mm-4419 === Subject: Re: $13.2 Million For Math, Science, English I watched one of the awarded grants being written. The individuals put in a >lot of work and thought. I think the idea is that many of our regular kids >need to see themselves as potential AP students. I couldn't disagree more strongly. Only a small percentage of our > students have the aptitude and the attitude required for AP level > study. I teach these kids every day and the typical regular student > is years behind most of them. The notion that there are all these > undiscovered scholars in regular ed is an myth. While examining some old issues of the American Mathematical MONTHLY, I found the following Note by J. H. Neelley of the Carnegie Institute of Technology [A generation of high school calculus, Dec. 1961, pp 1004-5]. Neelley concluded that high school calculus is largely a waste of time. According to Neelley: In 1956, the press, politicians and then the public began to demand that the high schools do the work equivalent to that done in Europe under different circumstances. This did create the high school accelerated programs. Under these programs the brightest students were taught Algebra I in 8th-grade and calculus in 12th- grade. Prior to 1957, Carnegie had less than 5 math majors in each freshman class. Subsequently the number of freshmen registering as math majors increased significantly, as indicated in the following table. (AP indicates students with high school calculus.) Number Registered as Freshman Total AP Reg as Total Reg as Math Majors @ Mar 1961 Year AP Math Majors Math Majors AP Total 1957 6 2 13 1 20 1958 48a 5 32 3 25 1959 33b 8 37 4 31 1960 54c 5 46 5 46 Notes: a--only 13 of these 48 completed first year college math b--only 18 0f these 33 ................................. c--only 17 of these 54 were still in first year math @ Mar 1961 High school calculus produced neither math majors nor success in the first college math course that these students took. Neelley reached the correct conclusion that high school calculus is a waste of time. For the vast majority of students, high school calculus leads to a dead end. This is as true today as it was at its inception. Some pundits would dismiss Neelley's data as outdated, which is not at all the case. In the June-July 1995 issue of the American Mathematical MONTHLY, is really making it more difficult to teach college level mathematics is the rush to have calculus taught in high school, ..., and at the price of not teaching basic algebra and geometry. Another excellent letter, by Joan Reinthaler, was published in the NOTICES, Pressure To Study Calculus in High School. This is available at: http://www.ams.org/notices/199911/commentary.pdf === Subject: Re: $13.2 Million For Math, Science, English > On Sep 11, 5:52 pm, J. Z. Al-Huriyeh Helium E= H(x) where x is in geometric expansion.. escaping solidification of heat condensed Helium as similar logic to stalagmites by insulation sheet will condense the escaping Helium into Lightweight Glass. The Creation of Lightest Weighing Glass (c)opyright 2007 by Seung Bum Kim See My Schematic at http://www.asiandb.com/community/iboard.pfm?code=freetalk&view=t&mode=view&p a ge=&num=904 === Subject: Roots to Power Series Compute the roots to any Power Series. http://mypeoplepc.com/members/jon8338/math/id2.html === Subject: MathPuzzle 194: Eleven squares Good evening from Ruurlo, The objective of my new puzzles is to press eleven different squares in a 5 x 6 matrix. Once more a puzzle, that can be solved with simple math and a bit of patience. Hopefully it is to your liking. Puzzle 193 led to a lot of confusion. The Limmens pyramide is just an introduction to the real question: How high a pyramid can be built with beer cases that also all can be piled up in a triangle, without coming one case short or leaving one cas out. Puzzle 187 has been closed now. All later puzzles are still open for solutions. Have fun with the new puzzle. Peter direct link: http://home.planet.nl/~p.j.hendriks/p194e.htm Please answer by email and not in this newsgroup. === Subject: The Treatment for Ostheoperosis (c)opyright 2007 by Seung Bum Kim Residual calcium deposits end up elongating as an excessive output where it is not a matter of building blocks. Where the Pelvis is, is also central to where the human form in the skeletal structure advances in ratioistic equilibrium. If like catalystic developments occur within the Pelvic unit it would operate as building blocks where the Pelvic bone is central for both upperbody and lowerbody.. by making the residual calcium directed into the Pelvis as building blocks hence not only enlarges the entire human skeletal structure, it enlarges it proportionally correct..as the femur, for instance, reaches its limit or else it must differentialize into a separate calcium outgrowth. The Treatment for Ostheoperosis (c)opyright 2007 by Seung Bum Kim === Subject: Interfacing Data Unit (c)opyright 2007 by Seung Bum Kim processed and through its processing it would be more interactive if the processed data had a concrete form as the interfacing output works on tones as operated by transistor intensities of blue, green, red light where the resistor is simultaneously there to gather capacitor readings where the values of the transistor per bitmap serve as simulated tangibility by the limits to the capacitor where the transistor that operates on a maximum level of intensity prevents synthesis of threshold values of intensity..and of which these are stored in bufferspace that makes the data entry perform operations on the interface data that reciprocates data to the memory chip in which the input is stored causing alterications upon the memory input based on interpretative analysis chips that regress the data of that which is in the interface unit into its binary equivalent. As such the data is set in boundaries where input operates on the level of the interfacing data. Interfacing Data Unit (c)opyright 2007 by Seung Bum Kim === Subject: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? How BIG do letters not to be written in the sand on the moon to be visible from earth by telescopes ? So that we visitors of the moon can proof to people on the earth that we or something has been there to anybody with a telescope ? ;) Good question for astronomers me thinks ? Need help from mathematicians ? Okdokie me include alt.math too :):):) (I call this idea a space marker ;) ) Bye, Skybuck. === Subject: Re: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? How BIG do letters not to be written in the sand on the moon to be visible > from earth by telescopes ? So that we visitors of the moon can proof to people on the earth that we > or something has been there to anybody with a telescope ? ;) Good question for astronomers me thinks ? Need help from mathematicians ? Okdokie me include alt.math too :):):) (I call this idea a space marker ;) ) Bye, > Skybuck. Perhaps a series of nuclear bomb blasts to carve out huge craters that will align to form some word! Double-A === Subject: Re: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? Bit drastic... No biggy ? ;) Bye, Skybuck. === Subject: Re: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? No grass on the moon. It might work if the underlying rock is a different shade than the surface dust, like at Nazca. No biggy ? ;) They would have to be big enough to be seen by the naked eye, otherwise the idiots will claim the pictures taken through a telescope are faked. Bye, > Skybuck. === Subject: Re: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? Bit drastic... > No grass on the moon. It might work if the underlying > rock is a different shade than the surface dust, like > at Nazca. No biggy ? ;) They would have to be big enough to be seen by > the naked eye, otherwise the idiots will claim > the pictures taken through a telescope are faked. Bye, > Skybuck. Perhaps a beacon could be placed on the Moon. A powerful laser beam could be aimed at the Earth, designed to spread out enough to be harmless, but it's flashing still visible to the naked eye. It could flash out Morse code for Kilroy was here! Double-A === Subject: Re: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? No grass on the moon. It might work if the underlying > rock is a different shade than the surface dust, like > at Nazca. No biggy ? ;) They would have to be big enough to be seen by > the naked eye, otherwise the idiots will claim > the pictures taken through a telescope are faked. Bye, > Skybuck. Perhaps a beacon could be placed on the Moon. A powerful laser beam > could be aimed at the Earth, designed to spread out enough to be > harmless, but it's flashing still visible to the naked eye. It could > flash out Morse code for Kilroy was here! Even better, wouldn't require defacing the moon surface. You could place it at the bottom of a deep crater, have it solar powered so that it only activates for a short time while the moon is full, like Arthur C. Clarke's The Sentinel (2001 to you Philistines). Double-A === Subject: Logarithm question Hi all, I know you can't subtract logarithmic values, but if you have 2 log numbers which you want to subtract, is it mathematically correct to alog them, subtract them, and then log again, so you get a single log value? Specifically, the question is regarding measuring noise of a machine in an environment, and I want to get the noise of the machine itself. Thx for the answers! === Subject: Re: Logarithm question > Hi all, I know you can't subtract logarithmic values, This is mathematical nonsense. Let x and y be positive real numbers. log(x) and log(y) are also real numbers. I can certainly subtract one from the other. log(x) - log(y) is a real number. > but if you have 2 log > numbers What is a log number??? There is no such thing. There is only real numbers. ALL real numbers are the logarithm of some other real number. The real number x is the logarithm of the real number e^x. There is NO SUCH THING as log number. which you want to subtract, is it mathematically correct to alog > them, What is alog them?? This is gibberish. >subtract them, and then log again, so you get a single log value? More gibberish. What is a single log value? What would a multiple log value be??? === Subject: Re: Logarithm question Hi all, I know you can't subtract logarithmic values, but if you have 2 log > numbers which you want to subtract, is it mathematically correct to alog > them, subtract them, and then log again, so you get a single log value? > Specifically, the question is regarding measuring noise of a machine in > an environment, and I want to get the noise of the machine itself. Thx for the answers! Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Useful things, symbols. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Logarithm question > > Hi all, > > I know you can't subtract logarithmic values, but if you have 2 log > numbers which you want to subtract, is it mathematically correct to alog > them, subtract them, and then log again, so you get a single log value? > Specifically, the question is regarding measuring noise of a machine in > an environment, and I want to get the noise of the machine itself. > > Thx for the answers! Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Are you saying that log x - log y = log(x - y) ? Try x = 2, y = 1. Useful things, symbols. When used correctly. === Subject: Re: Logarithm question > Hi all, > I know you can't subtract logarithmic values, but if you have 2 log > numbers which you want to subtract, is it mathematically correct to alog > them, subtract them, and then log again, so you get a single log value? > Specifically, the question is regarding measuring noise of a machine in > an environment, and I want to get the noise of the machine itself. > Thx for the answers! Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Are you saying that log x - log y = log(x - y) ? No I am not. I was asking the op if that was what _he_ was saying. > Try x = 2, y = 1. Useful things, symbols. When used correctly. I was trying to make sense of the op's but if you have 2 log numbers which you want to subtract, is it mathematically correct to alog them, subtract them, and then log again, so you get a single log value?. I translated that into symbols to point out to the op that it was wrong. I take it that your When used correctly. indicates that you have a different translation that is correct. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Who has been stealing Gabriel Agbasi's money using The 4th Dimension? Who has been stealing Gabriel Agbasi's money using The 4th Dimension? THIS IS NOT FICTION Use Sleep Programming to know everything Definitions What is sleep? Sleep is the act of switching one's attention ( perception / focus ) from the physical world to the 4th Dimension. What is programming? To program is to issue a command, that must be followed. What is a wish ? The word WISH is a special instruction used in sleep programming. It is the act of expressing your desire to do something, experience something, or to know something. What is The Truth? The Truth is the correct answer. What is a dream? A dream is an experience that occurs while a human being's attention is switched from the physical world to the 4th Dimension. What is the human brain? The human brain is a THOUGHT ENGINE. A machine that processes THOUGHT. Sleep programming is the idea of doing a physical act ( The act of speaking - issue the WISH COMMAND) that produces a 4th Dimensional effect. Our quest for knowledge is really a search for Truth. The idea of sleep programming, which is the ability of any human being acquiring any knowledge ( that has no errors in it ) by performing the physical act of expressing his / her desire by issuing the WISH COMMAND; to know The Truth for the branch of knowledge sought. Sleep programming is the crown jewel for all quests for knowledge. The human brain has the capacity for retaining over 100 trillion terabytes of information ( it's capacity is really infinite ). Any human being can sleep program from any age. Once a human being starts sleep programming, that human being's thoughts become full of knowledge that has no errors in it. Understand this truth, Sleep Programming has been around for thousands of years. In fact, it is known in some circles as THE SECRET. Once you apply The Truth you asked for to your life in the physical world, your life on earth will be peaceful. UNDERSTAND, YOU MUST ISSUE THE WISH COMMAND : SAY THIS OUT LOUD BEFORE GOING TO SLEEP: I WISH TO KNOW THE TRUTH: [ INSERT ANY QUESTION ON ANY BRANCH OF KNOWLEDGE SOUGHT HERE ]. TRUTH ABOUT THE KNOWLEDGE SOUGHT. Try this tonight: Program your sleep by saying the following out loud before you go to sleep tonight: I WISH TO KNOW THE TRUTH: WHO HAS BEEN STEALING GABRIEL AGBASI'S MONEY USING THE 4TH DIMENSION ? THEN GO TO SLEEP. === Subject: Just found this place to get replica watches

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=== Subject: Elements of engineering electromagnetics (6/e) by N.N. looking fo manual solution for this book === Subject: Re: solutions manual for differential eqns. > Woud it be possible to get the solutions manual for: Differential Equations with Boundary Value Problems, 2nd Ed., by > Polking, Arnold ? I don't know, would it? Contact the publisher, they'll be able to tell you. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: hi hi > i need the manual soltion for the book > classiacal mechanics 3rd edition for goldstein > plzzzzzzzz > my email > freedom9235@hotmail.com > no, you need englirsh manual to teached you to write a post for internet. Black pot? -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: hi > com... > hi > i need the manual soltion for the book > classiacal mechanics 3rd edition for goldstein > plzzzzzzzz > my email > freedom9235@hotmail.com > no, you need englirsh manual to teached you to > write a post for internet. Black pot? > Hey, englirsh is Bob Marlow's native language and he is very sensitive about that! > -- > He that giveth to the poor lendeth to the Lord, and > shall be repaid, > said Mrs Fairchild, hastily slipping a shilling into > the poor woman's > hand. === Subject: Re: Differentiation > Since x isn't t at least one of them's wrong. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Differentiation hello all! I'm having trouble with this problem: > Differentiate sin(t) + (pi)cos(t) Apply these two rules: d(f + g)/dt = df/dt + dg/dt d(af)/dt = a df/dt where a is a constant. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: JSH: Teaching false math You cannot care about mathematics and not be impressed with the > simplicity of what I call surrogate factoring, and you cannot keep > sticking your head in the sand and pretend that highly intelligent > people in the mathematical field can justify ignoring it. As I pile on the research results, I remove the ability for you to > honestly say you care about mathematics, and then ignore them. Years ago I thought my prime counting function What you should do is implement your _function_ as an _algorithm_ and apply it to some biggish numbers such as these: http://www.rsa.com/rsalabs/node.asp?id=2093. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Proposal: 1 year killfile of everyone who has asked for a solutions manual > [...] > I've mentioned this before, elsewhere, but: Some dictionaries will > include fake words as a test to see if anyone is indiscriminately > copying their definitions and re-publishing them elsewhere. The best- > known example of a fake word is Dord, which was given the > definition density. (They simply took the spaces out of D or d.) > I sometimes wonder whether publishing companies do the same thing --- > fake solutions --- to demonstrate that a copyright violation has > taken place. > Webster's > dictionary around 1934 was the result of human error... > cf.: > http://en.wikipedia.org/wiki/Dord > But how do you know that that isn't a fake entry, designed to catch > copiers, eh? I guess English language experts (working in lexicography) would be the ones to consult. For what it's worth, Merriam-Webster on-line has this: ghost word: an accidental word form never in established usage; especially : one arising from an editorial or typographical error or a mistaken pronunciation (as phantomnation or dord). [1] Admittedly, ghost word could be a fake entry, as far as I know. So a ghost word is an accidental word form. This is in contrast to the story behind 'esquivalience', as told in the New Yorker: http://www.newyorker.com/archive/2005/08/29/050829ta_talk_alford Esquivalience turned up later in other dictionaries. Erin McKean, the editor-in-chief of the second edition of NOAD said: ñItÍs interesting for us that we can see their methodology, or lack thereof. ItÍs like tagging and releasing giant turtles.î [1] ghost word. Webster's Third New International Dictionary, Unabridged. Merriam-Webster, 2002. http://unabridged.merriam-webster.com (17 Sep. 2007). David Bernier === Subject: Please help me...Find solution manual... hi,,, I'm a student how have trouble in solving the exercise problem in text book. I study the Theory of vibration with application..5th... In my university,,, I have no friend who can discuss the problems, since they have graduated.... I have to have a break because of my personal trouble... Therefore, to keep up with other students, I really need the solution manual... How can I get it? please,,,help me.....send mail...to oxjgu@hotmail.com === Subject: Re: Congruence relations and algebraic residues > So for instance, say you want to analyze x^2 + y^2 = z^2, to keep > things familiar. x+y+vz = 0(mod x+y+vz), so x+y=-vz(mod x+y+vz) and squaring both sides gives x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz) x = 100, y = 100, z = 1, v = 1 x^2 + 2xy + y^2 = 40000 v^2 z^2 (mod x+y+vz) = 1 (mod 201) = 1 I'm not a mathematician, but I understand basic algebra. Am I missing some advanced algebraic concept here, or is James' math this trivially flawed? (And what about the position of his parentheses, which I copied?) === Subject: Re: Congruence relations and algebraic residues > So for instance, say you want to analyze x^2 + y^2 = z^2, to keep > things familiar. > > x+y+vz = 0(mod x+y+vz), so > > x+y=-vz(mod x+y+vz) > > and squaring both sides gives > > x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz) Which means that (x+y)^2 - (vz)^2 is divisible by (x+y+vz). > x = 100, y = 100, z = 1, v = 1 > x^2 + 2xy + y^2 = 40000 > v^2 z^2 (mod x+y+vz) = 1 (mod 201) = 1 That's a confirming instance, because (40000-1) is divisible by x+y+vz = 201. It is correct that if a = b (mod m), then a^2 = b^2 (mod m). In this case we have (x+y)^2 - (vz)^2 = (x + y + vz)(x + y - vz), and therefore (x+y)^2 = (vz)^2 (mod x+y+vz). > I'm not a mathematician, but I understand basic algebra. Am I missing > some advanced algebraic concept here, or is James' math this trivially > flawed? (And what about the position of his parentheses, which I > copied?) JSH doesn't get much right, but the position of his parentheses and the argument you quoted is correct. Google for modular arithmetic. -- Dave Seaman Oral Arguments in Mumia Abu-Jamal Case heard May 17 U.S. Court of Appeals, Third Circuit === Subject: Re: Congruence relations and algebraic residues > > So for instance, say you want to analyze x^2 + y^2 = z^2, to keep > things familiar. > > x+y+vz = 0(mod x+y+vz), so > > x+y=-vz(mod x+y+vz) > > and squaring both sides gives > > x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz) Which means that (x+y)^2 - (vz)^2 is divisible by (x+y+vz). x = 100, y = 100, z = 1, v = 1 x^2 + 2xy + y^2 = 40000 > v^2 z^2 (mod x+y+vz) = 1 (mod 201) = 1 That's a confirming instance, because (40000-1) is divisible by x+y+vz = > 201. It is correct that if a = b (mod m), then a^2 = b^2 (mod m). In > this case we have (x+y)^2 - (vz)^2 = (x + y + vz)(x + y - vz), and > therefore (x+y)^2 = (vz)^2 (mod x+y+vz). I'm not a mathematician, but I understand basic algebra. Am I missing > some advanced algebraic concept here, or is James' math this trivially > flawed? (And what about the position of his parentheses, which I > copied?) JSH doesn't get much right, but the position of his parentheses and the > argument you quoted is correct. Google for modular arithmetic. today. === Subject: Re: Congruence relations and algebraic residues So for instance, say you want to analyze x^2 + y^2 = z^2, to keep > things familiar. x+y+vz = 0(mod x+y+vz), so x+y=-vz(mod x+y+vz) and squaring both sides gives x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz) x = 100, y = 100, z = 1, v = 1 x^2 + 2xy + y^2 = 40000 > v^2 z^2 (mod x+y+vz) = 1 (mod 201) = 1 I'm not a mathematician, but I understand basic algebra. Am I missing > some advanced algebraic concept here, or is James' math this trivially > flawed? (And what about the position of his parentheses, which I > copied?) Oops, never mind. I suppose the notation indicates that both sides are to be taken mod x+y+vz, so that part works at least. === Subject: Probability Random Variables and Stochastic Processes Solutions Manual, Papoulis Stochastic Processes Solutions Manual, Papoulis - does anyone have it? === Subject: Re: Highschool Math assignment question > I have this tricky question and i need some help > A tivetan monk leaves the monastery at 7 am and takes his usual path to the top of the mountain, arriving at 7 pm. The following morning, he starts at 7am at the top and takes the same path backm arriving at the monastery at 7pm. using a grpah maybe, can you show that there must be a point on the path that the monk will cross exactly the same time of day on both days? im not sure of what kind of graph i should be using > help will be much appreciated You don't use a graph. A graph would give a plausibility argument, but it would not be rigorous. Instead, you use the intermediate value theorem. === Subject: Re: Highschool Math assignment question I have this tricky question and i need some help > A tivetan monk leaves the monastery at 7 am and takes his usual path to the top of the mountain, arriving at 7 pm. The following morning, he starts at 7am at the top and takes the same path backm arriving at the monastery at 7pm. using a grpah maybe, can you show that there must be a point on the path that the monk will cross exactly the same time of day on both days? im not sure of what kind of graph i should be using > help will be much appreciated A graph of distance (vertical) against time (horizontal). You'll sketch two lines, one goes up (-ish) from (7 am, monastery) (= the origin) to (7 pm, mountain top) (= top right-hand corner), one goes down from (7 am, mountain top) to (7 pm, monastery). Even though the graphs refer to two different days, they are to be superimposed. No matter how the graphs wiggle about they will cross one another. I hope my use of words such as up and down does cause you to muddle up the the graph paper and the topography. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Highschool Math assignment question I hope my use of words such as up and down does cause you to muddle up > the the graph paper and the topography. Or, better, _doesn't_ cause etc. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: +1600 Solutions Manual to low cost hi if at all possible could you please send me the Advanced Mechanics of Materials (6th Ed., Boresi) + Ebook === Subject: fluid mechanics solution manual needed hello, I am looking for a solution manual to: Fluid Mechanics, sixth edition by Frank M. White can anyone help me? === Subject: Re: fluid mechanics solution manual needed hello, I am looking for a solution manual to: Fluid Mechanics, sixth edition by Frank M. White can anyone help me? Yes, the publisher. If it exists they will happily send it to you. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Cute elementary algebra problem >.... >Solve x^4 - 4x = 1 exactly. >.... if by 'exactly' you mean solving it in terms of radicals then the only >method i know of to do that is 'ferrari's method' , which you'd have to be >out of your mind to attempt as its hideously complicated. No it isn't in cases like this where the reducing cubic has an easy factor. It is possible to do this with only elementary algebra one would learn > in high school (at the latest) in the U.S.... Only by inspiration. (I've sometimes told students that the best method of integrating functions is integration by inspiration, where you can guess the answer and then just check it by differentiating.) Ferrari's method leads to the difference of squares (x^2 + 1)^2 - 2(x + 1)^2 = 0 which is easy to factorize. But finding that expression without Ferrari's help would involve more inspiration than most of us could hope for. Ken Pledger. === Subject: need solution manual of Semiconductor Physics and Devices Third Edition By Donald Neamen can u please uplaod the file of Semiconductor Physics and Devices Third Edition By Donald Neamen and send me link my alternative email address is mastkk1@hotmail.com thankyou === Subject: Mass Transfer Operations (3rd Ed., Treybal) solution needed!!!!!1 any1 got the solution manual for Mass Transfer Operations (3rd Ed., Treybal) let me know and how much?? my contact is windo367@gmail.com or nlegend@gmail.com === Subject: Vector Mechanics for Engineers: Statics (8th Ed., Ferdinand P. Beer) solutions manual Vector Mechanics for Engineers: Statics (8th Ed., Ferdinand P. Beer) solutions manual. Does any one know where I can get this one please e- mail me at kidkaffen@cox.net thank you === Subject: hi hi i need this book in pdf format and Instructor's Manual of MATHEMATICAL METHODS FOR PHYSICISTS, by George Arfken (4th or 5th edition). plzzzzzz === Subject: Re: Factoring and identities [...] > OK, you claim you can factor T with the use of at most 16 > surrogates, starting with x = floor(sqrt(T)), k = 2x, > n in the range nmin to nmax [usually -8 to +1], and > the surrogates S = 2k^2 + nT. > So, please show in detail how your method of factoring > works for > T = 9524208139. > Marcus. > [I have not tried surrogate factoring of this - > I am guessing it will not work as you describe, but > here is a clear chance to prove you are right.] ( 91571 )( 104009 ) And no it did not work well with that number, as it looped through 454 > surrogates, but it didn't factor them all as I have the program just > keep going if it can't factor the surrogate. I have an expectation of a high probability of factoring with certain > values of k and n within certain ranges, but that's the theory which I > haven't yet managed to demonstrate is true with an implementation. As for stepping through in detail for any particular factorization, > I've done that before and see no reason to repeatedly step through a > new example every time some poster demands it. Here instead is my favorite example, yet again: I arbitrarily chose T, k and n. T = 732367903, k=floor(T/30) = 24412263, n = -2 S = 2k^2 + nT = 1191915704826532 = ( 2^2 )( 7 )( 73 )( 583129014103 ) The program looped through factoring S, and in this case it worked > rather quickly which is why I like this example. > f_1 = 7/2 and f_2 = 85136836059038 Because it worked with a trivial factorization of the surrogate. Remember the base equation is (x+k)^2 = y^2 + 2k^2 + nT so I find x and y from factoring 2k^2 + nT, and if you solve for them > you have that if 4f_1*f_2 = 2k^2 + nT, then x = f_1 + f_2 - k, and y = f_1 - f_2, so y=-170273672118069/2 and x=170273623293557/2 so, x+y=-24412256, which has 223 as a factor, found by checking the > gcd with T. So it factored out the smallest prime factor which has the highest > probability of being found, but you can also claim that it'd have the > highest probability of being found with a random method as well. For the long term, looking forward to the next one or two years, I will be totally impressed if you write your own program (no copying) which can factor (10^71 - 1)/9 in less than one hour, without cheating (e.g. by giving the program the smallest prime factor). around 1984. [1], [2]. A software package called PARI/gp does it in about 40 minutes on my PC. David Bernier === Subject: Re: Factoring and identities confuse everyone with this message: > Let x=15, and T=10 > x mod T = remainder of 15/10 = 5 > 2(x mod T) = 2(5) = 10 > mod is not the remainder function. In Mathematica, C/C++, Pascal, and many other languages the symbol 'mod' is >the remainder function. If the symbol sequence 'x mod y' means something >else within the context of this discussion, then please elaborate. x mod y is the class of all x+k*y (k is integer). In z = x mod y equals sign is understood as membership sign (compare with big-O notation). Equivalently mod y could be understood as a sign that we actually do comparison in Z/y*Z ring, so z and x actually denote their equivalence classes. -- |Don't believe this - you're not worthless ,gr---------.ru |It's us against millions and we can't take them all... | ue il | |But we can take them on! | @ma | | (A Wilhelm Scream - The Rip) |______________| === Subject: Re: Intersections of circles > I'm stuck on a geometry problem that I don't know how to solve > efficiently. If anyone else is interested, here's my solution. I realized that my original description was a bit backwards. I want to maximize the radius of circles on my plane that gives at most N circles which 'contain this point', for all possible 'this' points. I initialize my best-solution-so-far to +INFINITY. I loop over all my points. The best-solution-so-far is the smaller of the best-solution-so-far, and half the distance between this point and it's Nth furtherest away neighbour. The solution is then the best-solution-so-far - 1/INFINITY. Nicholas Sherlock === Subject: Re: 1/. Factor Fermats method + divisibility.************example. > 1/. Factor Fermats method + > divisibility.************example. by Donald S. McDonald, Wellington, NZ. 2/. I bought Sharp PC1500A pocket computer from 2nd > hand shop about 10 yrs ago. > My Acorn RISC-os computer crashed on election night > 17/9/05, 2 yrs ago (Helen Clark PM.) I have hardly > used it [Acorn A5000.] at all since. 3/. My programs factor integers up to 2^52 (raised to > the power of), = 45e14 in 10 minutes It should take less than 10 milliseconds on any recent PC. My 1.5GHz laptop takes 2-3 milliseconds depending on the number. Use trial division by small primes (up to 100, say), followed by the Quadratic Sieve (best) or SQUFOF (slightly slower) === Subject: Evolving Forms Through Osmosis (c)opyright 2007 by Seung Bum Kim Provided that Gravitation builds up on a specific point in which a liquid forms osmosis, the constitution of the liquid is increased and gains mass that by itself forms an independent substance of the part of the element which by its descent into gravity resists the conservation process by Gr(Mass) in Set Boundary A[where osmotic bond maintains threshold before its eminent distinction] -> Mass+ > Boundary A. Osmosis is hence the phenomena of liquid that must constitute itself where osmosis doesn't exist as a satisfied state of liquid, where the segmented liquid cannot sustain itself in the bond that is outweighed by gravitational forces of the building droplet that exists in inertia due to its exact threshold where Gr(Mass)>the composite inclination to tranfusion of substance. The osmotic bond works on the ratio of the strength of the liquid ingredient : plus coefficient build up of Mass in a condition of incline which only at solid state would anything of liquid constitution remain without coefficiency build which is naturally in geometric generation of mass..the separation proves that things need to be integrated and from this integral requirement we also have evolving forms based particularly through osmosis. Evolving Forms Through Osmosis (c)opyright 2007 by Seung Bum Kim === Subject: Re: Logarithm question OK, I'll explain the real situation, Always a good plan! -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Logarithm question >OK, I'll explain the real situation, and you decide what operation I >need. When you measure sound, you get a number, eg 60 dB. Now, I want to >have the sound of a device alone, but when I measure the sound, I get >the sound of the device + sound of surrounding. Since log distribution >is not linear, I can't simply subtract them. So, what do I need to get >only the sound of the device? I have the sound of device + surrounding, >and I have the sound of the surrounding alone. The formula for the sound >is 20*log(P2/P1). So, I'm thinking I should divide this by 20, alog, and >then I have the P2/P1. So, I take the P2/P1 of the device+surrounding, >and subract the P2/p1 of the surrounding to get the sound of the device >alone. Then I log that and *20, and I get the sound of the device. Yes >or no? Yes. If by sound you meant SPL, the formula is 10*log 10[ (P2/P1)^2 ] but the idea is the same. You might also consider asking your question in a different newsgroup, maybe rec.audio.pro or some such. === Subject: Re: Logarithm question <4ip3pwcxooms$.1rlu2b0kvz2ru$.dlg@40tude.net > Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Useful things, symbols. Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same There is NO SUCH THING as 'antilog'. There is just the log function log(x) and ITS INVERSE exp(x). I have never seen the word 'antilog' in a mathematical paper. === Subject: Re: Logarithm question Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Useful things, symbols. Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same There is NO SUCH THING as 'antilog'. Why do you say such an absurd thing? Making such an assertion harms your credibility. > There is just the log function log(x) > and ITS INVERSE exp(x). I have never seen the word 'antilog' in a > mathematical paper. I don't doubt that you've never seen the term used in a paper. But so what? Do a Google search for antilog. David W. Cantrell === Subject: Re: Logarithm question <4ip3pwcxooms$.1rlu2b0kvz2ru$.dlg@40tude.net> <20070917093929.821$tI@newsreader.com There is NO SUCH THING as 'antilog'. Why do you say such an absurd thing? > Making such an assertion harms your credibility. Certainly the word exists. There is just the log function log(x) > and ITS INVERSE exp(x). I have never seen the word 'antilog' in a > mathematical paper. I don't doubt that you've never seen the term used in a paper. But so what? I've never heard it used by a mathematician either. I've never seen it in a math textbook either. I was under the impression that the subject being discussed is mathematics. Since mathematicians do not use the word, its use merely causes confusion when we try to teach the subject of logarithms as part of mathematics. > Do a Google search for antilog. The fact that non-mathematicians use such a term is not and should not be relevant in a mathematical discussion. The use of this term is (IMO) as bad as the artificial distinction that elementary school teachers place between proper and improper fractions. It only serves to make things more confusing than they need be. === Subject: Re: Logarithm question There is NO SUCH THING as 'antilog'. Why do you say such an absurd thing? > Making such an assertion harms your credibility. Certainly the word exists. There is just the log function log(x) and ITS INVERSE exp(x). > I have never seen the word 'antilog' in a mathematical paper. I don't doubt that you've never seen the term used in a paper. But so > what? I've never heard it used by a mathematician either. > I've never seen it in a math textbook either. Again, so what? That merely reflects your limited experience. > I was under the impression that the subject being discussed is > mathematics. It is, of course. > Since mathematicians do not use the word, its use merely causes confusion > when we try to teach the subject of logarithms as part of mathematics. Do a Google search for antilog. The fact that non-mathematicians use such a term is not and should not > be relevant in a mathematical discussion. The use of this term is (IMO) as bad as The use of the term antilogarithm began with Napier, who was also the first to use the term logarithm. See , for example. You're welcome to dislike the term antilogarithm -- and logarithm, too, for that matter -- if you wish. But that doesn't keep it from being a valid mathematical term. Both of the two recent mathematical dictionaries which I happened to consult had entries for antilogarithm. David W. Cantrell === Subject: Re: Logarithm question > There is NO SUCH THING as 'antilog'. Why do you say such an absurd thing? > Making such an assertion harms your credibility. Certainly the word exists. There is just the log function log(x) and ITS INVERSE exp(x). > I have never seen the word 'antilog' in a mathematical paper. I don't doubt that you've never seen the term used in a paper. But so > what? I've never heard it used by a mathematician either. > I've never seen it in a math textbook either. Again, so what? That merely reflects your limited experience. I was under the impression that the subject being discussed is > mathematics. It is, of course. Since mathematicians do not use the word, its use merely causes confusion > when we try to teach the subject of logarithms as part of mathematics. Do a Google search for antilog. The fact that non-mathematicians use such a term is not and should not > be relevant in a mathematical discussion. The use of this term is (IMO) as bad as The use of the term antilogarithm began with Napier, who was also the first > to use the term logarithm. See , for > example. You're welcome to dislike the term antilogarithm -- and logarithm, too, for > that matter -- if you wish. But that doesn't keep it from being a valid > mathematical term. Both of the two recent mathematical dictionaries which I > happened to consult had entries for antilogarithm. Further: even if it weren't a mathematical term that circumstance should not be used to attack the op; he probably isn't a mathematician and is just using a term he is familiar with. If the term is inappropriate (and I don't think it is) by all means tell him what term he should use. Actually, I think the important thing will turn out to be what base of logarithms he is using. I think it's base ten, so antilog x =/= exp x. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Logarithm question > Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Useful things, symbols. Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same There is NO SUCH THING as 'antilog'. There is just the log function > log(x) > and ITS INVERSE exp(x). I fancy the op is talking about logs base 10. Still, that doesn't have anything to do with his assumption that log is linear. > I have never seen the word 'antilog' in a > mathematical > paper. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Logarithm question <4ip3pwcxooms$.1rlu2b0kvz2ru$.dlg@40tude.net I fancy the op is talking about logs base 10. The base is irrelevant. It is just a constant scaling factor. Why do people try to make the log function mysterious??? At least for positive real numbers, it is quite simple. === Subject: Re: Logarithm question > OK, I'll explain the real situation, Always a good idea. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Logarithm question > Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Useful things, symbols. Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same Clearly antilog(logx) - antilog(log y) = x - y . If it doesn't, what on earth does antilog mean? > as log x - log y? It's clear that I can not subtract log numbers, > because 66 - 40 <> 26, since the log distribution is not linear. I think you mean that the log _function_ is not linear, i.e., generally log ax =/= a log x log(x + y) =/= log x + log y. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Logarithm question <4ip3pwcxooms$.1rlu2b0kvz2ru$.dlg@40tude.net> On 17 Sep., 12:02, Frederick Williams Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Useful things, symbols. Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same Clearly antilog(logx) - antilog(log y) = x - y . If it doesn't, what > on earth does antilog mean? as log x - log y? It's clear that I can not subtract log numbers, > because 66 - 40 <> 26, since the log distribution is not linear. As a matter of fact, 66 - 40 *is* 26. However, when having a sound source of 40dB and another (independant, e.g. noise) source of 26dB, then they do not combine to 66dB. Instead, we may assume that the 26dB are negligible compared to the 40dB and therefore the sum is still (almost exactly) 40dB. You need to ask 10^4.0 plus 10^2.6 is 10^what -- and in fact this is not much more than 10^4 (a better result would be 10^4.017, i.e. 40.17dB. Rule of thumb: If the difference is more than 10dB, you can safely ignore the lower sound source. > I think you mean that the log _function_ is not linear, i.e., generally log ax =/= a log x log(x + y) =/= log x + log y. -- > He that giveth to the poor lendeth to the Lord, and shall be repaid, > said Mrs Fairchild, hastily slipping a shilling into the poor woman's > hand. === Subject: analyzing primes for any natural N, potential primes are these numbers: 10N + 1, 10N + 3, 10N + 7 and 10N + 9 10N + 1 is prime if for all natural A and B: 1) N is not 10AB + A + B 2) N is not 10AB + 7A + 3B + 2 3) N is not 10AB + 9A + 9B + 8 10N + 3 is prime if for all natural A and B: 1) N is not 10AB + 3A + B 2) N is not 10AB + 9A + 7B 10N + 7 is prime if for all natural A and B: 1) N is not 10AB + 7A + B 2) N is not 10AB + 9A + 3B + 2 10N + 9 is prime if for all natural A nad B: 1) N is not 10AB + 9A + B 2) N is not 10AB + 3A + 3B 3) N is not 10AB + 7A + 7B + 4 === Subject: Re: analyzing primes This is almost surely crankish. Anyway > for any natural N, potential primes are these numbers: > 10N + 1, 10N + 3, 10N + 7 and 10N + 9 10N + 1 is prime if for all natural A and B: > 1) N is not 10AB + A + B > 2) N is not 10AB + 7A + 3B + 2 > 3) N is not 10AB + 9A + 9B + 8 > Conditions not necessary, as 331 = 10*30*1+30+1 Conditions not sufficient, as 21 is not prime, yet it is easy to check that for all A, B between 1 and 2, 21 is not of this form, and it is clear that greater A and/or B will give results higher than 21 Crank eliminated, you can now return to your normal activities > 10N + 3 is prime if for all natural A and B: > 1) N is not 10AB + 3A + B > 2) N is not 10AB + 9A + 7B 10N + 7 is prime if for all natural A and B: > 1) N is not 10AB + 7A + B > 2) N is not 10AB + 9A + 3B + 2 10N + 9 is prime if for all natural A nad B: > 1) N is not 10AB + 9A + B > 2) N is not 10AB + 3A + 3B > 3) N is not 10AB + 7A + 7B + 4 === Subject: Re: analyzing primes [...] > for any natural N, potential primes are these numbers: > 10N + 1, 10N + 3, 10N + 7 and 10N + 9 10N + 1 is prime if for all natural A and B: > 1) N is not 10AB + A + B > 2) N is not 10AB + 7A + 3B + 2 > 3) N is not 10AB + 9A + 9B + 8 > Conditions not necessary, as 331 = 10*30*1+30+1 Prime 10N + 1 = 331 -> N = 33 -> seems that for N = 33 conditions 1) , 2) and 3) are met and the statement is not disproved. > Conditions not sufficient, as 21 is not prime, yet it is easy to check > that for all A, B between 1 and 2, 21 is not of this form, and it is > clear that greater A and/or B will give results higher than 21 Non-Prime 10N + 1 = 21 -> N = 2 = 10AB + 7A + 3B + 2 with A=B=0 . This contradicts condition 2) for Primes and thus does not disprove the statement (whereby the need of discussing whether 0 is a natural number might arise). > Crank eliminated Not necessarily and not sufficiently. >, you can now return to your normal activities OP might be able to do so regardless crank-elimination-concerns. might because maybe maths is one of OP's normal activities. How could s/he return in this case? Ulrich === Subject: Re: analyzing primes Ulrich Diez a .8ecrit : [...] > for any natural N, potential primes are these numbers: > 10N + 1, 10N + 3, 10N + 7 and 10N + 9 > 10N + 1 is prime if for all natural A and B: > 1) N is not 10AB + A + B > 2) N is not 10AB + 7A + 3B + 2 > 3) N is not 10AB + 9A + 9B + 8 > Conditions not necessary, as 331 = 10*30*1+30+1 Prime 10N + 1 = 331 -> N = 33 -> seems that for N = 33 conditions > 1) , 2) and 3) are met and the statement is not disproved. Yes, that was too fast. 1), for instance, implies 10N+1=100AB+10A+10B+1=(10A+1)(10B+1) is not prime, and the other forms are similar (2) factor as (10A+3)(10B+7), 3)as (10a+9)(10b+9), etc. > > Conditions not sufficient, as 21 is not prime, yet it is easy to check > that for all A, B between 1 and 2, 21 is not of this form, and it is > clear that greater A and/or B will give results higher than 21 Non-Prime 10N + 1 = 21 -> N = 2 = 10AB + 7A + 3B + 2 with A=B=0 . Yes. Actually, I believe now the conditions given could be right (I didn't check), but as they are clearly useless... > This contradicts condition 2) for Primes and thus does not > disprove the statement (whereby the need of discussing whether 0 > is a natural number might arise). > > Crank eliminated Not necessarily and not sufficiently. > > , you can now return to your normal activities OP might be able to do so regardless crank-elimination-concerns. > might because maybe maths is one of OP's normal activities. > How could s/he return in this case? Ulrich > === Subject: Re: analyzing primes > Crank eliminated, you can now return to your normal > activities I feel good with your state of vigilance. Fernando. === Subject: Re: analyzing primes <46ebd647$0$21143$7a628cd7@news.club-internet.fr> On 15 Sep, 14:56, Denis Feldmann Anyway for any natural N, potential primes are these numbers: > 10N + 1, 10N + 3, 10N + 7 and 10N + 9 10N + 1 is prime if for all natural A and B: > 1) N is not 10AB + A + B > 2) N is not 10AB + 7A + 3B + 2 > 3) N is not 10AB + 9A + 9B + 8 Conditions not necessary, as 331 = 10*30*1+30+1 Read more carefully: 10*331 + 1 is not prime. --- J K Haugland http://home.no.net/zamunda === Subject: Re: analyzing primes >Yes, but not in the way you think, more a presenting->trivialities-as-new-discoveries-type crank (reminds me >of Vincenzo Librandi). Hi Haugland, which the sense! (quale il senso !) Vincenzo Librandi vincenzo.librandweoz@alice.it === Subject: Re: analyzing primes jankrihau@hotmail.com a .8ecrit : > On 15 Sep, 14:56, Denis Feldmann This is almost surely crankish. Yes, but not in the way you think, more a presenting-trivialities-as- > new-discoveries-type crank (reminds me of Vincenzo Librandi). > > Anyway > for any natural N, potential primes are these numbers: > 10N + 1, 10N + 3, 10N + 7 and 10N + 9 > 10N + 1 is prime if for all natural A and B: > 1) N is not 10AB + A + B > 2) N is not 10AB + 7A + 3B + 2 > 3) N is not 10AB + 9A + 9B + 8 > Conditions not necessary, as 331 = 10*30*1+30+1 Read more carefully: 10*331 + 1 is not prime. Yes, I understood after having posted. But are those condition enough for primality? Probably, as they test for all reminders modulo 10, but I have not the heart to check... OK, the guy is indeed trying to reinvent the wheel, but as long as he doesn't boast of crashing the stock market overnight... --- > J K Haugland > http://home.no.net/zamunda === Subject: Re: Manual I have the following solutions manuals in pdf. If you need any of them, send me email at modernbooks(at)hotmail.com , replace (at) with @ I accept paypal payments only * Solutions manual for Analysis and Design of Analog Integrated Circuits, 4th Ed., by Gray,Hurst, Lewis, Meyer * Solutions manual for Analytical Mechanics, 7th Edition, by Fowels, Cassiday * Solutions manual for An Interactive Introduction to Mathematical Analysis, by Jonathan Lewin * Solutions manual for An Introduction to the Mathematics of Financial Derivatives, 2nd Ed.,by Neftci [ISBN:0125153929] * Solutions manual for Antenna Theory, 2nd Ed., by Balanis * Solutions manual for Antennas for all Applications, 3rd Edition, Kraus, Marhefka * Solutions manual for Applied Linear Statistical Models, 5th Ed., by Neter (Selected Sol.) * Solutions manual for Applied Numerical Analysis, 6th Edition, by Gerald, Wheatley * Solutions manual for Applied Numerical Methods with MATLAB for Engineers and Scientists,1st Ed,. by Chapra * Solutions manual for Applied Statistics and Probability for Engineers, 3rd Ed., by Montgomery, Runger (Selected Solutions) * Solutions manual for Applied Strength of Materials, 4th Edition, by Mott * Solutions manual for A Transition to Advanced Mathematics, 5th Edition, by Smith, Eggen,Andre * Solutions manual for Automatic Control Systems, 8th Edition, by Kuo, Golnaraghi * Solutions manual for A Course in Game Theory by Osborne, Rubinstein * Solutions manual for A Course in Algebraic Number Theory by Cohen * Solutions manual for Adaptive Filter Theory, 4th Edition, by Haykin * Solutions manual for Adaptive Control, 2nd. Ed., by Astrom, Wittenmark * Solutions manual for Advanced Engineering Mathematics, 8th Editoin, by Erwin Kreyszig (even solutions) * Solutions manual for Advanced Engineering Mathematics, 9th Edition, by Erwin Kreyszig (even solutions) * Solutions manual for Advanced Macroeconomics, 1st Ed., by David Romer * Solutions manual for Advanced Mathematical Concepts Precalculus With Applications by Holliday [ISBN: 0028341759] * Solutions manual for Advanced Modern Engineering Mathematics, 3rd Ed., by G. James * Solutions manual for A First Course In Differential Equations, 7th Edition, by Zill, Cullen * Solutions manual for Analog Integrated Circuit Design, 1st Ed., by Johns, Martin (text ebook and solution manual) * Solutions manual for Basic Business Statistics: Concepts and Applications, 10th Ed., by Berenson, Krehbiel, Levine (chap1-18) * Solutions manual for Basic Engineering Circuit Analysis, 7th Ed., by J. David Irwin * Solutions manual for Basic Engineering Circuit Analysis, 8th Ed., by J. David Irwin, Nelms (Missing a chapter or 2) * Solutions manual for Bioprocess Engineering Principles by Doran * Solutions manual for Calculus: Study and Solutions Guide, Vol. 1, 7th Ed., by Larson,Hostetler, Edwards * Solutions manual for Chemical and Engineering Thermodynamics, 3rd Ed., Stanley I. Sandler * Solutions manual for Chemical Engineering Volume 1, 6th Edition, by Richardson, Coulson,Backhurst, Harker * Solutions manual for College Physics, Volume 1: 7th Edition, by Serway, Faugh * Solutions manual for College Physics, Volume 2: 7th Edition, by Serway, Faughn * Solutions manual for Communications Systems, 4th Ed., by Haykin * Solutions manual for Communications Systems Engineering, 2nd Edition, by Proakis * Solutions manual for Computational Techniques for Fluid Dynamics by Srinivas, Fletcher * Solutions manual for Computer Networks, 4th Ed., by Andrew S. Tanenbaum * Solutions manual for Computer Networks: A Systems Approach, 3rd Edition, by Davie * Solutions manual for Control Systems Engineering, 4th Ed., by Norman Nise * Solutions manual for Corporate Finance, 6th Edition, by Ross * Solutions manual for C++ How to Program: Intro Object-Oriented Design with the UML, 3rd Ed., by Deitel, Nieto * Solutions manual for Calculus Early Transcendental, 5th Ed., by James Stewart * Solutions manual for Calculus - Early Transcendentals, 7th Ed., by Anton, Bivens, Davis * Solutions manual for Calculus: Graphical, Numerical, Algebraic, 3rd Ed., Waits, Finney,Demana, Kennedy * Solutions manual for Calculus: Multivariable, 5th Edition, by James Stewart * Solutions manual for Calculus: Single Variable, Early Transcendental, 5th Edition, by James Stewart * Solutions manual for Calculus, Single and Multivariable, 3rd Ed., by Hughes-Hallett,McCallum * Solutions manual for Device Electronics for * Solutions manual for Integrated Circuits 3rd Edition by Muller * Solutions manual for Differential Equations with Boundary Value Problems, 2nd Ed., by Polking, Arnold * Solutions manual for Digital And Analog Communication Systems 7th Ed., Leon W. Couch * Solutions manual for Digital Communications, 4th Edition, by Proakis * Solutions manual for Digital Communications: Fundamentals and Applications, 2nd Ed, Skylar * Solutions manual for Digital Design, 4th Edition, by Mano, Ciletti * Solutions manual for Digital Image Processing, 2nd Edition, by Gonzalez, Woods * Solutions manual for Digital Integrated Circuits, 2nd Ed., by Rabaey (Solutions ONLY for Chapters 3, 5, 6, 10) * Solutions manual for Digital Signal Processing: A Computer Based Approach, 1st Ed., by Mitra * Solutions manual for Digital Signal Processing: A Computer Based Approach, 2nd Ed., by S.Mitra * Solutions manual for Digital Signal Processing: A Computer Based Approach, 3rd Ed., by S.Mitra * Solutions manual for Digital Signal Processing: Principles, Algorithms and Applications,3rd Edition, by Proakis * Solutions manual for Discrete Time Signal Processing, 2nd Edition, Oppenheim * Solutions manual for Dynamics of Mechanical Systems by C.T.F. Ross * Solutions manual for Data and Computer Communications, 8th Edition by Stallings * Solutions manual for Database Management Systems, 3rd Ed., by Ramakrishnan, Gehrke (Sol.for Chapters 2-21, odd only) * Solutions manual for Design of Analog CMOS Integrated Circuits, 1st Edition, by Razavi Design of Analysis of Experiments, 6th Edition, Montgomery (missing chapter 6-8) * Solutions manual for Design of Machinery, 3rd Ed by Robert L. Norton * Solutions manual for Design With Operational Amplifiers and Analog Integrated Circuits, 2nd Ed., by Sergio Franco * Solutions manual for Design With Operational Amplifiers and Analog Integrated Circuits, 3rd Ed., by Sergio Franco * Solutions manual for Elementary Principles of Chemical Processes, 3rd Ed., by Felder,Rousseau * Solutions manual for Elements of Chemical Reaction Engineering, 3rd Ed., by H. Scott Fogler * Solutions manual for Engineering and Chemical Thermodynamics, by Koretsky [ISBN:0471385867] (No sol. for chapt 6) * Solutions manual for Engineering Circuit Analysis, 6th Edition, Hyat * Solutions manual for Engineering Electromagnetics, 6th Ed W. Hayt, J. Buck * Solutions manual for Engineering Electromagnetics, 7th Ed., Hayt, Buck * Solutions manual for Engineering Fluids Mechanics 7th Edition by Crowe * Solutions manual for Engineering Fluids Mechanics 8th Edition by Crowe * Solutions manual for Engineering Mathematics, 4th Ed., by John Bird * Solutions manual for Engineer Mechanics: Dynamics, 4th Ed., by Bedford * Solutions manual for Engineering Mechanics: Dynamics, 10th Ed., by Russell C. Hibbeler * Solutions manual for Engineering Mechanics: Dynamics 11th Ed. by Hibbeler * Solutions manual for Engineering Mechanics: Dynamics 5th Ed. by Meriam, Kraige * Solutions manual for Engineering Mechanics: Statics, 4th Edition - A. 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Contact me at modernbooks@hotmail.com === Subject: Polysigned , R x C , integer domain in n dimensions --> tommy notation worth some considerations (copied from another thread) > On Sep 13, 4:42 pm, tommy1729 it. > rather R x C defines a 3rd dimension rather than a > number system... > logical since if algebraicly closed R x R then R x > R = C > and therefore R x C = R x R x R = R^3 as expected. > if you find this confusing , unbelievable or > incomplete > i can assure you I am pretty sure timothy agrees > that R x C is to simple to describe P4. Dimensionally yes. The catch is we have a product to > consider and > there is no automatically defined product in the RxC > space. So an > arbittrary definition of a componentwise product > comes up but this > choice is arbitrary and is not equivalent though the > great Gene Ward > Smith did try to pronounce it so. I am open to a > fixup but I have > tried myself and it seems to be more of an iterative > self-similar > solution that is needed. Something like a Taylor > series approximation. I like your statement above Tommy and your set > theoretic style of > saying it beneath here. We are very near the domain > of information and > the issue of orthogonality in an informational domain > versus a purely > geometrical domain may have subtleties to do with > independence and > dependence. As much as the Cartesian thinker wants it > his way it may > not be so in actuality. Representationally there can > be no qualms that > there is informational equivalence. The Cartesian > product is forgone > in the polysign domains and so a slimmer family of > systems is exposed. and that R x C simply implies integer domain in 3 > dimensions. > i have already discussed P4 alot with timothy true > email and even in this forum. > i can also assure that i will start a topic soon > about such issues wich will go more into detail ... > if timothy doesnt do it before me of course. > (note : this remainds me of certain bogus > proofs/disproofs to various set theorems making the > same dimensional mistakes... my own set theory > included , however this is a bit of topic and any set > theory is independent of the polysigned numbers ) > thinking in 3d or 4d is hard and counterintuitive > at times, not to mention hard to imagine or draw. in fact , as far as i know only perelmann can > visualize the 4th dimension... In this regard it is an open problem. > a very intresting one... > unfortunately we live far away , else i would have > invited you to discuss it. > tommy1729 I hear you Tommy. But here we have a neutral open > forum with a sense > of justice. looking back at certain replies by certain individuals, i dont find much justice... i have even been critized for not explicitly defining an infinite series as not meaning 1 + 0 + 0 + 0 + 0 + ... not to mention the insults for not accepting cantor. well if you read sci.math often you will have noticed this behaviour before ... This way is even better yet since I can > scratch my balls > and you needn't know. spare me the details... but the point is that we have a long history of discussing this, and certain things are easier to explain if we meet ... since we are outside of the realm of standard notation ( dont accept cartesian and cantor as gods math ) and are introducing new concepts for which no symbols have been invented yet ( most symbols refer to cartesian or cantor for historical reasons ) and are not able to say a lot in little time , or make a drawing to clarify; i think a personal meeting will benefit us both. -Tim > to the heart of the reply: P4 belongs to C x R. but so do other number systems ( as pointed out above already ). so P4 is not neccesarily equal to C x R. to give an example of a number system that is not P4 yet is C x R : beresford numbers ( 3d - complex ) also 3d beresford is essentially the cartesian version of complex in 3d. a = 1 a^2 = 1 bc = 1 b^2 = c c^2 = b and we have the 6 cartesian axes a , -a , b , -b , c , -c which make a 3rd dimension. in in the work of professor beresford he proves beresford is isomorphic to R x C. logical since a = 1 a^2 = 1 bc = 1 b^2 = c c^2 = b are actually P3 and R = P2 so beresford is isomorphic to P2 x P3 => R x C. yet beresford =/= P4 !!! furthermore beresford has a matrixrepresentation and P4 has not ... or at least a very different one (still open problem , yet probably P4 has no matrix representation ) so since P4 AND BERESFORD are both CONSISTANT --> we must recognize that there are at least 2 kinds of integer domains in 3 dimensions that belong to R x C. in fact 3rd dimension is isomorphic to R x C , as pointed out in the very beginning of this post by me. so we can simplify: --> we must recognize that there are at least 2 kinds of integer domains in 3 dimensions we must also recognize that using cartesian , cantor and R x C logic we dont get an accurate view of dimensionality or integer domains... writing R x C does not define how many fundamental 3 dimensional integer domains exist. worse ; it implies only 1 since all 3rd dimensions are R x C and cannot be for instance C x C or R x R x R ( different from R x C ) and this gets even worse for higher dimensions... defining the amount of integer domains in 6D as ( we know we need at least one C to be integer domain ) R x R x R x R x C R x R x C x C C x C x C is not working at all !!! this lists 3 wich is totally incorrect ! if i am not mistaken there is even another 3D number ( also integer domain ) P4 has 4 signs / vectors / axes ( and less than 4 is inconsistant in 3d ) beresford has 6 signs / vectors / axes ( and more than 6 is inconsistant in 3d ; overdefined space) so we have ( tommy notation ) P4 = (3,4) beresford = (3,6) but there is a number between 4 and 6 ! assuming it is not equivalent to P4 or beresford ( still open question ) 3D tommynumbers = (3,5) which brings questions similar to prevouis P4 and dimensional number questions the tommy notation for an integer domain in 'a' dimensions is (a,b) R+ / P0 -> (0,1) real / P1 -> (1,2) complex / P3 -> (2,3) P4 -> (3,4) 3D tommynumbers -> (3,5) Beresford 3D complex -> (3,6) P5 -> (4,5) 4D tommynumbers -> (4,6) relativity numbers -> (4,7) bicomplex / Beresford 4D complex -> (4,8) P6 -> (5,6) 5D tommynumbers -> (5,7) .. of course for b there is the rule that b exists if and only if a < b for all a b =< 2a for all a >= 3 even much stronger !! ; every fundamental integer domain ( with fundamental meaning P3 and C are the same and is also equal to integer domains that are P3 in disguise by using for instance 3 different vectors which also make up the complex plane ; and the analogue in higher dimensions ) is equal to (a,b) with a < b for all a b =< 2a for all a >= 3 using the differential equations from prof beresford (easily reconstructable too) one can perform computations in almost any integer domain ... yet many questions are still open ... tommy1729 === Subject: Re: Polysigned , R x C , integer domain in n dimensions --> tommy notation > worth some considerations (copied from another thread) On Sep 13, 4:42 pm, tommy1729 it. > rather R x C defines a 3rd dimension rather than a > number system... > logical since if algebraicly closed R x R then R x > R = C > and therefore R x C = R x R x R = R^3 as expected. > if you find this confusing , unbelievable or > incomplete > i can assure you I am pretty sure timothy agrees > that R x C is to simple to describe P4. Dimensionally yes. The catch is we have a product to > consider and > there is no automatically defined product in the RxC > space. So an > arbittrary definition of a componentwise product > comes up but this > choice is arbitrary and is not equivalent though the > great Gene Ward > Smith did try to pronounce it so. I am open to a > fixup but I have > tried myself and it seems to be more of an iterative > self-similar > solution that is needed. Something like a Taylor > series approximation. I like your statement above Tommy and your set > theoretic style of > saying it beneath here. We are very near the domain > of information and > the issue of orthogonality in an informational domain > versus a purely > geometrical domain may have subtleties to do with > independence and > dependence. As much as the Cartesian thinker wants it > his way it may > not be so in actuality. Representationally there can > be no qualms that > there is informational equivalence. The Cartesian > product is forgone > in the polysign domains and so a slimmer family of > systems is exposed. and that R x C simply implies integer domain in 3 > dimensions. > i have already discussed P4 alot with timothy true > email and even in this forum. > i can also assure that i will start a topic soon > about such issues wich will go more into detail ... > if timothy doesnt do it before me of course. > (note : this remainds me of certain bogus > proofs/disproofs to various set theorems making the > same dimensional mistakes... my own set theory > included , however this is a bit of topic and any set > theory is independent of the polysigned numbers ) > thinking in 3d or 4d is hard and counterintuitive > at times, not to mention hard to imagine or draw. in fact , as far as i know only perelmann can > visualize the 4th dimension... In this regard it is an open problem. > a very intresting one... > unfortunately we live far away , else i would have > invited you to discuss it. > tommy1729 I hear you Tommy. But here we have a neutral open > forum with a sense > of justice. looking back at certain replies by certain individuals, i dont find much justice... i have even been critized for not explicitly defining an infinite series as not meaning 1 + 0 + 0 + 0 + 0 + ... not to mention the insults for not accepting cantor. well if you read sci.math often you will have noticed this behaviour before ... This way is even better yet since I can scratch my balls > and you needn't know. spare me the details... but the point is that we have a long history of discussing this, and certain things are easier to explain if we meet ... since we are outside of the realm of standard notation ( dont accept cartesian and cantor as gods math ) and are introducing new concepts for which no symbols have been invented yet ( most symbols refer to cartesian or cantor for historical reasons ) and are not able to say a lot in little time , or make a drawing to clarify; i think a personal meeting will benefit us both. -Tim to the heart of the reply: P4 belongs to C x R. but so do other number systems ( as pointed out above already ). so P4 is not neccesarily equal to C x R. to give an example of a number system that is not P4 yet is C x R : beresford numbers ( 3d - complex ) also 3d beresford is essentially the cartesian version of complex in 3d. a = 1 > a^2 = 1 > bc = 1 > b^2 = c > c^2 = b and we have the 6 cartesian axes a , -a , b , -b , c , -c which make a 3rd dimension. in in the work of professor beresford he proves beresford is isomorphic to R x C. logical since a = 1 > a^2 = 1 > bc = 1 > b^2 = c > c^2 = b are actually P3 and R = P2 so beresford is isomorphic to P2 x P3 => R x C. yet beresford =/= P4 !!! furthermore beresford has a matrixrepresentation and P4 has not ... or at least a very different one (still open problem , yet probably P4 has no matrix representation ) so since P4 AND BERESFORD are both CONSISTANT --> we must recognize that there are at least 2 kinds of integer domains in 3 dimensions that belong to R x C. in fact 3rd dimension is isomorphic to R x C , as pointed out in the very beginning of this post by me. so we can simplify: --> we must recognize that there are at least 2 kinds of integer domains in 3 dimensions we must also recognize that using cartesian , cantor and R x C logic we dont get an accurate view of dimensionality or integer domains... writing R x C does not define how many fundamental 3 dimensional integer domains exist. worse ; it implies only 1 since all 3rd dimensions are R x C and cannot be for instance C x C or R x R x R ( different from R x C ) and this gets even worse for higher dimensions... defining the amount of integer domains in 6D as ( we know we need at least one C to be integer domain ) R x R x R x R x C R x R x C x C C x C x C is not working at all !!! this lists 3 wich is totally incorrect ! if i am not mistaken there is even another 3D number ( also integer domain ) P4 has 4 signs / vectors / axes > ( and less than 4 is inconsistant in 3d ) > beresford has 6 signs / vectors / axes > ( and more than 6 is inconsistant in 3d ; overdefined space) so we have ( tommy notation ) P4 = (3,4) > beresford = (3,6) but there is a number between 4 and 6 ! assuming it is not equivalent to P4 or beresford ( still open question ) 3D tommynumbers = (3,5) which brings questions similar to prevouis P4 and dimensional number questions the tommy notation for an integer domain in 'a' dimensions is (a,b) R+ / P0 -> (0,1) > real / P1 -> (1,2) > complex / P3 -> (2,3) > P4 -> (3,4) > 3D tommynumbers -> (3,5) > Beresford 3D complex -> (3,6) > P5 -> (4,5) > 4D tommynumbers -> (4,6) > relativity numbers -> (4,7) > bicomplex / Beresford 4D complex -> (4,8) > P6 -> (5,6) > 5D tommynumbers -> (5,7) > .. of course for b there is the rule that b exists if and only if a < b for all a > b =< 2a for all a >= 3 even much stronger !! ; every fundamental integer domain ( with fundamental meaning P3 and C are the same and is also equal to integer domains that are P3 in disguise by using for instance 3 different vectors which also make up the complex plane ; and the analogue in higher dimensions ) is equal to (a,b) > with a < b for all a > b =< 2a for all a >= 3 using the differential equations from prof beresford (easily reconstructable too) one can perform computations in almost any integer domain ... yet many questions are still open ... tommy1729 Hi Tommy. I do not follow the meaning of (a,b) here. So I don't really understand the numerical pairing of the instances you give. It looks to me as if some are off by one but again I do not understand. All that I need is D = n - 1 . A P0 though is such a large concept. In a sensorial way P0 puts all at an origin with no extension whatsoever. This is a metaphysical concept of universality and unconditional contact whereby all are none. I do not see any math to be done here but it does pose that the same entity in P1 may be ordered and that there may be a P1 mapping of the universe though relativity on it poses quite a structural incoherency. Until we reach higher dimension coherency is lacking. This is somewhat brane theory and what I will argue is that Lisa Randall and her compatriots should come down in dimension and treat electromagnetics from its foundation as a progressive structure. Sorry if this does not make sense to you but the 3D math of maxwells theory finds natural support within the tatrix which also supports spacetime naturally. Such a double coherency is unlikely to be a mistake. One would think then that the relation would be easy but just as the existence of this polysign math has gone overlooked for a long time and perhaps will be overlooked for a long time to come so it may be even more time to get to a reformulation of Maxwell's equations. The relation will be puzzlingly simplistic. I think about it every day and I feel it but the translation is not here yet. I will be happy for anyone that can do it. Of course there is the hope that gravitation will be here and perhaps in the P1/P2 layer but I do not believe that this awareness will be sound guidance. It should probably come in hindsight no different than the realization that P1 is zero dimensional comes that same way. -Tim === Subject: Transporting the arithmetic. Le N be the set of natural numbers and be S a set. If f: N -> S is a bijective map, we can transport the usual algebraic structure ( N, +, *) to S in the usual manner and as a consequence we have defined on two different sets ( eventually N=S ) the same arithmetic. Example: S={ (a,n): n e N } . Transported operations on S: (a) Sum: (a,n) +' (a,n')= (a,n+n') (b) Product: (a,n)*' (a,n')= (a, n*n') Apparently, this does not add anything relevant to Peano Arithmetic unless we are able to choose S adequately. Fernando. === Subject: A not so easy problem about concavity Let X in or equal to R^n be a nonempty, convex and compact set, and consider the function f: X x X --> R defined by f(u,v) := ((a*b)/2)*|v|^2 - (a/2)*|u|^2 - (1/2)*|v-g(u)|^2, where g:X --> X is twice continuously differentiable, b is a parameter in the interval (0,1), a is a positive parameter, and |.| denotes the Euclidean norm. My question is: is it possible to find values for the parameters a and b such that f is strictly concave on u and v? Paul === Subject: how to factor gaussian integers ? how does one efficiently factor a gaussian integer ?? plz dont tell me what a gaussian prime is and how to find them ; i already know this. but despite knowing about gaussian primes, no efficient factoring method for gaussian integers ?? how about other integer domains ? factoring polysigned numbers ( coefficients integer , not real ) factoring a + b i + c j + d k with a , b , c , d integers and the number being bicomplex it is known that the integers have no easy factoring method as of 9-15-07... but worse , i dont see any other integer domain that has an easy factoring method !!!! worse ( this sounds horrible ) factoring has been somewhat overvalued and it is not well understood in any domain , or there is no good algoritm in any integer domain... almost sounds like a factoring lie which comes ugly close to JSH statements ... (not that jSH has a good method of course) does any integer domain factor easily ? tommy1729 === Subject: Re: how to factor gaussian integers ? days. My association with the Department is that of an alumnus. >how does one efficiently factor a gaussian integer ?? plz dont tell me what a gaussian prime is and how to find them ; i >already know this. but despite knowing about gaussian primes, no efficient factoring >method for gaussian integers ?? No more efficient method than for integers, because if you can factor gaussian integers, then by letting a be a rational integer, factor it as a gaussian integer, and by pairing off the conjugate non-rational gaussian primes, which can be done easily, you get a factorization of a into rational primes. If you have a method for factoring regular integers, then do the following to factor Gaussian integers: Given a+bi. (i) Compute d = gcd(a,b); factor d into integers, then factor the rational primes (by writing them as sums of two squares) that are not gaussian primes to get a factorization of this factor. So we may assume that gcd(a,b)=1. (ii) If gcd(a,b)=1, then compute a^2+b^2, and factor it into rational primes. Divide the primes into three groups: 2, those congruent to 1 modulo 4, and those congruent to 3 modulo 4: a^2 + b^2 = 2^r * p_1^{s_1}*...*p_n^{s_n}* q_1^{t_1}*...*q_m^{t_m}. with r>=0, s_i,t_j positive integers, p_i pairwise distinct primes congruent to 1 modulo 4; q_i pairwise distinct primes congruent to 3 modulo 4. (iii) The exponents t_i will all be even. Write each p_i as a sum of two squares, p_j = (d_j^2 + f_j^2). Then p_i factors in Gaussian primes as (d_j + i*f_j)(d_j - i*f_j). (iv) (a+bi)(a-bi) = a^2+b^2. Each of a+bi and a-bi will be exactly divisible by q_j^{t_j/2}. 2 will not divide a+bi (since gcd(a,b)=1), but 1+i may; check. (v) The remaining factors will be up to s_i factors of d_j + if_j or d_j - if_j in some combination. Again, not hard to figure out which and how many. >does any integer domain factor easily ? Oh, yes. Any integral domain with finitely many primes has very easy factoring algorithms, and there are plenty of those. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: Re: how to factor gaussian integers ? > <8970362.1189855862584.JavaMail.jakarta@nitrogen.mathf > orum.org>, >how does one efficiently factor a gaussian integer > ?? plz dont tell me what a gaussian prime is and how to > find them ; i >already know this. but despite knowing about gaussian primes, no > efficient factoring >method for gaussian integers ?? No more efficient method than for integers, because > if you can factor > gaussian integers, then by letting a be a rational > integer, factor it > as a gaussian integer, and by pairing off the > conjugate non-rational > gaussian primes, which can be done easily, you get a > factorization of > a into rational primes. If you have a method for factoring regular integers, > then do the > following to factor Gaussian integers: Given a+bi. (i) Compute d = gcd(a,b); factor d into integers, > then factor the > rational primes (by writing them as sums of two > squares) that are not > gaussian primes to get a factorization of this > factor. So we may > assume that gcd(a,b)=1. (ii) If gcd(a,b)=1, then compute a^2+b^2, and factor > it into rational > primes. Divide the primes into three groups: 2, those > congruent to 1 > modulo 4, and those congruent to 3 modulo 4: a^2 + b^2 = 2^r * p_1^{s_1}*...*p_n^{s_n}* > n}* q_1^{t_1}*...*q_m^{t_m}. with r>=0, s_i,t_j positive integers, p_i pairwise > distinct primes > congruent to 1 modulo 4; q_i pairwise distinct primes > congruent to 3 > modulo 4. (iii) The exponents t_i will all be even. Write each > p_i as a sum of > two squares, p_j = (d_j^2 + f_j^2). Then p_i factors > in Gaussian > primes as (d_j + i*f_j)(d_j - i*f_j). (iv) (a+bi)(a-bi) = a^2+b^2. Each of a+bi and a-bi > will be exactly > divisible by q_j^{t_j/2}. 2 will not divide a+bi > (since gcd(a,b)=1), > but 1+i may; check. (v) The remaining factors will be up to s_i factors > of d_j + if_j or > d_j - if_j in some combination. Again, not hard to > figure out which > and how many. it seems intresting. and i like berkeley :-) but somewhere you lost me ... im sorry its weekend :-) could you give an example maybe to factor 89703621729 + 11898558625841729 i > >does any integer domain factor easily ? Oh, yes. Any integral domain with finitely many > primes has very easy > factoring algorithms, and there are plenty of those. i meant efficient rather than easily ... -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes by Bill > Bill Watterson) i like that quote :-) Arturo Magidin > magidin-at-member-ams-org > tommy1729 === Subject: Re: how to factor gaussian integers ? > despite knowing about gaussian primes, no efficient factoring > method for gaussian integers ?? You write as if there were only two extremes, efficient and not-efficient, with nothing between, and no variation within either extreme. That's not the case. Efficiency is a scale. Think in terms of computational cost. Suppose you have the problem domain graded in some way as a parameter of N. Then you might have an algorithm with computational cost N or N^2 or N^3 or exp(N) or sqrt(N) or log(N) or N*log(N) etc. So the first thing you need to do is decide how to grade the problem domain for gaussian integers, for example the largest absolute value of coordinates, or the absolute value of the number, or the number of digits in the representation, etc., call that N, then describe the algorithm you know of with smallest asymptotic computational cost, state what that computational cost is as function of N, then ask for suggestions for another algorithm with smaller computational cost. === Subject: Re: how to factor gaussian integers ? > how does one efficiently factor a gaussian integer ?? plz dont tell me what a gaussian prime is and how to find them ; i already know this. but despite knowing about gaussian primes, no efficient factoring method for gaussian integers ?? Dario Alpern from Buenos Aires, Argentina has a gaussian integer factoring program, which can be tested here: http://dmoz.com.ar/GAUSSIAN.HTM Also, the source code in Java can be downloaded from the same page. The applet on page: http://dmoz.com.ar/GAUSSPR.HTM can be used to explore the gaussian primes. Note: 1997 = (34 + 29i)*(34 - 29i), and I'm not sure how one does to find this. http://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares, if p>2 is a prime in the ring Z, and p == 1 (mod 4), then p = x^2 + y^2 , some x,y in Z. So p = (x+iy)(x-iy). For euclidean domains that can be obtained as rings of integers in some quadratic field, (but nothing about efficient algorithms), see: http://www.math.niu.edu/~rusin/known-math/95/euclidean.dom David Bernier === Subject: Re: how to factor gaussian integers ? > how does one efficiently factor a gaussian integer ?? > plz dont tell me what a gaussian prime is and how to find them ; i > already know this. > but despite knowing about gaussian primes, no efficient factoring > method for gaussian integers ?? Dario Alpern from Buenos Aires, Argentina has a gaussian integer > factoring program, > which can be tested here: > http://dmoz.com.ar/GAUSSIAN.HTM Also, the source code in Java can be downloaded from the same page. The applet on page: > http://dmoz.com.ar/GAUSSPR.HTM > can be used to explore the gaussian primes. Note: 1997 = (34 + 29i)*(34 - 29i), and I'm not sure how one does to > find this. http://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares, > if p>2 is a prime in the ring Z, and p == 1 (mod 4), then p > = x^2 + y^2 , some x,y in Z. > So p = (x+iy)(x-iy). For the p = x^2 + y^2, there's Brillhart's algorithm. cf.: K.S. Williams, ``Some Refinements of an Algorithm of Brillhart, pp. 409-410 http://mathstat.carleton.ca/~williams/papers/pdf/202.pdf The first step is to solve z^2 == -1 (mod p) , with 03 is a prime conguent to 3 mod 4, how often is x+iy be a gaussian prime? > For euclidean domains that can be obtained as rings of integers in some > quadratic field, > (but nothing about efficient algorithms), see: http://www.math.niu.edu/~rusin/known-math/95/euclidean.dom David Bernier === Subject: Re: solutions manual urgent!! I have the following solutions manuals in pdf. If you need any of them, send me email at modernbooks at hotmail dot com I accept paypal payments only * Solutions manual for Analysis and Design of Analog Integrated Circuits, 4th Ed., by Gray,Hurst, Lewis, Meyer * Solutions manual for Analytical Mechanics, 7th Edition, by Fowels, Cassiday * Solutions manual for An Interactive Introduction to Mathematical Analysis, by Jonathan Lewin * Solutions manual for An Introduction to Database Systems, 8th * Solutions manual for An Introduction to the Mathematics of Financial Derivatives, 2nd Ed.,by Neftci [ISBN:0125153929] * Solutions manual for Antenna Theory, 2nd Ed., by Balanis * Solutions manual for Antennas for all Applications, 3rd Edition, Kraus, Marhefka * Solutions manual for Applied Linear Statistical Models, 5th Ed., by Neter (Selected Sol.) * Solutions manual for Applied Numerical Analysis, 6th Edition, by Gerald, Wheatley * Solutions manual for Applied Numerical Methods with MATLAB for Engineers and Scientists,1st Ed,. by Chapra * Solutions manual for Applied Statistics and Probability for Engineers, 3rd Ed., by Montgomery, Runger (Selected Solutions) * Solutions manual for Applied Strength of Materials, 4th Edition, by Mott * Solutions manual for A Transition to Advanced Mathematics, 5th Edition, by Smith, Eggen,Andre * Solutions manual for Automatic Control Systems, 8th Edition, by Kuo, Golnaraghi * Solutions manual for A Course in Game Theory by Osborne, Rubinstein * Solutions manual for A Course in Algebraic Number Theory by Cohen * Solutions manual for Adaptive Filter Theory, 4th Edition, by Haykin * Solutions manual for Adaptive Control, 2nd. Ed., by Astrom, Wittenmark * Solutions manual for Advanced Engineering Mathematics, 8th Editoin, by Erwin Kreyszig (even solutions) * Solutions manual for Advanced Engineering Mathematics, 9th Edition, by Erwin Kreyszig (even solutions) * Solutions manual for Advanced Macroeconomics, 1st Ed., by David Romer * Solutions manual for Advanced Mathematical Concepts Precalculus With Applications by Holliday [ISBN: 0028341759] * Solutions manual for Advanced Modern Engineering Mathematics, 3rd Ed., by G. James * Solutions manual for A First Course In Differential Equations, 7th Edition, by Zill, Cullen * Solutions manual for Analog Integrated Circuit Design, 1st Ed., by Johns, Martin (text ebook and solution manual) * Solutions manual for Basic Business Statistics: Concepts and Applications, 10th Ed., by Berenson, Krehbiel, Levine (chap1-18) * Solutions manual for Basic Engineering Circuit Analysis, 7th Ed., by J. David Irwin * Solutions manual for Basic Engineering Circuit Analysis, 8th Ed., by J. David Irwin, Nelms (Missing a chapter or 2) * Solutions manual for Bioprocess Engineering Principles by Doran * Solutions manual for Calculus: Study and Solutions Guide, Vol. 1, 7th Ed., by Larson,Hostetler, Edwards * Solutions manual for Chemical and Engineering Thermodynamics, 3rd Ed., Stanley I. Sandler * Solutions manual for Chemical Engineering Volume 1, 6th Edition, by Richardson, Coulson,Backhurst, Harker 5th Ed, by Marion, Thornton * Solutions manual for College Physics, Volume 1: 7th Edition, by Serway, Faugh * Solutions manual for College Physics, Volume 2: 7th Edition, by Serway, Faughn * Solutions manual for Communications Systems, 4th Ed., by Haykin * Solutions manual for Communications Systems Engineering, 2nd Edition, by Proakis * Solutions manual for Computational Techniques for Fluid Dynamics by Srinivas, Fletcher * Solutions manual for Computer Networks, 4th Ed., by Andrew S. 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White * Solutions manual for Fluid Mechanics and Thermodynamics of Turbomachinery, 5th Ed., by S. L. Dixon [ISBN: 0750678704] * Solutions manual for Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., by Cengel & Cimbala * Solutions manual for Fluid Mechanics with Engineering Applications, 10th Edition, by Finnemore * Solutions manual for Fundamentals of Aerodynamics, 3rd Edition, by J. D. Anderson, Jr. * Solutions manual for Fundamentals of Applied Electromagnetics, 1st Ed., 2001 Media Edition, by Ulaby * Solutions manual for Geometry, 04 Edition, by McGraw-Hill [ISBN: 0078296374] * Solutions manual for Guide to Energy Management, 5th Edition, by Pawlik * Solutions manual for Heat Transfer: A Practical Approach - 2nd Edition by Cengel * Solutions manual for Hydraulics in Civil and Environmental Engineering, 4th Ed., by Andrew Chadwick Applications by Oz Shy Introduction to Algorithms, 2nd Ed by Cormen, Leiserson (Selected Sol.) * Solutions manual for Introduction To Chemical Engineering Thermodynamics, 7th Ed., by Van Ness, Smith, Abbott * Solutions manual for Introduction to Electric Circuits, 6th Ed., by Dorf, Svoboda * Solutions manual for Introduction to Electric Circuits, 7th Ed., by Dorf, Svoboda * Solutions manual for Introduction to Electrodynamics, 3rd Ed. by David Griffiths * Solutions manual for Introduction to Fluid Mechanics - 5th Ed. by Fox.. * Solutions manual for Introduction to Fluid Mechanics - 6th Ed by Fox, McDonald... * Solutions manual for Introduction to Linear Algebra, 3rd Ed., by Gilbert Strang * Solutions manual for Introduction to Linear Algebra, 5th Ed., Arnold, Johnson, Riess * Solutions manual for Introduction to Probability by Grinstead, Snell (odd solutions only, not just answers but step by step solutions) * Solutions manual for Introduction to Quantum Mechanics, 2nd Ed. by Griffiths * Solutions manual for Introdution to Solid State Physics, 8th Edition by Kittel * Solutions manual for Introduction to Statistical Quality Control, 4th Edition, by Montgomery * Solutions manual for Introduction to Thermal Systems Engineering: Thermodynamics, Fluid Mechanics, and Heat Transfer by Moran, Shapiro, Munson, DeWitt * Solutions manual for Introduction to Thermal Systems Engineering, by Moran, Shapiro * Solutions manual for Linear Algebra, by J. 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Norton * Solutions manual for Managerial Accounting, 11th Ed., by Noreen, Brewer, Garrison * Solutions manual for Materials Science and Engineering: An Introduction, 6th Ed. by Callister * Solutions manual for Matrix Analysis and Applied Linear Algebra by Carl Meyer * Solutions manual for MC68HC11: An Introduction: Software/Hardware Interf, 2nd Ed, by Huang * Solutions manual for Mechanical Engineering Design, 7th Ed. by Mischke, Shigley * Solutions manual for Mechanical Vibrations, 3rd Edition, by S. S. Rao (99% same as 4th Edition, No Solutions for Chapters 6, 9, and 12) * Solutions manual for Mechanics of Fluids, 8th Ed., by Bernard Massey * Solutions manual for Mechanics of Fluids, 4th Ed., Irving H. Shames * Solutions manual for Mechanics of Fluids, 8th Ed., by Bernard Massey * Solutions manual for Mechanics of Materials - 3rd Ed. by Beer, Johnston, Dewolf * Solutions manual for Mechanics of Materials - 6th Ed. by Hibbeler * Solutions manual for Mechanics of Materials, 6th Edition by James M. Gere (missing small portion, section 8.5) * Solutions manual for Mechanics of Materials, 6th Ed., by Sturges, Morris, Riley (part of Chapt 2 is missing but only #1 thru #60) * Solutions manual for Mechanics of Solids by C.T.F. Ross Microeconomic Analysis, 3rd Ed., by H. Varian (Ans. to Exercises: Ch. 1- Ch.25) * Solutions manual for Microeconomic Theory, by Mas-Colell, Whinston, Green * Solutions manual for Microelectronic Circuit Analysis and Design, 3rd Edition, by D. 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Pozar * Solutions manual for Microwave Transistor Amplifiers: Analysis and Design, 2nd Ed., by Guillermo Gonzalez Miller & Freund's Probability and Statistics for Engineers, 7th Edition, Johnson, Miller * Solutions manual for Modern Compressible Flow, 3rd Edition, by Anderson * Solutions manual for Modern Control Engineering, 3rd Edition, by Ogata * Solutions manual for Modern Control Engineering, 4th Edition, by Ogata * Solutions manual for Modern Digital and Analog Communication Systems, 3rd Ed., by Lathi * Solutions manual for Modern Control Systems, 9th Ed., by Richard C. 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Sze * Solutions manual for Semiconductor Physics And Devices -3rd Ed. by D. Neamen * Solutions manual for Separation Process Principles, 2nd Ed., Seader, Henley * Solutions manual for Signal Processing and Linear Systems by Lathi * Solutions manual for Signals and Systems, 2nd Edition, by Haykin, Van Veen * Solutions manual for Signals and Systems, 2nd Edition, Oppenheim, Willsky, Hamid, Nawab * Solutions manual for Signals and Systems: Analysis Using Transform Methods and MATLAB, 1st Ed., by M. J. Roberts * Solutions manual for Signals, Systems, and Transforms, 3rd Ed., by Charles L. Phillips, Eve A. Riskin, John M. Parr * Solutions manual for Shigley's Mechanical Engineering Design, 8th Ed. by Budynas, Nisbett (No Sol. for Chapt 18 & 19) * Solutions manual for Simply C#: An Application-Driven Tutorial Approach, by Deitel, Hoey (Chapters 1-32) * Solutions manual for Soil Mechanics: Concepts and Applications, 2nd Ed., by Powrie * Solutions manual for Solid State Electronic Devices - 5th Ed by Streetman * Solutions manual for Solid State Electronic Devices - 6th Ed by Streetman * Solutions manual for Statics and Mechanics of Materials: An Integrated Approach, 2nd Ed., by Riley, Sturges, Morris * Solutions manual for Structural Analysis, 5th Edition, by Hibbeler * Solutions manual for University Physics 11th Edition by Young.. * Solutions manual for Vector Mechanics: Statics 7th Edition by Beer * Solutions manual for Vector Mechanics: Dynamics, 7th Ed., by Beer, Johnston, Staab, Clausen * Solutions manual for Vibrations and Stability: Advanced Theory, Analysis, and Tools, 7th Ed., by Thomsen * Solutions manual for Wireless Communications: Principles and Practice, 2nd Ed, by Rappaport * Solutions manual for Theory and Design for Mechanical Measurements, 4th Ed., Beasley, Figliola * Solutions manual for Thermal Physics, 2nd Edition, by Charles Kittel * Solutions manual for Thermal Physics, by Ralph Baierlein * Solutions manual for Thermodynamics: An Engineering Approach, 5th Ed., by Cengel, Boles (Missing solutions #118-149 of Chapter 7) * Solutions manual for Thermodynamics: An Engineering Approach, 6th Ed., by Cengel, Boles * Solutions manual for The Science and Engineering of Materials, 4th Ed., by Donald R. Askeland, Pradeep P. Phule * Solutions manual for Thomas' Calculus, Early Trans., Part 1, 10th Ed. by Thomas, Weir, Hass, Giordano * Solutions manual for Thomas' Calculus: Part 2, 10th Ed. (Multivariable, chs. 8-13), by Thomas, Weir, Hass, Giordano * Solutions manual for Thomas' Calculus, Early Trans., Part 1, 11th Ed. by Thomas, Weir, Hass, Giordano * Solutions manual for Thomas' Calculus: Part 2, 11th Ed. (Multivariable, chs. 11-16), by * Solutions manual for Thomas, Weir, Hass, Giordano * Solutions manual for Transport Phenomena, 1st Edition, by R. Byron Bird * Solutions manual for Transport Phenomena, 2nd Ed., by Bird. Contact me at modernbooks@hotmail.com === Subject: Re: NEED SOLUTION MANUAL I have the following solutions manuals in pdf. If you need any of them, send me email at modernbooks at hotmail dot com I accept paypal payments only * Solutions manual for Analysis and Design of Analog Integrated Circuits, 4th Ed., by Gray,Hurst, Lewis, Meyer * Solutions manual for Analytical Mechanics, 7th Edition, by Fowels, Cassiday * Solutions manual for An Interactive Introduction to Mathematical Analysis, by Jonathan Lewin * Solutions manual for An Introduction to Database Systems, 8th * Solutions manual for An Introduction to the Mathematics of Financial Derivatives, 2nd Ed.,by Neftci [ISBN:0125153929] * Solutions manual for Antenna Theory, 2nd Ed., by Balanis * Solutions manual for Antennas for all Applications, 3rd Edition, Kraus, Marhefka * Solutions manual for Applied Linear Statistical Models, 5th Ed., by Neter (Selected Sol.) * Solutions manual for Applied Numerical Analysis, 6th Edition, by Gerald, Wheatley * Solutions manual for Applied Numerical Methods with MATLAB for Engineers and Scientists,1st Ed,. by Chapra * Solutions manual for Applied Statistics and Probability for Engineers, 3rd Ed., by Montgomery, Runger (Selected Solutions) * Solutions manual for Applied Strength of Materials, 4th Edition, by Mott * Solutions manual for A Transition to Advanced Mathematics, 5th Edition, by Smith, Eggen,Andre * Solutions manual for Automatic Control Systems, 8th Edition, by Kuo, Golnaraghi * Solutions manual for A Course in Game Theory by Osborne, Rubinstein * Solutions manual for A Course in Algebraic Number Theory by Cohen * Solutions manual for Adaptive Filter Theory, 4th Edition, by Haykin * Solutions manual for Adaptive Control, 2nd. Ed., by Astrom, Wittenmark * Solutions manual for Advanced Engineering Mathematics, 8th Editoin, by Erwin Kreyszig (even solutions) * Solutions manual for Advanced Engineering Mathematics, 9th Edition, by Erwin Kreyszig (even solutions) * Solutions manual for Advanced Macroeconomics, 1st Ed., by David Romer * Solutions manual for Advanced Mathematical Concepts Precalculus With Applications by Holliday [ISBN: 0028341759] * Solutions manual for Advanced Modern Engineering Mathematics, 3rd Ed., by G. James * Solutions manual for A First Course In Differential Equations, 7th Edition, by Zill, Cullen * Solutions manual for Analog Integrated Circuit Design, 1st Ed., by Johns, Martin (text ebook and solution manual) * Solutions manual for Basic Business Statistics: Concepts and Applications, 10th Ed., by Berenson, Krehbiel, Levine (chap1-18) * Solutions manual for Basic Engineering Circuit Analysis, 7th Ed., by J. David Irwin * Solutions manual for Basic Engineering Circuit Analysis, 8th Ed., by J. David Irwin, Nelms (Missing a chapter or 2) * Solutions manual for Bioprocess Engineering Principles by Doran * Solutions manual for Calculus: Study and Solutions Guide, Vol. 1, 7th Ed., by Larson,Hostetler, Edwards * Solutions manual for Chemical and Engineering Thermodynamics, 3rd Ed., Stanley I. Sandler * Solutions manual for Chemical Engineering Volume 1, 6th Edition, by Richardson, Coulson,Backhurst, Harker 5th Ed, by Marion, Thornton * Solutions manual for College Physics, Volume 1: 7th Edition, by Serway, Faugh * Solutions manual for College Physics, Volume 2: 7th Edition, by Serway, Faughn * Solutions manual for Communications Systems, 4th Ed., by Haykin * Solutions manual for Communications Systems Engineering, 2nd Edition, by Proakis * Solutions manual for Computational Techniques for Fluid Dynamics by Srinivas, Fletcher * Solutions manual for Computer Networks, 4th Ed., by Andrew S. Tanenbaum * Solutions manual for Computer Networks: A Systems Approach, 3rd Edition, by Davie * Solutions manual for Control Systems Engineering, 4th Ed., by Norman Nise * Solutions manual for Corporate Finance, 6th Edition, by Ross * Solutions manual for C++ How to Program: Intro Object-Oriented Design with the UML, 3rd Ed., by Deitel, Nieto * Solutions manual for Calculus Early Transcendental, 5th Ed., by James Stewart * Solutions manual for Calculus - Early Transcendentals, 7th Ed., by Anton, Bivens, Davis * Solutions manual for Calculus: Graphical, Numerical, Algebraic, 3rd Ed., Waits, Finney,Demana, Kennedy * Solutions manual for Calculus: Multivariable, 5th Edition, by James Stewart * Solutions manual for Calculus: Single Variable, Early Transcendental, 5th Edition, by James Stewart * Solutions manual for Calculus, Single and Multivariable, 3rd Ed., by Hughes-Hallett,McCallum * Solutions manual for Device Electronics for * Solutions manual for Integrated Circuits 3rd Edition by Muller * Solutions manual for Differential Equations with Boundary Value Problems, 2nd Ed., by Polking, Arnold * Solutions manual for Digital And Analog Communication Systems 7th Ed., Leon W. 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Askeland, Pradeep P. Phule * Solutions manual for Thomas' Calculus, Early Trans., Part 1, 10th Ed. by Thomas, Weir, Hass, Giordano * Solutions manual for Thomas' Calculus: Part 2, 10th Ed. (Multivariable, chs. 8-13), by Thomas, Weir, Hass, Giordano * Solutions manual for Thomas' Calculus, Early Trans., Part 1, 11th Ed. by Thomas, Weir, Hass, Giordano * Solutions manual for Thomas' Calculus: Part 2, 11th Ed. (Multivariable, chs. 11-16), by * Solutions manual for Thomas, Weir, Hass, Giordano * Solutions manual for Transport Phenomena, 1st Edition, by R. Byron Bird * Solutions manual for Transport Phenomena, 2nd Ed., by Bird. Contact me at modernbooks@hotmail.com === Subject: Re: solution manual mccabe I have the following solutions manuals in pdf. If you need any of them, send me email at modernbooks at hotmail dot com I accept paypal payments only * Solutions manual for Analysis and Design of Analog Integrated Circuits, 4th Ed., by Gray,Hurst, Lewis, Meyer * Solutions manual for Analytical Mechanics, 7th Edition, by Fowels, Cassiday * Solutions manual for An Interactive Introduction to Mathematical Analysis, by Jonathan Lewin * Solutions manual for An Introduction to Database Systems, 8th * Solutions manual for An Introduction to the Mathematics of Financial Derivatives, 2nd Ed.,by Neftci [ISBN:0125153929] * Solutions manual for Antenna Theory, 2nd Ed., by Balanis * Solutions manual for Antennas for all Applications, 3rd Edition, Kraus, Marhefka * Solutions manual for Applied Linear Statistical Models, 5th Ed., by Neter (Selected Sol.) * Solutions manual for Applied Numerical Analysis, 6th Edition, by Gerald, Wheatley * Solutions manual for Applied Numerical Methods with MATLAB for Engineers and Scientists,1st Ed,. by Chapra * Solutions manual for Applied Statistics and Probability for Engineers, 3rd Ed., by Montgomery, Runger (Selected Solutions) * Solutions manual for Applied Strength of Materials, 4th Edition, by Mott * Solutions manual for A Transition to Advanced Mathematics, 5th Edition, by Smith, Eggen,Andre * Solutions manual for Automatic Control Systems, 8th Edition, by Kuo, Golnaraghi * Solutions manual for A Course in Game Theory by Osborne, Rubinstein * Solutions manual for A Course in Algebraic Number Theory by Cohen * Solutions manual for Adaptive Filter Theory, 4th Edition, by Haykin * Solutions manual for Adaptive Control, 2nd. 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Scott Fogler * Solutions manual for Engineering and Chemical Thermodynamics, by Koretsky [ISBN:0471385867] (No sol. for chapt 6) * Solutions manual for Engineering Circuit Analysis, 6th Edition, Hyat * Solutions manual for Engineering Electromagnetics, 6th Ed W. Hayt, J. Buck * Solutions manual for Engineering Electromagnetics, 7th Ed., Hayt, Buck * Solutions manual for Engineering Fluids Mechanics 7th Edition by Crowe * Solutions manual for Engineering Fluids Mechanics 8th Edition by Crowe * Solutions manual for Engineering Mathematics, 4th Ed., by John Bird * Solutions manual for Engineer Mechanics: Dynamics, 4th Ed., by Bedford * Solutions manual for Engineering Mechanics: Dynamics, 10th Ed., by Russell C. Hibbeler * Solutions manual for Engineering Mechanics: Dynamics 11th Ed. by Hibbeler * Solutions manual for Engineering Mechanics: Dynamics 5th Ed. by Meriam, Kraige * Solutions manual for Engineering Mechanics: Statics, 4th Edition - A. 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Norton * Solutions manual for Managerial Accounting, 11th Ed., by Noreen, Brewer, Garrison * Solutions manual for Materials Science and Engineering: An Introduction, 6th Ed. by Callister * Solutions manual for Matrix Analysis and Applied Linear Algebra by Carl Meyer * Solutions manual for MC68HC11: An Introduction: Software/Hardware Interf, 2nd Ed, by Huang * Solutions manual for Mechanical Engineering Design, 7th Ed. by Mischke, Shigley * Solutions manual for Mechanical Vibrations, 3rd Edition, by S. S. Rao (99% same as 4th Edition, No Solutions for Chapters 6, 9, and 12) * Solutions manual for Mechanics of Fluids, 8th Ed., by Bernard Massey * Solutions manual for Mechanics of Fluids, 4th Ed., Irving H. 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If a given segment is graduated by Plancklengths, and an observer @ is moving to the right as follows: --> |-|-|-|-|-|-|-|-|-|-|-|-@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| a b Clearly, observer @ will percieve graduations in region a as being rarified and graduations in region b as being enriched. Just like redshift / blueshift. If the graduations are fixed, then there is no reason why you would not expect a Doppler shift. The equivalent scenario is the situation where a staionary observer on the Earth percieves length as being probabilistically rarified or enriched as illustrated by: Sun|-------------Earth-----------------|Mars a b Observer on Earth thinks that length is rarified in region a, and enriched in region b. Equivalence Principle can be justified this way from first principles using probabilistic ideas, thereby unifying GR and QM. Allowing length to be taken as being probabilistic allows unification of QM and GR and it is practically elementary. Let |~~~~~~| represent length which is everywhere probabilistic. Let |---------| represent well behaved length, some bounded subset of R1. Let |###| represent an uncertainty. expected value of expected value of (a)*(~b) (a + ~b) |~~~~~~~~~~~~~~| = |-----------| + |#######| This : |~~~~~~~~~~~~~~| is EQUIVALENT to this : |-----------| + |#######| because the EXPECTED VALUE of this : |~~~~~~~~~~~~~~| is EQUAL to the EXPECTED VALUE of this : |-----------| + |#######| consider a length : |-----------------------------------| Any such length may be regarded as being chopped up into little Plancklengths as follows : |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| If space is taken as begin geometrically originless, then theses apparent graduations are all blurred and the length : |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| may be LEGITIMATELY regarded as being indistinguishable from : |-----------------------------------| , where the graduations are simply blurred because their position is taken as being indeterminate. That's right folks, you heard it here first : Expected Value(1) = Expected Value(2) |-|-|-|-|-|-|-|-|-|-|-|-|-|-| = |---------------------|. Space may therefore be treated as if it were either continuous or discrete. Either way is fine. Both ways will work. E(1) = expected value of length segment |---------------------------------------------------------| E(1) E(2) |---------------------------|------------------------------| E(1) E(2) E(3) |-----------------|--------------------|--------------------| E(1) E(2) E(3) E(4) |------------|--------------|---------------|----------------| E(1) E(2) E(3) E(4) E(5) |---------|----------|------------|-------------|-------------| By hypothesis: for all scales m,n __m__ ___n__ Length = E(a) = E(b) / / /______/ /______/ a=1 b=1 In other words, the expected value of length should be the same on all scales for a given piece of length. This explains conservation of energy. This same damned thing can be expressed using something like a Harris Integral. There are discrete and continuous arguments to achieve this same result. And then one can introduce a scalar coordinate system something like this : E(1,1) |---------------------------------------------------------| E(2,1) E(2,2) |---------------------------|------------------------------| E(3,1) E(3,2) E(3,3) |-----------------|--------------------|--------------------| E(4,1) E(4,2) E(4,3) E(43,4) |------------|--------------|---------------|----------------| E(5,1) E(5,2) E(5,3) E(5,4) E(5,5) |---------|----------|------------|-------------|-------------| Expanding on this idea a little, our Scalar Coordinate System can be written as follows : E(1,-1) E(1,1) |---------------------------|------------------------------| E(2,-2) E(2,-1) E(2,1) E(2,2) |-------------|--------------|---------------|----------------| E(3,-3) E(3,-2) E(3,-1) E(3,1) E(3,2) E(3,3) |--------|---------|----------|----------|----------|-----------| .....etc etc. This approach can be generalized to any number of dimensions, and any number of scales relative to the reference scale E(1). I'm not sure if this is the best possible way to characterize things because it would be nice to have a space which is geometrically originless, but I'm not sure that it is possible to write functions without a coordinate system, and the usage of coordinates implies that you will have an origin somewhere, at least for now. === Subject: Re: Equivalence Principle and how to prove it probabilistically. paraphrasing http://en.wikipedia.org/wiki/Doppler_effect For waves that travel through a medium (sound, ultrasound, etc...) the relationship between observed frequency f' and emitted frequency f is given by f ' = [ v / (v +/- v_s) ] * f where v is the speed of waves in the medium v_s is the velocity of the source f ' is observed frequency f is emitted frequency For waves that travel at the speed of light, such as radio waves, the relationship between observed frequency f ' and emitted frequency f is given by: delta f = fv / c = v / lambda f ' = f + fv / c where f is the transmitted frequency v is the velocity of the transmitter relative to the receiver in meters/second: positive when moving towards one another, negative when moving away c is the speed of light in a vacuum 3*10^8 m/s lambda is the wavelength of the transmitted wave subject to change What we need to show is that This: --> |-|-|-|-|-|-|-|-|-|-|-|-@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| is equivalent to This : Sun|-------------Earth-----------------|Mars using probabilistic methods and notions of length as described below. If a given segment is graduated by Plancklengths, and an observer @ is > moving to the right as follows: -- |-|-|-|-|-|-|-|-|-|-|-|-@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| a b Clearly, observer @ will percieve graduations in region a as being > rarified and graduations in region b as being enriched. Just like redshift > / blueshift. If the graduations are fixed, then there is no reason > why you would not expect a Doppler shift. The equivalent scenario is the situation where a > staionary observer on the Earth percieves length as being probabilistically > rarified or enriched as illustrated by: Sun|-------------Earth-----------------|Mars > a b Observer on Earth thinks that length is rarified in region a, and enriched > in region b. > Equivalence Principle can be justified this way from first principles using > probabilistic > ideas, thereby unifying GR and QM. Allowing length to be taken as being probabilistic allows unification of QM > and GR and it is practically elementary. > Let |~~~~~~| represent length which is everywhere probabilistic. > Let |---------| represent well behaved length, some bounded subset of R1. > Let |###| represent an uncertainty. > expected value of expected value of > (a)*(~b) (a + ~b) > |~~~~~~~~~~~~~~| = |-----------| + |#######| > This : > |~~~~~~~~~~~~~~| > is EQUIVALENT to this : > |-----------| + |#######| > because the EXPECTED VALUE of this : > |~~~~~~~~~~~~~~| > is EQUAL to the EXPECTED VALUE of this : > |-----------| + |#######| consider a length : > |-----------------------------------| > Any such length may be regarded as being chopped up into little > Plancklengths as follows : > |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| > If space is taken as begin geometrically originless, then theses apparent > graduations are all blurred and the length : > |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| > may be LEGITIMATELY regarded as being indistinguishable from : |-----------------------------------| , where the graduations > are simply blurred because their position is taken as > being indeterminate. > That's right folks, you heard it here first : Expected Value(1) = Expected Value(2) > |-|-|-|-|-|-|-|-|-|-|-|-|-|-| = |---------------------|. > Space may therefore be treated as if it were either continuous or discrete. > Either way is fine. Both ways will work. > E(1) = expected value of length segment > |---------------------------------------------------------| > E(1) E(2) > |---------------------------|------------------------------| > E(1) E(2) E(3) > |-----------------|--------------------|--------------------| > E(1) E(2) E(3) E(4) > |------------|--------------|---------------|----------------| > E(1) E(2) E(3) E(4) E(5) > |---------|----------|------------|-------------|-------------| > By hypothesis: for all scales m,n > __m__ ___n__ > > Length = E(a) = E(b) > / / > /______/ /______/ > a=1 b=1 > In other words, the expected value of length should be the same on all > scales for a given piece of length. This explains conservation of energy. This same damned thing can be expressed using something like a Harris > Integral. > There are discrete and continuous arguments to achieve this same result. And then one can introduce a scalar coordinate system something like this : > E(1,1) > |---------------------------------------------------------| > E(2,1) E(2,2) > |---------------------------|------------------------------| > E(3,1) E(3,2) E(3,3) > |-----------------|--------------------|--------------------| > E(4,1) E(4,2) E(4,3) E(43,4) > |------------|--------------|---------------|----------------| > E(5,1) E(5,2) E(5,3) E(5,4) E(5,5) > |---------|----------|------------|-------------|-------------| > Expanding on this idea a little, our Scalar Coordinate System can be > written as follows : > E(1,-1) E(1,1) > |---------------------------|------------------------------| > E(2,-2) E(2,-1) E(2,1) E(2,2) > |-------------|--------------|---------------|----------------| > E(3,-3) E(3,-2) E(3,-1) E(3,1) E(3,2) E(3,3) > |--------|---------|----------|----------|----------|-----------| > .....etc etc. > This approach can be generalized to any number of dimensions, and any number > of scales relative to the reference scale E(1). > I'm not sure if this is the best possible way to characterize things because > it would be nice to have a space which is geometrically originless, but I'm > not sure that it is possible to write functions without a coordinate system, > and the usage of coordinates implies that you will have an origin somewhere, > at least for now. > === Subject: Re: Equivalence Principle and how to prove it probabilistically. The claim is that the _sequence_ of expected values using scalar coordinates e.g. E(3,-3) E(3,-2) E(3,-1) E(3,1) E(3,2) E(3,3) |--------|---------|----------|----------|----------|-----------| will be _same_ whether one considers : moving -> -> observer |-|-|-|-|-|-|-|-|-|-|-|-@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| or : stationary observer Sun|-------------Earth-----------------|Mars > paraphrasing http://en.wikipedia.org/wiki/Doppler_effect For waves that travel through a medium (sound, ultrasound, etc...) the > relationship between observed frequency f' and emitted frequency f is given > by f ' = [ v / (v +/- v_s) ] * f where > v is the speed of waves in the medium > v_s is the velocity of the source > f ' is observed frequency > f is emitted frequency > For waves that travel at the speed of light, such as radio waves, the > relationship between observed frequency f ' and emitted frequency f is given > by: delta f = fv / c = v / lambda f ' = f + fv / c where > f is the transmitted frequency > v is the velocity of the transmitter relative to the receiver in > meters/second: positive when moving towards one another, negative when > moving away > c is the speed of light in a vacuum 3*10^8 m/s > lambda is the wavelength of the transmitted wave subject to change What we need to show is that This: > -- |-|-|-|-|-|-|-|-|-|-|-|-@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| > is equivalent to This : > Sun|-------------Earth-----------------|Mars using probabilistic methods and notions of length as described below. > If a given segment is graduated by Plancklengths, and an observer @ is > moving to the right as follows: -- |-|-|-|-|-|-|-|-|-|-|-|-@-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| a b Clearly, observer @ will percieve graduations in region a as being > rarified and graduations in region b as being enriched. Just like > redshift > / blueshift. If the graduations are fixed, then there is no reason > why you would not expect a Doppler shift. The equivalent scenario is the situation where a > staionary observer on the Earth percieves length as being > probabilistically > rarified or enriched as illustrated by: Sun|-------------Earth-----------------|Mars > a b Observer on Earth thinks that length is rarified in region a, and > enriched > in region b. > Equivalence Principle can be justified this way from first principles > using > probabilistic > ideas, thereby unifying GR and QM. Allowing length to be taken as being probabilistic allows unification of > QM > and GR and it is practically elementary. > Let |~~~~~~| represent length which is everywhere probabilistic. > Let |---------| represent well behaved length, some bounded subset of R1. > Let |###| represent an uncertainty. > expected value of expected value of > (a)*(~b) (a + ~b) > |~~~~~~~~~~~~~~| = |-----------| + |#######| > This : > |~~~~~~~~~~~~~~| > is EQUIVALENT to this : > |-----------| + |#######| > because the EXPECTED VALUE of this : > |~~~~~~~~~~~~~~| > is EQUAL to the EXPECTED VALUE of this : > |-----------| + |#######| consider a length : > |-----------------------------------| > Any such length may be regarded as being chopped up into little > Plancklengths as follows : > |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| > If space is taken as begin geometrically originless, then theses apparent > graduations are all blurred and the length : > |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| > may be LEGITIMATELY regarded as being indistinguishable from : |-----------------------------------| , where the graduations > are simply blurred because their position is taken as > being indeterminate. > That's right folks, you heard it here first : Expected Value(1) = Expected Value(2) > |-|-|-|-|-|-|-|-|-|-|-|-|-|-| = |---------------------|. > Space may therefore be treated as if it were either continuous or > discrete. > Either way is fine. Both ways will work. > E(1) = expected value of length segment > |---------------------------------------------------------| > E(1) E(2) > |---------------------------|------------------------------| > E(1) E(2) E(3) > |-----------------|--------------------|--------------------| > E(1) E(2) E(3) E(4) > |------------|--------------|---------------|----------------| > E(1) E(2) E(3) E(4) E(5) > |---------|----------|------------|-------------|-------------| > By hypothesis: for all scales m,n > __m__ ___n__ > > Length = E(a) = E(b) > / / > /______/ /______/ > a=1 b=1 > In other words, the expected value of length should be the same on all > scales for a given piece of length. This explains conservation of energy. This same damned thing can be expressed using something like a Harris > Integral. > There are discrete and continuous arguments to achieve this same result. And then one can introduce a scalar coordinate system something like this > : > E(1,1) > |---------------------------------------------------------| > E(2,1) E(2,2) > |---------------------------|------------------------------| > E(3,1) E(3,2) E(3,3) > |-----------------|--------------------|--------------------| > E(4,1) E(4,2) E(4,3) E(43,4) > |------------|--------------|---------------|----------------| > E(5,1) E(5,2) E(5,3) E(5,4) E(5,5) > |---------|----------|------------|-------------|-------------| > Expanding on this idea a little, our Scalar Coordinate System can be > written as follows : > E(1,-1) E(1,1) > |---------------------------|------------------------------| > E(2,-2) E(2,-1) E(2,1) E(2,2) > |-------------|--------------|---------------|----------------| > E(3,-3) E(3,-2) E(3,-1) E(3,1) E(3,2) E(3,3) > |--------|---------|----------|----------|----------|-----------| > .....etc etc. > This approach can be generalized to any number of dimensions, and any > number > of scales relative to the reference scale E(1). > I'm not sure if this is the best possible way to characterize things > because > it would be nice to have a space which is geometrically originless, but > I'm > not sure that it is possible to write functions without a coordinate > system, > and the usage of coordinates implies that you will have an origin > somewhere, > at least for now. > === Subject: Re: Topology with topological property. > X : metrizable space. > X~Y : homeomorphic. > then Y : metrizable space. > is this possible ? > Homeomorphism f:(X,d) -> Y ==> D(x,y) = d(f^-1(x), f^-1(y) metric for Y. > Yes, I see. > 1) D(x,y) >= 0, > D(x,y) =0 <=> x=y. > 2) D(x,y) = D(y,x) > 3) D(x,z) <= D(x,y) + D(y,z) > Yes, D is a metric. > But wait, there's more to do. Besides noting g = f^-1, > is a bijection from Y to X. You have to show > f:(X,d) -> (Y,D) > g:(Y,D) -> (X,d) > are contiguous, that f is bicontinuous bijection. > Yes, exact... > Since X~Y, > there is a bijection function f : X -> Y with (X,d). > Define D(x,y) = d(f^-1(x), f^-1(y) metric for Y. > I must show that f and f^-1 are continuous. > Since B_d(x, e) subset f^-1{B_D(f(x), e)} for any e>0, > Since B_D(f(x), e) subset f{B_d(x, e)} for any e>0, > f and f^-1 are continuous. > Please excuse me, I have mislead you. By the above with = instead of subset, > you'd shown X and (Y,D) are homeomorphic. > Since X and Y are homeomorphic, > Y is homeomorphic to (Y,D). > Consequently to conclude the problem that Y is metrizable, > we need to use the result of the problem, that a space > homeomorphic to a metric space is metrizable. Whoops! -- > A space (X,tau) is metrizable when there's a metric d > such that (X,tau) and (X,d) have the same topology. Thus what's needed to be shown is the topological space > Y and the metric space (Y,D) have the same topology. Yes, I try again. 1) X is said to be metrizable if there exists a metric d on the set X that induces the topology of X. 2) If d is a metric on the set X, then the collection of all e-balls B_d(x,e) for x in X and e>0, is a basis for a topology on X, called the metric topology induced by d. so, I must show that Y is the topology that induced by metric D. D(x,y) = d(f^-1(x), f^-1(y)) metric for Y. Ok, Let's go. Since X~Y, there is a homeomorphism f : (X,d) -> Y. Define D(x,y) = d(f^-1(x), f^-1(y)) metric for Y. For any open set V of Y, f^-1(V) = U is a open set of X. I can express that U = Union_{x in A} B_d(x, e_x) for some set A by basis definition. so, V = f(U) = Union_{x in A} B_D(f(x), e_x). Because, f(B_d(x,e)) = B_D(f(x),e). so, B = {B_D(y, e) | y in Y, e >0}is a basis of Y. so, Y is the metric topology induced by D. so, there exists a metric D on the set Y that induces the topology of Y. so, Y is metrizable. Really End ? === Subject: Re: It is irrational to put rational and irrational numbers on the same line. > , > Imaginary numbers have their own axis. > If irrational numbers cannot be derived from > rational numbers; why are > they said to be inbetween? >Because between any two rationals lies at least one > irrational and >between any two irrationals lies at lest one > rational. > The only things that lies between rationals are > square roots of > rationals. >In the set of reals, both the set of rationals and > the set of irratinals >are dense. >http://en.wikipedia.org/wiki/Dense_set > ~v~~ pi = sqrt( a/b ), a e N*, b e N* a= ? , b = ? Fernando. a=22^2 b=7^2 ? === Subject: Re: It is irrational to put rational and irrational numbers on the same line. > pi = sqrt( a/b ), a e N*, b e N* a= ? , b = ? Fernando. a=22^2 > b=7^2 ? It is good to know that pi is rational. Fernando. === Subject: Re: It is irrational to put rational and irrational numbers on the same line. > Imaginary numbers have their own axis. If irrational numbers cannot be derived from rational numbers; why are > they said to be inbetween? **************************************************** > Inbetween? In between...what? Who says that? What does in between > mean in this context? > I think there're a few things you should clarify to yourself, and then > to us, in order to make your question clearer. > Tonio GO TO DICTIONARY.COM : lookup 'inbetween' ********************************************************* Thanx a lot for the tip. Inwww.dictionary.com, I found that inbetween > is -noun Also, in-be.87tween.87er. 1. a person or thing that is between two > extremes, two contrasting conditions, etc.: yeses, noes, and in- > betweens; a tournament for professional, amateur, and in-between. > 2. a person who handles the intermediary steps, as in a manufacturing > or sales process. > -adjective 3. being between one thing, condition, etc., and another: a > coat for in-between weather. numbers cannot be derived from rational numbers; why are > they said to be inbetween? ....so again I ask: who says that > irrationals are inbetween? And whoever asks this he means in between > WHAT? Tonio- > Sorry, irrational numbers are NOT fractions. ****************************************************************** I'm sorry you're sorry, though I can't see what are you sorry for. Anyway, who claimed such a thing and where? Finally, I supose you meant fractions of integer numbers, otherwise one could say Sqrt(2) = 2/Sqrt(2) is a fraction... Tonio === Subject: FRee E-Books www.ebooks.adrianhosting.com === Subject: Re: FRee E-Books On Sep 15, 8:50 am, ralqara...@gmail.com R continuous F:A->R defined by F(x)= Int {[a1,x^1]X....[an,x^n]} f What is D_i_F(x),for x in the interior of A? === Subject: Re: help with differentiation > Hello everyone, need help > the question is: > A=[a1,b1]X...X[an,bn] > f:A->R continuous > F:A->R defined by > F(x)= Int {[a1,x^1]X....[an,x^n]} f What is D_i_F(x),for x in the interior of A? > For D_i, fix all variables except x_i. Then what happens? Can you do the case n=1? === Subject: Re: help with differentiation <150920071023253799%anniel@nym.alias.net.invalid > Hello everyone, need help > the question is: > A=[a1,b1]X...X[an,bn] > f:A->R continuous > F:A->R defined by > F(x)= Int {[a1,x^1]X....[an,x^n]} f What is D_i_F(x),for x in the interior of A? For D_i, fix all variables except x_i. Then what happens? Can you do the case n=1? I don't know how should I look on it,can you explain it to me please? === Subject: Re: help with differentiation <150920071023253799%anniel@nym.alias.net.invalid > Hello everyone, need help > the question is: > A=[a1,b1]X...X[an,bn] > f:A->R continuous > F:A->R defined by > F(x)= Int {[a1,x^1]X....[an,x^n]} f What is D_i_F(x),for x in the interior of A? For D_i, fix all variables except x_i. Then what happens? Can you do the case n=1? I don't know ,how should I look on it,can you explain it to me please? === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz > On Sep 13, 9:01 pm, Chris Menzel > said: On Sep 12, 3:04 pm, Chris Menzel The assertion for every prime there is a larger > one does not tell us > in advance how much work we have to do to find > that next prime. With > the successor axiom, we know exactly how much > work we have to do to > find the next number. That is a significant > difference. It is a difference (significant?) between the two > *algorithms* for > computing the next number in the two relevant > series. But (a) both > algorithms are in and of themselves very simple (my > point in the > previous post) and (b) simple or not, it's not even > clear what bearing > the complexity of those algorithms has on the > question of whether there > is a significant conceptual difference between > the two general > propositions in question, which in and of > themselves don't say anything > at all about algorithms and computation time started this thread, > I'm taking that view that mathematics may be defined > as the science of > phenomena observable in the world of computation (and > that all of the > mathematics that has the potential to be applied fits > within the scope > of that definition). So algorithms and computation > time are the > fundamental concepts on which mathematics is built. > Nonsense. If mathematics were identical to computation, mathematicians would be superfluous and mathematics defined as what computers do. The naivete in this view is so obvious that I would not bother to comment except for the perception that so many unschooled in mathematics share this belief. Tom === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz I'm taking the view that mathematics should be viewed as a science. This implies (at least) two things: 1) Statements should be falsifiable 2) We are especially interested in the world we can actually observe.<< By viewed, I take it you mean treated. Mathematics cannot be treated as a science without being a science. One cannot observe mathematical objects in any more context than one observes the letter c or the number 7. Symbols, rules and syntax allow us to map phenomena to language. Meaning, however, is independent of language in the same way that the map is independent of the territory it represents. Perhaps, you would say,no map could be drawn unless some explorer actually measured the physical terrain by some arbitrary standard--a meter stick or the length of his footsteps. Consider that if this were true (it isn't), Einstein would not have had available to him the Riemannian geometry--the 4-dimensional Pythagorean Theorem that cannot be actually observed--by which to pattern general relativity. Or consider the probabilistic models by which we explain the actual observation of the behavior of light (and thus, quantum mechanics) in Thomas Young's 2-slit experiment. One doesn't observe the path important to understanding the theory. Some little research and reflection will inform you that MOST of what we objectively know cannot be actually observed. A falsifiable scientific theory has the status of theory because it has been subjected to repeated experimental attempts to show that it doesn't hold under such and such conditions. Therefore, Newtonian mechanics is falsified as a unified theory of gravity because it fails at relativistic speeds and distances. What many here have failed to grasp, though, is that Newton's MATHEMATICAL theory is complete in its domain--indeed, we still rely on Newtonian mechanics to land on the moon. Scientific theories are never true in the sense that mathematical theorems are true. Failure to falsify does not make a theory more true. Solid theories in physical science--e.g., quantum mechanics and Darwin's theory of common ancestry--have been subjected and continue to be subjected to the most rigorous attempts to prove them wrong. These theories are considered true because they have not been shown false; after all, the next experiment--in principle though not likely--could be the falsifying result. As Karl Popper said, a scientific theory cannot be true in the absolute sense; a theory approaches verisimilitude (truth likeness) as it meets the test to falsify. It cannot be verified. Mathematical theorems, OTOH, are explicitly confined to the domain in which they are true, and the results derived therefrom explicitly live in the range of that domain. We have no idea, and cannot in principle have an idea, of an absolute limit to mathematics; it is not, however, such a puny limit as that which we actually observe. This we know, to a moral certainty. Mathematics is a liberal art. Until one grasps this fact, one does not know enough mathematics to have a view of what one personally believe mathematics should be. Tom >To say that we are not especially interested in whether the successor axiom holds when we go outrageously far beyond what we can actually observe is not the same as placing a restriction on the successor axiom. At most, it's placing a mild restriction on what we care to talk about. In hindsight, I probably shouldn't have brought it up; the idea that statements should be falsifiable is main idea we've been talking about. === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz I don't want to make any restriction on the successor axiom. But there > is a significant conceptual difference between the successor axiom and > the assertion that for every prime there is a larger prime. > The successor axiom *defines* the integers. It tells us how to build > up the integers. It doesn't involve any search to find the next > integer. But the assertion that for every prime there is a larger one > tells us that if we start searching for prime numbers larger than some > given prime number, we will eventually find one. It takes work to find > that next prime number. That is a very substantial conceptual > difference, and it justifies treating the two statements in different > ways. > Perhaps i miss something fundamental, but i don't see this at all. > It seems that you 'accept' the successor axiom, but with an 'obviously > needed additional common sense restriction'. Other people would say in > such a case that they don't accept the full successor axiom (abo is an > example). > So, from where i'm standing, you feel that it makes sense to not accept > the successor axiom, but to call it true 'in all fairness', nonetheless. > Ok, fine. That's not quite right. I'm taking the view that mathematics should be > viewed as a science. This implies (at least) two things: 1) Statements should be falsifiable > 2) We are especially interested in the world we can actually observe. To say that we are not especially interested in whether the successor > axiom holds when we go outrageously far beyond what we can actually > observe is not the same as placing a restriction on the successor > axiom. At most, it's placing a mild restriction on what we care to > talk about. In my ears, that still means 'when DP claims the successor axiom to be true, then he merely means the weaker version of it that is restricted to feasibly small numbers'. That is, you at least -interpret- the statement to mean something weaker than the meaning it has for most modern mathematicians, who make a -formalism-, and then believe the formalism. At the very least (i assume), you want external influences (such as common sense or resource bounds) to influence what conclusions still count as reasonable conclusions from a set of axioms. Correct? Wouldn't that completely change the role that formalism plays in mathematics? (They are no longer viewed/treated as completely self-contained, i.e. objective and unambiguously defined.) > In hindsight, I probably shouldn't have brought it up; the idea that > statements should be falsifiable is main idea we've been talking > about. Ok, and point well taken, but the dirt is in the details. I mean, finding out the precise consequences for mathematical formalisms is where things get interesting, isn't? > But... why doesn't that go through anymore for the statement 'for every > natural, there is a larger prime'? Or even, now that i'm at it, for > 'there is an infinitude of primes'? It can be called true 'in all > fairness', provided you mean the statement in a certain common sense way > only. As I believe I've said previously, we would be justified in saying > informally that there is an infinitude of primes as long as we > would be capable of providing the essential bounds on where a larger > prime is to be found when asked for them. Ok. -- Herman Jurjus === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz > So what have we shown? We have shown that once it is acknowledged that > mathematics does have a strong connection to truth and reality--i.e. > that mathematicians are implicitly making that claim that their > theorems are true--then Godel's theorem (the claim that no formal > system can prove its own consistency) is merely an immediate and > rather trivial implication of the notion of proof. In fact there are formal systems which can prove their own consistency, for instance, second-order (PA Successor Axiom). So there must be something incorrect in your reasoning that it is immediate and [a] rather trivial implication. Where is it? > So why do we believe that mathematics is consistent? And is it even > sensible to try to reason about the consistency of mathematics? For > starters, consider this: we simply cannot believe that we have the > ability to reason consistently about the possibility that we lack the > ability to reason consistently (and you might want to read that a > second time). So simply by virtue of the empirical fact that we are > self-aware, and aware that we can reason, we are compelled to believe > that we have the ability to reason consistently, but we cannot > reason about our own consistency, since we cannot reason about the > possibility that we lack the ability to reason consistently! So if > mathematics were innate--that is, if mathematics were part of our own > model of who we are--then we would be forced to agree that we must > believe that mathematics is consistent, and that we cannot reason > about its consistency. But that's precisely the case! One could also reason like this: (1) Mathematics which is connected to reality must be true. (2) Anything true is obviously consistent (3) Therefore mathematics must be consistent. But that's trivial (although very handwavey), and only pushes all the difficulty into determining what is mathematics and what is not. True mathematics is consistent is not very informative, and not the answer to any question that anyone (other than yourself?) is asking. Also, just some more questions... I think that you agree now that (SA) Every natural number has a successor is not falsifiable. Suppose someone proves that SA <==> phi. Is this logical equivalence falsifiable? Is that part of mathematics? === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz <46e4ff8b$0$25476$ba620dc5@text.nova.planet.nl> <46e6bb45$0$25501$ba620dc5@text.nova.planet.nl> <46e79d8d$0$25475$ba620dc5@text.nova.planet.nl> <46ea7221$0$25493$ba620dc5@text.nova.planet.nl> The successor axiom does explicitly -mention- a concrete bound, namely > 'n+1'. So, the successor axiom is more like 'for every n, there's a > prime between n and n!+1', then it is like 'for every n, there's a > prime larger than n'. Actually, that's not so. The Successor Axiom does not explicitly mention a concrete bound. It says: (x)(there exists y)Sx,y The assertion (x)(y)(y = x+ 1 ==> (there exists z)(z <= y & Sx,y) has different truth conditions than the Successor Axiom. The assertion (x)(there exists y)(there exists z)(y = x + 1 & z <= y & Sx,z) is equivalent to the Successor Axiom. But while z is bounded, y is not. === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz <46e4ff8b$0$25476$ba620dc5@text.nova.planet.nl> <46e6bb45$0$25501$ba620dc5@text.nova.planet.nl> <46e79d8d$0$25475$ba620dc5@text.nova.planet.nl> On Sep 14, 11:50 pm, david petry Many constructivists have commented on the difficulty of explaining > the constructivist view to classically trained mathematicians. It > always appears to the classical mathematicians that the > constructivists lack clarity and haven't thought enough about the > topic. Even if this were true, and so? That's like my child showing me some picture scrawl and saying that many artists have been misunderstood. I guess I'm supposed to make the connection that *you* are a constructivist, and that you belong to the happy category many constructivists. perfectly coherent position, if this is in fact your position, and coheres (imho) much better with your dictum about truth and reality. === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > This is because the > meaning of symbols created in math only refer to other symbols, or > to the actions of symbol processing systems. Really??? Let's take two simple examples. 1) The symbol i (which I thought referred ambiguously to one of the > square root of minus one). One of the square root of minus one? I can't parse that. Is that a > typo or are you saying something I just don't understand? Sorry, it is inded a typo for square roots. (But if I'd have said > i refers to *the* square root of minus one, some bright spark here > would have immediately pointed out that there are two, one the > negation of the other, but no way to distinguish which is which -- but > let that pass!!!). What symbol or action of a symbol > processing system does that refer to? It refers to the symbols: the square root of minus one. Nonsense. If i referred to those symbols then we could truly say > things like i contains 21 letters, i is only understood by English > speakers and i is never mentioned in Spanish mathematics books. Yes, quite true. My use of the word symbols does create confusion here. > Look, if I introduce C as a symbol for Curt Welch (shorthand I can > use in my diary perhaps), then C refers to YOU not your name. When I > write I met C, I'm saying I met YOU, not some symbols!!! Sure, again, very true. > Exactly similarly, when we introduce i into mathematician speak as a > short symbol for the square root of minus one, i refers to the > square root of minus one, not to symbols. I when I say i^5 = i I am > talking about the square root of minus one not some symbols of > English. And this is where all the problems begin. When we talk about me being the referent of the symbol Curt, there is no confusion about what the referent of the symbol is. It's a simple objective object which is easy to agree on it's location and form. But when you start talking as the square root of minus was something real, like my fat body is real, we have a problem. Where exactly is this thing you called i, or the square root of minus one? All you did there, was give it another name. You called it i, then you called it the square root of minus one. When in the universe can I find the true referent of these symbols? > 2) What about |R (which I thought referred to the set of real > numbers). What symbol or action of a symbol processing system does > that refer to? I don't know the symbol |R, MY best attempt as an ascii version of blackboard R. Still don't know what you mean, but it's not important. > but if it means the set of real numbers then > |R is a symbol which refers to the the symbols: the set of all real > numbers. ..... So the words > the set of all real numbers is a reference to the infinite list of > infinite strings of digits: The first is wrong, while we might buy the second. They are *entirely* > different claims. For obviously the symbols : the set of all real > numbers are NOT an infinite list of infinite strings! (They are at > most one list of 21 symbols!!!!) At root, you are confusing use and mention, as the logicians say. I was not making that mistake at all. But I was failing to communicate what I was thinking. In mathematics, what exactly is the referent of those words? We can claim the referent of the symbol i is the _concept_ of the square root of minus one and everyone understands that. But what are we really saying when we say that? What is a concept? How exactly is it defined? The bottom line is that there is no universal agreement on what a concept is. For example, if you are materialist, you believe that the only stuff that exists in the universe is the stuff the physicists tell us about. Matter. So where in all this matter, can we find the concept of the square root of minus one located? How do we define it in physical terms, like we can define what the referent of the symbols C is in physical terms? The reason I was saying the referent of the the symbol i is other symbols, is because the only thing that exists in the world which we can point to as the referent of i, is other language generated by mathematicians. Now, to address the fact that i is not a referent to the 21 letters the square root of minus one. The brain acts as a sensory data classification device. I would argue this is it's only real function in terms of the parts of the brain which produce our intelligent behavior, and all our mathematical behavior. But that's another debate. For now, we just need to understand that one very important function it has, is the ability to classify sensory patterns. What I mean by this, is that there are a virtually infinite number of patters our brains will classify as a dog. There are all sorts of smells, and sounds, and visual images that we will classify as being a dog. There area nearly infinite number of different pictures we could all classify as being a picture of a dog. The brain learns how to identify sensory data as dog data though training and experience. So, when we talk about the abstract concept of a dog what exactly are we making reference to? We are making reference to either the brain hardware which is responsible for defining this huge set of different sensory dog patterns, or we are talking about the huge set of sensory patterns themselves, or we are talking about the part of the part of the universe which is responsible for the creation of all those different patterns (all the dogs in the world and all the drawings of dogs, and all the pictures of dogs, etc.). Which of those things we are making reference to when we talk about the concept of a dog is normally never made clear in our casual use of the word concept. As a matter of fact, most people probably haven't even thought about the word concept in terms of the pattern matching powers of the brain, even though that is actually the source of all concepts - they are nothing but sensory patterns detected by the brain. So, what is the referent of the symbol i in math? We can talk as you seem to have done above, and pretend that the squire root of minus one is something that actually exists, like Curt exists or Dogs exist. This is what we are generally trained to do. But doing that is playing fast and loose with the concept of exists, and this is exactly where math drifts off from reality without anyone realizing it has drifted away from reality. What exists, is only pattern matching hardware in our brain, which is able to classify many different _real_ sensory patterns, into the concept of i. Just like the light coming from a real dog is translated into nerve impulses by the eye, and then patterns of those nerve impulses are recognized by the brain as being dog patterns, there are also many different real sensory patterns in the world we can see, and classify as the mathematical concept of i. There are all the different ways we can write or print the symbol i, there is the sequence of words: the square root of negative one, there are the spoken versions of these same words which reach our brain though the ear, there is the square root symbol with a -1 inside it. there is the square root symbol with the symbols 3-4 inside it. Just like with the millions of different sensory data patterns our brain will classify as dog patterns there are also millions of different sensory patterns which we will classify as i patterns. So, if a physical dog is the referent of the word dog, what is the referent of the mathematical symbol i? What are all the sensory patterns which will cause our brain to activate it's i concept pattern detector? The only things I know the brain will respond to for i, is lots of other language symbols. It makes no difference if those symbols are spoken, or written in a Usenet message, or a book, or if the symbols are produced in our thoughts, the only things I know of that will trigger the concept of i in my brain, is other symbols. That is why I said the referent of i, was other symbols. Now, you responded as if the referent of the symbol I was a real thing we were talking about. That type of thought is fairly standard in math. We are taught to talk as if these mathematical things, like i, and like 1, and like the set of real numbers, were all things that existed, just like Curt, and Dog exists. But the problem is, they don't exist in that way. And the fact that about 999 out of 1000 mathematicians don't understand this, is a big part of what this thread is about. It's a big part of what david is complaining about. In an attempt to fix this, david has chosen to believe that the action of a computer (or a human doing computations) is in fact the referent of all of math concepts and symbols. He has choose to believe that all mathematical concepts need to be grounded back to the action of a computation machine of some type. Or at least, all mathematical concepts worth talking about have to be grounded that way. The truth about what math is however is not so simple. The referents of most mathematical symbols are more symbols (aka language) and not computers. Though you were very correct in calling me out about saying the referent of I was the words the square root of negative one, a better way to say what I was trying to express, is something like, the referent of the concept of i, is all the different ways we can say something that means, the square root of minus i, to us. In other words, all the sensory patterns that our brain will classify as the concept of i. So, the problem david has, I think, is that he believes any mathematical concept that can't be grounded back to the behavior of a computation machine, should not be considered valid math. But, I think what mathematicians are actually doing, is considering it fair game, to ground any new concept, to other language concepts, and never worry if those concepts are rounded back to the reality of a computing machine or not. So, if you don't ground all concepts of math back to computing machines, what happens? What type of trouble can you get into? I don't know the answer to that. and I don't really care, because I'm not a mathematician. I use the parts of math I understand and which I find useful, and I don't worry about the rest. But there are a few things I see as potential danger points. If for example, you create circular definitions where a set of mathematical concepts are grounded to each other, but none of them are grounded to reality, we risk creating a reality that exists only in the words (a fairy tale), and which has no useful connection to the reality we actually exist in. But, how do we know if this reality we have created with the circular definitions can't be grounded to reality? After all, there is much about reality we still might not understand, so a concept in math that doesn't currently seem to fit reality, might turn out to be the analytical tool that will help us see our reality in a new way one day. I can agree with david in that it's a worthy pursuit to try and understand where mathematical concepts fit reality, and where it creates a fairy tale. And his attempt to ground math to the operation of computing machines might be a useful approach. I believe others have looked at doing that as well. But I don't agree with him that everything in math must be grounded to the operation of computing machines, or that this is maths true definition, which he seems to try and claim it is. Math is the work of understanding the implications of anything we can express with any type of formal language we can create and play with. However, when you look at what math seems to be working with, a lot of it seems to start with just what david is talking about, the ability of computing machines to manipulate symbols, and then it drifts more into the ability of a language machine (humans) to produce language which describes other language which only at the lowest levels, is tied to the behavior of computing machines. But them it seems to ignore a few real constraints of these language machines, like the fact that they can't produce an infinite number of symbols, or you can't invert a finite set in an infinite universe, or other things like that. I don't know here it all leads, and if I were a mathematician, it's likely I would want to explore just that, and see if it were possible to formally segment the concepts of mathematics into the ideas that do apply to our reality, and the ideas that don't. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz > Many constructivists have commented on the difficulty of explaining > the constructivist view to classically trained mathematicians. It > always appears to the classical mathematicians that the > constructivists lack clarity and haven't thought enough about the > topic. Can you name a few constructivists who have commented thus? As far as I have noticed, real constructivists don't encounter the problems you have. I have often wondered why. -- I was driving down the interstate through Winslow, Arizona, I had Seven Vices on my mind -- Sloth and Avarice, Fornication, Television, Whiskey, Beer and Wine. -- Austin Lounge Lizards === Subject: Re: Godel's proof, truth, reality, self-awareness, and all that jazz Many constructivists have commented on the > difficulty of explaining > the constructivist view to classically trained > mathematicians. It > always appears to the classical mathematicians that > the > constructivists lack clarity and haven't thought > enough about the > topic. Can you name a few constructivists who have commented > thus? As far as I have noticed, real constructivists don't > encounter the > problems you have. I have often wondered why. > -- > I was driving down the interstate through Winslow, > Arizona, > I had Seven Vices on my mind -- > Sloth and Avarice, Fornication, Television, > Whiskey, Beer and Wine. -- Austin Lounge > Lizards Indeed. DP's concrete objects are not the point of constructive proofs, which are concerned merely with that group of statements that can be constructed in a finite number of steps. I think what DP among others has failed to understand, is that constructivists are still mathematicians. Many researchers, particularly analysts who are concerned with the nature of the continuum, could be considered constructivists in some sense; e.g., Dedekind, Weierstrass, Weyl. Brouwer's own Intuitionist program grew out of a desire for more rigor in his early nonconstructive results, even as important and well regarded as they were. I think part of the problem may be a failure to understand what constitutes a proof. Proof theory itself has rules for compelling correspondence between theorem and proof, and some strategies are more compelling than others. Constructivists found that strategies relying on double negation (proving a statement true by assuming it is false)are logically equivalent to proving A by showing it is not-not-A. Because A and not-not-A are identical, the tautology is unconvincing to many who would rather see the proof proceed by some other strategy. In fact, Errett Bishop, I'm told, did show that most so-called classical theorems can be proved constructively. Tom === Subject: finding limits? This fraction: (8x^4 + 2x) / (3x^4 - 7) has the limit 8/3 when x goes to infinity. The is easily realized if its divided by x^4. But if I were to apply the rules for each part of the fraction I would start with the parts in the numerator and conclude that it goes to infinity since: 8x^4 and 2x goes to infinity for x going to infinity. Next the denominator would also go to infinity since 3x^4 goes to infinity and infinity minus 7 equals infinity. Therefore I would have a new fraction: inf/inf = 1 ..why does it only give the correct result when dividing through with x^4? === Subject: Re: finding limits? > This fraction: (8x^4 + 2x) / (3x^4 - 7) has the limit 8/3 when x goes to infinity. The is easily realized if its > divided by x^4. You mean top and bottom are divided by x^4. > But if I were to apply the rules for each part of the fraction I would start > with the parts in the numerator and conclude that it goes to infinity > since: 8x^4 and 2x goes to infinity for x going to infinity. Next the denominator would also go to infinity since 3x^4 goes to infinity > and infinity minus 7 equals infinity. Therefore I would have a new > fraction: inf/inf = 1 No, oo/oo is undefined just as 0/0 is undefined. Here's an easier example: x^2/x, which gives inf/inf = 1. But obviously x^2/x = x -> oo as x -> oo. > ..why does it only give the correct result when dividing through with x^4? === Subject: Re: finding limits? On 15 Sep, 18:06, The World Wide Wade divided by x^4. You mean top and bottom are divided by x^4. But if I were to apply the rules for each part of the fraction I would start > with the parts in the numerator and conclude that it goes to infinity > since: 8x^4 and 2x goes to infinity for x going to infinity. Next the denominator would also go to infinity since 3x^4 goes to infinity > and infinity minus 7 equals infinity. Therefore I would have a new > fraction: inf/inf = 1 No, oo/oo is undefined just as 0/0 is undefined. Here's an easier > example: x^2/x, which gives inf/inf = 1. But obviously x^2/x = x - oo as x -> oo. ..why does it only give the correct result when dividing through with x^4? Is Cauchy's epsilon-delta treatment of limits cumpletely outmoded? === Subject: Re: finding limits? > Next the denominator would also go to infinity since 3x^4 goes to infinity > and infinity minus 7 equals infinity. Therefore I would have a new > fraction: inf/inf = 1 ..why does it only give the correct result when dividing through with x^4? What is / exactly? A function perhaps? What's its domain? Is inf in that domain? Kiuhnm === Subject: Re: finding limits? > This fraction: (8x^4 + 2x) / (3x^4 - 7) has the limit 8/3 when x goes to infinity. The is easily realized if its > divided by x^4. But if I were to apply the rules for each part of the fraction I would start > with the parts in the numerator and conclude that it goes to infinity > since: 8x^4 and 2x goes to infinity for x going to infinity. Next the denominator would also go to infinity since 3x^4 goes to infinity > and infinity minus 7 equals infinity. Therefore I would have a new > fraction: inf/inf = 1 inf/inf != 1. Logically speaking, if lim_x->oo f(x) = oo, then f(x) grows without bound. Here, both the denominator and the numerator grow without bound. The actual limit depends on whether or not the numerator grows faster than the denominator, vice versa, or they grow only by a constant factor of one another. L'H.99pital's Rule must be applied here to give the correct answer. It states that when the limit is +/-inf/inf or 0/0, the limit can be determined by taking the limit of the derivative. Repeated applications: (32x^3 + 2) / (12x^3) (96x^2) / (36x^2) = 96/36 = 8/3, which is the correct answer. === Subject: Re: finding limits? > This fraction: (8x^4 + 2x) / (3x^4 - 7) has the limit 8/3 when x goes to infinity. The is easily realized if its > divided by x^4. But if I were to apply the rules for each part of the fraction I would start > with the parts in the numerator and conclude that it goes to infinity > since: 8x^4 and 2x goes to infinity for x going to infinity. Next the denominator would also go to infinity since 3x^4 goes to infinity > and infinity minus 7 equals infinity. Therefore I would have a new > fraction: inf/inf = 1 inf/inf != 1. Logically speaking, if lim_x->oo f(x) = oo, then f(x) > grows without bound. Here, both the denominator and the numerator grow > without bound. The actual limit depends on whether or not the numerator > grows faster than the denominator, vice versa, or they grow only by a > constant factor of one another. L'H.99pital's Rule must be applied here to give the correct answer. Certainly not. In fact it's almost always the case that an indeterminate form can be handled better, with more understanding, without L'H.99pital's Rule. > It > states that when the limit is +/-inf/inf or 0/0, the limit can be > determined by taking the limit of the derivative. Repeated applications: (32x^3 + 2) / (12x^3) > (96x^2) / (36x^2) = 96/36 = 8/3, which is the correct answer. === Subject: Re: finding limits? > This fraction: > > (8x^4 + 2x) / (3x^4 - 7) > > has the limit 8/3 when x goes to infinity. The is easily realized if its > divided by x^4. > > But if I were to apply the rules for each part of the fraction I would > start with the parts in the numerator and conclude that it goes to > infinity since: > > 8x^4 and 2x goes to infinity for x going to infinity. > > Next the denominator would also go to infinity since 3x^4 goes to > infinity and infinity minus 7 equals infinity. Therefore I would have a > new fraction: > > inf/inf = 1 inf/inf != 1. Logically speaking, if lim_x->oo f(x) = oo, then f(x) > grows without bound. Here, both the denominator and the numerator grow > without bound. The actual limit depends on whether or not the numerator > grows faster than the denominator, vice versa, or they grow only by a > constant factor of one another. L'H.99pital's Rule must be applied here to give the correct answer. It > states that when the limit is +/-inf/inf or 0/0, the limit can be > determined by taking the limit of the derivative. Repeated applications: (32x^3 + 2) / (12x^3) > (96x^2) / (36x^2) = 96/36 = 8/3, which is the correct answer. Ok but still the hack to divide through with x^4 is also correct right? Dividing through with x^4 gives: (8 + 2/x^3)/(3 - 7/x^4) The operands containing x goes to 0 for x going to infinity and I therefore also get the correct answer 8/3. Or is that illegal? === Subject: Re: finding limits? > Ok but still the hack to divide through with x^4 is also correct right? > Dividing through with x^4 gives: (8 + 2/x^3)/(3 - 7/x^4) The operands containing x goes to 0 for x going to infinity and I therefore > also get the correct answer 8/3. Or is that illegal? You are correct there. === Subject: Re: finding limits? > Ok but still the hack to divide through with x^4 is also correct right? > Dividing through with x^4 gives: > > (8 + 2/x^3)/(3 - 7/x^4) > > The operands containing x goes to 0 for x going to infinity and I > therefore also get the correct answer 8/3. Or is that illegal? You are correct there. What if the case is infinity/integer (still assuming that I don't know of L'Hopital): 5x^3 + 2x - 13 / 7x - 4 I then divide through with x and get: (5x^2 + 2 - 13/x) / (7 - 4/x) this gives: infinity/7 I would conclude that the end result is infinity but I can't find any lemmas in my books that confirms this. === Subject: Re: finding limits? > What if the case is infinity/integer (still assuming that I don't know of > L'Hopital): I would conclude that the end result is infinity but I can't find any lemmas > in my books that confirms this. Yes, the answer is infinity. k/infinity = 0 k/0 = infinity infinity/k = infinity k^infinity = infinity unless k = 1 infinity^k = infinity unless k = 0 infinity-k = infinity These seven: infinity/infinity 0/0 infinity*0 1^infinity infinity^0 0^0 infinity - infinity can be reduced to various forms; all others can be reduced to 0 or infinity (i.e., infinitesimally small or grows without bound). === Subject: Re: finding limits? > What if the case is infinity/integer (still assuming that I don't know > of L'Hopital): I would conclude that the end result is infinity but I can't find any > lemmas in my books that confirms this. Yes, the answer is infinity. > k/infinity = 0 > k/0 = infinity I suppose you could be thinking about the _unsigned_ infinity of the one-point extension of the reals; if so, then you could say k/0 = unsigned infinity unless k = 0. But normally, in this context, the two-point extension of the reals is used. Then we cannot determine whether the desired limit is -infinity or +infinity unless we know the sign of k and also how the denominator approaches 0. Henceforth, I assume that the two-point extension is being used, so that we're distinguishing between -infinity and +infinity. > infinity/k = infinity Again, more information is needed: For finite k, +infinity/k = -infinity if k < 0, +infinity if k > 0 -infinity/k = +infinity if k < 0, -infinity if k > 0 > k^infinity = infinity unless k = 1 Rather, k^+infinity = +infinity if k > 1, 0 if 0 < k < 1 k^-infinity = 0 if k > 1, +infinity if 0 < k < 1 > infinity^k = infinity unless k = 0 Rather, +infinity^k = +infinity if k > 0, 0 if k < 0 David W. Cantrell > infinity-k = infinity These seven: > infinity/infinity > 0/0 > infinity*0 > 1^infinity > infinity^0 > 0^0 > infinity - infinity can be reduced to various forms; all others can be reduced to 0 or > infinity (i.e., infinitesimally small or grows without bound). === Subject: Re: star people <_nTFi.199232$p7.103748@fe2.news.blueyonder.co.uk The universal skeptic wishes to > claim truth for a theory that > denies man's ability to arrive > at truth, and this puts the > skeptic in the unenviable > position of uttering > nonsense. > Kewl! You really should take on L. Zick the next > time he sobers up enough to > start a new blossom thread. Well I have been reading the construction of the OP and trying to come up with something clever. But here is a frontal assault on skepticism that does not hold broad merit and I will explain why. Perhaps instead of 'universal skepticism' the author should have used 'absolutist skepticism' and then the self contradiction would be more clearly stated. For a long time I have drawn a clear line between the skeptic and the pessimist. I won't go into great detail here but essentially my view on skepticism is that the skeptic is capable of treating problems as open. It is this willingness to go vacuously into a problem that defines the skeptic. All human knowledge is constructed and it is the skeptic who can see the layers and while a dim view may be taken on the whole it is the skeptic who see the brightest places in the pile. We deal in such an accumulation of knowledge that the need to dismiss much of it for clear thought is self evident. We humans have limited awareness. The more things we cram in and have gears spinning for in our minds independent of one another the more toxic the system becomes. A possibility of collapsing the accumulation and finding a better route should be the driving force of every skeptic. This may mean that the careful skeptic has little to say because of his own uncertainties. There is a factor of stifling here that probably needs to be balanced with a willingness to err. Fundamental understanding may be wrong. For the human to know is merely to believe strongly. We start from a blank state and what we learn along the way is all done by mimicry especially if you believe in the accuracy of human knowledge. Variations must be taken to progress. I suppose the non- skeptic grows a twig out at the end of the tree and hopes for lots of sunlight. Perhaps this is a good role, but the branch that you sprout from is extremely critical to your growth and so on down to the roots of the tree. As an accumulant in a system one who isolates themselves up in the canopy seeking food will be different than the one who walks the ground at the base of the tree studying its roots and witnessing its structure. My simplest statement on galathaea's puzzle is that we are free to construct and if anything this act of construction deserves more practice and encouragement. To weigh such constructions on their consequences alone is not enough. A slender and clean basis will be preferred as well. In this regard modern physics falls short. The collage of constructions that have accumulated are toxic to the skeptic. -Tim === Subject: Re: Equivalence of N and R? it, you can't work with it, you can't communicate it _excepting_ > this information is encoded in a finite formula. That is true, but the mere fact that you can't work with it or > communicate it > IS NOT SUFFICIENT TO STOP IT from EXISTING! If something is existing there must be a domain in which it is existing. I think we are agreed about the fact that the extension of infinite information can't exist in the domain of the mind. So you must take a platonic standpoint to argue for the existence of unthinkable things? Or you take the formalistic standpoint and then you have to think that the extension of an idea is irrelevant? > If you allow that it does not exist, you get far WORSE consequences > than > admitting the existence of something we can't communicate or work > with, > namely, INCONSISTENCY. Why inconsistency? I feel that the problem lies herein, that the domain, in which the mathematics (e.g. ZFC) is working in, is not definable proper? But is this a real problem? And is Goedel's proof not showing that the domain of mathematics (e.g. ZFC) isn't definable proper (incompleteness)? Albrecht Storz === Subject: Re: Treating Magnitude as Fundamental On 15 Sep., 03:46, Timothy Golden BandTechnology.com If I get it right, you essentilly start with P, the set of positive > reals (or possibly > the set of positive rationals or of some other subfield F of R), for me I call this magnitude(x). add symbols s_0, ..., s_{n-1} such that > 1) s_0 = 1 > 2) s_i * s_j = s_k if i+j = k mod n > 3) s_0 + ... + s_{n-1} = 0 > Because of 3), s_0 has an additive inverse and we may therefore forget > that we started > only with P instead of R (or Q or F). > Thus we can use standard notation to see that you construct the ring > TIM_n(F) := F[s_0,s_1,...,s_{n-1}]/(s_0-1, s_0+...+s_{n-1}, s_i s_j > - s_k | i+j=k mod n) > for some natural number n>0. > Since s_i = (s_1)^i, this can be simplified to > TIM_n(F) = F[s]/(1+s+...+s^{n-1}, s^n-1) = F[s]/(1+s+...+s^{n-1}) > Especially, > TIM_1(F) = F[s]/(1) = 0 > TIM_2(F) = F[s]/(1+s) = F > TIM_3(F) = F[s]/(1+s+s^2) I'm sorry to say that I do not understand the expressions above. I > think it is the usage of the slash mark '/' that I don't understand. Here: ring modulo ideal. > If A is a ring and I is an ideal then A/I is the ring with elements of > the > form a+I, a in A and operations inherited from A. > If X is an unknown and f a polynomial in X, then you can view > A[X]/(f) as the ring of polynomials in X with coefficients i nA > subject to the condition > that f is zero. For F=R, we find TIM_3(F) = R[s]/(1+s+s^2) = C by mapping s |-> (third > root of unity). Here is the point that I noted above. This mapping is only in > hindsight. The polysign construction is freestanding. It is directly a > consequence of the generalization of sign wrt the superposition and > product that P3 and C correspond; nothing more. A transform must be > established but this is merely a frame of reference. In general, we have a surjective ring homomorphism TIM_n(R) -> C by > mapping > s|->(n^th root of unity), but this is not an isomorphism. > If we take F=Q instead, the standard map TIM_n(Q) -> C, s |-> n^th > root of unity > is injective (i.e. an isomorphism with its image field Q[n^th root of > unity]) > because we have (n-1)-dimensional Q-spaces on both sides -- provided > that n is prime > (we need that phi(n)=n-1). Did I get it right so far? > hagman I hope you got that right. The construction is very direct under my > own rendition and the indirect nature of your language is difficult > for me. The truth is that sign can be generalized so no different than > you teach a grade schooler about two signs out of thin air you can > also teach them three signs. Well, '-' is nto out of thin air, it comes from the desire to solve x > +a=0. Beyond this there are a series of > consequences and open problems that deserve study and I do welcome > your own rendition though I may criticize it as elitist jargon. I did not mean to imply anything about didactic aspects of multisign > theory, I just wanted to find out what your construct is in standard > terms. I don't mean offense to you. It seems that you truly think this way and > there are others like you so I am deeply grateful of your attention > carry on. There is a puzzle in expressing the P4 product in terms of > RxC that seems to be coming up again here and if you like a challenge > I welcome this pursuit. Well, apparently the mapping > f: TIM_4(R) -> RxC > s |-> (-1,i) > presented in another post is a ring isomorphism: > It defines a ring homomoprhism because > (1,1) + (-1,i) + (-1,i)^2 + (-1,i)^3 = (1,1)+(-1,i)+(1,-1)+(-1,- > i)=(0,0) > and from > f((s^2+1)/2) = (1,0) > f((1+s)/(1+i)) = (0,1) > f((1+s)/(1-i)) = (0,i) > we see that f is an isomorphism between 3dimensional R-vector spaces. -Tim The puzzle was a matter of trying to decipher the P4 product behavior > in terms of RxC by defining an equivalent product in RxC. The canonical multiplication in RxC is given by (a,b) * (a',b') = (a*a',b*b') and that corresponds to your P4 muktiplication via the isomorphism presented. > So for > instance if I write: > ( - 1 # 1 )( + 2 * 3 # 2) > I could transform these into RxC and get the same result upon > transfoming the resultant back into P4: > * 2 # 3 - 2 + 2 * 3 # 2 Indeed, if I got it right, '#' corresponds to the usual positive numbers '-' times '-' is '+' '-' times '+' is '*' '-' times '*' is '#' Thus (note that the meaning of '+','-','*' is the usual one on the right hand side: -1 (=s) |-> (-1,i) #1 (=1) |-> (1,1) hence - 1 # 1 |-> (0,i+1) +2 (=2s^2) |-> (2,-2) *3 (=3s^3) |-> (-3,-3i) #2 (=2) |-> (2,2) hence + 2 * 3 # 2 |-> (1,-3i) We have (0,i+1)(1,-3i) = (0,3-3i). Is this the same as * 2 # 3 - 2 + 2 * 3 # 2 (or shorter: * 5 # 5 - 2 + 2)? Well, *5 (=5s^3) |-> (-5,-5i) #5 (=5) |-> (5,5) -2 (=2s) |-> (-2,2i) +2 (=2s^2) |-> (2,-2) hence * 2 # 3 - 2 + 2 * 3 # 2 = * 5 # 5 - 2 + 2 |-> (0,3-3i) as expected. > There are two subproblems to this: how are the reference frames > connected and then what is the product definition. The product definition in RxC is the canonical one, i.e. component- wise. The product definition in R[s] is the canonical one, i.e. from definition of polynomials. > Could it be that if > this does not exist then we can draw a conclusion? RxC sounds pretty > flat. Probably a clean interpretation includes some dimensional mixing > so the RxC product is somewhat complicated. Anyhow such interpretation > is what is sought but as Tommy has said higher up it doesn't matter. > Dimensionality is there and again as Tommy says whether you call it > RxRxR ... in which case the product is not the canonical (i.e. component- wise) product. That's okay, but requires us to describe the product somewhat lengthy. > or RxC also does not matter. P4 is a geometry in the usual sense > though it a nonorthogonal basis. -Tim === Subject: Re: Treating Magnitude as Fundamental > On 15 Sep., 03:46, Timothy Golden BandTechnology.com > If I get it right, you essentilly start with P, the set of positive > reals (or possibly > the set of positive rationals or of some other subfield F of R), for me I call this magnitude(x). add symbols s_0, ..., s_{n-1} such that > 1) s_0 = 1 > 2) s_i * s_j = s_k if i+j = k mod n > 3) s_0 + ... + s_{n-1} = 0 > Because of 3), s_0 has an additive inverse and we may therefore forget > that we started > only with P instead of R (or Q or F). > Thus we can use standard notation to see that you construct the ring > TIM_n(F) := F[s_0,s_1,...,s_{n-1}]/(s_0-1, s_0+...+s_{n-1}, s_i s_j > - s_k | i+j=k mod n) > for some natural number n>0. > Since s_i = (s_1)^i, this can be simplified to > TIM_n(F) = F[s]/(1+s+...+s^{n-1}, s^n-1) = F[s]/(1+s+...+s^{n-1}) > Especially, > TIM_1(F) = F[s]/(1) = 0 > TIM_2(F) = F[s]/(1+s) = F > TIM_3(F) = F[s]/(1+s+s^2) I'm sorry to say that I do not understand the expressions above. I > think it is the usage of the slash mark '/' that I don't understand. Here: ring modulo ideal. > If A is a ring and I is an ideal then A/I is the ring with elements of > the > form a+I, a in A and operations inherited from A. > If X is an unknown and f a polynomial in X, then you can view > A[X]/(f) as the ring of polynomials in X with coefficients i nA > subject to the condition > that f is zero. For F=R, we find TIM_3(F) = R[s]/(1+s+s^2) = C by mapping s |-> (third > root of unity). Here is the point that I noted above. This mapping is only in > hindsight. The polysign construction is freestanding. It is directly a > consequence of the generalization of sign wrt the superposition and > product that P3 and C correspond; nothing more. A transform must be > established but this is merely a frame of reference. In general, we have a surjective ring homomorphism TIM_n(R) -> C by > mapping > s|->(n^th root of unity), but this is not an isomorphism. > If we take F=Q instead, the standard map TIM_n(Q) -> C, s |-> n^th > root of unity > is injective (i.e. an isomorphism with its image field Q[n^th root of > unity]) > because we have (n-1)-dimensional Q-spaces on both sides -- provided > that n is prime > (we need that phi(n)=n-1). Did I get it right so far? > hagman I hope you got that right. The construction is very direct under my > own rendition and the indirect nature of your language is difficult > for me. The truth is that sign can be generalized so no different than > you teach a grade schooler about two signs out of thin air you can > also teach them three signs. Well, '-' is nto out of thin air, it comes from the desire to solve x > +a=0. Beyond this there are a series of > consequences and open problems that deserve study and I do welcome > your own rendition though I may criticize it as elitist jargon. I did not mean to imply anything about didactic aspects of multisign > theory, I just wanted to find out what your construct is in standard > terms. I don't mean offense to you. It seems that you truly think this way and > there are others like you so I am deeply grateful of your attention > carry on. There is a puzzle in expressing the P4 product in terms of > RxC that seems to be coming up again here and if you like a challenge > I welcome this pursuit. Well, apparently the mapping > f: TIM_4(R) -> RxC > s |-> (-1,i) > presented in another post is a ring isomorphism: > It defines a ring homomoprhism because > (1,1) + (-1,i) + (-1,i)^2 + (-1,i)^3 = (1,1)+(-1,i)+(1,-1)+(-1,- > i)=(0,0) > and from > f((s^2+1)/2) = (1,0) > f((1+s)/(1+i)) = (0,1) > f((1+s)/(1-i)) = (0,i) > we see that f is an isomorphism between 3dimensional R-vector spaces. -Tim The puzzle was a matter of trying to decipher the P4 product behavior > in terms of RxC by defining an equivalent product in RxC. The canonical multiplication in RxC is given by > (a,b) * (a',b') = (a*a',b*b') > and that corresponds to your P4 muktiplication via the isomorphism > presented. So for > instance if I write: > ( - 1 # 1 )( + 2 * 3 # 2) > I could transform these into RxC and get the same result upon > transfoming the resultant back into P4: > * 2 # 3 - 2 + 2 * 3 # 2 Indeed, if I got it right, > '#' corresponds to the usual positive numbers > '-' times '-' is '+' > '-' times '+' is '*' > '-' times '*' is '#' > Thus (note that the meaning of '+','-','*' is the usual one on the > right > hand side: > -1 (=s) |-> (-1,i) > #1 (=1) |-> (1,1) > hence - 1 # 1 |-> (0,i+1) > +2 (=2s^2) |-> (2,-2) > *3 (=3s^3) |-> (-3,-3i) Let's consider just the above line where you say that *3 -> (-3, 3i ) and let us try to see if you've maintained distance. Clearly | *3 | = 3 . But | ( -3, -3i ) | = sqrt( 9 + 9 ) = 4.89 so this is not consistent. These magnitudes should match. I'm still open to your interpretation and it looks like you are pretty careful so I'll way to hear what you have to say about this. -Tim > #2 (=2) |-> (2,2) > hence + 2 * 3 # 2 |-> (1,-3i) > We have (0,i+1)(1,-3i) = (0,3-3i). > Is this the same as * 2 # 3 - 2 + 2 * 3 # 2 (or shorter: * 5 # 5 - 2 > + 2)? > Well, > *5 (=5s^3) |-> (-5,-5i) > #5 (=5) |-> (5,5) > -2 (=2s) |-> (-2,2i) > +2 (=2s^2) |-> (2,-2) > hence * 2 # 3 - 2 + 2 * 3 # 2 = * 5 # 5 - 2 + 2 |-> (0,3-3i) as > expected. There are two subproblems to this: how are the reference frames > connected and then what is the product definition. The product definition in RxC is the canonical one, i.e. component- > wise. > The product definition in R[s] is the canonical one, i.e. from > definition > of polynomials. Could it be that if > this does not exist then we can draw a conclusion? RxC sounds pretty > flat. Probably a clean interpretation includes some dimensional mixing > so the RxC product is somewhat complicated. Anyhow such interpretation > is what is sought but as Tommy has said higher up it doesn't matter. > Dimensionality is there and again as Tommy says whether you call it > RxRxR ... in which case the product is not the canonical (i.e. component- > wise) > product. That's okay, but requires us to describe the product > somewhat lengthy. or RxC also does not matter. P4 is a geometry in the usual sense > though it a nonorthogonal basis. -Tim === Subject: Re: Treating Magnitude as Fundamental On Sep 13, 4:51 am, Timothy Golden BandTechnology.com > I'm pretty sure that the anonymous author has a memory mismatch. I > think he is actually talking about information from Gene Ward Smith. > The only thread that I find Robin Chapman on is > which was very early on. And I really don't think that I've spoken > with Chapman about four-signed whereas Gene has made the author's > specific claim and I have refuted it near OK, I finally found the thread. The problem is that for some reason, the thread can't be found in Google Groups. I had to go to mathforum.org to find the thread. http://mathforum.org/kb/message.jspa?messageID=512617&tstart=0 In this post, Chapman first mentions the isomorphism between P4 and R x C. He makes a mistake regarding P4 (he thought that +1 was the P4 identity, not #1), but Will Twentyman was there to correct him. Still, Chapman arrived at the conclusion that P4 is isomorphic to R x C. http://mathforum.org/kb/message.jspa?messageID=512624&tstart=0 > What is the meaning of the four-signed numbers with their simple > arithmetical product? Does it have an equivalent like complex math? > This is a puzzle I hope you will work on. Since the three-signed > numbers match complex math exactly then there may be some value to > these four-signed numbers for three-dimensional space, the space we > all seem to live in. So now we ask, is P4 isomorphic to R x C? hagman: > Well, apparently the mapping > f: TIM_4(R) -> RxC > s |-> (-1,i) > presented in another post is a ring isomorphism: (snip) > f((1+s)/(1+i)) = (0,1) > f((1+s)/(1-i)) = (0,i) > we see that f is an isomorphism between 3dimensional R-vector spaces. But hold on a minute. For hagman presents f as a function with domain TIM_4(R) (or P4 in Tim's own notation) and codomain R x C. But then the argument of f is given as (1+s)/(1+i) and (1+s)/(1-i) -- which are not necessarily in the domain of f at all! For in each case, 1+s, an element of TIM_4(R) (or #1-1 in P4) is divided by the _complex_ numbers 1+i and 1-i. How does one divide a P4 number by a complex number, anyway? Indeed, by using only TIM_4(R) notation, we see that the elements of TIM_4(R) are (equivalence classes of) _real_ polynomials (due to the R in TIM_4(R)). But one cannot divide a _real_ polynomial 1+s by a _complex_ number like 1+i or 1-i and expect to have a _real_ polynomial as a result! At first I thought that hagman had successfully proved but now I see the flaw in the proof. If hagman can patch this flaw, then Chapman will be proved correct after all, and P4 is isomorphic to R x C, and I would conjecture that Pn is isomorphic to the Cartesian product of copies of R and C (i.e., R^r x C^c with r+2c = n-1) for all n. But if the flaw cannot be fixed, then Tim and t-1729 would be proved correct, and Chapman would be wrong, and that P4 is indeed a distinct ring from R x C. And my own claim earlier in this thread that P5 is isomorphic to C x C would also be wrong, since it is a generalization of Chapman's claim. === Subject: Re: Treating Magnitude as Fundamental So now we ask, is P4 isomorphic to R x C? As i said before, Tim is using a generating set of four vectors, which are forming a tetrahedron with edges of equal length. Robin is using four vectors, which are forming a tetra, but not with edges of equal length. With friendly greetings Hero === Subject: Re: Treating Magnitude as Fundamental > On Sep 13, 4:51 am, Timothy Golden BandTechnology.com I'm pretty sure that the anonymous author has a memory mismatch. I > think he is actually talking about information from Gene Ward Smith. > The only thread that I find Robin Chapman on is > which was very early on. And I really don't think that I've spoken > with Chapman about four-signed whereas Gene has made the author's > specific claim and I have refuted it near OK, I finally found the thread. The problem is that for some > reason, the thread can't be found in Google Groups. I had to > go to mathforum.org to find the thread. http://mathforum.org/kb/message.jspa?messageID=512617&tstart=0 In this post, Chapman first mentions the isomorphism between > P4 and R x C. He makes a mistake regarding P4 (he thought > that +1 was the P4 identity, not #1), but Will Twentyman was > there to correct him. Still, Chapman arrived at the conclusion > that P4 is isomorphic to R x C. > http://mathforum.org/kb/message.jspa?messageID=512624&tstart=0 What is the meaning of the four-signed numbers with their simple > arithmetical product? Does it have an equivalent like complex math? > This is a puzzle I hope you will work on. Since the three-signed > numbers match complex math exactly then there may be some value to > these four-signed numbers for three-dimensional space, the space we > all seem to live in. So now we ask, is P4 isomorphic to R x C? hagman: Well, apparently the mapping > f: TIM_4(R) -> RxC > s |-> (-1,i) > presented in another post is a ring isomorphism: (snip) f((1+s)/(1+i)) = (0,1) > f((1+s)/(1-i)) = (0,i) > we see that f is an isomorphism between 3dimensional R-vector spaces. Hm, see below. But hold on a minute. For hagman presents f as a function > with domain TIM_4(R) (or P4 in Tim's own notation) and > codomain R x C. But then the argument of f is given as > (1+s)/(1+i) and (1+s)/(1-i) -- which are not necessarily > in the domain of f at all! For in each case, 1+s, an > element of TIM_4(R) (or #1-1 in P4) is divided by the > _complex_ numbers 1+i and 1-i. How does one divide a > P4 number by a complex number, anyway? Indeed, by using only TIM_4(R) notation, we see that the > elements of TIM_4(R) are (equivalence classes of) _real_ > polynomials (due to the R in TIM_4(R)). But one cannot > divide a _real_ polynomial 1+s by a _complex_ number > like 1+i or 1-i and expect to have a _real_ polynomial > as a result! At first I thought that hagman had successfully proved > but now I see the flaw in the proof. If hagman can > patch this flaw, then Chapman will be proved correct > after all, and P4 is isomorphic to R x C, and I would > conjecture that Pn is isomorphic to the Cartesian > product of copies of R and C (i.e., R^r x C^c with > r+2c = n-1) for all n. But if the flaw cannot be fixed, then Tim and t-1729 > would be proved correct, and Chapman would be wrong, > and that P4 is indeed a distinct ring from R x C. And > my own claim earlier in this thread that P5 is > isomorphic to C x C would also be wrong, since it is a > generalization of Chapman's claim. I might have been typing too fast and thereby intrduced complex arithmetic on the left hand side. To recap, let TIM_4(R) = R[s]/(1+s+s^2+s^3). This is a 3dimensional R-vector space and so is RxC, in fact both are R-algebras by virtue of their multiplications. We can define an R-algebra homomorphism R[s] -> RxC by sending s to (-1,i). Since (1,1) + (-1,i) + (-1,i)^2 + (-1,i)^3 = (0,0) holds in RxC, this homomorphism factors and defines a R-algebra homomorphism f: TIM_4(R) -> RxC To show that f is an isomorphism, it is sufficient that f is an isomorphism when viewed as a linear map, i.e. to show that the image is a 3dimensional vector space. But we have f(1) = (1,1) f(s) = (-1,i) f(s^2) = (1,-1) and these three vectors are R-linearly independent. Therefore, f is an isomorphism between TIM_4(R) and RxC. By simple transfer, we obtain an isomorphism between P4 and RxC if we send #1 to (1,1), -1 to (-1,i), +1 to (1,-1) and *1 to (-1,-i). I've not checked but assume that the proof by Chapman was essentially identical to this one. hagman === Subject: Re: Treating Magnitude as Fundamental > On Sep 13, 4:51 am, Timothy Golden BandTechnology.com I'm pretty sure that the anonymous author has a memory mismatch. I > think he is actually talking about information from Gene Ward Smith. > The only thread that I find Robin Chapman on is > which was very early on. And I really don't think that I've spoken > with Chapman about four-signed whereas Gene has made the author's > specific claim and I have refuted it near OK, I finally found the thread. The problem is that for some > reason, the thread can't be found in Google Groups. I had to > go to mathforum.org to find the thread. http://mathforum.org/kb/message.jspa?messageID=512617&tstart=0 In this post, Chapman first mentions the isomorphism between > P4 and R x C. He makes a mistake regarding P4 (he thought > that +1 was the P4 identity, not #1), but Will Twentyman was > there to correct him. Still, Chapman arrived at the conclusion > that P4 is isomorphic to R x C. > http://mathforum.org/kb/message.jspa?messageID=512624&tstart=0 What is the meaning of the four-signed numbers with their simple > arithmetical product? Does it have an equivalent like complex math? > This is a puzzle I hope you will work on. Since the three-signed > numbers match complex math exactly then there may be some value to > these four-signed numbers for three-dimensional space, the space we > all seem to live in. So now we ask, is P4 isomorphic to R x C? hagman: Well, apparently the mapping > f: TIM_4(R) -> RxC > s |-> (-1,i) > presented in another post is a ring isomorphism: (snip) f((1+s)/(1+i)) = (0,1) > f((1+s)/(1-i)) = (0,i) > we see that f is an isomorphism between 3dimensional R-vector spaces. But hold on a minute. For hagman presents f as a function > with domain TIM_4(R) (or P4 in Tim's own notation) and > codomain R x C. But then the argument of f is given as > (1+s)/(1+i) and (1+s)/(1-i) -- which are not necessarily > in the domain of f at all! For in each case, 1+s, an > element of TIM_4(R) (or #1-1 in P4) is divided by the > _complex_ numbers 1+i and 1-i. How does one divide a > P4 number by a complex number, anyway? Indeed, by using only TIM_4(R) notation, we see that the > elements of TIM_4(R) are (equivalence classes of) _real_ > polynomials (due to the R in TIM_4(R)). But one cannot > divide a _real_ polynomial 1+s by a _complex_ number > like 1+i or 1-i and expect to have a _real_ polynomial > as a result! At first I thought that hagman had successfully proved > but now I see the flaw in the proof. If hagman can > patch this flaw, then Chapman will be proved correct > after all, and P4 is isomorphic to R x C, and I would > conjecture that Pn is isomorphic to the Cartesian > product of copies of R and C (i.e., R^r x C^c with > r+2c = n-1) for all n. But if the flaw cannot be fixed, then Tim and t-1729 > would be proved correct, and Chapman would be wrong, > and that P4 is indeed a distinct ring from R x C. And > my own claim earlier in this thread that P5 is > isomorphic to C x C would also be wrong, since it is a > generalization of Chapman's claim. Nice work digging up this thread. As you can see I wasn't in tune with the discussion between Chapman and Twentyman just as I am not in tune with this current analysis. Yet your analysis of hagman is helping me see a bit. Anyway upon a full definition we will have a set of rules that establish a reference frame for transformation and a product definition in representation HagP4 = ( a, b, c ) where a is real, and b+ic is complex. This is just a restatement of RxC and these a,b,c coordinates are returned by a function HagP4( z ) where z is a four-signed value. Since the product is not defined for this domain it must be established. In this way the differentiation between RxRxR and RxC is negligible. Two instances of HagP4 (a1,b1,c1), (a2,b2,c2) can be combined by a product HagProduct( (a1,b1,c1), (a2,b2,c2)) = (a3,b3,c3) and will be consistent with this unindexed four-signed representation: ( - a + b * c # d )( - e + f * g # h ) = + ae * af # ag - ah * be # bf - bg + bh # ce - cf + cg * ch - de + df * dg # dh which is simply the distribution of terms. If such a 3D math were well known then it would be presented alongside quaternions wouldn't it? So this is a new construction and especially of interest because it is so well behaved algebraically in any dimension(unlike existing math). The extensibility of such a product would also be highly desirable such that one is not fumbling for the P6 version. If such a mapping does not exist cleanly then clearly the native tribe is Polysign. You can pose with your Cartesian equivalent if it exists however its existence is already so obscured that its value is questionable. Yes it is clear that the P4 product is rotational in nature and nearly mimics a primitive RxC product, but some discrepancy is present. I am open to a solution but I don't think it is nearly so easy as you are thinking. Perhaps there is an argument in the terms of orthogonal components here and their supposed independence versus the nonorthogonal components and their inherent dependence. In effect a small tweak in any of the P4 components of one operand throws the product out of whack in all of its resultants. Mimicing this behavior in orthogonal coordinates will require dimensional mixing and I believe it will be a iterated form of solution that is cleanest. This is somewhat already established in my study of the problem and the graphical result: http://bandtechnology.com/PolySigned/Deformation/T3P4DifferenceStudy.gif which is the self similar difference of the simplect version of RxC product where independence is maintained between R and C within the product: (a1,b1,c1) (a2,b2,c2) = ( a1a2, b1b2-c1c2, b1c2+b2c1). I do not believe that this is HagProduct(). I have forgone the determination of reference frame but it is covered at my website. It is pretty straightforward to establish the reference frame. Any claim of isomorphism can be proven when instantiated. A claim of existence has been forwarded by several now: Robin Chapman, Gene Ward Smith, perhaps lwal, and now perhaps hagman. Yet none have instantiated such a product. So we wait for Hag_P4() and HagProduct(). -Tim === Subject: Re: Moebius strips embedded in R^3 Ok, suppose that now we ask the Moebius strips to be diffeomorphic. Is it possible to have an uncountable number of them (pairwise disjoint and diffeomorphic) in the space? === Subject: Re: JSH: State of the research (Sigh.) There you go again. You equate the opposition you receive from those on sci.math with the opposition of the entire world-wide mathematical community. Okay, maybe some on sci.math got the Southwest Journal of Pure and Applied Mathematics to drop your paper after it had been initially accepted for publication. That does not mean -- it simply cannot mean -- that the entire world-wide mathematical community was involved in getting your paper dropped. Mathematicians in places such as China and India, Russia and Brazil had absolutely nothing to do with it. It was just a handful of Americans. Mathematicians in places such as China and India, Russia and Brazil don't even know who you are. That, of course, is because you focus your activities exclusively on sci.math. There's only 7997 people who currently subscribe to sci.math. 7997 people hardly constitutes the entire world-wide mathematical community. If you really want to address the entire world-wide mathematical community, then address it. But enough with this childishness that by posting on sci.math your addressing it. You're not. === Subject: Re: JSH: State of the research > That, of course, is because you focus your activities exclusively on > sci.math. There's only 7997 people who currently subscribe to > sci.math. 7997 people hardly constitutes the entire world-wide > mathematical community. There are certainly more than 7997 mathematicians in the world, but why do you think there are 7997 people who currently subscribe to sci.math? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: JSH: State of the research <877imr4zcs.fsf@huxley.huxley.fi> On Sep 15, 12:23?pm, Aatu Koskensilta sci.math. There's only 7997 people who currently subscribe to > sci.math. 7997 people hardly constitutes the entire world-wide > mathematical community. There are certainly more than 7997 mathematicians in the world, but > why do you think there are 7997 people who currently subscribe to > sci.math? It's the number of registered Google Groups users who have sci.math in their subscription list. But, as Jesse points out, not every sci.math reader is a member of Google Groups. Needless to say, it is possible to register in Google using more than one alias, as long as those aliases have different legitimate e-mail addresses. If you are a registered Google Groups user, you will have a My groups link, from which you can get the member counts if you display your subscription as a list instead of a grid: Group New items Members Role Last visit alt.folklore.computers 0 421 Member 12 minutes ago alt.math 1 526 Member 12 minutes ago alt.math.recreational 0 387 Member 13 minutes ago comp.lang.python 2 11045 Member 8 minutes ago comp.programming 0 3142 Member 6 minutes ago rec.humor.oracle 0 172 Member 47 hours ago rec.puzzles 0 1008 Member 5 minutes ago sci.math 3 7998 Member 4 minutes ago True But Unproven - 0 12 Group Longer than a week the Collatz Conjecture - owner As you can see, it's already changed to 7998. -- > Aatu Koskensilta (aatu.koskensi...@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: JSH: State of the research <877imr4zcs.fsf@huxley.huxley.fi On Sep 15, 12:23?pm, Aatu Koskensilta why do you think there are 7997 people who currently subscribe to > sci.math? It's the number of registered Google Groups users > who have sci.math in their subscription list. But, as Jesse points out, not every sci.math reader > is a member of Google Groups. It's also worth pointing out that not everyone who reads sci.math through GG is subscribed (I wasn't until recently, for example). === Subject: Re: JSH: State of the research > It's the number of registered Google Groups users > who have sci.math in their subscription list. Right. But that is not the number of people subscribed to sci.math. There is no way to determine that figure. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: JSH: State of the research > > That, of course, is because you focus your activities exclusively on > sci.math. There's only 7997 people who currently subscribe to > sci.math. 7997 people hardly constitutes the entire world-wide > mathematical community. There are certainly more than 7997 mathematicians in the world, but > why do you think there are 7997 people who currently subscribe to > sci.math? In 2002, Dave Rusin did a test, inviting readers to click on a link (once). After a week, the counter had slightly surpassed 3900. cf.: Obviously, the math. background of sci.math readers varies a lot. David Bernier P.S. Going the other way, how many pure or applied mathematicians have read sci.math in the past week? === Subject: Re: JSH: State of the research > There's only 7997 people who currently subscribe to sci.math. 7997 > people hardly constitutes the entire world-wide mathematical > community. Where on earth do you get this figure? Usenet is bigger than Google Groups, you know. There really is no reliable measure for how many folks read sci.math. (Nonetheless, your main claim is correct: sci.math readership is not synonymous with the worldwide mathematics community.) -- Jesse F. Hughes [I]t's the damndest thing. There's something wrong with every last one of you, and I *never* thought that was a possibility. But now I feel it's the only reasonable conclusion. --JSH sees some sorta light === Subject: Re: JSH: State of the research >Surrogate factoring preferentially yanks out small prime factors. And >it doesn't seem to care much how big the number is when it does that >yanking. Nothing special there. Trial factorisation also preferentially yanks out small prime factors and doesn't seem to care much how big the number is. I would expect SF to preferentially find small primes. On average half of the numbers you try in the GCD have a factor of 2, one third of them have a factor of 3, one fifth of them have a factor of 5 and so on. The larger the prime, the smaller proportion of numbers you are trying will have that prime as a factor. Hence SF is likely to find small prime factors before it finds large ones. rossum === Subject: Re: JSH: State of the research > Whether you realize it or not the preamble with any factoring research > that I do is looking for some way to show it's trivial. JSH, is there a solution manual for your work? === Subject: wholesale Nike shoes,Jordan shoes,Air force ones,Adidas shoes air nike shox air max air jordan air bape shoes gucci shoes Welcome to our company. www.lead-nike.com We can provide nike jordan series Air max and shox series etc. we can not only provide the top quality shoes but also best service,prompt delivery. if you interested time and we hope to hear from u. Email: lead_nike@yahoo.com.cn MSN: lead-nike@hotmail.com === Subject: Algebra: field of radicals Is there a name for the field generated by {p^(1/n): p a positive prime, n a positive integer}? How does one go about determining whether elements in this field have square roots there? For example, is sqrt(2 sqrt(2) - 1) in this field? I suspect there is an area of algebra that addresses this matter. My abstract algebra is quite rusty. Pointers? -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Algebra: field of radicals > Is there a name for the field generated by {p^(1/n): p a positive > prime, n a positive integer}? How does one go about determining whether > elements in this field have square roots there? For example, is > sqrt(2 sqrt(2) - 1) in this field? I read quasi's reply. He suggested looking at extension fields (say F ) of Q by adjoining roots { (p_i)^(1/n_i) : i = 1, .... M}. Let 'a' be the number and the (simplified) problem is to determine whether 'a' has a square root in F. Isn't this equivalent to whether x^2 -a is irreducible in F[x]? Maybe it is, but I'm not familiar with extension fields as general as F above. David Bernier > I suspect there is an area of algebra that addresses this matter. My > abstract algebra is quite rusty. Pointers? > === Subject: Re: Algebra: field of radicals > Is there a name for the field generated by {p^(1/n): p a positive > prime, n a positive integer}? How does one go about determining whether > elements in this field have square roots there? For example, is > sqrt(2 sqrt(2) - 1) in this field? >I read quasi's reply. He suggested looking at extension fields (say F ) >of Q by adjoining roots >{ (p_i)^(1/n_i) : i = 1, .... M}. Let 'a' be the number and the >(simplified) problem >is to determine whether 'a' has a square root in F. Isn't this equivalent to whether x^2 -a is irreducible in F[x]? Yes. >Maybe it is, but I'm not familiar with extension fields as general as F >above. Well, that's the problem. How do you determine the irreducibility of elements of F[x]? If F was finite dimensional over Q, it would be straightforward. quasi === Subject: Re: Algebra: field of radicals >Is there a name for the field generated by {p^(1/n): p a positive >prime, n a positive integer}? In other words, the field generated by the n'th roots of p where n ranges over the positive integers and p ranges over all primes? Thus, for example, the field contains 2^(1/3), 5^(1/2), 7^(1,6), etc as well as all sums, differences, products and quotients of such elements, right? >How does one go about determining whether elements in this field have square roots there? With great difficulty. If you actually believe that the specified element does have a square root in the field, guess the form for a finite set of generators (which primes, which roots) and solve the problem in the associated finite extension of Q. On the other hand, if the given element doesn't have a square root in the field, you probably have little or no chance of proving it. > For example, is sqrt(2 sqrt(2) - 1) in this field? Probably not. But of course, that's just an intuitive guess, not a true probability. Assuming I'm right, what's the likelihood an elementary proof? Close to zero. >I suspect there is an area of algebra that addresses this matter. Field Theory, Ring Theory, Constructive Algebra (especially Groebner bases). >My abstract algebra is quite rusty. Pointers? Yes -- give up -- you have essentially no chance. The difficulty derives from the fact that these fields are not finitely generated. Trying to prove that the higher dimensional subfields don't introduce any lower dimensional subfields (that weren't already there using lower degree generators) will probably require the development of a specialized Galois theory, tailored to this exact problem. Unless it's a vital question for you, it's probably not worth the effort. For one thing, it could take months, even years, to work out the needed machinery. Worse, the attempt might fail. As an analogy, consider the calculus problem of finding an elementary expression, if any, for the antiderivative of a given elementary function. What theory resolves that problem? Differential Galois Theory, but it's far from easy. On the other hand, if you just want to get an understanding of the basic concepts and issues, start by studying Field Theory. Two recommended books Fields & Rings Kaplansky Algebraic Extensions of Fields McCarthy quasi === Subject: Need a guitar? Free guitars here!!!!!! http://freeguitars.blogspot.com/ === Subject: Re: Need a guitar? > Free guitars here!!!!!! People in this newsgroup are in bigger need of violins to be playing when they whine about their beats. === Subject: Coin toss problem Hi everyone !!! I need some little help with this problem, just to get me started.. A biased coin is tossed until heads appears for the first time. If the probability of a head for the coin is 2/5, the expected number of tosses needed to get heads for the first time is given by: The summation sign (from n=1 to infinity) of (n*(2/5)*(3/5)^(n-1)) Compute the expected value. === Subject: Re: Coin toss problem > Hi everyone !!! I need some little help with this problem, just to get > me started.. A biased coin is tossed until heads appears for the first time. If > the probability of a head for the coin is 2/5, the expected number > of tosses needed to get heads for the first time is given by: The summation sign (from n=1 to infinity) of (n*(2/5)*(3/5)^(n-1)) Compute the expected value. > You might consider evaluating that series with standard methods. Or do as follows: Let E(X) denote the expected number of tosses needed. The first toss will either produce heads (with 2/5) and under this restriction, the expected number is exactly 1. Or the first toss will produce tails (with 3/5) and under this restriction, the expected number of tosses needed additional to the first one is E(X). Therefore E(X) = 2/5 * 1 + 3/5 * (1+E(X)) hagman === Subject: Re: State of the research > Whether you realize it or not the preamble with any factoring research > that I do is looking for some way to show it's trivial. Like with my first attempts at factoring algorithms back about 5 years > ago, I was just working on extensions of ideas used by Fermat. Later I had the concept of surrogate factoring as an idea from a > question: could you factor one number using another? No, idiot. You would have to produce two *different* numbers, one of which divides (factors) the other. No two such numbers can exist - a result proven at the turn of the century by Hilbert. === Subject: Re: State of the research > Whether you realize it or not the preamble with any factoring research > that I do is looking for some way to show it's trivial. Like with my first attempts at factoring algorithms back about 5 years > ago, I was just working on extensions of ideas used by Fermat. Later I had the concept of surrogate factoring as an idea from a > question: could you factor one number using another? how do you factor without using another number ? > And method after method after method failed as I'd figure out or > others would figure out that it was something trivial where the > underlying relations were often about random or some kind of sieving > that would not be earth-shattering in terms of impact on the problem. So the first year of the life of the latest surrogate factoring > research where after over 3 years of searching I realized I only > needed to add one variable k, where k = 2x mod T, where T is the > target to factor, was really about finding some way to trivialize the > research. the mod function takes substantially more time than simple divide function And it survived that year plus. Now it is clear that what I call surrogate factoring IS a new way to > factor and is as fundamental in mathematics as methods related to > congruence of squares, and there is no way to show it is just trivial, > meaning that like methods before it, there is the possibility of a > growing body of research that continually improves it. where is the growing body? Except I somewhat accidentally discovered how rapidly it can be > improved while typing in some numbers when I watched it factor a 100+ > bit number, so already it is far beyond methods based on congruence of > squares at this point in its life, and has behavior more like Dixon's. any 100 bit number? or just a certain one? > Surrogate factoring preferentially yanks out small prime factors. And > it doesn't seem to care much how big the number is when it does that > yanking. The mathematics is very fundamental as I've only added k = 2x mod T, > to x^2 = y^2 mod T, so you have a basis in very rudimentary equations, > and now after a year I am certain that it cannot be shown to be > trivial. So it's about time, effort and the natural maturation of an idea. Previous factoring methods took hundreds of years to reach maturity, > but that was without modern computing technology, modern mathematical > technique, modern problem solving technique, and trillions of dollars > flowing behind an encryption standard that could be made obsolete > motivating highly intelligent people to work very hard. not really, extended gold codes were developed quickly in the 50's and need no factoring for encryption/decryption there are several thousand solid codes in use today the worlds banking community uses them, and none of them require any factoring at all. > Past history with my prior research indicates that modern > mathematicians have taken an absolute position of holding against my > research in denial--no matter what. Even the destruction of a modern electronic mathematical journal had it says you withdrew the paper, it was not published, what happened? > But factoring research has the potential of breaking them like people > before who have often been broken by taking absolute positions against > more powerful forces. Consider Chinese in the Boxer Rebellion who > thought that painting themselves magically could stop bullets. They were high on opium, that is what the rebellion was about. > And in this case, breaking the absolutism of the mathematical > community is unlikely to happen without changing the economic > landscape of the entire world. as stated above, banking systems do not use factoring for encryption/decryption, just very long codes and keys. > There is no way that I can see that mathematicians ignoring this > research and waiting until it matures is helpful for my own country, > currently the dominant world power. But it is the decision of the modern mathematical and cryptography > community that holds sway here as the world trust you. They trust you, so that is how you have the power to decide the fate > of the world. They sure don't trust me. I'm the crackpot with yet another idea > claiming it's important. actually you missed the train 25 years ago, it left the station in the early 1980's > If you all say it's not, and you're wrong, then the maturation process > of surrogate factoring can happen mostly in the dark, and the world > instead of facing a relatively new idea that has a distance to go in > order to be as powerful as it can be, can instead face a fully matured > factoring method--known because it is unleashed. unleashed or excreted ? > The issue here is ignorance in the now, and full realization later. I agree, get a basic crypto book and read up, most systems are not dependent upon factoring a large number, only a few, like RSA > Or an end to the absolute position taken by the world's mathematical > community against my research now, versus later. Fight me on this and you can wake up in a few years to a totally > changed world order where you helped create it, dashing the now > dominant countries to the ground on their fateful and naive trust in > your honesty about your discipline. all this for a 2x mod T ? > History shows that in these situations, your choice is usually against > your own best interest, which is why history is so interesting, as > empires fall not just on the decisions of the world leaders, but on > the seemingly minor ones of people at the fulcrum point. And this time, in this history to be, the lever is surrogate > factoring, and I assure you that with years of research under my belt > I now even more firmly believe that it can move the world. The challenge to me is to balance the needs of the many against the > wants of the few. As I consider the livelihoods of mathematicians around the world, and > the savings of people around the world, including in my own country, > against the survival of the human race depending on ever forward > progress in our knowledge, science and technology. And I have no choice but to sacrifice the few against the needs of the > many. I will sacrifice your actual lives against the needs of the > future, against the children yet to be born. sure, I agree with you, but you may want to read a book or two on Crypto first, and learn that factoring large numbers has little to do with any of the security of our world's banking and finical systems In fact there is shareware available that can tripple encode messages, uses three complex codes sequentially, and there is no factoring approch that will break it, ever. your point is mute. As far as I'm concerned if you make the wrong choice, you simply > killed yourselves. > James Harris > === Subject: Re: The inverse function of a function of two variables I finally found the solution in a little bit different way from what you have suggested (thank you so much anyway): v/u=sin(y)/cos(y), then y=arctan(v/u) put y=arctan(v/u) back to either original eq (let me do u), we have u=exp(x)*cos(arctan(v/u)) you can draw a right triangle with base u and height v, so cos(arctan(v/u))=u/sqrt(u2+v2) we have x=ln(sqrt(u2+v2)) Ans: x=ln(sqrt(u2+v2); y=arctan(v/u). === Subject: Polysign source code tarball http://bandtechnology.com/Polysign.cvs.tar.gz is pretty well developed. I'm not sure what your standards are but this source is very old school. The makefile is of poor quality but easy to read. There may be issues with case sensitivity of file names. Also dependencies are imperfect here so if a header changes a clean build is advised. Wherever the compiler complains almost certainly the solution will be very simple. For graphics libgd and other related subroutines are necessary. This code will run fine on cygwin or linux. If you don't know C++ but would like an excuse to learn some this may be a good project for you since here you see a lightly structured code body with no GUI complications. This is a math research library. It is good for some more than polysign since the projection code is done in Cartesian math. So doing a 10D animation is not a big deal but as you study the code you will see that projection does not take into account occlusion and this direction will certainly be taken in a future version of this code. This software is gpl'd with a text file residing in the project source code conspicuously. Individual code modules are not labelled. Please respect the GPL. You should have cvs installed and learn a bit of cvs overhead but that will be good for you if you don't know it already. -Tim === Subject: Re: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? > Bit drastic... > No grass on the moon. It might work if the underlying > rock is a different shade than the surface dust, like > at Nazca. > No biggy ? ;) > They would have to be big enough to be seen by > the naked eye, otherwise the idiots will claim > the pictures taken through a telescope are faked. > Bye, > Skybuck. > Perhaps a beacon could be placed on the Moon. A powerful laser beam > could be aimed at the Earth, designed to spread out enough to be > harmless, but it's flashing still visible to the naked eye. It could > flash out Morse code for Kilroy was here! Double-A I read a beautiful Science Fiction story a long time ago where an exploding natrium gas bomb was used to direct a high velocity gas stream trough a metal mask to generate an COCACOLA advert across the face of the moon. Maybe that would convince some doubting characters. Ask a real scientist whether that is possible. === Subject: Re: Space marker on the moon idea: Writing Big Letters on The Moon for Proof of our visit ;) How big is big enough ? Way to high tech. What if laser breaks huh ? :) Should be there for a long time... Though what if meteors hit the ing land mark ? Darn ! Ok good point about no grass but eum.. Simply make little carvings... Surely that would create enough shades ???? :) Gje... Maybe the moon needs anti-meteor protection system to protect the proof we were there :):):):) Bye, Skybuck. === Subject: Re: Just found this place to get replica watches What is Prestige Replica store? Prestige make some very good cookware. -- He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Logarithm question Supersedes: 26, since the log distribution is not linear. === Subject: Re: Logarithm question Are you asking if this is legitimate: log x - log y = log(antilog(log x) - antilog(log y)) The rhs of which is log(x - y). Useful things, symbols. Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same > as log x - log y? It's clear that I can not subtract log numbers, > because 66 - 40 <> 26, since the log distribution is not linear. > Are you sure you are asking that? antilog(logX)-antilog(logY) = x-y not log(x-y), I think you want log[antilog(logX)-antilog(logY)] which is = log(x-y) Mike === Subject: Re: Logarithm question Supersedes: <9o2yvda5jdmi.4liplee9vlxj.dlg@40tude.net> Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same > as log x - log y? It's clear that I can not subtract log numbers, > because 66 - 40 <> 26, since the log distribution is not linear. >Are you sure you are asking that? antilog(logX)-antilog(logY) = x-y not log(x-y), > I think you want log[antilog(logX)-antilog(logY)] which is = log(x-y) OK, I'll explain the real situation, and you decide what operation I need. When you measure sound, you get a number, eg 60 dB. Now, I want to have the sound of a device alone, but when I measure the sound, I get the sound of the device + sound of surrounding. Since log distribution is not linear, I can't simply subtract them. So, what do I need to get only the sound of the device? I have the sound of device + surrounding, and I have the sound of the surrounding alone. The formula for the sound is 20*log(P2/P1). So, I'm thinking I should divide this by 20, alog, and then I have the P2/P1. So, I take the P2/P1 of the device+surrounding, and subract the P2/p1 of the surrounding to get the sound of the device alone. Then I log that and *20, and I get the sound of the device. Yes or no? === Subject: Re: Logarithm question Mike transported these words from his/hers keyboard to our eyes: > Hmm, actually, I'm asking if antilog(logx) - antilog(log y) is the same > as log x - log y? It's clear that I can not subtract log numbers, > because 66 - 40 <> 26, since the log distribution is not linear. >Are you sure you are asking that? antilog(logX)-antilog(logY) = x-y not log(x-y), > I think you want log[antilog(logX)-antilog(logY)] which is = log(x-y) OK, I'll explain the real situation, and you decide what operation I need. When you measure sound, you get a number, eg 60 dB. Now, I want to have the sound of a device alone, but when I measure the sound, I get the sound of the device + sound of surrounding. Since log distribution is not linear, I can't simply subtract them. So, what do I need to get only the sound of the device? I have the sound of device + surrounding, and I have the sound of the surrounding alone. The formula for the sound is 20*log(P2/P1). So, I'm thinking I should divide this by 20, alog, and then I have the P2/P1. So, I take the P2/P1 of the device+surrounding, and subract the P2/p1 of the surrounding to get the sound of the device alone. Then I log that and *20, and I get the sound of the device. Yes or no? -- Oh no, yo soy un narcotrafficante columbiano! === Subject: Intersections of circles Hey all, I'm stuck on a geometry problem that I don't know how to solve efficiently. I have a set of circles on an x,y plane. All the circles have the same radius. No matter where I place a point on this plane, I need the number of circles which contain this point to be no larger than a given maximum. Actually, I want to find this maximum number. So, if none of my circles intersect, the answer will be 1 (An arbitrary point on the plane is contained by at most one circle). If a pair of circles intersect at two points, the answer would be two. If four circles all enclose one area, the answer will be four. Four circles could also intersect such that any area is enclosed by at most two circles, so the answer would be two, or a couple of other ways. At the moment, I'm constructing a graph where the nodes represent the circles, and I add a bidirectional edge between two nodes when the circles at the two nodes overlap each other (There is an area that they both enclose). Then, I find the largest clique in the graph. I think this will give me an upper bound on the maximum number of times a point could be contained by my circles. I can see that I could compute the points of intersections between all the circles in the clique. If one of these points was contained by or was on the boundary of all the circles in the clique, then I would know that there is a point enclosed the same number of times as the number of circles in the clique. But I'm hoping that somebody knows a faster solution. Nicholas Sherlock === Subject: Re: Intersections of circles > I'm stuck on a geometry problem that I don't know how to solve > efficiently. I have a set of circles on an x,y plane. All the circles > have the same radius. No matter where I place a point on this plane, I > need the number of circles which contain this point to be no larger > than a given maximum. Actually, I want to find this maximum number. What do you mean by contain this point? The point lius on the circle, within the circle, or on or within the circle? The maximum of what? The number of circles which contain this point, for all possilbe 'this' points? > So, if none of my circles intersect, the answer will be 1 (An > arbitrary point on the plane is contained by at most one circle). > If a pair of circles intersect at two points, the answer would be two. > If four circles all enclose one area, the answer will be four. Four > circles could also intersect such that any area is enclosed by at most > two circles, so the answer would be two, or a couple of other ways. > Is the point on, within, or on or within the circle? > At the moment, I'm constructing a graph where the nodes represent the > circles, and I add a bidirectional edge between two nodes when the > circles at the two nodes overlap each other (There is an area that > they both enclose). Then, I find the largest clique in the graph. I > think this will give me an upper bound on the maximum number of times > a point could be contained by my circles. > You are jumping the gun. > I can see that I could compute the points of intersections between all > the circles in the clique. If one of these points was contained by or > was on the boundary of all the circles in the clique, then I would > know that there is a point enclosed the same number of times as the > number of circles in the clique. But I'm hoping that somebody knows a > faster solution. > Consider just two tangent circles with same radius that intersect in exactly one point. The maximum numbers of circles a point can be within is one. So what are you considering, circles, closed disks or open disks? === Subject: Re: Intersections of circles The point lius on the circle, within the circle, or > on or within the circle? Strictly within. > The maximum of what? The number of circles which > contain this point, for all possilbe 'this' points? Yep :). > So what are you considering, circles, closed disks or open disks? I think I am considering open disks. Nicholas Sherlock === Subject: 1/. Factor Fermats method + divisibility.************example. 1/. Factor Fermats method + divisibility.************example. by Donald S. McDonald, Wellington, NZ. 2/. I bought Sharp PC1500A pocket computer from 2nd hand shop about 10 yrs ago. My Acorn RISC-os computer crashed on election night 17/9/05, 2 yrs ago (Helen Clark PM.) I have hardly used it [Acorn A5000.] at all since. 3/. My programs factor integers up to 2^52 (raised to the power of), = 45e14 in 10 minutes, ... and powers of integers 2^31. Usually much faster; a few seconds. I programmed the Sharp pocket computer in BASIC to factor numbers. And I have 2 x my 1-step iterative methods on Casio FX-82 MS schools scientific calculator. As previously 4/. I study the Lotto numbers. 5/. On Saturday 15/9/07 at 8pm NZST , the 'lucky wheel serial number' from Draw #1058, Levin, was: A = 747 5266 33.41 20 XX 6/. I entered the first 9 digits and by trial found factors 7*19. I then referred to pocket computer to obtain the complete prime factorisation. A = 7*19 * 1409 * 3989. == ........ 7 * 19 * 1 409 * 3 989 = 747 526 633 7/. None of this set of prime factors divides 41 or 4120, above. However, factor 9 divides the segments 66,33,747,54 and factor 3 divides 21. 8/. I recognised that factor 1409 is approximately ~= 700*2+19/2. And factor 3989 ~= 1900*2. 9/. This implies (?) that, if the prime factors are combined in (2 groups of 2) THEN wheel number, A, is a product of 2 similar factors. Hence, well suited .. for Fermat's method of factorising. 10/. Therefore, I completed the calculation on Casio FX 82 schools scicalc. X squared - A = Y^2 = 576 squared = 24^4 = 3^4* 2^12th. X = 27 347. sqrt((27 347^2) - 747 526 633) = 576. 11/. That is A = X^2 - Y^2 = (X+Y)*(X-Y) = 27,923*26,771. 12/. Check - 7 divides 1771, but not 25,ooo. Therefore, 7 should divide 27923 = 28,91,7 - 27,923 = 994= 7*142. 13/. In fact, (a bit curious,) A = 7* 10 6789 519, including the increasing sequence 6-7-8-9, string. 14/. A ~= 95 76 95 19 + 1102 = 95 +19 -38. Also verifies factor 19. 15/. I completed the initial draft of this letter 1 hr after live draw on TV2. 905 pm SAT.