mm-3999 === Subject: Re: Need Advice - - just passed PreCalculus with an A+, now moving on to Calculus w/ Applications My question is, since we are starting on derivatives, and Limits, > what is good way to study? I may have slight weakness with partial > fraction decomposition, I don't want to drop this class because of > confusement, and/or teaching style of professor. . . My experience of teaching is approximately zero, but on the other hand I've done a lot of self-directed study, so I'll venture a few avuncular remarks. The introductury calculus course is a somewhat inhomogeneous array of notions and methods. If it doesn't look entirely coherent, that's because it isn't. Don't get disoriented by all those epsilons and deltas. Try to keep the main ideas visible, as an aid to navigation, so to speak. The notion of limit is fundamental. Some say it should be taught in high school. Two varieties of limit appear in calculus: the limit of a function at a point, and the limit of a sequence at infinity. Both involve infinite processes (speaking loosely) which go like for any delta (out of infinitely many) there exists an epsilon such that .... Infinite processes are new to most first-year people, which might account for some of the difficulty. About series, terminology to the contrary, there is no such thing as an infinite sum. The sum of a series (if it exists) is simply the limit at infinity (if it exists) of a sequence of partial sums, each of which has only finitely many summands. When it comes to proving something about a limit, e.g. proving that a given series converges, two issues: -- you generally need to think something up, e.g. another series which majorizes the first and converges. -- several equally good answers are often possible. Differential calculus is concerned with the _local_ approximation of a given _function_ by a _linear_ function. Given say a function f: Rto R, the derivative of f at a point x can be thought of (or defined as) a linear function L: R to R such that the function | f(y) - f(x) - L(y-x) | / |y-x| approaches zero as y approaches x. Thus f(y)-f(x) is approximately L(y-x) locally. This definition works equally well in any number of variables. But in one variable, the linear functions L are one-to-one with their slopes, and it is traditional to refer to the slope of L, instead of the function L itself, as the derivative. In several variables a linear mapping is described by a matrix, and the total derivative is a corresponding matrix of somewhat arbitrary partial derivatives. The chain rule in several variables is quite intelligible in terms of the linear mappings which approximate the two given mappings. This simple linear algebra stuff dissolves some of the apparent differences between calculus in one variable and in several. A lot of infinitesimal thingees can be modelled with finite thingees; you might find it enlightening sometimes to do some modelling of this kind without passing to any limit. E.g. (A+a)(B+b) = AB +aB + Ba +ab. If a is small in comparison with A, and b is small in comparison with B, then ab is doubly small, and if we ignore it, we get a picture of the product rule for derivatives. Likewise ( f(x+2h) - 2f(x+h) + f(x) ) / h^2 is usually (e.g. when f is analytic) approximately the second derivative of f at x, when h is small. Mixed partial derivatives can be modelled in a similar way. The mean value theorem, and the slightly stronger mean value inequality, are used in many proofs, and should be mastered. Don't ignore math books that were written for (or written by) physicists and engineers. I've often found them helpful, and refreshing too. LH === Subject: Re: All solutions manuals are in pdf format or Microsoft Word format!! > All solutions manuals are in pdf format or Microsoft Word format. I have tried my best to put up with the collections of my solutions > manual. > Here is the list that only part of my collections: > I will put some time to make up the full list daily. > If you cant find the solution manual listed below, don't give up, feel > free to email me To get the solutions manual, just email me with a price you want to > get the solutions manual you want. > To see samples, just email me. My email is edisonee(at)gmail.com Solutions Manuals .NET 2.0 Interoperability Recipes by Bruce Bukovics > A Course in Game Theory by Martin J. 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Rivest, Clifford Stein > Introduction to Antenna Theory by Palacios > Introduction to electric circuits 6th Ed. by Dorf-Svaboda > Introduction to electrodynamics 3rd Ed. by David Griffiths > Introduction to Fluid Mechanics 5ed. by Fox > Introduction to Fluid Mechanics 6th Ed. by Fox > Introduction to Linear Algebra 3th Ed. by Gilbert Strang > Introduction to Probability by Charles M. Grinstead and J. laurie > Snell > Introduction to Quantum Mechanics 2ed Griffiths, David J. > Introduction to Software Engineering by R. Mall > Introduction to VLSI circuits and Systems by John P. Uyemura > Introduction to Wireless Systems by P. M. Shankar > Investment Analysis and Portfolio Management by Reilly Brown > Linear Algebra and its Applications 3rd Ed. by Lay > Linear Algebra by Jim Hefferon > Linear circuit analysis by R. A. DeCarlo and P. Lin > Linear Systems and Signals by B P Lathi > Logic Computer Design Fundamentals 2th Ed. by Morris Mano (selected > problems) > Managerial Accounting 11th Ed. > Material Science and Engineering an Introduction 5th Ed. by Callister > Material Science and Engineering an Introduction 6th Ed. by Callister > Materials & Processing in Manufacturing 9th Ed. by Demargo > Mathematical Methods for Physics and read more ?... Please zip all the files in a folder and mail me the stuff at joe.vineet@gmail.com please..... === Subject: Prenex form How can I turn: (Q(c) -> Ax(P(x,c))) AND EuR(u) into prenex normal form? Ax is for all x... and Eu is There is a u .. TNX!!! === Subject: Re: Prenex form simply put the quantors in front of the formula? === Subject: Re: Spin (Pauli) matrices > And the rest looks easy, but: The equations are nonlinear. > TxT**-1=TyT**-1TzT**-1-...ah, I see. Doesn't matter. > Good to know for doing the 3*3 case. > Uhm, is there anything that can be done *directly* with > the commutator equations? E.g. for the 2*2 case it's > also true that x^2+y^2+z^2=((-3/4,0),(0,-3/4)). Well, the commutator relations simply tell us that we have a representation of a certain Lie algebra, but undoubtedly you knew about that. Objects like x^2+y^2+z^2 emerge from the universal enveloping algebra. Use Casimir element as a buzzword for searching more material. IIRC in the 3x3 case (when x, y, z may be the infinitesimal generators for rotations about the respective axes) x^2+y^2+z^2=-2*identity. x^2+y^2+z^2 isn't always a scalar matrix. For example you can build a system of 5x5 matrices, where you have the corresponding 2x2 and 3x3 blocks along the diagonal. Jyrki PS To honor your .sig I should quote a passage from e.g. Tannh.8auser, but, alas, I can't. === Subject: Mobile 3GP Videos, Mobile Games, Mobile secrets www.entertainmentvenues.org venues for live entertainment, mobile sms, Mobile SMS Bomber,send bomb to your friends mobile, mobile themes, free nokia themes, Mobile 3GP Videos, Mobile Games, Mobile secrets, Online Games, free wallpapers, Msn Stuff and many more. === Subject: Preservation of the tridiagonal structure - QR algorithm I've been trying to demonstrate the following fact unsuccesfully: let A be a symmetric real matrix (hence Hermitian), and T its Hessenberg form; thus T is tridiagonal and Hermitian. Let T=QR be a QR factorization of T, where Q is a unitary and R a upper-triangular matrix. Under these assumptions, D=RQ is a Hermitian tridiagonal matrix. Now, if T is hermitian, then it follows from the similarity equation that D is Hermitian (D=Q^-1*T*Q). And here I get stuck. Any hints? === Subject: Re: Two questions on dedekind domain! > Let S bet the set of all ideals not invertible. Assume S is nonempty. > Then since R is noetherian, there exist maximal ideal in S. Denote it > M. If we can show M is prime ideal, we get contradiction and so we can > prove R is dedekind. But do you know that R is noetherian? What precisely are you trying to prove? And what are you assuming? I took it from your original post (if it was you) that you were trying to show that if every prime ideal in an integral domain R is invertible then R is a Dedekind domain. As has been pointed out, there are several definitions of Dedekind domain, the original one being that every non-zero ideal is a product of prime ideals, or (a slight variant) a unique product of prime ideals. You would have to specify, it seems to me, exactly what definition you are starting from. Incidentally, if you really assume throughout that R is noetherian all the problems become much simpler. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Two questions on dedekind domain! <10678368.1180952809095.JavaMail.jakarta@nitrogen.mathforum.org> Let S bet the set of all ideals not invertible. Assume S is nonempty. > Then since R is noetherian, there exist maximal ideal in S. Denote it > M. If we can show M is prime ideal, we get contradiction and so we can > prove R is dedekind. But do you know that R is noetherian? > What precisely are you trying to prove? > And what are you assuming? I took it from your original post (if it was you) > that you were trying to show that if every prime ideal > in an integral domain R is invertible > then R is a Dedekind domain. As has been pointed out, there are several definitions > of Dedekind domain, the original one being > that every non-zero ideal is a product of prime ideals, > or (a slight variant) a unique product of prime ideals. You would have to specify, it seems to me, > exactly what definition you are starting from. Incidentally, if you really assume throughout that R is noetherian > all the problems become much simpler. -- > Timothy Murphy > e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie > tel: +353-86-2336090, +353-1-2842366 > s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland The definition of dedekind domain is used with any of the nine statement written above since I alreay know they are actually equivalent statement. If you wish to define dedekind domain, then define it as the original one of them, we come to prove that R is dedekind in the sense as the origianl one. As Hagen demonstrated, that R is noetherian is proved by using Cohen's thm. So now we can start at the point R is noetherian. === Subject: Re: Cartesian Geometry Question Nice group you have here. :-) > The question I have been pondering really has little to do with > mathematics, but I feel the answer may be related to mathematics > because I accidentally observed the same relational disconnect in > Cartesian geometry a few years ago. If I can better understand the > answer to the math question, I may be able to apply certain properties > to the original question. > QUESTION: What properties do irrational numbers possess that allow > them to coexist on a line with rational numbers without destroying the > integrity of the line? > They fit in the gaps between rationals. > That there are gaps can be shown by exhibiting increasing infinite > sequences of rationals having rational upper bounds but no rational > exact upper bound. > But geometrically we expect there to be a point as the exact bound of > the corresponding sequence of rational points. Does that mean the progression of points on a line is absolutely > linear with infinitely small graduations along the line where no point > can ever technically permeate or truly touch another point? Two (different) points have a positive distance and therefore cannot touch. Also note that for a point on a geometric line the property of being rational or not is not intrinsic, it depends on origin an unit length chosen (i.e. on the ratio of lengths) Imagine someone with a cm-based ruler, someone with an inch-based ruler and someone from another planet with a totally unrelated ruler. === Subject: Re: Cartesian Geometry Question > On Jun 4, 3:38 am, Voice of Reason --> QUESTION: What properties do irrational numbers possess that allow > them to coexist on a line with rational numbers without destroying the > integrity of the line? > -- > If by integrity do you mean, recognizing only one operation or an > operation set viz. division/addition/multiplication between/among > rational numbers to yield other rational numbers? > If other operations and combinations are allowed among rationals,lots > of irrationals can be generated vying for accommodation on the line of > integers. > But then the 'purity' or 'integrity' is gone, that is what you mean? 7 > + 1/3 is rational, but 7^(1/3), log(7)to base 3 etc. are not rational > and so on? > Narasimham Multiplication of numbers is repeated addition.In fact,along with > division,multiplication,addition and subtraction operations between > whole numbers are also included to form new rational numbers. E.g., m > and n are + ve integers, m [n + 1/m^5 /(n-1)]/(n - m)^3 is rational, > but .. m [n + m^1/n /(n-1)]/(n - m)^1/n for (non-integral 1/n) is > irrational. Exponentiation is repeated multiplication. But then why has the more > powerful fractional exponentiation been a barred operation in forming > further real numbers? After that, even transcendental numbers have a place there, as sin(2) > = 2 - 2^3/3! etc. by an infinite series but up of positive integers. IMHO classical rational number definition/formation can be extended or > included to give fractional exponents also a berth on the line of > rationals as reals, as needs from such generalization in mode of > number forming, to remedy (what looks to me as a) hitherto compromised > situation. It stands to at least some reason. Narasimham- Hide quoted text - - Show quoted text - By pointing out the flawed nature of my question and explaining why it was mathematically flawed, you have aided me in discovering the answer to the metaphysical question that prompted me to notice what I mistakenly thought was a problematic mathematical disconnect. It is funny how being corrected in one field can lead to self-correction in another. Again, thank you all for being so kind and helpful. === Subject: Re: An exact simplification challenge - 18 <4665548d.96849271@news.individual.de> TR> value(combine(convert(%,'Int'))); TR> TR> 0 TR> No myths involved, just facts. I fully agree with you. It's best to rely upon the facts, not myth. So, can we hear Mathematica experts? How simple is to zero these elliptics in Mathematica? > Hello computer algebra souls, > Is there a simplifier who can invent and display a string > of CAS commands to reduce this elliptic expression > EllipticK(sqrt(3/4)) - sqrt(4/3)*EllipticK(sqrt(4/3)) - > I*EllipticK(sqrt(1/4)) Maple can do: convert(%,'Int'); > combine(%); > value(%); No myths involved, just facts. -- > Thomas Richard > Maple Support > Scientific Computers GmbHhttp://www.scientific.de === Subject: Re: An exact simplification challenge - 18 <4665548d.96849271@news.individual.de> Hello. I have tried to simplify the involving elliptics to zero within Mathematica, but with no success. It is a hard task! Anyway in Mma 5.2 you could try In[93]:= Developer`ZeroQ[EllipticK[3/4] - Sqrt[4/3]*EllipticK[4/3] - I*EllipticK[1/4]] Out[93]= True But ZeroQ uses a combination of symbolic transformations and randomized numerical evaluation. So I guess this is not a desired solutions since may be numerical procedures take place! Dimitris / Vladimir Bondarenko : TR> value(combine(convert(%,'Int'))); > TR TR> 0 > TR> No myths involved, just facts. I fully agree with you. It's best to rely upon the > facts, not myth. So, can we hear Mathematica experts? How simple is > to zero these elliptics in Mathematica? > Hello computer algebra souls, > > Is there a simplifier who can invent and display a string > of CAS commands to reduce this elliptic expression > > EllipticK(sqrt(3/4)) - sqrt(4/3)*EllipticK(sqrt(4/3)) - > I*EllipticK(sqrt(1/4)) > > Maple can do: > > convert(%,'Int'); > combine(%); > value(%); > > No myths involved, just facts. > > -- > Thomas Richard > Maple Support > Scientific Computers GmbHhttp://www.scientific.de === Subject: Unlock your Mobile Phone Unlock your mobile phone for free. Also watch television and sport online for free. http://homepage.ntlworld.com/louise.randall41 === Subject: Infinite Dimensional Polysign Numbers > Timothy , have you ever considered infinite > dimensional polysigned numbers ? I have considered them some. One neat thing is that the unit vector angle approaches ninety degrees. In effect the higher the sign the closer we get to the traditional Cartesian system. Yet all of the missing negative portions are balanced by one extra unit vector n = D + 1 . The exact angle is pi - arccos( 1 / ( n - 1 ) ) . Beyond this the progression P1 + P2 + P3 + P4 + ... has physical correspondence with a natural breakpoint in product behavior beyond P3. This progression is infinite in its construction. However to claim that physical reality carries the entirety of this progression to infinity one would need a physics model which I do not have. Either way the natural progression itself as a pure mathematical construct is infinite. It supports spacetime via the product operation behavior. This is suggestive of a physics model built with a product The infinite dimensional system has another fascinating behavior. When we take a distance we see that d d = x0 x0 + x1 x1 + x2 x2 + x3 x3 ... so in effect we can find infinite distances who are measured to be one unit away in every dimension: x0 = 1, x2 = 1, x3 = 1, ... d d = 1 + 1 + 1 + ... At this level large distance could imply high dimension. I guess if you want a pocket universe then infinite dimension is a good place to start based simply on the distance behavior alone. Getting 3D physical correspondence out of that basis becomes the challenge. Playing games on say a ten dimensional space that yield a three dimensional one are much more suspicious than going whole hog on it. If you play the ten game then you are merely left asking why ten? which comes up when you ask why three? which is an upward progression. Neatly enough in the polysign progression P1 P2 P3 | P4 P5 ... whichever side of the breakpoint bar you take you can still get three dimensions. In other words P4 could be of importance on its own. The bar is a unique point in the sequence even just from summation behavior: 0D 1D 2D | 3D 4D 5D ... The trouble I have with trying P4 as a standalone is how to take the P1,P2,P3 portion and make a coherent basis out of it while that portion screams spacetime. -Tim === Subject: Re: Infinite Dimensional Polysign Numbers > Timothy , have you ever considered infinite > dimensional polysigned numbers ? > > I have considered them some. One neat thing is that > the unit vector angle approaches ninety degrees. In > effect the higher the sign the closer we get to the > traditional Cartesian system. Yet all of the missing > negative portions are balanced by one extra unit > vector > n = D + 1 . > The exact angle is > pi - arccos( 1 / ( n - 1 ) ) . yep i found that too. > > Beyond this the progression > P1 + P2 + P3 + P4 + ... > has physical correspondence with a natural breakpoint > in product behavior beyond P3. This progression is > infinite in its construction. However to claim that > physical reality carries the entirety of this > progression to infinity one would need a physics > model which I do not have. Either way the natural > progression itself as a pure mathematical construct > is infinite. It supports spacetime via the product > operation behavior. This is suggestive of a physics > model built with a product i did not claim that. im not even sure that's true. > > If we allow that humans are backward in their > thinking then perhaps we should consider that the > natural state starts at infinity and allow an inverse > accretion toward the singularity zero. Is this an > identical system? Could there be some slender > consequence to inverse accretion? It does look like a > continuum is exposed but that sense is no different > than standard calculus is it? Going upward from zero > we could get to 2531 and say good enough with the > infinity doesn't seem to get us anywhere at all at > first glance. Perhaps it is an opportunity to use > exponentials and work back in fifths say or tenths. > Then zero looks a long way away but we could say good > enough at some point. It looks like the natural > bidirectional symmetric system is not integer in > both directions, or maybe rather that the integer > should be considered a nonlinear concept. Hah-hah. > > The infinite dimensional system has another > fascinating behavior. When we take a distance we see > that > d d = x0 x0 + x1 x1 + x2 x2 + x3 x3 ... > so in effect we can find infinite distances who are > measured to be one unit away in every dimension: > x0 = 1, x2 = 1, x3 = 1, ... > d d = 1 + 1 + 1 + ... > At this level large distance could imply high > dimension. > Equivalently short distances imply thinly populated > coordinates which is somewhat like the observed > density of matter. Physical correspondence is fairly > broken here but if we allow for a product space then > some weird things can happen especially in the > theorems that show subspaces are important. I don't > understand that stuff yet but it is good stimulus. I > do understand that the division problem exposes the > subspace structure and so to reverse the product > should expose that substructure. > > I guess if you want a pocket universe then infinite > dimension is a good place to start based simply on > the distance behavior alone. snip Getting 3D physical > correspondence out of that basis becomes the > challenge. not for me ?! i can make any dimensional basis for the infinite dimensional polysigned numbers if dimension basis is 2 or more. that was the idea. Playing games on say a ten dimensional > space that yield a three dimensional one are much > more suspicious than going whole hog on it. If you > play the ten game then you are merely left asking > why ten? > which comes up when you ask > why three? > which is an upward progression. > > Neatly enough in the polysign progression > > P1 P2 P3 | P4 P5 ... > > whichever side of the breakpoint bar you take you can > still get three dimensions. In other words P4 could > be of importance on its own. The bar is a unique > point in the sequence even just from summation > behavior: > 0D 1D 2D | 3D 4D 5D ... > The trouble I have with trying P4 as a standalone is > how to take the P1,P2,P3 portion and make a coherent > basis out of it while that portion screams spacetime. > > > -Tim you do have a wild imagination maybe you should learn to take things step bye step. your very imaginative and speculative. maybe focus a bit more. no offense. also note that i meant uncountable infinity , hoping you know what that means. tommy1729 === Subject: Re: Infinite Dimensional Polysign Numbers <21867006.1181081421426.JavaMail.jakarta@nitrogen.mathforum.org > Timothy , have you ever considered infinite > dimensional polysigned numbers ? > I have considered them some. One neat thing is that > the unit vector angle approaches ninety degrees. In > effect the higher the sign the closer we get to the > traditional Cartesian system. Yet all of the missing > negative portions are balanced by one extra unit > vector > n = D + 1 . > The exact angle is > pi - arccos( 1 / ( n - 1 ) ) . yep i found that too. > Beyond this the progression > P1 + P2 + P3 + P4 + ... > has physical correspondence with a natural breakpoint > in product behavior beyond P3. This progression is > infinite in its construction. However to claim that > physical reality carries the entirety of this > progression to infinity one would need a physics > model which I do not have. Either way the natural > progression itself as a pure mathematical construct > is infinite. It supports spacetime via the product > operation behavior. This is suggestive of a physics > model built with a product i did not claim that. > im not even sure that's true. | A B | = | A || B | is the behavior which indicates spacetime congruence. These || operations are distance or magnitude in the traditional sense. Under the real(P2) and complex(P3) numbers (and P1) distance is conserved. When we take these three alone: P1 P2 P3 we see spacetime congruence. In P4 and up this general behavior is broken. Distance under product is not well behaved in P4+. Yet the usual algebraic field properties do hold if additional nonzero division exceptions are permitted. Associative and commutative properties are upheld in P4+ so to say that the math is badly designed is not quite appropriate. Instead if we adopt this math then we have a natural basis for spacetime. We then need to develop a product based approach to physics. This is fairly consistent with the classical force equations. Anyhow under the polysign numbers algebra (at least using just product and sum) can take place in any dimension. The product reversal (division) is very difficult in the higher signs and so if we do consider ouselves to be in a product space of the natural progression P1 P2 P3 P4 ... then retrieving the higher dimension components will be more difficult than the lower (P3-) components. Perhaps even impossible. It is the breakpoint of the product behavior | A B | = | A || B | which indicates support for spacetime. Furthermore the unidirectional zero-dimensional P1 is congruent with time and this feature has been left out of traditional mathematics. -Tim > If we allow that humans are backward in their > thinking then perhaps we should consider that the > natural state starts at infinity and allow an inverse > accretion toward the singularity zero. Is this an > identical system? Could there be some slender > consequence to inverse accretion? It does look like a > continuum is exposed but that sense is no different > than standard calculus is it? Going upward from zero > we could get to 2531 and say good enough with the > infinity doesn't seem to get us anywhere at all at > first glance. Perhaps it is an opportunity to use > exponentials and work back in fifths say or tenths. > Then zero looks a long way away but we could say good > enough at some point. It looks like the natural > bidirectional symmetric system is not integer in > both directions, or maybe rather that the integer > should be considered a nonlinear concept. Hah-hah. > The infinite dimensional system has another > fascinating behavior. When we take a distance we see > that > d d = x0 x0 + x1 x1 + x2 x2 + x3 x3 ... > so in effect we can find infinite distances who are > measured to be one unit away in every dimension: > x0 = 1, x2 = 1, x3 = 1, ... > d d = 1 + 1 + 1 + ... > At this level large distance could imply high > dimension. > Equivalently short distances imply thinly populated > coordinates which is somewhat like the observed > density of matter. Physical correspondence is fairly > broken here but if we allow for a product space then > some weird things can happen especially in the > theorems that show subspaces are important. I don't > understand that stuff yet but it is good stimulus. I > do understand that the division problem exposes the > subspace structure and so to reverse the product > should expose that substructure. > I guess if you want a pocket universe then infinite > dimension is a good place to start based simply on > the distance behavior alone. snip Getting 3D physical > correspondence out of that basis becomes the > challenge. not for me ?! i can make any dimensional basis for the infinite dimensional polysigned numbers if dimension basis is 2 or more. Well if you'd like to discuss this I am open to that discussion. That is why I am here. I don't really understand the statement that you have made. It seems to me that the polysign numbers form dimension rather than the other way around. That is why P1 has been overlooked. That the complex numbers follow in the progression directly after the real numbers with no additional rules is another strong gain. When we talk traditionally of an infinite dimensional space we have built that space from a one dimensional basis. So I think your value 2 could be 1. that was the idea. Playing games on say a ten dimensional > space that yield a three dimensional one are much > more suspicious than going whole hog on it. If you > play the ten game then you are merely left asking > why ten? > which comes up when you ask > why three? > which is an upward progression. > Neatly enough in the polysign progression > P1 P2 P3 | P4 P5 ... > whichever side of the breakpoint bar you take you can > still get three dimensions. In other words P4 could > be of importance on its own. The bar is a unique > point in the sequence even just from summation > behavior: > 0D 1D 2D | 3D 4D 5D ... > The trouble I have with trying P4 as a standalone is > how to take the P1,P2,P3 portion and make a coherent > basis out of it while that portion screams spacetime. > -Tim you do have a wild imagination > maybe you should learn to take things step bye step. > your very imaginative and speculative. > maybe focus a bit more. This criticism is devoid of content. As you say, equations please. I would much prefer you to point out the things which you see as weak or unbelievable and why. Yes, this is speculative. But your original question is about infinite dimension and that concept is speculative especially if we are referring to reality. I do think it is an interesting problem. Informationally a single infinite dimensional point is a large burden. I don't mean to discourage the concept. Perhaps this is reason to try the concept on a different basis than the reals or any continuum type basis. Since the polysign has a product it should also be pointed out that the rotational effect of the product can never be instantiated on the infinite sign system. P3: ( + 2 )( + 4 ) = - 8 This single sign resultant from higher signs cannot be demonstrated in P(inf). I don't really know what the consequences of that are but it is no different than calculus type problems. We can let n approach infinity and instantiate large n so one would hope we could get to a large enough n and say good enough. -Tim no offense. also note that i meant uncountable infinity , hoping you know what that means. tommy1729 === Subject: Re: Infinite Dimensional Polysign Numbers >> > snip > > Getting 3D physical > > correspondence out of that basis becomes the > challenge. > > not for me ?! > > i can make any dimensional basis for the infinite > dimensional polysigned numbers if dimension basis is > 2 or more. > > Well if you'd like to discuss this I am open to that > discussion. That > is why I am here. > I don't really understand the statement that you have > made. It seems > to me that the polysign numbers form dimension rather > than the other > way around. That is why P1 has been overlooked. > That the complex numbers follow in the progression > directly after the > real numbers with no additional rules is another > strong gain. When we > talk traditionally of an infinite dimensional space > we have built that > space from a one dimensional basis. So I think your > value 2 could be > 1. > no? since complex already requires 2 dimensions and the reals are not algebraicly closed whereas the complex are. can you plot the complex in 1 dimension ?? dont think so. and therefore any extension is also at least 2 dimensional. > > you do have a wild imagination > maybe you should learn to take things step bye > step. > your very imaginative and speculative. > maybe focus a bit more. > > This criticism is devoid of content. As you say, > equations please. > I would much prefer you to point out the things which > you see as weak > or unbelievable and why. i just did above > Yes, this is speculative. But your original question > is about infinite > dimension and that concept is speculative especially > if we are > referring to reality. I do think it is an interesting > problem. > Informationally a single infinite dimensional point > is a large burden. > I don't mean to discourage the concept. Perhaps this > is reason to try > the concept on a different basis than the reals or > any continuum type > basis. Since the polysign has a product it should > also be pointed out > that the rotational effect of the product can never > be instantiated on > the infinite sign system. > P3: ( + 2 )( + 4 ) = - 8 > This single sign resultant from higher signs cannot > be demonstrated in > P(inf). I don't really know what the consequences of > that are but it > is no different than calculus type problems. We can > let n approach > infinity and instantiate large n so one would hope we > could get to a > large enough n and say good enough. > i would prefer to visualise things however the forum does not allow that :( > -Tim > > > no offense. > > also note that i meant uncountable infinity , > hoping you know what that means. do you understand that ? > tommy1729 > > === Subject: Re: Infinite Dimensional Polysign Numbers <32253125.1181149083536.JavaMail.jakarta@nitrogen.mathforum.org > snip > Getting 3D physical > correspondence out of that basis becomes the > challenge. > not for me ?! > i can make any dimensional basis for the infinite > dimensional polysigned numbers if dimension basis is > 2 or more. > Well if you'd like to discuss this I am open to that > discussion. That > is why I am here. > I don't really understand the statement that you have > made. It seems > to me that the polysign numbers form dimension rather > than the other > way around. That is why P1 has been overlooked. > That the complex numbers follow in the progression > directly after the > real numbers with no additional rules is another > strong gain. When we > talk traditionally of an infinite dimensional space > we have built that > space from a one dimensional basis. So I think your > value 2 could be > 1. no? since complex already requires 2 dimensions and the reals are not algebraicly closed whereas the complex are. > can you plot the complex in 1 dimension ?? > dont think so. > and therefore any extension is also at least 2 dimensional. > you do have a wild imagination > maybe you should learn to take things step bye > step. > your very imaginative and speculative. > maybe focus a bit more. > This criticism is devoid of content. As you say, > equations please. > I would much prefer you to point out the things which > you see as weak > or unbelievable and why. i just did above > Yes, this is speculative. But your original question > is about infinite > dimension and that concept is speculative especially > if we are > referring to reality. I do think it is an interesting > problem. > Informationally a single infinite dimensional point > is a large burden. > I don't mean to discourage the concept. Perhaps this > is reason to try > the concept on a different basis than the reals or > any continuum type > basis. Since the polysign has a product it should > also be pointed out > that the rotational effect of the product can never > be instantiated on > the infinite sign system. > P3: ( + 2 )( + 4 ) = - 8 > This single sign resultant from higher signs cannot > be demonstrated in > P(inf). I don't really know what the consequences of > that are but it > is no different than calculus type problems. We can > let n approach > infinity and instantiate large n so one would hope we > could get to a > large enough n and say good enough. i would prefer to visualise things however the forum does not allow that :( > -Tim > no offense. > also note that i meant uncountable infinity , > hoping you know what that means. do you understand that ? > tommy1729 I guess I got confused Tommy. I still don't understand where you are going with 2D as a special space. Originally you said i can make any dimensional basis for the infinite dimensional polysigned numbers if dimension basis is 2 or more. I was thinking that when we call it 2D we mean that it is 1D + 1D and so the 2D is traditionally constructed from the 1D, as are all higher dimensional spaces. The polysign construction gets dimension by increasing sign instead and maps directly to the simplex coordinate system. Are you thinking that four-signed numbers are actually 2D? -Tim === Subject: Re: @ timothy golden Funny timing Tommy. I just posted a response. === Subject: Re: coequalizer + coproduct --> cocomplete >> I am trying to prove that a category with coequalizers and coproducts >> is cocomplete (has colimits). >> My idea is that a colimit is always a quotient of a coproduct. Good idea. >> Somehow, I need to express this quotient using coequalizers. I can >> handle the very simple case of a colimit of a diagram with 2 objects >> and a single morphism between them but cannot go further. >> Any suggestions/hints ? In Categories for the Working Mathematician by Mac Lane there is a > proof that a category that has equalisers for all parallel pairs of > arrows and all small products (including a terminal object) then that > category is complete (it's on page 113 in the second edition). The > proof you are looking for is dual to the aforementioned proof I > think. Bingo. -- Jesse F. Hughes Like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as it might seem. -- Alan MacKay === Subject: Re: ED CONRAD WINS NOBEL PRIZE -- Turns on Light in Dark Ages of Corrupt Science -- Hip, Hip, Hurrah!. neilist how about posting some math too ? HA HA HA HA...... I have some interesting mathematical thoughts, but they are too small for this margin ... er ... I mean newsgroup. When Harris/Bassam stop posting and rantings (what, they have to DIE to go away?), I'll post some math which, hopefully, others won't consider . Remember Gauss' motto: Few but ripe! (quoted in E.T. Bell's Men of Mathematics) === Subject: =?iso-8859-1?q?Re:_looking_for_Fr=E9chet's_1906_Ph=2ED=2E_dissertation_intro d ucing_metric_spaces?= > Does anyone know of any online archives where I can > find Maurice Fr.8echet's 1906 Ph.D. dissertation > Sur quelques points du calcul fonctionnel ? It is > supposed to be the first paper to introduce the concept > of a metric space (even though the term metric space > itself is due to Hausdorff). I think that Fr.8echet's 1906 dissertation may have > been published in the same year in the journal > Rendiconti del Circolo Mathematico di Palermo, > volume 22, pages 1-74. Does anyone know where I > could find this version of Fr.8echet's work? I have asked basically this same question about 3 weeks > ago in another post. I hope I don't annoy anyone by > asking the same question twice. I don't believe this journal is on-line, at least not freely available. Most large U.S. state universities will have it, if you're in the U.S. However, assuming you can read French, you may find much of the motivation and approach difficult to follow, unless you have a good knowledge of the point set theory of the time. For example, limit points and closed sets and the Cantor-Bendixson theorem are central concepts, while the notion of an open set may not even appear. Also, much of the motivation springs from generalizing and organizing these notions that were then beginning to be applied to collections of curves and collections of functions, which Volterra and others had been introducing in their study of the calculus of variations and (what we now know as) the beginnings of functional analysis. The following will be very useful if you want to study Frechet's thesis: Angus E. Taylor, A study of Maurice Fr.8echet I [II] {III}, Archive for History of Exact Sciences 27 (1982) [34 (1985)] {37 (1987)}, 233-295 [279-380] {25-76}. Michael Bernkopf, The development of function spaces with particular reference to their origins in the integral equation theory, Archive for History of Exact Sciences 3 (1966), 1-96. C. E. Aull and R. Lowen (editors), Handbook of the History of General Topology, Kluwer Academic Punlishers, 3 volumes, 1997 & 1998 & 2001. Dave L. Renfro === Subject: =?iso-8859-1?q?Re:_looking_for_Fr=E9chet's_1906_Ph=2ED=2E_dissertation_intro d ucing_metric_spaces?= > I don't believe this journal is on-line, at least not > freely available. I would be willing to settle for a non-freely available one. I do have access to a number of (regrettably) non-free archives (such as JSTOR) through my university. > However, assuming > you can read French, you may find much of the motivation > and approach difficult to follow, Despite the quote in my post from Abel (... one should study the masters and not the pupils), in practice I don't follow this. That is, I think that rather than learning from the master that introduced a concept while that concept was still in an immature state, it is better to learn from a pupil that has mastered the concept after the concept reached maturity and can connect it in a natural illuminating way, with a large complex of other mathematical ideas (G.H. Hardy). For example, rather than learning about metric spaces from Frechet's 1906 dissertation, I would think it more expedient to learn about them from Rudin's Principles of Mathematical Analysis or some other standard text. The reason that I look for these old classic papers is not so much to learn mathematical concepts from them, but rather to reference them in the stuff I write about to give myself and potential readers some idea of where and when and how a link in the web of mathematics came to be. I think knowing such background information is helpful and enlightening, even though not directly related to the concept itself. And I guess the reason why I like to give online web links to these sources is it kind of gives readers a sense of empowerment. It gives them the sense that the classic math papers that made history are not just available to researchers in a ~900 year old Oxford University or an ivy league school, but are available to anyone with an internet connection in any country no matter how impoverished. I know that there is more and more intellectual material available online. And I don't like to be a whiner. But to be honest, it is a little difficult for me to understand how papers that have revolutionized mathematics (like Frechet's 1906 paper introducing metric spaces or Legesgue's 1902 paper introducing Lebesgue integration) can remain so elusive and difficult to access. This is intellectual history that has changed the thinking of mankind. Shouldn't that mean something to someone enough to make it easily accessible to all? What if the U.S.'s Bill of Rights was as difficult to access? England's Magna Carta? The Code of Hammurabi? Enough whining from me. I wish you the best in your work. Dan Greenhoe === Subject: =?iso-8859-1?q?Re:_looking_for_Fr=E9chet's_1906_Ph=2ED=2E_dissertation_intro d ucing_metric_spaces?= > For example, rather than learning about metric spaces > from Frechet's 1906 dissertation, I would think it more > expedient to learn about them from Rudin's Principles > of Mathematical Analysis or some other standard text. The reason that I look for these old classic papers is > not so much to learn mathematical concepts from them, > but rather to reference them in the stuff I write about > to give myself and potential readers some idea of where > and when and how a link in the web of mathematics came > to be. I think knowing such background information is > helpful and enlightening, even though not directly > related to the concept itself. I didn't think you intended to learn about metric spaces from Frechet's paper. My point (perhaps not very well expressed) was that even if you know quite a bit about metric spaces, you might still find it difficult to follow Frechet's paper (which involves quite a bit more than just introducing metric spaces). > I know that there is more and more intellectual material available > online. And I don't like to be a whiner. But to be honest, it > is alittle difficult for me to understand how papers that have > revolutionized mathematics (like Frechet's 1906 paper introducing > metric spaces or Legesgue's 1902 paper introducing Lebesgue > integration) can remain so elusive and difficult to access. This > is intellectual history that has changed the thinking of mankind. > Shouldn't that mean something to someone enough to make it easily > accessible to all? What if the U.S.'s Bill of Rights was as > difficult to access? England's Magna Carta? The Code of Hammurabi? Well, as anyone who has been following my posts since 1999 will know, this isn't something you need to convince me about! I've gone to great lengths to archive in sci.math quite a bit of mathematical history. Often, I'll just cite the relevant papers, but even then I try to give as complete bibliographic information as I can (complete journal titles, full author names, MR & Zbl & JFM citation codes, etc.). The ruler function Biography of Luzin Algebra word problems from two 1850's U.S. texts Ten all-time most influential math books Dave L. Renfro === Subject: =?iso-8859-1?q?Re:_looking_for_Fr=E9chet's_1906_Ph=2ED=2E_dissertation_intro d ucing_metric_spaces?= > ... > I've gone to great lengths to archive in sci.math quite > a bit of mathematical history. ... The ruler function --- I have looked at some of your previous posts and they are impressive. Actually, a few weeks ago I was looking for information about the ruler function and found that it originated with Thomae's 1875 text. Although I can't remember for sure, I think I may have found some of the information I was looking for from one of your posts: Dan Greenhoe === Subject: =?windows-1253?Q?Re:_looking_for_Fr=E9chet's_1906_Ph.D._dissertation_intr?= =?windows-1253?Q?oducing_metric_spaces?= > Well, as anyone who has been following my posts since 1999 will > know, this isn't something you need to convince me about! > I've gone to great lengths to archive in sci.math quite > a bit of mathematical history. Often, I'll just cite the > relevant papers, but even then I try to give as complete > bibliographic information as I can (complete journal titles, > full author names, MR & Zbl & JFM citation codes, etc.). That's why you've earned the well-deserved title of Moveable Mathematics Encyclopedia of sci.math. It should also be noted for the benefit of those who are relatively new to sci.math that Dave has *greatly* aided some of us individually by providing references for research material which would otherwise be very difficult to obtain [*] or cause the author to expend tremendous amounts of time looking for it. Because it is reasonable to assume that Dave has performed similar service to other mathematicians and because his interests are very diverse, I'd easily nominate him as one of the top most valuable contributors of sci.math. [*] for example, most of the tetration references which I've used have come from Dave and I needed them at a time when I didn't have access to JSTOR or to the corresponding journals themselves. > Dave L. Renfro -- I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/ ---------------------------------------------------------- There's ALWAYS a mistake somewhere === Subject: Why The Simplex Coordinate System? The simplex coordinate system reveals a new means of deriving dimension. Rather than place your simplex in a Cartesian space please try to swap them. There is a slim redundancy whose consequences are more revealing when the simplex is taken as the basis. The nuances are simple and so simple as to be overlooked by existing mathematics. The finest display of isotropy is found in the simplex coordinate system. In n dimensions we simply take n+1 vertices and arrange them such that they are of equal distance to one another. At the center of this arrangement we denote an origin and from that origin we have now established unit vectors to the vertices. This is the simplex coordinate system. Here is a 2D version: http://bandtechnology.com/PolySigned/Lattice/P3Lattice.png One of the most astounding features of the new coordinate system is that it is unidirectional. The means of return is always by traveling in the other directions. This feature is one of the nuances of the simplex coordinate system that standard geometry lacks. - 1 + 1 = 0 . If this were the only consequence then we might dismiss the rearrangement as trivial. However the consequences do extend further. The real arithmetic product contains a generalized form on the simplex structure. The complex numbers are yielded directly when we extend upward in dimension via the simplex. The procedure entails defining sign as the unit vectors of the simplex. The two systems are in perfect correspondence since we see that the sum of the unit vectors of the simplex is zero. This is the logical generalization of sign. Consider that a third sign '*' could exist. The generalizable form that can be lent from the real numbers comes for the symmetrical - 1 + 1 = 0 which then yields - 1 + 1 * 1 = 0 . Thus the 2D simplex is arrived at with three unit vectors (-,+,*). The product operation is fairly obvious and you will find that it exactly matches the complex numbers http://bandtechnology.com/PolySigned/ThreeSignedComplexProof.html The product and sum operations are extensible to any dimension including zero. The generalization of sign is directly tied to the simplex coordinate system geometry by the sum of the unit (sign) vectors both of which obey Sum over s ( s x ) = 0 where s is sign or simplex unit vector and x is magnitude. Beneath the real numbers is a unidirectional space that is zero dimensional. This space has correspondence with time and has been overlooked since the real numbers are traditionally treated as the geometrical basis. Upon accepting the simplex or polysign as primitive the usage of the real number and the Cartesian product look questionable. Is the Cartesian product necessary? Because the polysign arithmetic generates dimension it is not necessary in the new system. http://bandtechnology.com/PolySigned/PolySigned.html -Tim === Subject: having difficulty understanding Annihilator By definition, annihilator of Y is the set of all linear functionals l that vanish on a subspace Y of linear space X. Does this mean annihilator contains set with l(y)=0 i.e. every value is O or I am wrong? How does one show if any linear function is in the annihilator of a subspace. Do we tackle this problem by looking at which linear functions will give O. === Subject: Re: having difficulty understanding Annihilator On Tue, 05 Jun 2007 10:33:34 EDT, quantperson Does this mean annihilator contains set with l(y)=0 i.e. every value is O or I am wrong? You aren't stating it very precisely. A linear functional f is in the annihilator of Y if for all y in Y, f(y) = 0. > How does one show if any linear function is in the annihilator >of a subspace. Do we tackle this problem by looking at which > linear functions will give O. Assuming that O really means 0 (zero) (the number zero, not the vector O), yes. Just check that f(y) = 0 for all y in Y. --Lynn