 distinct pairs all cross-checked back to the undirected generator [CapitalThorn]le and nothing wrong with the random generator. This square law equates with a baseline [CapitalThorn]gue of about 1 in 320,000. Triples 12 dig - 527 14 dig - 10 16 dig - 0 Quads 12 dig - 37 14 - 1 Quins 12 dig - 4 14 dig - 0 I only added homozygosity ot the subest 3........ which had 422,260 pro[CapitalThorn]les and in the original sorted [CapitalThorn]le and had 2 separate 9 locus matched pairs 8 Ôlocus' - 9 pairs 7 Ôlocus' - 94 pairs 6 Ôlocus' - 1,135 pairs At the moment I've added HZ randomly across all loci and alleles. If the random Ôadder' fell on a HZ pair then no change. Originally 83,816 pro[CapitalThorn]les 33.... increased to 102,641 pro[CapitalThorn]les 33..... [ some 34.... , 35... etc became 33.... , the altered allele always taking the value of the [CapitalThorn]rst ] After randomly adding in 22.8 per cent excess homozygosity these results emerged 9 locus - 1 match 8 locus - 9 7 locus - 135 6 locus - 1,520 For triplet matches in both 3..... subsets Original , 12 dig - 4, 14 dig - 0 Added HZ, 12 dig - 65, 14 dig - 1 All very odd Checking the 9 locus matches . The one with added HZ was a new pair. One each of the previous pair had one locus changed in each so destroyed the original matches. I cannot [CapitalThorn]nd any errors in processing and assume this was just one of those statistical hickups. Hopefully if I process the remaining non-3 pro[CapitalThorn]les a different picture will emerge. Intuitively I would have thought it made a lot off difference. Although 9 matches in each of the 8 locus cases , only 4 of these 9 were the same in both sets What they aren't telling you about DNA pro[CapitalThorn]les and what Special Branch don't want you to know. http://www.nutteing2.freeservers.com/dnapr.htm or nutteingd in a search engine email nonarevers@yahoo.co.....uk (remove 4 of 5 dots) === Subject: Re: Minimum total value spanning trees 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21: > We know how to [CapitalThorn]nd a minimum spanning tree and we know how to [CapitalThorn]nd > the path of minimum cost from one vertex to another vertex. However, > what if we wanted to [CapitalThorn]nd the tree which has the lowest sum of costs > to travel from each vertex to every other vertex. ... > if this problem has already been addressed (my > professor, Prof. Dave Perkins, says that it has not been attempted to This is apparently known as the minimum routing cost spanning tree or minimum average distance spanning tree. It's NP-complete but can be ef[CapitalThorn]ciently approximated. Googling on either of those phrases will [CapitalThorn]nd plenty of references. -- David Eppstein http://www.ics.uci.edu/~eppstein/ Univ. of California, Irvine, School of Information & Computer Science === Subject: www.dividi.com.hr www.dividi.com.hr === Subject: Re: Is Mathematica Student Edition fully functional >I think the real problem with mathematica is the Draconian licensing > scheme put in place. For instance, any discovery made while using > mathematica is Wolfram Institute property.--he owns the copyright! > What a greedy outrage!!! So all research institutes and all > colleges and professors with aim to publish and all corporations who > are developing new products and all students with a creative future > to expect wouldn't want that arrogant error prone piece of soft... > sware, even for free, because> they'd be mortgaging their future > potential to the big bad Wolfram. What if somebody discovers a > theorem with the help of the big bad Wolfram? Does that mean that the > big bad Wolfram owns the theorem? >Yes--you technically need Wolfram's permission to publish your result. >friend with maple or matlab to con[CapitalThorn]rm their calculations. Then they >forget all about mathematica. :) I'd like to see where you found the statement that technically you need Wolfram's permission to publish. I've searched and I cannot [CapitalThorn]nd it. I have a couple of failing brain cells that seem to think they once saw a statement that said you needed to cite Mathematica if you published a result obtained from Mathematica. I think the wording was about the same as you would need to cite any other published material if you published some information from it. But I cannot [CapitalThorn]nd that now, I have searched for that and I'd like to [CapitalThorn]nd that. I even sent an email to Wolfram asking them about this. Their response could only provide the citation information I'd already found: http://support.wolfram.com/mathematica/reference/general/ citing.html You can reference Mathematica in your published work just as you would reference a book or any other publication. The standard citation elements for Mathematica are: * Mathematica 5.0 Author: Wolfram Research, Inc. Title: Mathematica Edition: Version 5.0 Publisher: Wolfram Research, Inc. Place of publication: Champaign, Illinois ... So, does anyone else remember reading the statement about needing to cite Mathematica when you publish results obtained from it? And please provide the original source for your statement that all work done with is owned by Wolfram and that permission is needed before publishing. === Subject: Re: Is Mathematica Student Edition fully functional === Subject: Re: Is Mathematica Student Edition fully functional I think the real problem with mathematica is the Draconian licensing > > scheme put in place. For instance, any discovery made while using > > mathematica is Wolfram Institute property.--he owns the copyright! > What a greedy outrage!!! So all research institutes and all colleges and > professors with aim to publish and all corporations who are developing new > products and all students with a creative future to expect wouldn't want > that arrogant error prone piece of soft... sware, even for free, because > they'd be mortgaging their future potential to the big bad Wolfram. > What if somebody discovers a theorem with the help of the big bad Wolfram? > Does that mean that the big bad Wolfram owns the theorem? What's happen with your quoting software? Too much acid? Can you get it to come down? Like, you know, appear normal? >Yes--you technically need Wolfram's permission to publish your >result. They're crazy, attempting to be at least as obnoxious and as malicious as MicroSoft. The big bad Wolfram doesn't have Mother Goose's permission to use any of her characters, much less make them a company logo. >a friend with maple or matlab to con[CapitalThorn]rm their calculations. Then >they forget all about mathematica. :) Indeed, even employers wouldn't want any workers anywhere near it. ---- === Subject: Re: Is Mathematica Student Edition fully functional I think the real problem with mathematica is the Draconian licensing > > scheme put in place. For instance, any discovery made while using > > mathematica is Wolfram Institute property.--he owns the copyright! > What a greedy outrage!!! So all research institutes and all colleges and > professors with aim to publish and all corporations who are developing new > products and all students with a creative future to expect wouldn't want > that arrogant error prone piece of soft... sware, even for free, because > they'd be mortgaging their future potential to the big bad Wolfram. > What if somebody discovers a theorem with the help of the big bad Wolfram? > Does that mean that the big bad Wolfram owns the theorem? Yes--you technically need Wolfram's permission to publish your result. friend with maple or matlab to con[CapitalThorn]rm their calculations. Then they forget all about mathematica. :) Ôcid Ôooh === Subject: Re: Great-circle radius of ellipsoid > Tying this all up succinctly, I would de[CapitalThorn]ne this choice for a > best [CapitalThorn]t spherical arcradius for a (simple biaxial) spheroid thusly: > ************************************************************ > * The Gratispherical Arcradius of a biaxial ellipsoid * > * is the arcradius of the transverse graticular midpoint: * > * * > * Gr = a * [1 - (cos{45} * sin{45} * sin{Oz})^2]^.5, * > * = a * [1 - (.5 * sin{Oz})^2]^.5, * > * = [.25 * (3a^2 + b^2)]^.5; * > * * > ************************************************************ The convasation was intresting yet still I am not confedant about sqr(a)*sqr(b) and Math Freaks r(u,v) is to complacated but this avarage seems suf[CapitalThorn]ceing. Best day, Kiyo === Subject: Melissa Rae Dumont - December 23rd 1984 D U M O N T 4 21 13 15 14 20 = 87 Melissa stopped and provided stats for her family today, she is a nubile sweety. 87+ Dad 2 4 56 93/273 +321 87+ Mom 20 5 59 140/225 822 87+ Sis 9 5 78 129/236 7751 189 Melissa 23 12 84 358/8 10171 Melissa 78 Rae 24 Dumont 87 Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 15 21 76 9 29 16 34 123 10 31 18 55 199 11 37 20 89 322 12 41 <-13th-> 21 <-13th-> 144 <-13th-> 521 <-13th-> 13 --- --- -- 238 154 <-Lamentations 91 The family was born on days of the month adding to 54 (13 plus the 13th prime), Bible Book 13 opens with 54 (13 plus the 13th prime) verses while Bible Book 54 (13 plus the 13th prime) contains 113 verses. The females were born in years adding to 143 (11x13). Melissa's name begins with the 13th letter of the alphabet, her [CapitalThorn]rst name adds to 78 (6x13). She has 13 letters in her common name. Her square valued letters add to 13, her unrepeated letters add to 113, her consonants add to 132. She was born in 84 (the [CapitalThorn]rst 13 primes minus the [CapitalThorn]rst 13 non-primes). Her names average 63 (Exodus 13). Melissa is 19.42 years old, pretty as there are 942 verses in Bible Book 13. The older sister was born in 78 (6x13). Mom and Melissa were born in 59 and 84, these are the 17th prime and 61st non-prime, together for 78 (6x13). The females were born in years adding to 221 (13x17). Mom was born in 59, corresponding to James with 108 verses (the primes in prime positions up to the 13th prime). The parents were born in 56 and 59, together these Bible Books contain 154 verses (the [CapitalThorn]rst 13 non-primes). Lucas 1 3 4 7 11 18 29 -- 73 <-the Lucas numbers up to 29 add to the 73 verses of Bible Book 29 J O E L <-Bible Book 29 10 15 5 12 = 42 <-29th non-prime C O P P E R <-29th element 3 15 16 16 5 18 = 73 <-Book 29 and is the Lucas numbers up to 29, there is a copper riding a horse on the 1973 Canadian 25 cent piece C E N T <-made out of 29th element 3 5 14 20 = 42 <-29th non-prime There are 29 chapters in Bible Book 13 and see that the older sister was born on day 129 (the primes up to 29 add to 129). The parents were born in years adding to 115 (the 73 verses of Book 29 plus the 29th non-prime). The kids were born in years adding to 162 (Deuteronomy 9 with 29 verses). Melissa and mom were born in years adding to 143, Ecclesiastes 7 contains 29 verses and brings Ecclesiastes up to 143 verses, pretty as this is the 109th non-prime while 109 is the 29th prime. Melissa was born 265 days after dad's birthday (First Samuel 29). The family name adds to 87 (29+29+29). Primes Non-Primes Numbers 2 1 1 3 4 2 5 6 3 7 8 4 11 9 5 13 10 6 17 12 7 19 14 8 23 15 9 29 16 10 31 18 11 37 20 12 41 21 13 43 22 14 47 24 15 53 25 16 59 26 17 61 27 18 67 28 19 71 30 20 73 32 21 79 33 22 83 <-23rd-> 34 <-23rd-> 23 --- --- --- 874 435 276 1-50 - Genesis 51-90 - Exodus 91-117 - Leviticus 118-153 - Numbers Mom was born on the 140th day of the year, it is 23 plus the 23rd prime (83) plus the 23rd non-prime (34), or simply 23+23p+23np, pretty that Bible chapter 140 would be Numbers 23. The parents were born on days of the year adding to 233. Mom and Melissa were together born with 233 days remaining in their years. Melissa was born on the 23rd, her unrepeated letters add to 113 (Leviticus 23). The family was born on days 2, 20, 9 and 23, together these Bible Books contain 4230 verses. Primes Non-Primes Numbers 2 1 1 3 4 2 5 6 3 7 8 4 11 9 5 13 10 6 17 12 7 19 14 8 23 15 9 29 16 10 31 18 11 37 20 12 41 21 13 43 22 14 47 24 15 53 25 16 59 <-17th-> 26 <-17th-> 17 --- --- --- 440 251 153 Mom was born in 59 (17th prime), the kids were born in months adding to 17, the family was born in months adding to 26 (17th non-prime) and in years adding to 277 (the 59th prime), prettier as the parents are 1277 days closer in age than the kids. Mom was born with 225 days remaining in the year, it is the 177th (59+59+59th non-prime). The kids were born in 78 and 84, these are the 57th and 61st non-primes (an average of 59). The sister was born with 236 (4x59) days remaining in the year. Melissa and her parents were born on days of the year adding to 591. The kids were born on days of the century adding to 59656. The consonants in Melissa's given names add to 81 (59th non-prime). Her vowels and consonants add to 57 and 132, these are the 41st and 100th non-primes (a difference of 59). Her primes and squares add to 74 and 13, these are the 53rd non-prime and the 6th prime, together for 59. Her odd and even valued letters add to 121 and 68, the former is 177% (59+59+59%) of the latter. Melissa's repeating letters add to 76, or 17 plus the 17th prime (59). Her Fibonacci valued letters add to 59 (the 17th prime). Mom's day, month and year of birth adds to 84 and she last gave birth in 84 (7 times the 7th non-prime). The family was born on days and in months and years adding to the 357 (7x17+7x17+7x17) verses of Daniel. Melissa was born 217 days after mom's birthday, it's the 170th non-prime while chapter 170 is Deuteronomy 17. Melissa's day, month and year of birth adds to 119 (7x17). Melissa's given names add to 102 (6x17 an is 17+17p+17np). Melissa's full name adds to 189 (the [CapitalThorn]rst 17 primes minus the [CapitalThorn]rst 17 non-primes). Her upsidedown and/or ßippababbles (her letters with upsidedown and/or ßippababble attributes... A, A, I, M, M, R, S and S) add together for 93 (Leviticus 3 with 17 verses). The parents are together 93.17 years old. Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 <-17 is the 7th prime 5 11 <- 11 while the primes up 6 13 to 7 add to 17 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 14 43 15 47 16 53 17 59 <- 59 <-the 7th prime in --- prime position 167 Esther Book 17 <-the 7th prime 17 is the 7th prime while the primes up to 7 add to 17. There are 7 primes in prime positions up to the 17th prime and they add to the 167 verse of Bible Book 17, Esther. Esther become Queen in Book 17 and Q is the 17th letter of the alphabet. Psalm 59 (the 17th prime) not only contains 17 verses, it is the 17th chapter in the Bible to contain the length of 17 verses. James is Book 59 (the 17th prime), it's 108 verses is the 17th prime short of the 167 verses of Book 17. Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 5 11 <- 11 6 13 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 --- 108 James Book 59 Leviticus begins with 17 verses and terminates at chapter 117 with 17+17 verses. There are 17 verses at chapters 1 and 3, and 59 (the 17 prime) verses at chapter 13, so the 17's and the 17th prime are at chapter numbers adding to 17 (1+3+13=17). The [CapitalThorn]rst 17 versed chapters in the Bible are at chapters 91 and 93, together for 184, or the 167 verses of Book 17 plus 17 more. Leviticus contains 859 verses, it ends in 59 (the 17th prime). The [CapitalThorn]rst 17's in the Bible surround chapter 92 (the 4x17th non-prime): Leviticus --------- 91 1 17 92 2 <-68th (4x17th) non-prime 93 3 17 94 4 95 5 96 6 97 7 98 8 99 9 100 10 101 11 102 12 103 13 59 <-17th prime 104 14 105 15 106 16 107 17 108 18 109 19 110 20 111 21 112 22 113 23 114 24 115 25 116 26 117 27 34 <-17+17 There are 89 chapters in The Gospels and 1189 chapters in the Bible. Melissa's full name adds to 189. The family was born on days of the century adding to 101891. The parents and the sister were together born 11.89 months into their years. The kids were born 571 days after mom's birthdays and 302 days after dad's birthdays, the former is 189% of the latter. Non-Primes 1 57 110 158 207 4 58 111 159 208 6 60 112 160 209 8 62 114 161 210 9 63 115 162 212 10 64 116 164 213 12 65 117 165 214 14 66 118 166 215 15 68 119 168 216 16 69 120 169 217 18 70 121 170 218 20 72 122 171 219 21 74 123 172 220 22 75 124 174 221 24 76 125 175 222 25 77 126 176 224 26 78 128 177 225 <-177th 27 80 129 178 226 28 81 130 180 228 30 82 132 182 230 32 84 133 183 231 33 85 134 184 232 34 86 135 185 234 35 87 136 186 235 36 88 138 187 236 38 90 140 188 237 39 91 141 189 238 40 92 142 190 240 42 93 143 192 242 44 94 144 194 243 45 95 145 195 244 46 96 146 196 245 48 98 147 198 246 49 99 148 200 247 50 100 150 201 248 51 102 152 202 249 52 104 153 203 250 54 105 154 204 252 55 106 155 205 253 56 108 156 206 254 Melissa is a 7 lettered name adding to 78. Her middle name adds to 24 (17 plus it's 7th prime position). The 7 different letters in her given names add to 77, while the letters missing from her given names exceed that by the 197 verses of Bible Book 28 (1 through 7). Her 7 vowels add to 57 (Exodus 7). Her odd valued letters add to 177% of her even valued letters (3 times the 17th prime). Her 14 (7+7) missing letters exceed her 12 (7th non-prime) different letters by 49 (7x7). Melissa's day, month and year of birth adds to 119 (7 times the 7th prime). She was born in the 12th month (7th non-prime) of year 84 (7 times the 7th non-prime). The older sister was born on the 28618th day of the century, 1 through 7 adds to 28 while Bible Book 7 contains 618 verses. When mom sees this, she will want to turn a seven foot tall tree into a decorated idol, and will tithe to a church that teaches her to adopt the [CapitalThorn]lthy practice. Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 <-9th-> 15 <-9th-> 21 <-9th-> 76 <-9th-> 9 --- -- -- --- -- 100 79 54 196 45 The parents were born in years adding to 115, The Samuels differ in length by 115 verses. The [CapitalThorn]rst of the kids was born on the 9th (First Samuel), the second on the 23rd (9th non-prime). The parents were born in months adding to 9, and mom was born in 59, prettier as 9 is the 5th non-prime. The sister was born on day 129. Dad was born in 56, it's the [CapitalThorn]rst 5 primes plus the [CapitalThorn]rst 5 non-primes. The kids were born in years adding to 162 (Deuteronomy 9) and on days of the year adding to 487 (Psalm 9). Primes Non-Primes 2 1 3 4 5 6 7 8 11 <-5th-> 9 -- -- 28 28 Mom was born on the 140th day of the year, and she is 822 days younger than me... there are 822 verses in Bible Book 14, pretty as 22 is the 14th non-prime while 14 is the 8th non-prime while 22 exceeds 8 by 14. The parents were born on days of the year adding to 233 (14th Fibonacci). Melissa and mom were born in years adding to 143, prettier as 43 is the 14th prime. Mom and Melissa were together born with 233 days remaining in their years (14th Fibonacci). The females were born in months adding to 22 (14th non-prime). The family was together born 22 days closer to the beginning of their years than to the end of their years (14th non-prime). The kids were together born 43.14 weeks after dad's birthdays. The parents are separated by 1143 days while Melissa and mom were born in years adding to 143. The parents and the kids were born in years adding to 115 and 162, the latter is 140% of the former, pretty as mom was born on the 140th day of the year. Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 15 21 76 9 29 16 34 123 10 31 18 55 199 11 37 20 89 322 12 41 21 144 521 13 43 <-14th-> 22 <-14th-> 233 <-14th-> 843 <-14th-> 14 --- --- --- ---- --- 281 176 609 2204 105 Dad was born in 56. The family was born on days 93, 140, 129 and 358, these are the 69th, 106th, 98th and the 287th non-primes, together for 560. The kids were born on days of the century adding to 59656. Non-Primes 1 57 110 158 207 255 303 351 4 58 111 159 208 256 304 352 6 60 112 160 209 258 305 354 8 62 114 161 210 259 306 355 9 63 115 162 212 260 308 356 10 64 116 164 213 261 309 357 12 65 117 165 214 262 310 358 <-287th 14 66 118 166 215 264 312 360 15 68 119 168 216 265 314 361 16 69 120 169 217 266 315 362 18 70 121 170 218 267 316 363 20 72 122 171 219 268 318 364 21 74 123 172 220 270 319 365 22 75 124 174 221 272 320 366 24 76 125 175 222 273 321 368 25 77 126 176 224 274 322 369 26 78 128 177 225 275 323 370 27 80 129 178 226 276 324 371 28 81 130 180 228 278 325 372 30 82 132 182 230 279 326 374 32 84 133 183 231 280 327 375 33 85 134 184 232 282 328 376 34 86 135 185 234 284 329 377 35 87 136 186 235 285 330 378 36 88 138 187 236 286 332 380 38 90 140 188 237 287 333 381 39 91 141 189 238 288 334 382 40 92 142 190 240 289 335 384 42 93 143 192 242 290 336 385 44 94 144 194 243 291 338 386 45 95 145 195 244 292 339 387 46 96 146 196 245 294 340 388 48 98 147 198 246 295 341 390 49 99 148 200 247 296 342 391 50 100 150 201 248 297 343 392 51 102 152 202 249 298 344 393 52 104 153 203 250 299 345 394 54 105 154 204 252 300 346 395 55 106 155 205 253 301 348 396 56 108 156 206 254 302 350 398 The kids are separated by 2420 days, it is 6.6 years, pretty that Bible chapter 242 would be First Samuel 6. The 12 (6+6) different letters in her full name add to 151 (the 6x6th prime). Dad was born in 56 and the kids today are together 16610 days old, pretty as it's 56 short of 16666. Today the family members are an average of 34.66 years old, and I meet Melissa when I am 17266 days old. Her middle name adds to the 24 chapters of Old Testament Books 6 and 10 (6th non-prime), her [CapitalThorn]rst name adds to 78 (6 times the 6th prime). She has 10 letters in her given names (6th non-prime). Melissa was born in the 6+6th month, and on the 23rd (6th prime plus the 6th non-prime), corresponding to Isaiah with 66 chapters. Isaiah 4, 12 and 20 (adds to 6x6) each contains 6 verses. 1-50 - Genesis 51-90 - Exodus 91-117 - Leviticus 118-153 - Numbers 154-187 - Deuteronomy 188-211 - Joshua 930-957 - Matthew 958-973 - Mark 974-997 - Luke 998-1018 - John 1019-1046 - Acts 1047-1062 - Romans 123 <-Numbers 6, it is three times the 13th prime (41+41+41), keeping in mind that 13 is the 6th prime 188 <-the opening chapter of Book 6 is 6x6x6 short of the 404 verses of Bible Book 66, it is the 6th prime squared (13x13) short of the 357 verses of Daniel (also in part about 666) 193 <-Book 6 chapter 6 is the 44th prime, while 44 is in turn 66.666...% of 66 211 <-the terminating chapter of Book 6 is approximately 66.6% of the 66th prime (317) 357 <-the opening chapter of Book 6 plus the 6th prime squared is the 357 verses of Daniel (in part about 666) 404 <-the 6th prime squared (13x13) plus the 6th prime squared (13x13) plus 66 adds to the 404 verses of Bible Book 66 1062 <-666 plus 6x66 is a combination of the 658 verses of Bible Book 6 plus the 404 verses of Bible Book 66, and is the terminating chapter of New Testament Book 6 1070 <-666 plus the 404 verses of Book 66 is the 1070 verses of Job (Book 6+6+6) 1213 <-Exodus terminates at chapter 90 (66th non- prime) with 1213 verses (the 198th or the 66+66+66th prime) 1292 <-the 658 verses of Book 6 plus twice the 66th prime (317) is the 1292 verses of Isaiah (the Book contains 66 chapters) The parents were born in 56 and 59, these are the 40th non-prime and the 17th prime (together for 57). The sister was born in 78 while Melissa adds to 78 (57th non-prime). The family was born on days and in months and years adding to the 357 (7x17+7x17+7x17) verses of Daniel. The kids were together born 571 days after mom's birthdays, or 81.57 weeks. The kids were born on days of the year adding to 487 (69.57 weeks). 389 <-77th prime 104 <-77th non-prime 77 <-77 --- 570 The Four 57's Genesis 41 -> 41 Leviticus 14 -> 104 Judges 9 -> 220 <-I dreamt of 220 roofs blown John 11 -> 1008 off homes in the Dakotas ---- 1373 <-220th prime Chapter 57 is Exodus 7 with 25 verses Book 57 is Philemon with 25 verses -- -- 41st non-prime 16th non-prime <-together for 57-> Major Books of End-Times Prophecy (Daniel and Revelation are in part about 666 while Isaiah contains 66 chapters): Daniel - 357 verses Revelation - 404 verses <-57 plus the 57th prime plus the 57th non-prime Isaiah - 1292 verses <-an average of 19.575757... verses per chapter Genesis 41 repeats the spoken seven 28 times, pretty as 1 through 7 adds to 28. We are warned in Genesis 41 to accumulate 7 years worth of food supplies in anticipation of several years of successive crop failures. Genesis 41 contains 57 verses, together for 98 (7x7+7x7). Pretty that Genesis 41 would contain 57 verses, for 57 is the 41st non-prime. Genesis 41 is a numerological marker in the Bible, I tried to tell people this in 1988 but they rolled their eyes back and smirked, people did not like my criticisms of their churches and used my interest in math as one of their reasons to arrest and torture me in psychiatric settings. I was arrested under the mental health act while the farmers were [CapitalThorn]ned for failing to get the grain to port fast enough. The Canadian government provided land for the railways, but then the railroad companies ripped out the tracks and sold the land, and used the money to purchase lucrative money making hotels. I was tortured for speaking while the people guilty of treason took their frequent trips to Europe, Asia and the Caribbean, while the Catholics sponsored Latin American and Asian Catholics to come to Canada and earn hard Western currency so they could tithe to the Catholic church. And they repeatedly raped Canadian taxpayers to the tune of many hundreds of millions of dollars to pay for costs involved in the visits of the pope (it is theft, just like the theft of the Egyptian obelisk situated at The Vatican). People used my interest in numbers as one of the reasons to support having me arrested and tortured, I show you people gems and year after year you spend billions of dollars on turning trees into idols while calling me insane (Second Chronicles 36:16). 187 Dar 17 2 57 48/317 00 Daryl 60 Shawn 65 Kabatoff 62 187 Marcia 6 8 80 219/147 8571 Marcia 45 Veronica 87 Acevedo 55 201 Melodee 7 4 74 97/268 6258 Melodee 59 Joy 50 Webster 92 240 Bethany 28 10 76 302/64 7193 Bethany 75 Ruth 67 Guenther 98 147 Natash 18 1 81 18/347 8736 Natash 63 Lyn 51 Isaac 33 214 Ali 12 1 83 12/353 9460 Alison 70 Janet 50 Gillespie 94 194 Lesley 27 5 83 147/218 9595 Lesley 78 Mae 19 Roberts 97 223 Camille 8 6 84 160/206 9973 Camille 55 Clarisse 86 Massey 82 221 Jessica 17 6 84 169/197 9982 Jessica 66 Vivian 77 Valois 78 288 Melinda 23 3 83 82/283 9530 Melinda 58 Janelle 59 Elaine 46 Joyce 58 Jarocki 67 If you people think you have the right to use my abusive parents as tools to have me arrested and tortured, then I think that I should have the right to ask women to marry me, or to marry Marcia and me. I have Scripture to support taking seven brides (Isaiah 4:1) and I have Scripture to support sleeping with Melinda Jarocki outside of wedlock (First Kings 1:1-5), while you people have a vast multitude of Scriptures condemning your decorated trees, phallic-capped churches and your violence against me for daring to point out your pagan traditions. Anyway, now that Melissa and her family sees evidence that their names are gifts from God, watch them start attending and tithing to churches that teach them to turn trees into decorated idols (maybe see Deuteronomy 12:2; 1 Kings 14:23; 2 Kings 16:4, 17:10; 2 Chronicles 28:4; Isaiah 57:5; Jeremiah 2:20, 3:6, 3:13, 10:3-4, 17:2 and Ezekiel 6:13). It just doesn't matter to you people that the Bible repeatedly condemns turning trees into idols (it is a violation of God's First and Second Commandments). And it just doesn't matter to you people that I was tortured for years after daring to identify the obelisks at The Vatican, The Whitehouse and on church roofs as being representations of penises (and in opposition to God's Second and Sixth Commandments). You people are compassionless turds, your only real compassion is for your traditions. You don't have an ounce of compassion for me or for anybody else being tortured by the psychiatric profession, all that truly matters to you is to go home for Christmas and stoop to the base of your mom's decorated tree (symbol of fertility that stays green all year) and collect a few presents. And then you feel guilty of your sin so you attend a church once a year, you go in December and give money to a preacher who tells you to stay away from idols, and he provides you with this less lesson while he standink next to a decorated tree and under his church steeple (an obelisk, an Egyptian representation of a penis). They worshipped and made representations of penises for these pagans believed it to have a creative or god-like force through it's reproductive ability, but Shawn knows that your penis is closer to your anus than to God. I don't have an ounce of respect for you and so don't require your permission to post your stats on the usenet and use you as an example to others, and look, here you are!!! There is a woman who hangs around Crocus Coop, she told me that the people are scared, they don't want to make waves against the psychiatric profession or anything else for they are afraid of being sent back to Hantelman at the U of S, or to City Hospital. I once heard a woman talk on the phone over at the mental health of[CapitalThorn]ce on Ave. P North, she cried and pleaded on the phone for a lengthy time, she was saying over and over that there was nothing wrong with her, how damaging the drugs being forced upon her were, the drugs were making her feel nauseous, she was horri[CapitalThorn]ed about the thought of continued forced treatment, she did not want to be on the drugs and stated over and over that there was nothing wrong with her and she did not need the drugs administered to her. She had no power, nobody cared about her being forced drugs against her will, nobody cared about her and nobody cared about me. And nobody cared about Jason Lee, he was in an abusive family relationship and then his parents forced him into the psychiatric mill, and he just eventually fell over and died. And Connie Glushyk's mom was ting on Connie and forcing Connie into the wards and then threatening to put Connie into a group home, well Connie just got a pass from the City Hall ward, walked across the street to her apartment in the high-rise, and jumped. The drugs make it so hard to verbalize arguments against what the psychiatrists are doing to you. It is a real horror show to think that you might get sent back to Hantelman, the threat of going back to Hantelman or to City Hospital looms for me and these people, once charged under the mental health act, they can take you and lock you up for 6 weeks until forced by law to release you. But then the NDP came to power provincially and immediately changed the laws and now can force medications upon people after the 6 weeks of forced treatment at Hantelman or City Hospital, and now thanx to the NDP, people who were horri[CapitalThorn]ed at the thoughts of being returned for a 6 week treatment, now get the drugs forced upon them every week of the year. And during this time, the NDP campaign colors included saffron, it's a colour holy to the Hindus, interesting as the NDP are now forcing people to be medicated 24/7/366 by Hindu psychiatrists. They would detain me for 6 weeks (during which time I would get to appeal twice to the middle-class Protestants and Catholics twice), and on the [CapitalThorn]nal day of the 6 weeks of treatment they would administer one last depot drug injection and release me, and the drugs remain in my body doing horrible things for months after that injection. It would take me months to get over the crap that they force upon me every 6 weeks, and then they would do it to me again and again and again and again. And now thanx to the NDP, people are having drugs forced upon them all the time. I was [CapitalThorn]rst arrested and tortured in 1988 when the NDP were not in power, but then the NDP came to power and took away more rights from people, the NDP have Hindus employed to force medications upon people against their will, the drugs remain in a person's body for months at a time, the NDP force people to be drugged every day of their lives. And now if some other party would come to power provincially in Saskatchewan and remove the NDP, they would leave the NDP law intact and continue to allow people to be drugged continuously. The NDP tortured me and tortures many in Saskatchewan, I don't want to live in Canada now and I certainly don't want to live in Canada if the NDP come to power nationally. I don't want to be here in Canada, and many other people don't want to be here either. There are people being forced medications upon them against their will and they would rather have a few dollars and be backpacking in Europe, they don't want to be living in poverty in Saskatchewan doped up and feeling nauseous, and then you people turn around and spend a few billion dollars annually turning trees into decorated idols. You don't care what the NDP and other governments are forcing these drugs upon people, you people care only about your tradions. I don't want to be in Canada and close to these places that I have been tortured. It was a horror show what they did to me, and year after year I see you people spending billions of dollars on turning trees into idols while calling me insane, and now some totally insensitive asshole starts speaking up in sk.forsale saying that he or she wouldn't want me babysitting his or her children, and that I make his or her skin crawl. You are God's children, and I have been babysitting all of you since 1988 when I started speaking up against the false traditions in your churches, and you make my skin crawl as well. Well Melissa thought it was all very nice and well and she paid me a dollar for my work (showing her evidence that her name was a gift from God), she then likely walked down the block and paid four dollars for a drink. Daryl Shawn Kabatoff Box 7134 Saskatoon Saskatchewan Canada S7K 4J1 Isaiah 45:4, Ephesians 3:15 - God gives you your name!!! === Subject: Re: Melissa Rae Dumont - December 23rd 1984 > people did not like my > criticisms of their churches and used my interest in math What you do is not related in any way, shape or form to mathematics. > And they repeatedly > raped Canadian taxpayers to the tune of many hundreds of millions of dollars > to pay for costs involved in the visits of the pope (it is theft, just like > the theft of the Egyptian obelisk situated at The Vatican). Religious debates aside, do you know how much tourist revenue visits by the Pope bring? Did you know the largest religion in Canada, in terms of census numbers, is Catholicism? l8r, Mike N. Christoff === Subject: Re: Melissa Rae Dumont - December 23rd 1984 > D U M O N T > 4 21 13 15 14 20 = 87 > Melissa stopped and provided stats for her family today, she is a nubile > sweety. Again? And you wonder why the below happens? > people did not like my > criticisms of their churches and used my interest in math as one of their > reasons to arrest and torture me in psychiatric settings. No, they noted your obsession with NUMEROLOGY (not math) and your [CapitalThorn]xation with posting personal information about young females on Usenet, and regarded you as a kook. > I was arrested under the mental health act Instead of spending hours trying to divine numerical signi[CapitalThorn]cance to essentially randomly chosen names, when are you going to put 2 and 2 together and realize that the main reason you're being locked up is because of your bizarre obsessions and cyber-stalking? > I was > tortured for speaking while the people guilty of treason took their frequent > trips to Europe, Asia and the Caribbean, while the Catholics sponsored Latin > American and Asian Catholics to come to Canada and earn hard Western > currency so they could tithe to the Catholic church. And they repeatedly > raped Canadian taxpayers to the tune of many hundreds of millions of dollars > to pay for costs involved in the visits of the pope (it is theft, just like > the theft of the Egyptian obelisk situated at The Vatican). People used my > interest in numbers as one of the reasons to support having me arrested and > tortured, I show you people gems and year after year you spend billions of > dollars on turning trees into idols while calling me insane (Second > Chronicles 36:16). Kook. > If you people think you have the right to use my abusive parents as tools > to have me arrested and tortured, then I think that I should have the right > to ask women to marry me, or to marry Marcia and me. And what if Marcia doesn't even want to be associated with you, much less marry you? >Well Melissa thought it was all very nice and > well and she paid me a dollar for my work (showing her evidence that her > name was a gift from God), she then likely walked down the block and paid > four dollars for a drink. And she certainly needed one after putting up with you. Sheesh. :O| === Subject: Re: PLUTO, CHARON, AND THE MOON, AND ... LYING ASTRONOMERS. >I believe that I have something new, unique, and hmm ... revolutionary >to say. > >http:// www.interlog.com/~wnowak /book > >I tried to talk to astronomers about all facts and my claims. >Unfortunately for political reasons they pretend that they do not >hear what I am trying to say. So I came here, since my story is maybe >more about mathematics than astronomy. >I also believe that mathematics is the only science politics free. > >I am looking for mathematician(s) that would support me. > >Please do not ignore me, because I have something very ( VERY ) >interesting to say. >You must spend 0.5-1 hour on my diagrams and tables ( no need to read >everything ), >in order to realize that this is not a graphite, but indeed much >closer to a diamond. > >I know that if you are a wise man, you will try to hear me out ( free >of prejudices ). > > > I read one page of chapter 10. > You can always [CapitalThorn]nd coincidences if you're looking for them. The > biggest coincidence would be if there were no coincidences. > Once I pulled into a parking lot, and noticed that the license > plates of the [CapitalThorn]rst three cars parked there all ended with the > same three numerical digits. The chance of that is about > one in one million, but who cares? > There is a difference between just coincidences, and prediction of additional coincidences > based on other coincidences. > To best example of what I am trying to say ( I use it in my book ) is the one from the movie > Casino, where Robert de Niro [CapitalThorn]red his employee based on the hypothesis that the > probability of the event that took place, was too low to take place in reality. > This is exactly my way of reasoning. > So, if you agree with R. de Niro way of reasoning, you should agree with me. Hey, eclipse-boy, since you're fond of citing Hollywood movies to justify your math, maybe you can explain why in Ladyhawke (1985), a total eclipse of the sun takes place the day after a full moon. And while you're at it, explain why in 2001: A Space Odessey (1968) the phase of the Earth as seen from the Moon changes back and forth from waxing to waning. > WN. === Subject: Re: PLUTO, CHARON, AND THE MOON, AND ... LYING ASTRONOMERS. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i522k1v11555; >I de[CapitalThorn]nitely do not have any ability in predicting future events. >All I want to say is that based on the EXTREMELY low probability of the astronomical event that I described, and that took place >in the sky during the total eclipse of the Sun ( completely ignored by astronomers ); >I discovered another completely INDEPENDENT ( in a sense that one could not inßuenced the other ) low probability system of >coincidences that exists for the same objects ( Pluto, Sun, Moon, Charon ). >The number of all possible sets of coincidences for these small number of objects is very limited; still these coincidences exist. >In such a case; the probability that both systems of coincidences are related is HIGHER, than the probability that they both >happened randomly. >So I created my interpretation. >It is as simple as this. >BTW, please see the movie CASINO ... 2 minutes long piece, where Robert de Niro [CapitalThorn]red his employee. >It is very expressive and self-explanatory. >WN. The coincidences may be interesting to some, but don't expect the astronomical community to agree with your interpretations, or show any interest in them. If you see a coincidence, and then look for more coincidences and [CapitalThorn]nd them, I don't think that is remarkable, because you can always [CapitalThorn]nd more coicidences if you keep looking, I believe. Even if the set of coincidences that you're talking about are remarkable, you can always [CapitalThorn]nd remarkable coincidences if you look in enough places. There was an email circulating recently to do with coincidences involving two former presidents of the US. Maybe you saw it. There were maybe 20 some coincidences relating the two. I would say that those were remarkable, but that's all they were - just remarkable coincidences. If you want to think of your coincidences as needing an interpretation, go ahead. Apparently astronomers don't see a need for your such an interpretation. === Subject: Re: PLUTO, CHARON, AND THE MOON, AND ... LYING ASTRONOMERS. >I de[CapitalThorn]nitely do not have any ability in predicting future events. >All I want to say is that based on the EXTREMELY low probability of the astronomical event that I described, and that took place >in the sky during the total eclipse of the Sun ( completely ignored by astronomers ); >I discovered another completely INDEPENDENT ( in a sense that one could not inßuenced the other ) low probability system of >coincidences that exists for the same objects ( Pluto, Sun, Moon, Charon ). >The number of all possible sets of coincidences for these small number of objects is very limited; still these coincidences exist. >In such a case; the probability that both systems of coincidences are related is HIGHER, than the probability that they both >happened randomly. >So I created my interpretation. >It is as simple as this. >BTW, please see the movie CASINO ... 2 minutes long piece, where Robert de Niro [CapitalThorn]red his employee. >It is very expressive and self-explanatory. >WN. > The coincidences may be interesting to some, but don't expect the > astronomical community to agree with your interpretations, or > show any interest in them. > If you see a coincidence, and then look for more coincidences and > [CapitalThorn]nd them, I don't think that is remarkable, because you can always > [CapitalThorn]nd more coicidences if you keep looking, I believe. Even if the > set of coincidences that you're talking about are remarkable, > you can always [CapitalThorn]nd remarkable coincidences if you look in enough > places. > There was an email circulating recently to do with coincidences > involving two former presidents of the US. Maybe you saw it. There > were maybe 20 some coincidences relating the two. I would say that > those were remarkable, but that's all they were - just remarkable > coincidences. > If you want to think of your coincidences as needing an > interpretation, go ahead. Apparently astronomers don't see a need > for your such an interpretation. I disagree, and that is way I spent so much time and effort on creation of my work. I have reasons to believe that my work directly, or indirectly triggered this scienti[CapitalThorn]c paper; http://arxiv.org/abs/astro-ph/0308467 please compare CONCLUSIONS ABOUT PLUTO ( please read just abstract ) against my work http://www.interlog.com/~wnowak/book The problem is that the diagram ( [CapitalThorn]g. 7 page # 16 ) is wrong. Quaoar should be positioned at mag. >= 19, con[CapitalThorn]rming that PLuto is unique, contrary to conclusions in the abstract, and in agreement with my theory, that PLUTO IS EXTREMELY UNIQUE. I asked authors very politely for the reason for this small mistake, and so far did not get the answer. I did not tell them anything about my work. WN. === Subject: Re: Proof of twin-prime conjecture? >Stephen Miller TOP-POSTED: > analytic number theory. Can anyone recommend a good book? >M. Ram Murty, Problems in Analytic Number Theory: Allow me to add: T. Apostol Intro to Analytic Number Theory, Vol 1. (Vol. II is a superb intro to modular forms & functions) X-mailer: xrn 9.02 === Subject: Re: Peano's space-[CapitalThorn]lling curve Mail-To-News-Contact: abuse@dizum.com >I deduce from your post that the unit line and the unit >square have different cardinality, That turns out not to be the case. The set of points in a line segment (of any length) has the same cardinality as the set of points in any two-dimensional [CapitalThorn]gure. And the same cardinality as the set of points in any three-dimensional [CapitalThorn]gure, and so on. Some preliminaries: To say that two sets, A and B, have the same cardinality means that it is possible to de[CapitalThorn]ne a function from one set to the other such that: - each element of A is mapped to a distinct element of B - each element of B has an element of A that is mapped to it This is called a bijection or a one to one and onto mapping. Now, it's possible that there can be other mappings between A and B that are not bijections. That does not matter. All that matters is that it is possible to de[CapitalThorn]ne one that is. To give an example based on [CapitalThorn]nite sets, let's look at some possible mappings with A = {a, b, c} and B = {x, y, z}. The [CapitalThorn]rst, and most obvious, is: f(a)=x, f(b)=y, f(c)=z This is a bijection. That shows that the sets A and B have the same cardinality. We could de[CapitalThorn]ne another bijection if we wanted: g(a)=z, g(b)=x, g(c)=y This also shows that the sets A and B have the same cardinality. We could also de[CapitalThorn]ne a mapping that isn't a bijection: h(a)=y, h(b)=y, h(c)=x The fact that a mapping exists that is not a bijection does not alter the fact that other mappings exist that are bijections, so sets A and B still have the same cardinality. Let's try a couple of in[CapitalThorn]nite sets now. Let's pick the integers, Z = { ..., -1, 0, 1, ... } and the even integers E = { ... -2, 0, 2, ... }. It's easy enough to create a mapping between them that isn't a bijection. De[CapitalThorn]ne a function i(x)=x that maps from E to Z. (When we start working with in[CapitalThorn]nite sets, it's better not to try to list all of the individual elements.) This maps every element of E to a distinct element of Z, but not every element of Z has an element of E mapped to it, so it's not a bijection. But, we can create a different function, j(x) = x/2 that also maps from E to Z. This one *is* a bijection. Since we can de[CapitalThorn]ne a bijection between E and Z, they have the same cardinality. Now, let's de[CapitalThorn]ne R to be the set of real numbers, and S to be the set of real numbers between -pi/2 and +pi/2. We can de[CapitalThorn]ne t(x) = tan(x), which maps every point in that tiny line segment to a distinct real number. What's more important is that every single element of R has an element of S mapped to it. So, t(x) is a bijection. Therefore, these two sets, S and T, have the same cardinality. Finally, let's look at mapping the points on a [CapitalThorn]nite line segment to the points on a square. For simplicity, I'm going to choose the line segment consisting of all points between 0.0 and 1.0, and a square with sides of the same length as the line segment. Every point on the line can be represented as a decimal number, represented as 0.abcdefghijklm... (There's a little issue here, with numbers ending in 999..., but those can be replaced with the equivalent non-repeating-nines representations.) Now, we take that number, and generate two numbers from it: 0.acegikm... and 0.bdfhjl... View those as x and y coordinates, and we've mapped every point on the segment to a point in the square. And, given any point (0.nprtvx..., 0.oqsuwy...) in the square, we can reverse this process to map it to a point, 0.nopqrstuvwxy..., on the segment. This gives us a bijection, so the set of points on the line segment and the set of points on the square have the same cardinality. Now, this is a really butt-ugly mapping. If you give me two points on the line segment, I can't easily tell you anything like how far apart their corresponding points in the square are. It doesn't (as far as I know) look anything like a Peano curve. But, it is a bijection, and that means that the two sets *do* have the same cardinality. > So what symbols > do mathies commonly use for each of these cardinalities? c, for continuum. > Is > the unit cube of a different cardinality, No, the cube has the same cardinality as the line segment and the square. > What about unit hypercubes of n=4 upwards? They also have the cardinality of the continuum, c. > It does leave me >with one problem; the nature of the bijection that maps >I^2 --> I Well, taking just the positive integers, you could do this: (1,1) 1 (1,2) 2 (2,1) 3 (1,3) 4 (2,2) 5 (3,1) 6 .. ... Or, if you want to get fancy, you could do something like: n(a,b) = (a+b-1)*(a+b-2)/2+b Again, just like with the points on the line segment and the square, these don't preserve order. But, they're still bijections. > . (Question: does a bijection f: always have an >inverse f^-1: Yes. > and if f^-1.f: (x) = f.f^-1: (x) = x is it >still called a bijection?) I'm not sure what you're asking here. A function that is a bijection always has an inverse. A function that has an inverse is not always a bijection. Does that help? > I can't grasp how this pair of digit >choosing functions can be regarded in the same light as, >say, f: x --> x^2 and g: x --> x^0.5, Why can't you grasp this? They are both functions. > and what differences, if any, exist. Well, you can't calculate the other functions on your calculator. But, that doesn't mean that they're not functions. -- Michael F. Stemper #include This message contains at least 95% recycled bytes. === Subject: Re: Peano's space-[CapitalThorn]lling curve Originator: grubb@lola >I deduce from your post that the unit line and the unit >square have different cardinality, which is what I thought I >knew before starting out on this quest :-) So what symbols >do mathies commonly use for each of these cardinalities? Is >the unit cube of a different cardinality, and if so what is >its symbol? What about unit hypercubes of n=4 upwards? So >many questions, so little time :-) We say that two sets A and B, have the same cardinality if there is a one-to-one function f:A->B which is also onto. We just need one such function to make the sets of the same cardinality. HOWEVER, there may be *other* functions g:A->B which fail to be one-to-one or onto. There may be functions which are one-to-one without being onto, and other functions which are onto without being being one-to-one. As long as there is *some* function which is both, the sets have the same cardinality. It turns out that [0,1] and [0,1]^2 have the same cardinality. This means that there is a function f:[0,1]->[0,1]^2 which is both one-to-one and onto. This may be a very nasty function (although there are fairly nice ones). It turns out that an f like this cannot be continuous. In other words, it is impossible to [CapitalThorn]nd a function f:[0,1]->[0,1]^2 which is one-to-one, onto *and* continuous. It *is* possible to [CapitalThorn]nd functions which have any two out of the three conditions though. Finding a function which is one-to-one and continuous, but not onto is easy. The Peano curve is a function which is onto and continuous, but not one-to-one. We have already talked about there being one that is one-to-one and onto without being continuous. >the nature of the bijection that maps >I^2 --> I . (Question: does a bijection f: always have an >inverse f^-1: and if f^-1.f: (x) = f.f^-1: (x) = x is it >still called a bijection?) Yes. A bijection f:A->B always has an inverse function f^-1:B->A. It turns out that f^-1 is a bijection also. In this case, f.f^-1 and f^-1.f are the identity functions on the appropriate sets. HOWEVER, whenever f:A->B is one-to-one, there is a function g:B->A such that g.f is the identity function on A. In this case, f.g will not be the identity function on B unless f is actually a bijection and g=f^-1. We cannot conclude from this alone that A and B have the same cardinality. ALSO, whenever f:A->B is onto, there is a function g:B->A with f.g the identity function on B. Remarkably, *but this fact requires the axiom of choice*. Once again, g.f will not be the identity function on A unless f is a bijection and g=f^-1. Happily, it turns out this if there are functions f_1:A->B and f_2:A->B with f_1 one-to-one and f_2 onto, then A and B have the same cardinality. So there is a function f:A->B that is *both* one-to-one and onto. However, f need not be the same as either f_1 or f_2. To prove this requires constructing f out of f_1 and f_2 (or showing in some other way that it exists). Yes, this fact requires the axiom of choice. I hope this clears some things up. There is a one-to-one and onto function from [0,1] to [0,1]^2 *and* there is a continous function from [0,1] to [0,1]^2 but these aren't the same functions! --Dan Grubb === Subject: Re: Peano's space-[CapitalThorn]lling curve >[...] >I hope this clears some things up. There is a one-to-one and >onto function from [0,1] to [0,1]^2 *and* there is a continous >function from [0,1] to [0,1]^2 but these aren't the same functions! That _might_ clear things up, but it doesn't seem likely - exactly this point has been explained to him several times. >--Dan Grubb ************************ David C. Ullrich === Subject: Re: Peano's space-[CapitalThorn]lling curve Originator: grubb@lola >I hope this clears some things up. There is a one-to-one and >onto function from [0,1] to [0,1]^2 *and* there is a continous >function from [0,1] to [0,1]^2 but these aren't the same functions! >That _might_ clear things up, but it doesn't seem likely - >exactly this point has been explained to him several times. *shrug* Is it any worse than I get when I teach calc? Occasionally a rewording makes a difference. --Dan Grubb === Subject: Re: Peano's space-[CapitalThorn]lling curve > [the usual] >You haven't told us what it is that you do. > Trollology. Is someone who responds to a troll a trollobite? Is the length of a trolling post measured in trollobytes? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Peano's space-[CapitalThorn]lling curve > I deduce from your post that the unit line and the unit > square have different cardinality, which is what I thought I > knew before starting out on this quest :-) So what symbols > do mathies commonly use for each of these cardinalities? Is > the unit cube of a different cardinality, and if so what is > its symbol? What about unit hypercubes of n=4 upwards? So > many questions, so little time :-) It has been explained to you that the unit line and the unit square have the same cardinality, which means that there is a bijection between them. Perhaps you are confused by the fact that there is a surjection from the line onto the square, but that this surjection is not an injection. Perhaps you think this means there are points left over in the line when we attempt to set up a bijection between the line and the square, and therefore the line must have a larger cardinality than the square. Er, no, it was the other way around, wasn't it? So what made you think that the square had a larger cardinality than the line, despite the fact that the points left over argument seems to indicate the opposite conclusion? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Melissa Rae Dumont - December 23rd 1984 D U M O N T 4 21 13 15 14 20 = 87 Melissa stopped and provided stats for her family today, she is a nubile sweety. 87+ Dad 2 4 56 93/273 +321 87+ Mom 20 5 59 140/225 822 87+ Sis 9 5 78 129/236 7751 189 Melissa 23 12 84 358/8 10171 Melissa 78 Rae 24 Dumont 87 Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 15 21 76 9 29 16 34 123 10 31 18 55 199 11 37 20 89 322 12 41 <-13th-> 21 <-13th-> 144 <-13th-> 521 <-13th-> 13 --- --- -- 238 154 <-Lamentations 91 The family was born on days of the month adding to 54 (13 plus the 13th prime), Bible Book 13 opens with 54 (13 plus the 13th prime) verses while Bible Book 54 (13 plus the 13th prime) contains 113 verses. The females were born in years adding to 143 (11x13). Melissa's name begins with the 13th letter of the alphabet, her [CapitalThorn]rst name adds to 78 (6x13). She has 13 letters in her common name. Her square valued letters add to 13, her unrepeated letters add to 113, her consonants add to 132. She was born in 84 (the [CapitalThorn]rst 13 primes minus the [CapitalThorn]rst 13 non-primes). Her names average 63 (Exodus 13). Melissa is 19.42 years old, pretty as there are 942 verses in Bible Book 13. The older sister was born in 78 (6x13). Mom and Melissa were born in 59 and 84, these are the 17th prime and 61st non-prime, together for 78 (6x13). The females were born in years adding to 221 (13x17). Mom was born in 59, corresponding to James with 108 verses (the primes in prime positions up to the 13th prime). The parents were born in 56 and 59, together these Bible Books contain 154 verses (the [CapitalThorn]rst 13 non-primes). Lucas 1 3 4 7 11 18 29 -- 73 <-the Lucas numbers up to 29 add to the 73 verses of Bible Book 29 J O E L <-Bible Book 29 10 15 5 12 = 42 <-29th non-prime C O P P E R <-29th element 3 15 16 16 5 18 = 73 <-Book 29 and is the Lucas numbers up to 29, there is a copper riding a horse on the 1973 Canadian 25 cent piece C E N T <-made out of 29th element 3 5 14 20 = 42 <-29th non-prime There are 29 chapters in Bible Book 13 and see that the older sister was born on day 129 (the primes up to 29 add to 129). The parents were born in years adding to 115 (the 73 verses of Book 29 plus the 29th non-prime). The kids were born in years adding to 162 (Deuteronomy 9 with 29 verses). Melissa and mom were born in years adding to 143, Ecclesiastes 7 contains 29 verses and brings Ecclesiastes up to 143 verses, pretty as this is the 109th non-prime while 109 is the 29th prime. Melissa was born 265 days after dad's birthday (First Samuel 29). The family name adds to 87 (29+29+29). Primes Non-Primes Numbers 2 1 1 3 4 2 5 6 3 7 8 4 11 9 5 13 10 6 17 12 7 19 14 8 23 15 9 29 16 10 31 18 11 37 20 12 41 21 13 43 22 14 47 24 15 53 25 16 59 26 17 61 27 18 67 28 19 71 30 20 73 32 21 79 33 22 83 <-23rd-> 34 <-23rd-> 23 --- --- --- 874 435 276 1-50 - Genesis 51-90 - Exodus 91-117 - Leviticus 118-153 - Numbers Mom was born on the 140th day of the year, it is 23 plus the 23rd prime (83) plus the 23rd non-prime (34), or simply 23+23p+23np, pretty that Bible chapter 140 would be Numbers 23. The parents were born on days of the year adding to 233. Mom and Melissa were together born with 233 days remaining in their years. Melissa was born on the 23rd, her unrepeated letters add to 113 (Leviticus 23). The family was born on days 2, 20, 9 and 23, together these Bible Books contain 4230 verses. Primes Non-Primes Numbers 2 1 1 3 4 2 5 6 3 7 8 4 11 9 5 13 10 6 17 12 7 19 14 8 23 15 9 29 16 10 31 18 11 37 20 12 41 21 13 43 22 14 47 24 15 53 25 16 59 <-17th-> 26 <-17th-> 17 --- --- --- 440 251 153 Mom was born in 59 (17th prime), the kids were born in months adding to 17, the family was born in months adding to 26 (17th non-prime) and in years adding to 277 (the 59th prime), prettier as the parents are 1277 days closer in age than the kids. Mom was born with 225 days remaining in the year, it is the 177th (59+59+59th non-prime). The kids were born in 78 and 84, these are the 57th and 61st non-primes (an average of 59). The sister was born with 236 (4x59) days remaining in the year. Melissa and her parents were born on days of the year adding to 591. The kids were born on days of the century adding to 59656. The consonants in Melissa's given names add to 81 (59th non-prime). Her vowels and consonants add to 57 and 132, these are the 41st and 100th non-primes (a difference of 59). Her primes and squares add to 74 and 13, these are the 53rd non-prime and the 6th prime, together for 59. Her odd and even valued letters add to 121 and 68, the former is 177% (59+59+59%) of the latter. Melissa's repeating letters add to 76, or 17 plus the 17th prime (59). Her Fibonacci valued letters add to 59 (the 17th prime). Mom's day, month and year of birth adds to 84 and she last gave birth in 84 (7 times the 7th non-prime). The family was born on days and in months and years adding to the 357 (7x17+7x17+7x17) verses of Daniel. Melissa was born 217 days after mom's birthday, it's the 170th non-prime while chapter 170 is Deuteronomy 17. Melissa's day, month and year of birth adds to 119 (7x17). Melissa's given names add to 102 (6x17 an is 17+17p+17np). Melissa's full name adds to 189 (the [CapitalThorn]rst 17 primes minus the [CapitalThorn]rst 17 non-primes). Her upsidedown and/or ßippababbles (her letters with upsidedown and/or ßippababble attributes... A, A, I, M, M, R, S and S) add together for 93 (Leviticus 3 with 17 verses). The parents are together 93.17 years old. Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 <-17 is the 7th prime 5 11 <- 11 while the primes up 6 13 to 7 add to 17 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 14 43 15 47 16 53 17 59 <- 59 <-the 7th prime in --- prime position 167 Esther Book 17 <-the 7th prime 17 is the 7th prime while the primes up to 7 add to 17. There are 7 primes in prime positions up to the 17th prime and they add to the 167 verse of Bible Book 17, Esther. Esther become Queen in Book 17 and Q is the 17th letter of the alphabet. Psalm 59 (the 17th prime) not only contains 17 verses, it is the 17th chapter in the Bible to contain the length of 17 verses. James is Book 59 (the 17th prime), it's 108 verses is the 17th prime short of the 167 verses of Book 17. Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 5 11 <- 11 6 13 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 --- 108 James Book 59 Leviticus begins with 17 verses and terminates at chapter 117 with 17+17 verses. There are 17 verses at chapters 1 and 3, and 59 (the 17 prime) verses at chapter 13, so the 17's and the 17th prime are at chapter numbers adding to 17 (1+3+13=17). The [CapitalThorn]rst 17 versed chapters in the Bible are at chapters 91 and 93, together for 184, or the 167 verses of Book 17 plus 17 more. Leviticus contains 859 verses, it ends in 59 (the 17th prime). The [CapitalThorn]rst 17's in the Bible surround chapter 92 (the 4x17th non-prime): Leviticus --------- 91 1 17 92 2 <-68th (4x17th) non-prime 93 3 17 94 4 95 5 96 6 97 7 98 8 99 9 100 10 101 11 102 12 103 13 59 <-17th prime 104 14 105 15 106 16 107 17 108 18 109 19 110 20 111 21 112 22 113 23 114 24 115 25 116 26 117 27 34 <-17+17 There are 89 chapters in The Gospels and 1189 chapters in the Bible. Melissa's full name adds to 189. The family was born on days of the century adding to 101891. The parents and the sister were together born 11.89 months into their years. The kids were born 571 days after mom's birthdays and 302 days after dad's birthdays, the former is 189% of the latter. Non-Primes 1 57 110 158 207 4 58 111 159 208 6 60 112 160 209 8 62 114 161 210 9 63 115 162 212 10 64 116 164 213 12 65 117 165 214 14 66 118 166 215 15 68 119 168 216 16 69 120 169 217 18 70 121 170 218 20 72 122 171 219 21 74 123 172 220 22 75 124 174 221 24 76 125 175 222 25 77 126 176 224 26 78 128 177 225 <-177th 27 80 129 178 226 28 81 130 180 228 30 82 132 182 230 32 84 133 183 231 33 85 134 184 232 34 86 135 185 234 35 87 136 186 235 36 88 138 187 236 38 90 140 188 237 39 91 141 189 238 40 92 142 190 240 42 93 143 192 242 44 94 144 194 243 45 95 145 195 244 46 96 146 196 245 48 98 147 198 246 49 99 148 200 247 50 100 150 201 248 51 102 152 202 249 52 104 153 203 250 54 105 154 204 252 55 106 155 205 253 56 108 156 206 254 Melissa is a 7 lettered name adding to 78. Her middle name adds to 24 (17 plus it's 7th prime position). The 7 different letters in her given names add to 77, while the letters missing from her given names exceed that by the 197 verses of Bible Book 28 (1 through 7). Her 7 vowels add to 57 (Exodus 7). Her odd valued letters add to 177% of her even valued letters (3 times the 17th prime). Her 14 (7+7) missing letters exceed her 12 (7th non-prime) different letters by 49 (7x7). Melissa's day, month and year of birth adds to 119 (7 times the 7th prime). She was born in the 12th month (7th non-prime) of year 84 (7 times the 7th non-prime). The older sister was born on the 28618th day of the century, 1 through 7 adds to 28 while Bible Book 7 contains 618 verses. When mom sees this, she will want to turn a seven foot tall tree into a decorated idol, and will tithe to a church that teaches her to adopt the [CapitalThorn]lthy practice. Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 <-9th-> 15 <-9th-> 21 <-9th-> 76 <-9th-> 9 --- -- -- --- -- 100 79 54 196 45 The parents were born in years adding to 115, The Samuels differ in length by 115 verses. The [CapitalThorn]rst of the kids was born on the 9th (First Samuel), the second on the 23rd (9th non-prime). The parents were born in months adding to 9, and mom was born in 59, prettier as 9 is the 5th non-prime. The sister was born on day 129. Dad was born in 56, it's the [CapitalThorn]rst 5 primes plus the [CapitalThorn]rst 5 non-primes. The kids were born in years adding to 162 (Deuteronomy 9) and on days of the year adding to 487 (Psalm 9). Primes Non-Primes 2 1 3 4 5 6 7 8 11 <-5th-> 9 -- -- 28 28 Mom was born on the 140th day of the year, and she is 822 days younger than me... there are 822 verses in Bible Book 14, pretty as 22 is the 14th non-prime while 14 is the 8th non-prime while 22 exceeds 8 by 14. The parents were born on days of the year adding to 233 (14th Fibonacci). Melissa and mom were born in years adding to 143, prettier as 43 is the 14th prime. Mom and Melissa were together born with 233 days remaining in their years (14th Fibonacci). The females were born in months adding to 22 (14th non-prime). The family was together born 22 days closer to the beginning of their years than to the end of their years (14th non-prime). The kids were together born 43.14 weeks after dad's birthdays. The parents are separated by 1143 days while Melissa and mom were born in years adding to 143. The parents and the kids were born in years adding to 115 and 162, the latter is 140% of the former, pretty as mom was born on the 140th day of the year. Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 15 21 76 9 29 16 34 123 10 31 18 55 199 11 37 20 89 322 12 41 21 144 521 13 43 <-14th-> 22 <-14th-> 233 <-14th-> 843 <-14th-> 14 --- --- --- ---- --- 281 176 609 2204 105 Dad was born in 56. The family was born on days 93, 140, 129 and 358, these are the 69th, 106th, 98th and the 287th non-primes, together for 560. The kids were born on days of the century adding to 59656. Non-Primes 1 57 110 158 207 255 303 351 4 58 111 159 208 256 304 352 6 60 112 160 209 258 305 354 8 62 114 161 210 259 306 355 9 63 115 162 212 260 308 356 10 64 116 164 213 261 309 357 12 65 117 165 214 262 310 358 <-287th 14 66 118 166 215 264 312 360 15 68 119 168 216 265 314 361 16 69 120 169 217 266 315 362 18 70 121 170 218 267 316 363 20 72 122 171 219 268 318 364 21 74 123 172 220 270 319 365 22 75 124 174 221 272 320 366 24 76 125 175 222 273 321 368 25 77 126 176 224 274 322 369 26 78 128 177 225 275 323 370 27 80 129 178 226 276 324 371 28 81 130 180 228 278 325 372 30 82 132 182 230 279 326 374 32 84 133 183 231 280 327 375 33 85 134 184 232 282 328 376 34 86 135 185 234 284 329 377 35 87 136 186 235 285 330 378 36 88 138 187 236 286 332 380 38 90 140 188 237 287 333 381 39 91 141 189 238 288 334 382 40 92 142 190 240 289 335 384 42 93 143 192 242 290 336 385 44 94 144 194 243 291 338 386 45 95 145 195 244 292 339 387 46 96 146 196 245 294 340 388 48 98 147 198 246 295 341 390 49 99 148 200 247 296 342 391 50 100 150 201 248 297 343 392 51 102 152 202 249 298 344 393 52 104 153 203 250 299 345 394 54 105 154 204 252 300 346 395 55 106 155 205 253 301 348 396 56 108 156 206 254 302 350 398 The kids are separated by 2420 days, it is 6.6 years, pretty that Bible chapter 242 would be First Samuel 6. The 12 (6+6) different letters in her full name add to 151 (the 6x6th prime). Dad was born in 56 and the kids today are together 16610 days old, pretty as it's 56 short of 16666. Today the family members are an average of 34.66 years old, and I meet Melissa when I am 17266 days old. Her middle name adds to the 24 chapters of Old Testament Books 6 and 10 (6th non-prime), her [CapitalThorn]rst name adds to 78 (6 times the 6th prime). She has 10 letters in her given names (6th non-prime). Melissa was born in the 6+6th month, and on the 23rd (6th prime plus the 6th non-prime), corresponding to Isaiah with 66 chapters. Isaiah 4, 12 and 20 (adds to 6x6) each contains 6 verses. 1-50 - Genesis 51-90 - Exodus 91-117 - Leviticus 118-153 - Numbers 154-187 - Deuteronomy 188-211 - Joshua 930-957 - Matthew 958-973 - Mark 974-997 - Luke 998-1018 - John 1019-1046 - Acts 1047-1062 - Romans 123 <-Numbers 6, it is three times the 13th prime (41+41+41), keeping in mind that 13 is the 6th prime 188 <-the opening chapter of Book 6 is 6x6x6 short of the 404 verses of Bible Book 66, it is the 6th prime squared (13x13) short of the 357 verses of Daniel (also in part about 666) 193 <-Book 6 chapter 6 is the 44th prime, while 44 is in turn 66.666...% of 66 211 <-the terminating chapter of Book 6 is approximately 66.6% of the 66th prime (317) 357 <-the opening chapter of Book 6 plus the 6th prime squared is the 357 verses of Daniel (in part about 666) 404 <-the 6th prime squared (13x13) plus the 6th prime squared (13x13) plus 66 adds to the 404 verses of Bible Book 66 1062 <-666 plus 6x66 is a combination of the 658 verses of Bible Book 6 plus the 404 verses of Bible Book 66, and is the terminating chapter of New Testament Book 6 1070 <-666 plus the 404 verses of Book 66 is the 1070 verses of Job (Book 6+6+6) 1213 <-Exodus terminates at chapter 90 (66th non- prime) with 1213 verses (the 198th or the 66+66+66th prime) 1292 <-the 658 verses of Book 6 plus twice the 66th prime (317) is the 1292 verses of Isaiah (the Book contains 66 chapters) The parents were born in 56 and 59, these are the 40th non-prime and the 17th prime (together for 57). The sister was born in 78 while Melissa adds to 78 (57th non-prime). The family was born on days and in months and years adding to the 357 (7x17+7x17+7x17) verses of Daniel. The kids were together born 571 days after mom's birthdays, or 81.57 weeks. The kids were born on days of the year adding to 487 (69.57 weeks). 389 <-77th prime 104 <-77th non-prime 77 <-77 --- 570 The Four 57's Genesis 41 -> 41 Leviticus 14 -> 104 Judges 9 -> 220 <-I dreamt of 220 roofs blown John 11 -> 1008 off homes in the Dakotas ---- 1373 <-220th prime Chapter 57 is Exodus 7 with 25 verses Book 57 is Philemon with 25 verses -- -- 41st non-prime 16th non-prime <-together for 57-> Major Books of End-Times Prophecy (Daniel and Revelation are in part about 666 while Isaiah contains 66 chapters): Daniel - 357 verses Revelation - 404 verses <-57 plus the 57th prime plus the 57th non-prime Isaiah - 1292 verses <-an average of 19.575757... verses per chapter Genesis 41 repeats the spoken seven 28 times, pretty as 1 through 7 adds to 28. We are warned in Genesis 41 to accumulate 7 years worth of food supplies in anticipation of several years of successive crop failures. Genesis 41 contains 57 verses, together for 98 (7x7+7x7). Pretty that Genesis 41 would contain 57 verses, for 57 is the 41st non-prime. Genesis 41 is a numerological marker in the Bible, I tried to tell people this in 1988 but they rolled their eyes back and smirked, people did not like my criticisms of their churches and used my interest in math as one of their reasons to arrest and torture me in psychiatric settings. I was arrested under the mental health act while the farmers were [CapitalThorn]ned for failing to get the grain to port fast enough. The Canadian government provided land for the railways, but then the railroad companies ripped out the tracks and sold the land, and used the money to purchase lucrative money making hotels. I was tortured for speaking while the people guilty of treason took their frequent trips to Europe, Asia and the Caribbean, while the Catholics sponsored Latin American and Asian Catholics to come to Canada and earn hard Western currency so they could tithe to the Catholic church. And they repeatedly raped Canadian taxpayers to the tune of many hundreds of millions of dollars to pay for costs involved in the visits of the pope (it is theft, just like the theft of the Egyptian obelisk situated at The Vatican). People used my interest in numbers as one of the reasons to support having me arrested and tortured, I show you people gems and year after year you spend billions of dollars on turning trees into idols while calling me insane (Second Chronicles 36:16). 187 Dar 17 2 57 48/317 00 Daryl 60 Shawn 65 Kabatoff 62 187 Marcia 6 8 80 219/147 8571 Marcia 45 Veronica 87 Acevedo 55 201 Melodee 7 4 74 97/268 6258 Melodee 59 Joy 50 Webster 92 240 Bethany 28 10 76 302/64 7193 Bethany 75 Ruth 67 Guenther 98 147 Natash 18 1 81 18/347 8736 Natash 63 Lyn 51 Isaac 33 214 Ali 12 1 83 12/353 9460 Alison 70 Janet 50 Gillespie 94 194 Lesley 27 5 83 147/218 9595 Lesley 78 Mae 19 Roberts 97 223 Camille 8 6 84 160/206 9973 Camille 55 Clarisse 86 Massey 82 221 Jessica 17 6 84 169/197 9982 Jessica 66 Vivian 77 Valois 78 288 Melinda 23 3 83 82/283 9530 Melinda 58 Janelle 59 Elaine 46 Joyce 58 Jarocki 67 If you people think you have the right to use my abusive parents as tools to have me arrested and tortured, then I think that I should have the right to ask women to marry me, or to marry Marcia and me. I have Scripture to support taking seven brides (Isaiah 4:1) and I have Scripture to support sleeping with Melinda Jarocki outside of wedlock (First Kings 1:1-5), while you people have a vast multitude of Scriptures condemning your decorated trees, phallic-capped churches and your violence against me for daring to point out your pagan traditions. Anyway, now that Melissa and her family sees evidence that their names are gifts from God, watch them start attending and tithing to churches that teach them to turn trees into decorated idols (maybe see Deuteronomy 12:2; 1 Kings 14:23; 2 Kings 16:4, 17:10; 2 Chronicles 28:4; Isaiah 57:5; Jeremiah 2:20, 3:6, 3:13, 10:3-4, 17:2 and Ezekiel 6:13). It just doesn't matter to you people that the Bible repeatedly condemns turning trees into idols (it is a violation of God's First and Second Commandments). And it just doesn't matter to you people that I was tortured for years after daring to identify the obelisks at The Vatican, The Whitehouse and on church roofs as being representations of penises (and in opposition to God's Second and Sixth Commandments). You people are compassionless turds, your only real compassion is for your traditions. You don't have an ounce of compassion for me or for anybody else being tortured by the psychiatric profession, all that truly matters to you is to go home for Christmas and stoop to the base of your mom's decorated tree (symbol of fertility that stays green all year) and collect a few presents. And then you feel guilty of your sin so you attend a church once a year, you go in December and give money to a preacher who tells you to stay away from idols, and he provides you with this less lesson while he standink next to a decorated tree and under his church steeple (an obelisk, an Egyptian representation of a penis). They worshipped and made representations of penises for these pagans believed it to have a creative or god-like force through it's reproductive ability, but Shawn knows that your penis is closer to your anus than to God. I don't have an ounce of respect for you and so don't require your permission to post your stats on the usenet and use you as an example to others, and look, here you are!!! There is a woman who hangs around Crocus Coop, she told me that the people are scared, they don't want to make waves against the psychiatric profession or anything else for they are afraid of being sent back to Hantelman at the U of S, or to City Hospital. I once heard a woman talk on the phone over at the mental health of[CapitalThorn]ce on Ave. P North, she cried and pleaded on the phone for a lengthy time, she was saying over and over that there was nothing wrong with her, how damaging the drugs being forced upon her were, the drugs were making her feel nauseous, she was horri[CapitalThorn]ed about the thought of continued forced treatment, she did not want to be on the drugs and stated over and over that there was nothing wrong with her and she did not need the drugs administered to her. She had no power, nobody cared about her being forced drugs against her will, nobody cared about her and nobody cared about me. And nobody cared about Jason Lee, he was in an abusive family relationship and then his parents forced him into the psychiatric mill, and he just eventually fell over and died. And Connie Glushyk's mom was ting on Connie and forcing Connie into the wards and then threatening to put Connie into a group home, well Connie just got a pass from the City Hall ward, walked across the street to her apartment in the high-rise, and jumped. The drugs make it so hard to verbalize arguments against what the psychiatrists are doing to you. It is a real horror show to think that you might get sent back to Hantelman, the threat of going back to Hantelman or to City Hospital looms for me and these people, once charged under the mental health act, they can take you and lock you up for 6 weeks until forced by law to release you. But then the NDP came to power provincially and immediately changed the laws and now can force medications upon people after the 6 weeks of forced treatment at Hantelman or City Hospital, and now thanx to the NDP, people who were horri[CapitalThorn]ed at the thoughts of being returned for a 6 week treatment, now get the drugs forced upon them every week of the year. And during this time, the NDP campaign colors included saffron, it's a colour holy to the Hindus, interesting as the NDP are now forcing people to be medicated 24/7/366 by Hindu psychiatrists. They would detain me for 6 weeks (during which time I would get to appeal twice to the middle-class Protestants and Catholics), and on the [CapitalThorn]nal day of the 6 weeks of treatment they would administer one last depot drug injection and release me, and the drugs remain in my body doing horrible things for months after that injection. It would take me months to get over the crap that they force upon me every 6 weeks, and then they would do it to me again and again and again and again. And now thanx to the NDP, people are having drugs forced upon them all the time. I was [CapitalThorn]rst arrested and tortured in 1988 when the NDP were not in power, but then the NDP came to power and took away more rights from people, the NDP have Hindus employed to force medications upon people against their will, the drugs remain in a person's body for months at a time, the NDP force people to be drugged every day of their lives. And now if some other party would come to power provincially in Saskatchewan and remove the NDP, they would leave the NDP law intact and continue to allow people to be drugged continuously. The NDP tortured me and tortures many in Saskatchewan, I don't want to live in Canada now and I certainly don't want to live in Canada if the NDP come to power nationally. I don't want to be here in Canada, and many other people don't want to be here either. There are people being forced medications upon them against their will and they would rather have a few dollars and be backpacking in Europe, they don't want to be living in poverty in Saskatchewan doped up and feeling nauseous, and then you people turn around and spend a few billion dollars annually turning trees into decorated idols. You don't care what the NDP and other governments are forcing these drugs upon people, you people care only about your tradions. I don't want to be in Canada and close to these places that I have been tortured. It was a horror show what they did to me, and year after year I see you people spending billions of dollars on turning trees into idols while calling me insane, and now some totally insensitive asshole starts speaking up in sk.forsale saying that he or she wouldn't want me babysitting his or her children, and that I make his or her skin crawl. You are God's children, and I have been babysitting all of you since 1988 when I started speaking up against the false traditions in your churches, and you make my skin crawl as well. Well Melissa thought it was all very nice and well and she paid me a dollar for my work (showing her evidence that her name was a gift from God), she then likely walked down the block and paid four dollars for a drink. Daryl Shawn Kabatoff Box 7134 Saskatoon Saskatchewan Canada S7K 4J1 Isaiah 45:4, Ephesians 3:15 - God gives you your name!!! === Subject: Fibonacci spirals in nature by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52IXOW05046; In website www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/[CapitalThorn]bnat.html# spiral the number of spirals in clockwise and anticlockwise directions in a sunßower/pinecones is Fib(n+1) and Fib(n), integer n could be around 10. There is no polar symmetry. Is it archimadean, logarathmic or what sort of spiral? What are their two polar equations involving golden ratio? TIA for interesting information... (There was a similar post by someone recently, but was not traceable). === Subject: Whats the Edgeworth expansion of a random vector? I have a statistical question regarding on the Edgeworth expansion of a random vector. Assume x is a d-dimensional random vector, and whats the expression of its Edgeworth expansion up to order 4 about its best Gaussian approximation? I only get the formulation for a scalar random variable. So anyone can help me out on this problem? Fred === Subject: How to study Circulay Strings Hello to all: I am a programmer looking for advice on how to study a particular to the notion of Equivalence Relation on a Set but I am not very knowledgable in this area and am looking for insights from others to point me in the right direction. The Problem ----------- A Circular String is a string with a circular boundary condition. The string wraps around so there's no start and end. Let S be a string of length N and Dn be the digits of that string. Then S = D1D2D3...Dn (n=N). For a binary string, D is in {0,1}. If we set N=2, then there are 4 possible strings (00, 01, 10, 11). Now the boundary condition induces a kind of Rotational Symmetry on the string, since there's no start and end, so 10=01 (just see the string as a discrete ring of length 2 and rotate). Set Theory ---------- I think that we can use Set Theory to abstract this problem as follow: Let T be the set of all states. The cartesian product of T with itself (U=T1xT2...xTn with n=length of the string), represents the set of all possible strings . Now the Circular Boundary Condition we imposed on our string can be seen (I think) as an equivalence relation on U. This means that the Circular Boundary Condition partitions the set U into disjoints union of subsets and those subsets are called Equivanence Classes. Questions --------- I need to [CapitalThorn]nd the numbers of such Equivalence Classes for different program to do this by looking at all strings but the number of permutations grows pretty fast and as a result, I'm restricted to very simple strings -> small N and |T|. Is it possible to analytically [CapitalThorn]nd the number of Equivalence Classes for a speci[CapitalThorn]c Set T and String Length N? Which Mathematical Aparatus/Theory should I learn to gain insights into the consequesces of having this Equivalence Relation and the structure of those subsets? Any Help would be very much appreciated Alain Lavoie === Subject: Re: How to study Circulay Strings >I am a programmer looking for advice on how to study a particular >to the notion of Equivalence Relation on a Set but I am not very >knowledgable in this area and am looking for insights from others to >point me in the right direction. >The Problem >----------- >A Circular String is a string with a circular boundary condition. >The string wraps around so there's no start and end. >Let S be a string of length N and Dn be the digits of that string. >Then S = D1D2D3...Dn (n=N). >For a binary string, D is in {0,1}. If we set N=2, then there are 4 >possible strings (00, 01, 10, 11). Now the boundary condition induces >a kind of Rotational Symmetry on the string, since there's no start >and end, so 10=01 (just see the string as a discrete ring of length 2 >and rotate). >Set Theory >---------- >I think that we can use Set Theory to abstract this problem as follow: > Let T be the set of all states. The cartesian product of T with >itself (U=T1xT2...xTn with n=length of the string), represents the set >of all possible strings . Now the Circular Boundary Condition we >imposed on our string can be seen (I think) as an equivalence relation >on U. This means that the Circular Boundary Condition partitions the >set U into disjoints union of subsets and those subsets are called >Equivanence Classes. >Questions >--------- >I need to [CapitalThorn]nd the numbers of such Equivalence Classes for different >program to do this by looking at all strings but the number of >permutations grows pretty fast and as a result, I'm restricted to very >simple strings -> small N and |T|. Is it possible to analytically >[CapitalThorn]nd the number of Equivalence Classes for a speci[CapitalThorn]c Set T and String >Length N? Most equivalence classes are going to be of size N, because there are N different ways to rotate the string. However, if a string repeats in period M, where M|N and MWhich Mathematical Aparatus/Theory should I learn to gain insights >into the consequesces of having this Equivalence Relation and the >structure of those subsets? Combinitorics. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Equating differential equations I am looking for some help or hint on this problem. I have two differential equations and want them to have the same exact solution. For example: y' = x + a * y [1] where a is a constant and x is a known input variable. Now say I have the following equation: y' = x + z + b * y [2] where b is a constant (different than a), x is the same known input variable and z is a variable that I can manipulate. I want to solve for z, such as both [1] and [2] have the same y solution. Does anybody have any idea of how to solve this ? Equation [1] can be solved and their values are readily available (both y and y'). I thought that solving [2] for z and replacing y and y' from [1] was going to give me the solution, but apparently it is not working. I would really appreciate any help on this. Mariano Filippa m[CapitalThorn]lippa uol.com.ar === Subject: Re: Equating differential equations >I am looking for some help or hint on this problem. I have two differential >equations and want them to have the same exact solution. For example: >y' = x + a * y [1] >where a is a constant and x is a known input variable. Now say I have the >following equation: >y' = x + z + b * y [2] >where b is a constant (different than a), x is the same known input variable >and z is a variable that I can manipulate. By x is a known input variable do you mean the independent variable? In other words, is y' the derivative with respect to x? Or is y a function of some implicit variable t and do x and z depend on t? Whatever your symbols mean if they have the same solution then you must have x + ay == x + z + by so (a-b)y = z if that makes any sense. I think you need to state your problem more carefully. --Lynn === Subject: Re: Equating differential equations Sorry I forgot to mention that. It is a derivative of time. I am trying to translate a mechanical problem into equations. The equations are: dy/dt = x(t) + a * y(t) ; dy/dt = x(t) + z(t) + b * y(t); Now let me clarify this again. This corresponds to a mechanical system. The variable y is speed, and depends on an external force x and the system's characteristics a. I want to have the same speed response y, on a system that has the same external force x and different characteristics b. In order to have the same response, I introduced another force z. I want to solve for this force z to match the system response. I hope this is better. It comes from bond graph modeling. I know (from a mechanical point of view) that this system has a solution, but I also know that for some other con[CapitalThorn]gurations, there is no physical solution. I wanted to get a proof of this solving this problem. Matlab Simulink diagrams are available if anybody wants them. Mariano >I am looking for some help or hint on this problem. I have two differential >equations and want them to have the same exact solution. For example: >y' = x + a * y [1] >where a is a constant and x is a known input variable. Now say I have the >following equation: >y' = x + z + b * y [2] >where b is a constant (different than a), x is the same known input variable >and z is a variable that I can manipulate. > By x is a known input variable do you mean the independent variable? > In other words, is y' the derivative with respect to x? Or is y a > function of some implicit variable t and do x and z depend on t? > Whatever your symbols mean if they have the same solution then you > must have x + ay == x + z + by so (a-b)y = z if that makes any sense. > I think you need to state your problem more carefully. > --Lynn === Subject: Re: Equating differential equations >Sorry I forgot to mention that. It is a derivative of time. I am trying to >translate a mechanical problem into equations. >The equations are: >dy/dt = x(t) + a * y(t) ; >dy/dt = x(t) + z(t) + b * y(t); >Now let me clarify this again. This corresponds to a mechanical system. The >variable y is speed, and depends on an external force x and the system's >characteristics a. >I want to have the same speed response y, on a system that has the same >external force x and different characteristics b. In order to have the >same response, I introduced another force z. I want to solve for this >force z to match the system response. >I hope this is better. It comes from bond graph modeling. I know (from a >mechanical point of view) that this system has a solution, but I also know >that for some other con[CapitalThorn]gurations, there is no physical solution. I wanted >to get a proof of this solving this problem. >Matlab Simulink diagrams are available if anybody wants them. >Mariano Then, as I said before, if these two equations are to have the same solution, their right sides must be equal: x(t) + a * y(t) == x(t) + z(t) + b * y(t) and z(t) = (a - b) * y(t) So solve your [CapitalThorn]rst equation: y(t) = y(0)exp(at) + exp(at)*int[0..t]exp(-au) x(u)du and use it in the equation above for z(t). --Lynn === Subject: Re: Jai and the fake Einstein quote by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52J3ec08242; I have all information on the bogus sentence by Einstein: its origin as well as the actual printed advice of Einstein on astrology, which is of course negative. Provide me with an e-mail address. Denis Hamel === Subject: Asymptotic behavior of an integral I am interested in the asymptotic behavior, as the real parameter u -> 0, of the function f(u) de[CapitalThorn]ned by the integral f(u) = int {2, in[CapitalThorn]nity} [ e^(i x u) -1 ] / (x [Log(x)]^2) dx. Here Log(.) is the natural log. Can anyone suggest the leading term in this limit (and how to get it); i.e., f(u) ~ g(u), where g(u) = ? alan === Subject: Huffman Code by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52JOCH10328; Can anyone help me with this question?? A signal has 8 states A,B,C...H, which have the probabilities given below. Construct a huffman code for this source, with a diagram showing the code reductions, and evaluate the ef[CapitalThorn]ciency of your code. State Probability A 0.27 B 0.26 C 0.22 D 0.15 E 0.05 F 0.02 G 0.02 H 0.01 === Subject: Re: Huffman Code > Can anyone help me with this question?? > A signal has 8 states A,B,C...H, which have the probabilities given below. Construct a huffman code for this source, with a diagram showing the code reductions, and evaluate the ef[CapitalThorn]ciency of your code. > State Probability > A 0.27 > B 0.26 > C 0.22 > D 0.15 > E 0.05 > F 0.02 > G 0.02 > H 0.01 Did you think about looking for usenet newsgroups with compression in their name, perhaps in the comp hierarchy? However, we don't tend to do people's homework for them over there either. Consider your course ßunked unless you begin to pull your socks up. Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: Huffman Code >Can anyone help me with this question?? >A signal has 8 states A,B,C...H, which have the probabilities given below. Construct a huffman code for this source, with a diagram showing the code reductions, and evaluate the ef[CapitalThorn]ciency of your code. >State Probability >A 0.27 >B 0.26 >C 0.22 >D 0.15 >E 0.05 >F 0.02 >G 0.02 >H 0.01 There are examples worked out in the book. Why would _this_ particular example worked out help you more than reading the examples in the book? (What's that, you didn't want to know how to do it, you were just hoping for something you could hand in? Sorry, I didn't consider that possibility.) ************************ David C. Ullrich === Subject: Re: Huffman Code >Can anyone help me with this question?? >A signal has 8 states A,B,C...H, which have the probabilities given below. Construct a huffman code for this source, with a diagram showing the code reductions, and evaluate the ef[CapitalThorn]ciency of your code. >State Probability >A 0.27 >B 0.26 >C 0.22 >D 0.15 >E 0.05 >F 0.02 >G 0.02 >H 0.01 > There are examples worked out in the book. Why > would _this_ particular example worked out help > you more than reading the examples in the book? > (What's that, you didn't want to know how to do it, > you were just hoping for something you could hand in? > Sorry, I didn't consider that possibility.) Let me help: A 01 B 011 C 0111 D 01111 E 011111 F 0111111 G 01111111 H 011111111 You'll have to do the hard part by yourself. === Subject: Re: Huffman Code >Can anyone help me with this question?? > >A signal has 8 states A,B,C...H, which have the probabilities given > below. Construct a huffman code for this source, with a diagram showing the > code reductions, and evaluate the ef[CapitalThorn]ciency of your code. > >State Probability >A 0.27 >B 0.26 >C 0.22 >D 0.15 >E 0.05 >F 0.02 >G 0.02 >H 0.01 > > > There are examples worked out in the book. Why > would _this_ particular example worked out help > you more than reading the examples in the book? > (What's that, you didn't want to know how to do it, > you were just hoping for something you could hand in? > Sorry, I didn't consider that possibility.) > Let me help: > A 01 > B 011 > C 0111 > D 01111 > E 011111 > F 0111111 > G 01111111 > H 011111111 > You'll have to do the hard part by yourself. I thought it was A 00001111 B 11001100 C 01010101 D 10101010 E 01010111 F 10011000 G 01100010 H 01010011 I 10101110 J 01011010 K 10000111 Where I, J and K arn't used? Huffman knows for sure. === Subject: Melissa Rae Dumont - December 23rd 1984 D U M O N T 4 21 13 15 14 20 = 87 Melissa stopped and provided stats for her family today, she is a nubile sweety. 87+ Dad 2 4 56 93/273 +321 87+ Mom 20 5 59 140/225 822 87+ Sis 9 5 78 129/236 7751 189 Melissa 23 12 84 358/8 10171 Melissa 78 Rae 24 Dumont 87 Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 15 21 76 9 29 16 34 123 10 31 18 55 199 11 37 20 89 322 12 41 <-13th-> 21 <-13th-> 144 <-13th-> 521 <-13th-> 13 --- --- -- 238 154 <-Lamentations 91 The family was born on days of the month adding to 54 (13 plus the 13th prime), Bible Book 13 opens with 54 (13 plus the 13th prime) verses while Bible Book 54 (13 plus the 13th prime) contains 113 verses. The females were born in years adding to 143 (11x13). Melissa's name begins with the 13th letter of the alphabet, her [CapitalThorn]rst name adds to 78 (6x13). She has 13 letters in her common name. Her square valued letters add to 13, her unrepeated letters add to 113, her consonants add to 132. She was born in 84 (the [CapitalThorn]rst 13 primes minus the [CapitalThorn]rst 13 non-primes). Her names average 63 (Exodus 13). Melissa is 19.42 years old, pretty as there are 942 verses in Bible Book 13. The older sister was born in 78 (6x13). Mom and Melissa were born in 59 and 84, these are the 17th prime and 61st non-prime, together for 78 (6x13). The females were born in years adding to 221 (13x17). Mom was born in 59, corresponding to James with 108 verses (the primes in prime positions up to the 13th prime). The parents were born in 56 and 59, together these Bible Books contain 154 verses (the [CapitalThorn]rst 13 non-primes). Lucas 1 3 4 7 11 18 29 -- 73 <-the Lucas numbers up to 29 add to the 73 verses of Bible Book 29 J O E L <-Bible Book 29 10 15 5 12 = 42 <-29th non-prime C O P P E R <-29th element 3 15 16 16 5 18 = 73 <-Book 29 and is the Lucas numbers up to 29, there is a copper riding a horse on the 1973 Canadian 25 cent piece C E N T <-made out of 29th element 3 5 14 20 = 42 <-29th non-prime There are 29 chapters in Bible Book 13 and see that the older sister was born on day 129 (the primes up to 29 add to 129). The parents were born in years adding to 115 (the 73 verses of Book 29 plus the 29th non-prime). The kids were born in years adding to 162 (Deuteronomy 9 with 29 verses). Melissa and mom were born in years adding to 143, Ecclesiastes 7 contains 29 verses and brings Ecclesiastes up to 143 verses, pretty as this is the 109th non-prime while 109 is the 29th prime. Melissa was born 265 days after dad's birthday (First Samuel 29). The family name adds to 87 (29+29+29). Primes Non-Primes Numbers 2 1 1 3 4 2 5 6 3 7 8 4 11 9 5 13 10 6 17 12 7 19 14 8 23 15 9 29 16 10 31 18 11 37 20 12 41 21 13 43 22 14 47 24 15 53 25 16 59 26 17 61 27 18 67 28 19 71 30 20 73 32 21 79 33 22 83 <-23rd-> 34 <-23rd-> 23 --- --- --- 874 435 276 1-50 - Genesis 51-90 - Exodus 91-117 - Leviticus 118-153 - Numbers Mom was born on the 140th day of the year, it is 23 plus the 23rd prime (83) plus the 23rd non-prime (34), or simply 23+23p+23np, pretty that Bible chapter 140 would be Numbers 23. The parents were born on days of the year adding to 233. Mom and Melissa were together born with 233 days remaining in their years. Melissa was born on the 23rd, her unrepeated letters add to 113 (Leviticus 23). The family was born on days 2, 20, 9 and 23, together these Bible Books contain 4230 verses. Primes Non-Primes Numbers 2 1 1 3 4 2 5 6 3 7 8 4 11 9 5 13 10 6 17 12 7 19 14 8 23 15 9 29 16 10 31 18 11 37 20 12 41 21 13 43 22 14 47 24 15 53 25 16 59 <-17th-> 26 <-17th-> 17 --- --- --- 440 251 153 Mom was born in 59 (17th prime), the kids were born in months adding to 17, the family was born in months adding to 26 (17th non-prime) and in years adding to 277 (the 59th prime), prettier as the parents are 1277 days closer in age than the kids. Mom was born with 225 days remaining in the year, it is the 177th (59+59+59th non-prime). The kids were born in 78 and 84, these are the 57th and 61st non-primes (an average of 59). The sister was born with 236 (4x59) days remaining in the year. Melissa and her parents were born on days of the year adding to 591. The kids were born on days of the century adding to 59656. The consonants in Melissa's given names add to 81 (59th non-prime). Her vowels and consonants add to 57 and 132, these are the 41st and 100th non-primes (a difference of 59). Her primes and squares add to 74 and 13, these are the 53rd non-prime and the 6th prime, together for 59. Her odd and even valued letters add to 121 and 68, the former is 177% (59+59+59%) of the latter. Melissa's repeating letters add to 76, or 17 plus the 17th prime (59). Her Fibonacci valued letters add to 59 (the 17th prime). Mom's day, month and year of birth adds to 84 and she last gave birth in 84 (7 times the 7th non-prime). The family was born on days and in months and years adding to the 357 (7x17+7x17+7x17) verses of Daniel. Melissa was born 217 days after mom's birthday, it's the 170th non-prime while chapter 170 is Deuteronomy 17. Melissa's day, month and year of birth adds to 119 (7x17). Melissa's given names add to 102 (6x17 an is 17+17p+17np). Melissa's full name adds to 189 (the [CapitalThorn]rst 17 primes minus the [CapitalThorn]rst 17 non-primes). Her upsidedown and/or ßippababbles (her letters with upsidedown and/or ßippababble attributes... A, A, I, M, M, R, S and S) add together for 93 (Leviticus 3 with 17 verses). The parents are together 93.17 years old. Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 <-17 is the 7th prime 5 11 <- 11 while the primes up 6 13 to 7 add to 17 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 14 43 15 47 16 53 17 59 <- 59 <-the 7th prime in --- prime position 167 Esther Book 17 <-the 7th prime 17 is the 7th prime while the primes up to 7 add to 17. There are 7 primes in prime positions up to the 17th prime and they add to the 167 verse of Bible Book 17, Esther. Esther become Queen in Book 17 and Q is the 17th letter of the alphabet. Psalm 59 (the 17th prime) not only contains 17 verses, it is the 17th chapter in the Bible to contain the length of 17 verses. James is Book 59 (the 17th prime), it's 108 verses is the 17th prime short of the 167 verses of Book 17. Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 5 11 <- 11 6 13 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 --- 108 James Book 59 Leviticus begins with 17 verses and terminates at chapter 117 with 17+17 verses. There are 17 verses at chapters 1 and 3, and 59 (the 17 prime) verses at chapter 13, so the 17's and the 17th prime are at chapter numbers adding to 17 (1+3+13=17). The [CapitalThorn]rst 17 versed chapters in the Bible are at chapters 91 and 93, together for 184, or the 167 verses of Book 17 plus 17 more. Leviticus contains 859 verses, it ends in 59 (the 17th prime). The [CapitalThorn]rst 17's in the Bible surround chapter 92 (the 4x17th non-prime): Leviticus --------- 91 1 17 92 2 <-68th (4x17th) non-prime 93 3 17 94 4 95 5 96 6 97 7 98 8 99 9 100 10 101 11 102 12 103 13 59 <-17th prime 104 14 105 15 106 16 107 17 108 18 109 19 110 20 111 21 112 22 113 23 114 24 115 25 116 26 117 27 34 <-17+17 There are 89 chapters in The Gospels and 1189 chapters in the Bible. Melissa's full name adds to 189. The family was born on days of the century adding to 101891. The parents and the sister were together born 11.89 months into their years. The kids were born 571 days after mom's birthdays and 302 days after dad's birthdays, the former is 189% of the latter. Non-Primes 1 57 110 158 207 4 58 111 159 208 6 60 112 160 209 8 62 114 161 210 9 63 115 162 212 10 64 116 164 213 12 65 117 165 214 14 66 118 166 215 15 68 119 168 216 16 69 120 169 217 18 70 121 170 218 20 72 122 171 219 21 74 123 172 220 22 75 124 174 221 24 76 125 175 222 25 77 126 176 224 26 78 128 177 225 <-177th 27 80 129 178 226 28 81 130 180 228 30 82 132 182 230 32 84 133 183 231 33 85 134 184 232 34 86 135 185 234 35 87 136 186 235 36 88 138 187 236 38 90 140 188 237 39 91 141 189 238 40 92 142 190 240 42 93 143 192 242 44 94 144 194 243 45 95 145 195 244 46 96 146 196 245 48 98 147 198 246 49 99 148 200 247 50 100 150 201 248 51 102 152 202 249 52 104 153 203 250 54 105 154 204 252 55 106 155 205 253 56 108 156 206 254 Melissa is a 7 lettered name adding to 78. Her middle name adds to 24 (17 plus it's 7th prime position). The 7 different letters in her given names add to 77, while the letters missing from her given names exceed that by the 197 verses of Bible Book 28 (1 through 7). Her 7 vowels add to 57 (Exodus 7). Her odd valued letters add to 177% of her even valued letters (3 times the 17th prime). Her 14 (7+7) missing letters exceed her 12 (7th non-prime) different letters by 49 (7x7). Melissa's day, month and year of birth adds to 119 (7 times the 7th prime). She was born in the 12th month (7th non-prime) of year 84 (7 times the 7th non-prime). The older sister was born on the 28618th day of the century, 1 through 7 adds to 28 while Bible Book 7 contains 618 verses. When mom sees this, she will want to turn a seven foot tall tree into a decorated idol, and will tithe to a church that teaches her to adopt the [CapitalThorn]lthy practice. Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 <-9th-> 15 <-9th-> 21 <-9th-> 76 <-9th-> 9 --- -- -- --- -- 100 79 54 196 45 The parents were born in years adding to 115, The Samuels differ in length by 115 verses. The [CapitalThorn]rst of the kids was born on the 9th (First Samuel), the second on the 23rd (9th non-prime). The parents were born in months adding to 9, and mom was born in 59, prettier as 9 is the 5th non-prime. The sister was born on day 129. Dad was born in 56, it's the [CapitalThorn]rst 5 primes plus the [CapitalThorn]rst 5 non-primes. The kids were born in years adding to 162 (Deuteronomy 9) and on days of the year adding to 487 (Psalm 9). Primes Non-Primes 2 1 3 4 5 6 7 8 11 <-5th-> 9 -- -- 28 28 Mom was born on the 140th day of the year, and she is 822 days younger than me... there are 822 verses in Bible Book 14, pretty as 22 is the 14th non-prime while 14 is the 8th non-prime while 22 exceeds 8 by 14. The parents were born on days of the year adding to 233 (14th Fibonacci). Melissa and mom were born in years adding to 143, prettier as 43 is the 14th prime. Mom and Melissa were together born with 233 days remaining in their years (14th Fibonacci). The females were born in months adding to 22 (14th non-prime). The family was together born 22 days closer to the beginning of their years than to the end of their years (14th non-prime). The kids were together born 43.14 weeks after dad's birthdays. The parents are separated by 1143 days while Melissa and mom were born in years adding to 143. The parents and the kids were born in years adding to 115 and 162, the latter is 140% of the former, pretty as mom was born on the 140th day of the year. Primes Non-Primes Fibonacci Lucas Numbers 2 1 0 1 1 3 4 1 3 2 5 6 1 4 3 7 8 2 7 4 11 9 3 11 5 13 10 5 18 6 17 12 8 29 7 19 14 13 47 8 23 15 21 76 9 29 16 34 123 10 31 18 55 199 11 37 20 89 322 12 41 21 144 521 13 43 <-14th-> 22 <-14th-> 233 <-14th-> 843 <-14th-> 14 --- --- --- ---- --- 281 176 609 2204 105 Dad was born in 56. The family was born on days 93, 140, 129 and 358, these are the 69th, 106th, 98th and the 287th non-primes, together for 560. The kids were born on days of the century adding to 59656. Non-Primes 1 57 110 158 207 255 303 351 4 58 111 159 208 256 304 352 6 60 112 160 209 258 305 354 8 62 114 161 210 259 306 355 9 63 115 162 212 260 308 356 10 64 116 164 213 261 309 357 12 65 117 165 214 262 310 358 <-287th 14 66 118 166 215 264 312 360 15 68 119 168 216 265 314 361 16 69 120 169 217 266 315 362 18 70 121 170 218 267 316 363 20 72 122 171 219 268 318 364 21 74 123 172 220 270 319 365 22 75 124 174 221 272 320 366 24 76 125 175 222 273 321 368 25 77 126 176 224 274 322 369 26 78 128 177 225 275 323 370 27 80 129 178 226 276 324 371 28 81 130 180 228 278 325 372 30 82 132 182 230 279 326 374 32 84 133 183 231 280 327 375 33 85 134 184 232 282 328 376 34 86 135 185 234 284 329 377 35 87 136 186 235 285 330 378 36 88 138 187 236 286 332 380 38 90 140 188 237 287 333 381 39 91 141 189 238 288 334 382 40 92 142 190 240 289 335 384 42 93 143 192 242 290 336 385 44 94 144 194 243 291 338 386 45 95 145 195 244 292 339 387 46 96 146 196 245 294 340 388 48 98 147 198 246 295 341 390 49 99 148 200 247 296 342 391 50 100 150 201 248 297 343 392 51 102 152 202 249 298 344 393 52 104 153 203 250 299 345 394 54 105 154 204 252 300 346 395 55 106 155 205 253 301 348 396 56 108 156 206 254 302 350 398 The kids are separated by 2420 days, it is 6.6 years, pretty that Bible chapter 242 would be First Samuel 6. The 12 (6+6) different letters in her full name add to 151 (the 6x6th prime). Dad was born in 56 and the kids today are together 16610 days old, pretty as it's 56 short of 16666. Today the family members are an average of 34.66 years old, and I meet Melissa when I am 17266 days old. Her middle name adds to the 24 chapters of Old Testament Books 6 and 10 (6th non-prime), her [CapitalThorn]rst name adds to 78 (6 times the 6th prime). She has 10 letters in her given names (6th non-prime). Melissa was born in the 6+6th month, and on the 23rd (6th prime plus the 6th non-prime), corresponding to Isaiah with 66 chapters. Isaiah 4, 12 and 20 (adds to 6x6) each contains 6 verses. 1-50 - Genesis 51-90 - Exodus 91-117 - Leviticus 118-153 - Numbers 154-187 - Deuteronomy 188-211 - Joshua 930-957 - Matthew 958-973 - Mark 974-997 - Luke 998-1018 - John 1019-1046 - Acts 1047-1062 - Romans 123 <-Numbers 6, it is three times the 13th prime (41+41+41), keeping in mind that 13 is the 6th prime 188 <-the opening chapter of Book 6 is 6x6x6 short of the 404 verses of Bible Book 66, it is the 6th prime squared (13x13) short of the 357 verses of Daniel (also in part about 666) 193 <-Book 6 chapter 6 is the 44th prime, while 44 is in turn 66.666...% of 66 211 <-the terminating chapter of Book 6 is approximately 66.6% of the 66th prime (317) 357 <-the opening chapter of Book 6 plus the 6th prime squared is the 357 verses of Daniel (in part about 666) 404 <-the 6th prime squared (13x13) plus the 6th prime squared (13x13) plus 66 adds to the 404 verses of Bible Book 66 1062 <-666 plus 6x66 is a combination of the 658 verses of Bible Book 6 plus the 404 verses of Bible Book 66, and is the terminating chapter of New Testament Book 6 1070 <-666 plus the 404 verses of Book 66 is the 1070 verses of Job (Book 6+6+6) 1213 <-Exodus terminates at chapter 90 (66th non- prime) with 1213 verses (the 198th or the 66+66+66th prime) 1292 <-the 658 verses of Book 6 plus twice the 66th prime (317) is the 1292 verses of Isaiah (the Book contains 66 chapters) The parents were born in 56 and 59, these are the 40th non-prime and the 17th prime (together for 57). The sister was born in 78 while Melissa adds to 78 (57th non-prime). The family was born on days and in months and years adding to the 357 (7x17+7x17+7x17) verses of Daniel. The kids were together born 571 days after mom's birthdays, or 81.57 weeks. The kids were born on days of the year adding to 487 (69.57 weeks). 389 <-77th prime 104 <-77th non-prime 77 <-77 --- 570 The Four 57's Genesis 41 -> 41 Leviticus 14 -> 104 Judges 9 -> 220 <-I dreamt of 220 roofs blown John 11 -> 1008 off homes in the Dakotas ---- 1373 <-220th prime Chapter 57 is Exodus 7 with 25 verses Book 57 is Philemon with 25 verses -- -- 41st non-prime 16th non-prime <-together for 57-> Major Books of End-Times Prophecy (Daniel and Revelation are in part about 666 while Isaiah contains 66 chapters): Daniel - 357 verses Revelation - 404 verses <-57 plus the 57th prime plus the 57th non-prime Isaiah - 1292 verses <-an average of 19.575757... verses per chapter Genesis 41 repeats the spoken seven 28 times, pretty as 1 through 7 adds to 28. We are warned in Genesis 41 to accumulate 7 years worth of food supplies in anticipation of several years of successive crop failures. Genesis 41 contains 57 verses, together for 98 (7x7+7x7). Pretty that Genesis 41 would contain 57 verses, for 57 is the 41st non-prime. Genesis 41 is a numerological marker in the Bible, I tried to tell people this in 1988 but they rolled their eyes back and smirked, people did not like my criticisms of their churches and used my interest in math as one of their reasons to arrest and torture me in psychiatric settings. I was arrested under the mental health act while the farmers were [CapitalThorn]ned for failing to get the grain to port fast enough. The Canadian government provided land for the railways, but then the railroad companies ripped out the tracks and sold the land, and used the money to purchase lucrative money making hotels. I was tortured for speaking while the people guilty of treason took their frequent trips to Europe, Asia and the Caribbean, while the Catholics sponsored Latin American and Asian Catholics to come to Canada and earn hard Western currency so they could tithe to the Catholic church. And they repeatedly raped Canadian taxpayers to the tune of many hundreds of millions of dollars to pay for costs involved in the visits of the pope (it is theft, just like the theft of the Egyptian obelisk situated at The Vatican). People used my interest in numbers as one of the reasons to support having me arrested and tortured, I show you people gems and year after year you spend billions of dollars on turning trees into idols while calling me insane (Second Chronicles 36:16). 187 Dar 17 2 57 48/317 00 Daryl 60 Shawn 65 Kabatoff 62 187 Marcia 6 8 80 219/147 8571 Marcia 45 Veronica 87 Acevedo 55 201 Melodee 7 4 74 97/268 6258 Melodee 59 Joy 50 Webster 92 240 Bethany 28 10 76 302/64 7193 Bethany 75 Ruth 67 Guenther 98 147 Natash 18 1 81 18/347 8736 Natash 63 Lyn 51 Isaac 33 214 Ali 12 1 83 12/353 9460 Alison 70 Janet 50 Gillespie 94 194 Lesley 27 5 83 147/218 9595 Lesley 78 Mae 19 Roberts 97 223 Camille 8 6 84 160/206 9973 Camille 55 Clarisse 86 Massey 82 221 Jessica 17 6 84 169/197 9982 Jessica 66 Vivian 77 Valois 78 288 Melinda 23 3 83 82/283 9530 Melinda 58 Janelle 59 Elaine 46 Joyce 58 Jarocki 67 If you people think you have the right to use my abusive parents as tools to have me arrested and tortured, then I think that I should have the right to ask women to marry me, or to marry Marcia and me. I have Scripture to support taking seven brides (Isaiah 4:1) and I have Scripture to support sleeping with Melinda Jarocki outside of wedlock (First Kings 1:1-5), while you people have a vast multitude of Scriptures condemning your decorated trees, phallic-capped churches and your violence against me for daring to point out your pagan traditions. Anyway, now that Melissa and her family sees evidence that their names are gifts from God, watch them start attending and tithing to churches that teach them to turn trees into decorated idols (maybe see Deuteronomy 12:2; 1 Kings 14:23; 2 Kings 16:4, 17:10; 2 Chronicles 28:4; Isaiah 57:5; Jeremiah 2:20, 3:6, 3:13, 10:3-4, 17:2 and Ezekiel 6:13). It just doesn't matter to you people that the Bible repeatedly condemns turning trees into idols (it is a violation of God's First and Second Commandments). And it just doesn't matter to you people that I was tortured for years after daring to identify the obelisks at The Vatican, The Whitehouse and on church roofs as being representations of penises (and in opposition to God's Second and Sixth Commandments). You people are compassionless turds, your only real compassion is for your traditions. You don't have an ounce of compassion for me or for anybody else being tortured by the psychiatric profession, all that truly matters to you is to go home for Christmas and stoop to the base of your mom's decorated tree (symbol of fertility that stays green all year) and collect a few presents. And then you feel guilty of your sin so you attend a church once a year, you go in December and give money to a preacher who tells you to stay away from idols, and he provides you with this less lesson while he standink next to a decorated tree and under his church steeple (an obelisk, an Egyptian representation of a penis). They worshipped and made representations of penises for these pagans believed it to have a creative or god-like force through it's reproductive ability, but Shawn knows that your penis is closer to your anus than to God. I don't have an ounce of respect for you and so don't require your permission to post your stats on the usenet and use you as an example to others, and look, here you are!!! There is a woman who hangs around Crocus Coop, she told me that the people are scared, they don't want to make waves against the psychiatric profession or anything else for they are afraid of being sent back to Hantelman at the U of S, or to City Hospital. I once heard a woman talk on the phone over at the mental health of[CapitalThorn]ce on Ave. P North, she cried and pleaded on the phone for a lengthy time, she was saying over and over that there was nothing wrong with her, how damaging the drugs being forced upon her were, the drugs were making her feel nauseous, she was horri[CapitalThorn]ed about the thought of continued forced treatment, she did not want to be on the drugs and stated over and over that there was nothing wrong with her and she did not need the drugs administered to her. She had no power, nobody cared about her being forced drugs against her will, nobody cared about her and nobody cared about me. And nobody cared about Jason Lee, he was in an abusive family relationship and then his parents forced him into the psychiatric mill, and he just eventually fell over and died. And Connie Glushyk's mom was ting on Connie and forcing Connie into the wards and then threatening to put Connie into a group home, well Connie just got a pass from the City Hall ward, walked across the street to her apartment in the high-rise, and jumped. The drugs make it so hard to verbalize arguments against what the psychiatrists are doing to you. It is a real horror show to think that you might get sent back to Hantelman, the threat of going back to Hantelman or to City Hospital looms for me and these people, once charged under the mental health act, they can take you and lock you up for 6 weeks until forced by law to release you. But then the NDP came to power provincially and immediately changed the laws and now can force medications upon people after the 6 weeks of forced treatment at Hantelman or City Hospital, and now thanx to the NDP, people who were horri[CapitalThorn]ed at the thoughts of being returned for a 6 week treatment, now get the drugs forced upon them every week of the year. And during this time, the NDP campaign colors included saffron, it's a colour holy to the Hindus, interesting as the NDP are now forcing people to be medicated 24/7/366 by Hindu psychiatrists. They would detain me for 6 weeks (during which time I would get to appeal twice to the middle-class Protestants and Catholics), and on the [CapitalThorn]nal day of the 6 weeks of treatment they would administer one last depot drug injection and release me, and the drugs remain in my body doing horrible things for months after that injection. It would take me months to get over the crap that they force upon me every 6 weeks, and then they would do it to me again and again and again and again. And now thanx to the NDP, people are having drugs forced upon them all the time. I was [CapitalThorn]rst arrested and tortured in 1988 when the NDP were not in power, but then the NDP came to power and took away more rights from people, the NDP have Hindus employed to force medications upon people against their will, the drugs remain in a person's body for months at a time, the NDP force people to be drugged every day of their lives. And now if some other party would come to power provincially in Saskatchewan and remove the NDP, they would leave the NDP law intact and continue to allow people to be drugged continuously. The NDP tortured me and tortures many in Saskatchewan, I don't want to live in Canada now and I certainly don't want to live in Canada if the NDP come to power nationally. I don't want to be here in Canada, and many other people don't want to be here either. There are people being forced medications upon them against their will and they would rather have a few dollars and be backpacking in Europe, they don't want to be living in poverty in Saskatchewan doped up and feeling nauseous, and then you people turn around and spend a few billion dollars annually turning trees into decorated idols. You don't care what the NDP and other governments are forcing these drugs upon people, you people care only about your tradions. I don't want to be in Canada and close to these places that I have been tortured. It was a horror show what they did to me, and year after year I see you people spending billions of dollars on turning trees into idols while calling me insane, and now some totally insensitive asshole starts speaking up in sk.forsale saying that he or she wouldn't want me babysitting his or her children, and that I make his or her skin crawl. You are God's children, and I have been babysitting all of you since 1988 when I started speaking up against the false traditions in your churches, and you make my skin crawl as well. Well Melissa thought it was all very nice and well and she paid me $2.31 (77+77+77) for my work (showing her evidence that her name was a gift from God), she then likely walked down the block and paid four dollars for a drink. Daryl Shawn Kabatoff Box 7134 Saskatoon Saskatchewan Canada S7K 4J1 Isaiah 45:4, Ephesians 3:15 - God gives you your name!!! === Subject: A Postcard from erdos fan by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52KD5Y15323; Actually, all of you would be strangers to me, and so just take this half as a joke :) (The other half is serious; I think I did solve the prime number theorem.) How are you doing? Maybe I think I could solve the prime number theorem elementarily. The proof I came up with is so elegant (clear and crystal, simple) that I forgot my envelop on the way to the library (I don't have a printer, and I was going to go to the post of[CapitalThorn]ce right after I printed out my paper.). I really want to send the paper to Dr. Paul Erdos for his opinion, if he were alive. Well, everyone has to admit that he has left this world, though. Let's see what we can do, shaln't we? erdos fan P.S: Ha ha ha, SF, I stole your proof from your book! === Subject: Explicit non-measurable set etc by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52KD5b15327; Could someone tell me how to explicitly construct a subset of R which is not (Lebesgue) measurable? Also I'd like to better understand Tensors in General Relativity. I can mechanically compute the coordinate tranforms etc but I can't visualise what I am doing. Is there a visual way to interpret a metric and so on, even in just 2 dimensions? Rob === Subject: Re: Explicit non-measurable set etc >Could someone tell me how to explicitly construct a subset of R which is not (Lebesgue) measurable? If the Axiom of Determinateness is consistent, the answer must be no, for from AD it follows that all subsets are Lebesgue measurable. This is equivalent to the existence of a measurable cardinal (Solovay). -- 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: Explicit non-measurable set etc days. My association with the Department is that of an alumnus. >Could someone tell me how to explicitly construct a subset of R which >is not (Lebesgue) measurable? Depends on your de[CapitalThorn]nition of explicitly. Such a construction would invoke the Axiom of Choice, and thus is, in a very real sense, not an explicit construction... The usual nonmeasurable set is the Vitali set. De[CapitalThorn]ne an equivalence relation on the real numbers on [0,1] by letting x~y if and only if x-y is a rational number. This partitions [0,1] into equivalence classes. Using the Axiom of Choice, there exists a set V which contains exactly one element from each equivalence class. This set V is the nonmeasurable Vitali set. For any real number r, let V + r = {v + r : v in V}. CLAIM 1: if q and r are distinct rationals, then V+q is disjoint from V+r. Proof: Assume that x is in the intersection of V+q and V+r. Then there exist elements y and z in V such that y+q = x = z+r. Therefore, y-z = q-r which is a rational number, so y~z. Since y~z and V contains one and only one element from each equivalence class, we must have y=z. But then we have y+q = y+r, hence q=r, contradicting the choice of q and r. Therefore, V+q and V+r are distinct. CLAIM 2: If x is a real number in [0,1], then there exists a rational number q, -1<=q<=1, and an element v of V, such that x = v+q. Proof: Since x is in [0,1], there exists v in V such that x~v, as V contains an element in the equivalence class of x. Therefore, x-v is a rational. Since 0<= x <= 1 and 0<= v <=1, we have that -1<= x-v <= 1. Thus, letting q=x-v, we have the result. Therefore, [0,1] is contained in the disjoint union of V+q, with q ranging over all rationals between -1 and 1. Note also that V+q is contained in [-1,2] for every rational bewteen -1 and 1. Now assume that V is measurable. Since Lebesgue measure is sigma additive and invariant under translation, all sets V+q have the same measure. Since the disjoint union of all V+q with q rational between -1 and 1 is contained in [-1,2], which has measure 3, it follows that if V is measurable, then it must be of measure 0; otherwise, we would have 3 >= measure (union V+q) = sum measure(V+q) = sum (measure V). So if V is measurable, then it must be of measure 0. But on the other hand, we have that [0,1] is contained in the union of the V+q; so 1 <= sum(measure(V+q)) = sum (measure V). So if the measure of V is 0, then we have a contradiction. Therefore, V cannot possibly be Lebesgue measurable. -- 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: General Sum-Counting Result Below I generalize a result which I have posted two speci[CapitalThorn]c cases of before, for r= 1 and r =2. In One Sequence Or Another (counter-intuitive?) (r=1) Integer Occurs Same # Of Times In These Sequences (r=2) (Let me get some stuff out of the way [CapitalThorn]rst before the result.) Let r be a positive integer. Let each f_j(x), for 1<= j <= r, be ANY real -> real continuous monotonically increasing function. Let each g_j(x) be the inverse of f_j(x), ie. f_j(g_j(x)) =x for all x. And let each f and g be such that both f(1) and g(1) exist and are >= 0. Let q(n,m) be the number of ways that we can add n ßoor(f_j(k))'s, with n f's of distinct j's in the sum, so as to equal (m - k), where k is taken from 1 to m. (So, for example, we might have, for n = 3, m-1 = ßoor(f1(1)) + ßoor(f2(1)) + ßoor(f4(1)) = ßoor(f1(1)) + ßoor(f3(1)) + ßoor(f4(1)), m-2 = ßoor(f2(2)) + ßoor(f3(2)) + ßoor(f4(2)), m-k = no sums of 3 f's each, for k >= 3. So we have q(3,m) = 3, the number of sums with 3 f's each, if the sum-representations above are the only sums of this kind.) Now, If we let k = m - j_1 - j_2 -... - j_r + r - ceiling(max(g1(j_1),g2(j_2),...,g_r(j_r))), (max() is maximum) and we sum over all r-tuples of positive integers {j_1,j_2,..,j_r} which are such that k is a nonnegative integer, we have: --- / r-1 k / | | (-1) = --- k / k r --- n / q(n,m) (-1) --- n=0 ( q(0,m) = 1.) In linear-mode: sum{k} binomial(r-1,k) (-1)^k = sum{n=0 to r} q(n,m) (-1)^n. I may have made a mistake, and I realize it is relatively hard to understand what exactly I mean above. But I felt I should post this general result since I have been recently obsessed with the r=1 and r=2 cases. I used a simple generating function to prove this, and a more visual proof to prove the r=1 case, as noted in one of the links given above. (Is the result correct?) Leroy Quet === Subject: Re: Explicit non-measurable set etc by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i52KYLO17207; >Could someone tell me how to explicitly construct a subset of R which is not (Lebesgue) measurable? >Also I'd like to better understand Tensors in General Relativity. I can mechanically compute the coordinate tranforms etc but I can't visualise what I am doing. Is there a visual way to interpret a metric and so on, even in just 2 dimensions? >Rob look in Royden,i think they construct one === Subject: Re: cycle index of the group C_2 wr_k Sym_k Content-Length: 769 Originator: rusin@vesuvius >Where it's the autmorphism group of, say, k-dimensional cube, viewed as >a graph. The group operates on the edges of this cube. >The real problem I'm dealing with is the nxnx...xn-cube in k dimensions. > What do you mean by cycle index? Let $G$ be a group of permutations. For $piin G$, let $c_i(pi)$ be the number of cycles of length $i$ in the standard representation of $pi$ and let $k:=max{ iin N mid exists piin G: c_i(pi)not=0}$ be the length of the longest cycle of all permutations in $G$. Then the emph{cycle index} $P_GinNN_0[x_1,...,x_k]$ is de[CapitalThorn]ned as [ P_G(x_1,x_2,...,x_k)=frac{1}{|G|}sum_{piin G} x_1^{c_1(pi)}x_2^{c_2(pi)} cdots x_k^{c_k(pi)}. ] === Subject: Re: cycle index of the group C_2 wr_k Sym_k > Where it's the autmorphism group of, say, k-dimensional cube, viewed as > a graph. The group operates on the edges of this cube. > The real problem I'm dealing with is the nxnx...xn-cube in k dimensions. > For small n it's done by hand. Maybe somebody has aleady counted this > for general n so that we don't need to invent the weel twice. P.W.H. Lemmens, Polya Theory of Hypercubes, Geometriae Dedicata (64) p. 145-155. Gives a method for determining the cycle index for vertices, edges, planes, ... any subdimensional simplex. You'll have to generalize from 2^k to n^k (same conjugacy classes, and so same coef[CapitalThorn]cients and exponents, but modi[CapitalThorn]ed factors in terms. (The same reasoning goes for converting the cycle index for vertices to the cycle index for other dimensions). -- Mitch Harris (remove q to reply) === Subject: Re: cycle index of the group C_2 wr_k Sym_k > Where it's the autmorphism group of, say, k-dimensional cube, viewed > as a graph. The group operates on the edges of this cube. > The real problem I'm dealing with is the nxnx...xn-cube in k dimensions. > For small n it's done by hand. Maybe somebody has aleady counted this > for general n so that we don't need to invent the weel twice. > P.W.H. Lemmens, Polya Theory of Hypercubes, Geometriae Dedicata (64) p. > 145-155. > Gives a method for determining the cycle index for vertices, edges, > planes, ... any subdimensional simplex. > You'll have to generalize from 2^k to n^k (same conjugacy classes, and > so same coef[CapitalThorn]cients and exponents, but modi[CapitalThorn]ed factors in terms. > (The same reasoning goes for converting the cycle index for vertices to > the cycle index for other dimensions). -- Alex. PS. To email me, remove loeschedies from the email address given. === Subject: Re: cycle index of the group C_2 wr_k Sym_k > Where it's the autmorphism group of, say, k-dimensional cube, viewed > as a graph. The group operates on the edges of this cube. > The real problem I'm dealing with is the nxnx...xn-cube in k dimensions. > For small n it's done by hand. Maybe somebody has aleady counted this > for general n so that we don't need to invent the weel twice. > P.W.H. Lemmens, Polya Theory of Hypercubes, Geometriae Dedicata (64) p. > 145-155. > Gives a method for determining the cycle index for vertices, edges, > planes, ... any subdimensional simplex. > You'll have to generalize from 2^k to n^k (same conjugacy classes, and > so same coef[CapitalThorn]cients and exponents, but modi[CapitalThorn]ed factors in terms. > (The same reasoning goes for converting the cycle index for vertices to > the cycle index for other dimensions). -- Alex. PS. To email me, remove loeschedies from the email address given. === Subject: Re: cycle index of the group C_2 wr_k Sym_k > Where it's the autmorphism group of, say, k-dimensional cube, viewed > as a graph. The group operates on the edges of this cube. > The real problem I'm dealing with is the nxnx...xn-cube in k dimensions. > For small n it's done by hand. Maybe somebody has aleady counted this > for general n so that we don't need to invent the weel twice. > P.W.H. Lemmens, Polya Theory of Hypercubes, Geometriae Dedicata (64) p. > 145-155. > Gives a method for determining the cycle index for vertices, edges, > planes, ... any subdimensional simplex. > You'll have to generalize from 2^k to n^k (same conjugacy classes, and > so same coef[CapitalThorn]cients and exponents, but modi[CapitalThorn]ed factors in terms. > (The same reasoning goes for converting the cycle index for vertices to > the cycle index for other dimensions). === Subject: Simple Number-Picking Game Here is a highly unoriginal simple game. I am posting it to sci.math because I feel it might have some math-teaching potential. I am cross-posting it to rec.puzzles, as I do with all my games, because those reading rec.puzzles are a bunch of fun-lovers. :) Players (2) take turns picking a form, involving one or two variables, and an integer of that form. Each player, on his/her move, reveals the form and integer, and her/his opponent tries to come up with as many solutions as possible for the variables in a time-limit (such as a minute), and the opponent gets a point for every example of variables they can come up with. Examples are the best way for me to illustrate what I mean. Player 1 says: 32 is a square (S) plus a prime (P). Player 2 tries to determine as many S and P in a minute as possible. S = 9, P = 23, and S = 25, P=7, as an example. Or: Player 2 says: 9 is a power of 2 (N) minus a Fibonacci number (F). Player 1 might give: N = 64, F = 55 Each player who poses a form can use paper/pencil and calculator to get the integer of that form. And if no solution exists, because the posing player has made a math-error, the posing player loses automatically. But the goal of each posing player is pick a form with as few solutions as possible (but at least 1) so as to keep his/her opponent from easily getting many points. And, yes, pencil/paper and calculator are allowed. Each player should write down his solutions for the variables anyway. I would also suggest that the integer which is part of the posed problem be under some maximum value, which is to be decided by the players. Otherwise a player might say, 17726253434843783635 is the sum of 2 primes. ;) As for math-errors made by the player trying to come up with a set of variables, I do not believe these errors should automatically disqualify a player since this player is doing his/her math under a time-limit. I suggest the rule that if the player being challenged (with the other player's form and integer) makes a math-error, he/she simply does not get any points for that particular turn. Any other suggestions for rules? Be creative and have fun! (and do not be discouraged by my lame examples) Leroy Quet === Subject: PS:Simple Number-Picking Game >Here is a highly unoriginal simple game. >I am posting it to sci.math because I feel it might have some >math-teaching potential. >I am cross-posting it to rec.puzzles, as I do with all my games, because >those reading rec.puzzles are a bunch of fun-lovers. >Players (2) take turns picking a form, involving one or two variables, >and an integer of that form. >Each player, on his/her move, reveals the form and integer, and her/his >opponent tries to come up with as many solutions as possible for the >variables in a time-limit (such as a minute), and the opponent gets a >point for every example of variables they can come up with. >Examples are the best way for me to illustrate what I mean. >Player 1 says: >32 is a square (S) plus a prime (P). >Player 2 tries to determine as many S and P in a minute as possible. >S = 9, P = 23, >and >S = 25, P=7, as an example. >Or: >Player 2 says: >9 is a power of 2 (N) minus a Fibonacci number (F). >Player 1 might give: >N = 64, F = 55 >Each player who poses a form can use paper/pencil and calculator to get >the integer of that form. >And if no solution exists, because the posing player has made a >math-error, the posing player loses automatically. >But the goal of each posing player is pick a form with as few solutions >as possible (but at least 1) so as to keep his/her opponent from easily >getting many points. >And, yes, pencil/paper and calculator are allowed. Each player should >write down his solutions for the variables anyway. >I would also suggest that the integer which is part of the posed problem >be under some maximum value, which is to be decided by the players. >Otherwise a player might say, 17726253434843783635 is the sum of 2 >primes. I *thought* I had made the above large integer even (I did not check what I had written). So my example is ridiculous. But it brings up a good point. What is there to stop a player from simply always picking an odd integer (which is the sum of 2 and an odd prime) and saying it is the sum of 2 primes? His opponent could easily get the solution, but would be limited to a single point automatically. Now, this might not be much of an issue if players commonly come up with NO solutions otherwise while playing this game in-practice. For then a player would not want to pick a form and integer which would yield any solution easily anyway. I would suggest, however, that each form (sans the integer) be such that it has not been used previously in the game. Leroy >As for math-errors made by the player trying to come up with a set of >variables, >I do not believe these errors should automatically disqualify a player >since this player is doing his/her math under a time-limit. I suggest the >rule that if the player being challenged (with the other player's form >and integer) makes a math-error, he/she simply does not get any points >for that particular turn. >Any other suggestions for rules? >Be creative and have fun! (and do not be discouraged by my lame examples) >Leroy Quet === Subject: Re: Area under Receiver Operating Characteristic curve In brief: AUC = Phi(dprime / sqrt(2)) where AUC = area under the ROC curve Phi = standard N(0,1) Gaussain cumulative distribution function dprime is what it is and sqrt = 1.41421356... HTH, Vit D. > Is there a formula to express the area under an ROC curve as a function of === Subject: Re: The Meaning of Abstract >lossy compression is far from horribly sloppy. It's obviously >correct. Anyone that doesn't understand the reference doesn't >understand abstraction. >You can also think of abstraction as pattern matching or signal >[CapitalThorn]ltering. It's just different ways to talk about the same thing. >Well its not so very totally obviously correct to me; although i must >admit that i ßashed as such when i [CapitalThorn]rst read it. Thing is that an >abstraction is not located in time and space which is the most usual >criteria of something that is not physical. However, a compressed set >of marks is de[CapitalThorn]nitely located in time and space. > The belief that an abstraction is not located in time and space is just a > false, and useless belief. > We have the power to make stuff up that has never existed by combining > features from real things together in ways that we have never seen before. > Such as a ßying elephant. Our power to generate these ideas are limitless. > But, does the idea of a ßying elephant help you understand yourself or the > universe? Is it real? Well, the concept is real beacuse I just created > it for this post. And we all understand the concept so it's real in that > sense even if there are no ßying elephants in the world. > Is the belief that an abstraction is not located in time and space a ßying > elephant or is it real? It is in fact just a ßying elephant. > Show me an abstraction that doesn't exist in time and space. An ideal circle. Take an line segment and keeping the segment in the same plane, rotate it around one end, the shape that is made by the opposite end is the shape of an ideal circle. It is interesting to note that we can imagine andor recognize the shape without knowing the procedure. That shape (note the singular) is not located in time and space. If it is located there, please provide me the coordinates. > You can't do it because anything you can show me, by de[CapitalThorn]ntion has to exist > in time and space. All you can do is talk to me about ßying elephants and > say they are real and tell me that you believe they are real. I have no problem with calling a ideal circle, not real. > To say that > you believe something is real is to believe that you will one day see one, > but that you just haven't seen one yet. To say that something is real but > can never been seen, is your perosonal choice to believe that things which > can never be senseed, are in fact real. And if you belive that, then I > assume you believe not only that abstractions don't exist in time and > space, but that ßying elephants are also just as real. That last sentence does not follow, but then the whole paragraph is based upon the erroneous assumption that i believe that a ideal circle is real in the sense that it can be seen or sensed. Although the ideal circle cannot be sensed, we can manifest approximate instances of it that can be sensed. That is why we call the ideal circle an abstraction and why we make a special slot for it in our ontology. It exists in our ontology (in our culture and language) for the purpose of discussions, but it does not exist in time and space. > Our culture accepts fairly easilly the idea that things we can't sense are > real. This happens for many reasons. Mostly it happens because we put a > large amount of faith in second-hand knowledge. Bob said he saw a ßying > elephant, and I trust Bob, so even though I've never seen one, I too > believe they are real. Other people said they have seen time distort > according to Einstein's predictions so I believe it is real even though I > have never seen it happen. There is a big difference between a [CapitalThorn]ctional thing and a abstract thing. Both exist in our ontology. Neither of them can be found in time and space. Neither of them can be sensed. Both of them can be imagined. > Everyone knows that concepts exist in a dimension outside of time and space > so everyone believes it. Everyone is wrong. Or, more accurately, they are > just talking about ßying elephants because it's doesn't hurt to do so - > until you try to [CapitalThorn]gure out how the brain works and what consciousness is. > The other reason we accept the idea so easily is the fact that these things > happen inside our head - where we can sense them in our thoughts - but > where we can not tie them to our other sensory inputs from the physcial > world. When we see an action, and can tie that to a sound, and a smell, > and a taste, and touch, then we know the thing we sensed exists in the > world of all those senses. Yes it is reasonable to call those things that we sense and that others can sense as well as real. Those things that we make up as [CapitalThorn]ctions or that we just imagine are then termed not real. I would call them imaginary. They don't exist in some other dimension, they are artifacts of the behavior of our brains and are signs that we stick in our culture. We can play a language game about them and discuss them and if there are effective procedures for reproducing instances of them we can call them ideals or abstractions to differentiate them from [CapitalThorn]ctions which are related to no such effective procedures. I would think that even you would agree that all of that is fairly uncontroversial. Do you not? > But when we sense our thoughts, we see no connections to our other senses - > we can not hear, see, taste, or feel, our own thoughts. That makes us > believe our thoughts exist in a separte world from the world of the other 5 > senses. But, we have learned enough about the brain to know this is not > true. Eerything we think about is linked to brain activity, which we can > sense in physical world - even if most of us have never gotten the chance > to do that. > What we in fact have is a brain full of pattern recognition hardware. We > have pattern recognition hardware that can detect elephants, and hardware > which can detect ßying things. And if we ever saw a ßying elephant, we > would be able to detect it in a heartbeat. We would know what we were > looking at. > And we have trained our pattern recognition hardware for detecting > elephant to also respond to the word elephant. But we do not confuse > the real thing for the word because we know that elepahant in the context > of the hearing the word elephant is different than elepahnt in the > context of seeing a large animal. I have no major problems with those last three paragraphs. > It's the functioning of this pattern matching hardware which is creating > abstractions. Ok. > It's a lossy compression system for responding to some > aspects of the data, and ignoring other aspects. Our pattern matching > hardware is the de[CapitalThorn]ntion of all abstractions we know about. The > abstraction exists in the form of the hardware which produces it, which > is very real and physical and exists in time and space, and in the outputs > it generates each time it is used, which is also something that is very > real in time and space. If we base our ontology on process rather than substance then we could say that the process of our brains imagining an ideal (of abstracting) is a *subclass* of the process of making a lossy compression of marks. It is merely a *sub* class because there are compressions that do not have the particular features that are necessary for us to call it the process of constructing an abstraction, as others in this thread have pointed out. > The belief that an abstraction could be timeless is the belief that these > pattern matching machines could exist forever, and could exist without a > physcial form - both which are impossible - i.e., ßying elephants. There is an important distinction between saying that something exists forever and saying that for it time is irrelevant. I would like to use this example an an excuse to show how semiotics can help clarify some of these issues for us. Please refer to the diagram: Now let's discuss where the object of the sign that would be marked with the letters circle exists ? The diagram indicates that all objects exist in a plane labeled environment, language, culture. Well we already eliminated the possibility that it is a sensual object in the real environment, so it must be an object in language andor culture. Our friends from the EAB might say that it exists as contingencies in our language and our culture. And I would agree with them; that is most reasonable place for us to place it. You, on the other hand, might argue that it is an object in the squiggly line (which stands for cause and effect connections) that are drawn inside the interpreters which normally are found inside peoples heads. Let's see if that would work. Does the ideal circle go away if your head ceases to exist? I think not. Does the ideal circle go away if all such human heads cease to exist? I think not, for some alien creature could study the markings of our culture and recreate that concept from those. So it is in that culture that this object does exist. I rest my case. Do you see the use of semiotics now? patty === Subject: Re: The Meaning of Abstract > I guess I don't understand abstraction. > > Is abstract supposed to be a synonym for lossy compression? If > so, do you regard an mp3 ripped from a CD as an abstraction of the > original wave [CapitalThorn]le? > I would de[CapitalThorn]nitely understand an mp3 as an abstract representation of > the raw waveform data contained on a CD. Do you have a better example > for the point you are trying to make? Whether such an example exists is of interest for me, as well. > The act of [CapitalThorn]ltering a signal or lossily compressing a source is very > different to my mind than the mathematical (or philosophical) act of > abstraction. Evidently, what is obviously correct in your view is > horribly sloppy in mine. > I wouldn't go as far as to say it's obviously correct, however it lossy > compression is vaguely how I understand abstraction as well. Yes, I am trying to put it in mechanical terms so that we can build an AI that abstracts. > A few different questions: Does it have to be lossy? Can't abstraction > be lossless as well? If it is lossless, is it abstraction or merely a > different form representation? How can we universally quantify the level > of abstraction? In my opinion, it has to be lossy for it to be abstract. If it is not lossless, it is simply an algorithmic copy of the information (in requiring a decompressor program), ie. a full representation of the object. That is also quite important, as shown in Information Distance of Charles Bennett et al. You have stated an important question: lossy in what sense? I am working on this question, and I have a few mathematical ideas, but I do not know how well they will turn out. Immature as they are, I'd rather not speak of them. References to theoretical work would be appreciated. -- Eray Ozkural === Subject: Re: The Meaning of Abstract > I'm surprised that some people take abstract as a vague > common-sense concept. To me, it has a precise technical meaning: > lossy compression. Perhaps I'll write a textbook called Lossy Compression Algebra. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: The Meaning of Abstract >I'm surprised that some people take abstract as a vague common-sense >concept. To me, it has a precise technical meaning: lossy compression. >A program is abstract, because it *loses* the architectural details of >a computation, and it is concise. A blueprint of a house is abstract >because it *loses* the architectural and material properties of an >actual house, and it is concise. You are confusing abstract with abstraction. None of these is abstract, although they are abstractions. When understood, an abstract idea is a mental structure, which may or may not arise by the process of abstraction. It is likely to be better understood if it does not. -- 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: The Meaning of Abstract > I'm surprised that some people take abstract as a vague common-sense > concept. To me, it has a precise technical meaning: lossy compression. > Instantiation is most certainly a Platonist word which includes > counter-factuals in its meaning. An abstract entity represents > another entity in a purposeful way, it is a sign that points to > another object or sign. > Maybe we should all study semiotics instead of Platonist computer > science and mathematics! Maybe that is how one truly becomes a > hard-core materialist! I am a bit uncertain that all traces of Platonism are excluded in the explantion provided below or that abstract is treated as vague common-sense, but the comprehension compression is certainly loses nothing to the Tao*. Adding Abstract to Formal and Content Schemata: Results of Recent Work in Peircean Semiotics John W. Oller, Jr. http://members.door.net/arisbe/menu/library/aboutcsp/oller/ schemata.htm Abstract schemata, by contrast, concern everything that is contained within the meaning or de[CapitalThorn]nition of a symbol (including propositions, arguments, and discourses). They take all that possibly could be (as known through the symbols used) and relate it to whatever must be (provided the symbols are used truly). *As a result, the abstract level of the symbol (as Peirce showed) reaches from outside of time and space into the material world and yet is itself neither temporal nor spatial in its compass.* It comes nearer to our ideas of eternity, in[CapitalThorn]nity, continuity, and universality than to anything known through our senses in the material world. It involves what Peirce called a Ôthirdness' beyond the duality of opposing forces clashing in the material realm. Industrial strength supervening, Stephen === Subject: Re: The Meaning of Abstract > I guess I don't understand abstraction. > > Is abstract supposed to be a synonym for lossy compression? If > so, do you regard an mp3 ripped from a CD as an abstraction of the > original wave [CapitalThorn]le? > I would de[CapitalThorn]nitely understand an mp3 as an abstract representation of > the raw waveform data contained on a CD. Do you have a better example > for the point you are trying to make? > Most of us don't regard the relation between mp3 and wave [CapitalThorn]le as the > same as the relation between the concept of triangle and *all* of the > particular three-sided [CapitalThorn]gures with which we are familiar[1]. Isn't that called generalization, rather than abstraction? BTW, the common sense meanings of Ôgroup', Ôset', Ôclass' Ônumber', Ôsequence', Ô[CapitalThorn]eld', Ôcontradiction' are also rather different from their formal mathematical counterparts. One wonders how sloppy, irrelevant, or plainly wrong mathematics must look in the eyes of a non-mathie... ;-) Herman Jurjus === Subject: Re: The Meaning of Abstract <8765abdmqp.fsf@phiwumbda.org> <87y8n7c6iz.fsf@phiwumbda.org> <2i5fe6Fj2ju4U1@uni-berlin.de> Discussion, linux) > I guess I don't understand abstraction. > > Is abstract supposed to be a synonym for lossy compression? If > so, do you regard an mp3 ripped from a CD as an abstraction of the > original wave [CapitalThorn]le? > > I would de[CapitalThorn]nitely understand an mp3 as an abstract representation of > the raw waveform data contained on a CD. Do you have a better example > for the point you are trying to make? > Most of us don't regard the relation between mp3 and wave [CapitalThorn]le as the > same as the relation between the concept of triangle and *all* of the > particular three-sided [CapitalThorn]gures with which we are familiar[1]. > Isn't that called generalization, rather than abstraction? Maybe so. Bad example, I guess. > BTW, the common sense meanings of Ôgroup', Ôset', Ôclass' > Ônumber', Ôsequence', Ô[CapitalThorn]eld', Ôcontradiction' are also rather > different from their formal mathematical counterparts. > One wonders how sloppy, irrelevant, or plainly wrong > mathematics must look in the eyes of a non-mathie... But mathematicians don't claim that they've provided the correct technical de[CapitalThorn]nition for the common-sense usage. If one wants to use the term abstraction in a certain technical sense, then there is no controversy. If they claim, on the other hand, that this technical sense clari[CapitalThorn]es the common-sense usage, then that's a different matter. I've never heard a mathematician argue that Farmer Joe Bob's use of the term [CapitalThorn]eld is just a vague approximation of the correct and precise de[CapitalThorn]nition found in mathematics[1]. But lately, folks have responded to my intended reductio ad absurdums by agreeing with them, so maybe I should offer that example with trepidation. Footnotes: [1] Maybe some folk would argue that set, class or sequence are precisi[CapitalThorn]cations[2] of the common sense notions. In fact, the claim seems fairly plausible for sequence in particular. [2] A technical term of art. Honest. -- Jesse Hughes If you really think there's a bug you should report a bug. Maybe you're not using it properly... It turns out Luddites don't know how to use software properly, so you should look into that. -- Bill Gates === Subject: Re: The Meaning of Abstract > I'm surprised that some people take abstract as a vague common-sense > concept. To me, it has a precise technical meaning: lossy compression. > A program is abstract, because it *loses* the architectural details of > a computation, and it is concise. A blueprint of a house is abstract > because it *loses* the architectural and material properties of an > actual house, and it is concise. No it hasn't. Since real time programs lose no architectural details. since otherwise Internet wouldn't exist. And neither blueprints or houses or science would exist if real-time programs didn't exist. === Subject: Re: The Meaning of Abstract %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM% {l/{cT'{w X-Url: http://CurtWelch.Com/ > A simpler example. Good. Your other example was beyond me. I'm posting from the ai group, not the math group. I really don't know why this thread was cross posted to this set of groups. > Take the map N -> N taking 0 to 0 and n+1 to n. This is a lossy > compression scheme. Would you *really* want to call it an > abstraction? Sure, it's an abstraction. Just not a very useful one. But I agree, it's very unlikely I would ever use the world abstraction to describe it outside the conext of this type of discussion. In AI work, ideas of abstraction are a common themem. Understanding abstraction is key to understanding what the brain does. When you explore these things, you start to see how all these different ideas (and more - like data clasi[CapitalThorn]cation) are closely realated and very likely, only one problem to the brain hardware. When I [CapitalThorn]rst posted to this thread, I didn't notice the strange cross posting and thought the AI context was assumed. So I wasn't trying to debate the correct usage of the word abstraction but to simply debate the concept in the context of building smart machines. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com Webmaster for http://NewsReader.Com/ === Subject: Re: The Meaning of Abstract > A simpler example. >Good. Your other example was beyond me. I'm posting from the ai group, >not the math group. I really don't know why this thread was cross posted >to this set of groups. > Take the map N -> N taking 0 to 0 and n+1 to n. This is a lossy > compression scheme. Would you *really* want to call it an > abstraction? >Sure, it's an abstraction. So 41 is an abstraction derived from 42? That's nothing at all like abstraction as we know it - if this is an abstraction of that then any particular that should be an instance of this. Is 42 an instance of 41? (42 is a special case of 41 somehow?) >Just not a very useful one. But I agree, it's >very unlikely I would ever use the world abstraction to describe it >outside the conext of this type of discussion. Glad to hear _that_, at least. I can't imagine why 41 would count as an abstraction of 42 in _any_ context. >In AI work, ideas of abstraction are a common themem. Understanding >abstraction is key to understanding what the brain does. When you explore >these things, you start to see how all these different ideas (and more - >like data clasi[CapitalThorn]cation) are closely realated and very likely, only one >problem to the brain hardware. That's fascinating, but it doesn't give me any hint why it is that in AI it's appropriate to consider 42 a special case of 41. >When I [CapitalThorn]rst posted to this thread, I didn't notice the strange cross >posting and thought the AI context was assumed. So I wasn't trying to >debate the correct usage of the word abstraction Which is what the thread is about. >but to simply debate the >concept in the context of building smart machines. ************************ David C. Ullrich === Subject: Re: The Meaning of Abstract Discussion, linux) > A simpler example. > Good. Your other example was beyond me. I'm posting from the ai group, > not the math group. I really don't know why this thread was cross posted > to this set of groups. > Take the map N -> N taking 0 to 0 and n+1 to n. This is a lossy > compression scheme. Would you *really* want to call it an > abstraction? > Sure, it's an abstraction. Just not a very useful one. But I agree, it's > very unlikely I would ever use the world abstraction to describe it > outside the conext of this type of discussion. > In AI work, ideas of abstraction are a common themem. Understanding > abstraction is key to understanding what the brain does. I am doubtful that the brain's facility of abstraction has much to do with the metaphors which you claimed were obviously correct, so I'm not too sure that you have grasped that key to understanding yet. But I will not debate that point, since it is possible that your notion of abstraction is simply a technical term distinct from the common usage. > When you explore these things, you start to see how all these > different ideas (and more - like data clasi[CapitalThorn]cation) are closely > realated and very likely, only one problem to the brain hardware. [...] -- Jesse Hughes Well, if I can get [my proof of FLT accepted], then I hopefully get a book deal down the road, and maybe I get to go on ÔOprah'. James Harris, on the rewards of mathematical endeavours. === Subject: Re: The Meaning of Abstract Dave B. Held said: A blueprint is certainly more abstract than an actual house, but not because of what it is missing. It is abstraction because of what it has, which is the essential features of the house. Whether it speci[CapitalThorn]es the placement of all the electrical outlets or not is of no concern. Whether it conveys the proper layout of rooms and ßoors is. If I created a blueprint that located all the electrical outlets in the house but not any of the walls, according to your de[CapitalThorn]nition, it is an abstraction of the house, because I have *lost* detail, and created a concise version of the house. But merely losing detail is not the key feature. The key feature is identifying the details that we use to classify the concrete instances. Abstraction is a notion concerned with classes and types, not details and data representation. Such a de[CapitalThorn]nition is not suf[CapitalThorn]ciently abstract. You can make these analogies stronger here. A program is a concrete implementation of an algorithm--an abstract process (with certain properties... blah blah). Similarly, both houses and blueprints are concrete representations of a ßoorplan. In both cases, we can say that the abstract concept is the content, or meaning, of the representation (well, the house is a bit trickier than the blueprint in this case, but you handled it well, Dave). This leads to ideas about context sensitivity and language games, which I don't want to deal with now. Ôcid Ôooh === Subject: Integral Domain Concept Where Zn={0,1,...,n-1} under addition and multiplication modulo n, I'm asking myself if Zn is an integral domain, why n would have to be a prime number? An integral domain is commutative with no zero divisors and an element is prime if it is nonzero, nonunit, and if it divides ab then it divides either a or b. A prime number in Z is in fact the same as an irreducible element. I'm missing the connection between integral domain and prime n. Phil Holman === Subject: Re: Integral Domain Concept >Where Zn={0,1,...,n-1} under addition and multiplication modulo n, I'm >asking myself if Zn is an integral domain, why n would have to be a >prime number? An integral domain is commutative with no zero divisors >and an element is prime if it is nonzero, nonunit, and if it divides ab >then it divides either a or b. A prime number in Z is in fact the same >as an irreducible element. I'm missing the connection between integral >domain and prime n. >Phil Holman Pick a small non-prime n such as n = 6. Make a table with the multiplication rule for Z/nZ. Find two nonzero elements whose product is zero. Generalize to other non-prime n. -- John Adams served two terms as Vice President and one as President, but lost reelection. Later his son became President despite losing the popular vote. That son lost his reelection attempt badly. Now history is repeating itself. pmontgom@cwi.nl Microsoft Research and CWI Home: San Rafael, California === Subject: Re: Integral Domain Concept >Where Zn={0,1,...,n-1} under addition and multiplication modulo n, I'm >asking myself if Zn is an integral domain, why n would have to be a >prime number? Because another way to think of being prime is: an integer p is a prime if and only if p is not 0, 1, or -1, and when p divides the product n*m of integers, then p divides n or p divides m If a product x*y is 0 in Zn, that means that n divides the product of x and y as integers. To be an integral domain, you want xy = 0 ----> x=0 or y=0. So: xy = 0 -----> n divides xy x=0 ----> n divides x y=0 ----> n divides y. Since you want n divides xy ----> n divides x or n divides y this is what gives you that n should be a prime number. -- 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