Do you write ASM for the Pentium?:) > My calculus is self taught. Mine too. > Int(x*(e^(2x))/(2x+1)^2) The answer is: 1/4*(e^(2x))/(2x+1) + C I'm having a terrible time finding a way to simplify the integral > though. Since 2x appears twice in the integrand, I tried the substitution z = 2x+1 dx = dz/2 and ended up with 1/4e Int [ (z-1)e^z / z ] dz which is recognizable as (dismissing a constant or two) e^z / z. As you may know, the integrals of e^x / x and e^x / x^2 do not have a closed form , although we just saw that the difference between them does. LH ==== Math is a hobby for me. I've been reading up on Eigenvectors and Eigenvalues. It get the manipulations involved, but can't imagine the applications -- and the books I have don't help. Can people provide a few examples? Specific examples, if possible -- not just, they are used in electronics, or physics, or whatever, but rather, something like: M is the matrix which describes such-and-such physical property or transformation or process, It's eigenvectors V correspond to such and such property, and the eigenvalues of V and M indicate such-and-such. Steve O. ==== >Math is a hobby for me. I've been reading up on Eigenvectors and >Eigenvalues. It get the manipulations involved, but can't imagine the >applications -- and the books I have don't help. Can people provide a >few examples? Google uses a (big) matrix to describe hyperlinks between web pages and ranks them in order by solving the eigenvalues and -vectors. ==== > Math is a hobby for me. I've been reading up on Eigenvectors and > Eigenvalues. It get the manipulations involved, but can't imagine the > applications -- and the books I have don't help. Can people provide a > few examples? Specific examples, if possible -- not just, they are used in > electronics, or physics, or whatever, but rather, something like: M is the matrix which describes such-and-such physical property or > transformation or process, It's eigenvectors V correspond to such and > such property, and the eigenvalues of V and M indicate such-and-such. > One application is in differential equations. If you want to solve a system of linear equations in the form y' = A y where A is a nxn Matrix and y is a vector valued function you can use the eigenvectors of A to obtain the solutions. In particular, if a is an eigenvector of A with eigenvalue l, then y(x) = c e^(l x) a is a solution for any real or complex constant c. If the matrix A has a basis of eigenvectors (i.e. if A is diagonalizable) then any solution looks like y(x) = c_1 e^(l_1 x) a_1 + c_2 e^(l_2 x) a_2 + ... + c_n e^(l_n x) a_n where a_1, a_2, ... a_n are the eigenvectors to the eigenvalues l_1, l_2, ... l_n respectively and c_1, c_2, ... c_n are constants. This fact has also big implications for nonlinear differential equations. The keywords for this are linearization and stability . HTH, Michael. -- &&&&&&&&&&&&&&&&#@#&&&&&&&&&&&&&&&& Dr. Michael Ulm FB Mathematik, Universitaet Rostock michael.ulm@mathematik.uni-rostock.de ==== > Math is a hobby for me. I've been reading up on Eigenvectors and > Eigenvalues. It get the manipulations involved, but can't imagine the > applications -- and the books I have don't help. Can people provide a > few examples? Very useful for in solving simultaneous equations particularly with a large number of variables. I am from an electronics/physics background and used them extensively. Andy ==== I distinctly remember a usenet post quoting some member of British royalty saying, Algebra. Is that those pointy things? although a google search denies me. ==== Math is a hobby for me. I've been reading up on Eigenvectors and Eigenvalues. It get the manipulations involved, but can't imagine the applications -- and the books I have don't help. Can people provide a few examples? Specific examples, if possible -- not just, they are used in electronics, or physics, or whatever, but rather, something like: M is the matrix which describes such-and-such physical property or transformation or process, its eigenvectors V correspond to such and such property, and the eigenvalues of V and M indicate such-and-such. Steve O. ==== > Math is a hobby for me. I've been reading up on Eigenvectors and > Eigenvalues. It get the manipulations involved, but can't imagine the > applications -- and the books I have don't help. Can people provide a > few examples? Specific examples, if possible -- not just, they are used in > electronics, or physics, or whatever, but rather, something like: M is the matrix which describes such-and-such physical property or > transformation or process, its eigenvectors V correspond to such and > such property, and the eigenvalues of V and M indicate such-and-such. Imagine P being the matrix of transition probabilities from one state of a system to another of some system. P_ij is the probability that the system goes from state i to state j. The sum of each row is one: sum( P_ij, j = 1...n ) = 1. This is the transition matrix of a so-called Markov chain. Under certain circumstances the infinite matrix product limit converges such that limit( P_ij^(n), n-->infinity) = p_j for all i,j. where [ p_j, j=1...n ] is the limit vector of the probabilities of the system being in the different states. Here P_ij^(n) is the i,j element of the product matrix P^n, with the transition probabilities from state i to state j after n steps (as opposed to after 1 step as P_ij). In stead of calculating the limit, one can try to find the vector [ p_i ] of the probabilities of the initial states, such that these probabilities are not influenced by the evolution of the system, i.o.w. find the vector [ p_i ] such that sum( p_i * P_ij, i=1...n ) = p_j for all j, i.o.w. find an eigenvector with eigenvalue 1 of the transposed matrix P^t. This eigenvector with probabilities of the initial system being in the different states, does not change when the sytem evolves. Dirk Vdm ==== PM: > Specific examples, if possible -- not just, they are used in > electronics, or physics, or whatever, but rather, something like: One example is to describe states of polarized light. There are two eigenstates, say horizontally and vertically polaraized light. Any polarization state is represented as a sum of these two states multiplied by complex constants. That includes circular and elliptical polarization. Operations on these eigenvectors represent what happens when the light goes through a waveplate. Electron spins can also b represented by the same mathematics. An amusing series authored by Joseph Slepian some decades ago, probably in the 30s, maybe 40s appearing in the Transactions of the IEEE. You will need a good technical library to dig that out. It is about someone financed to a technical education by an uncle. The lad reduced the inventory of his uncle's custom fruit salad business by representing mixtures of fruit as vectors. The method depended upon making eigenvector combinations of fruit. Bill ==== > Math is a hobby for me. I've been reading up on Eigenvectors and > Eigenvalues. It get the manipulations involved, but can't imagine the > applications -- and the books I have don't help. Can people provide a > few examples? Specific examples, if possible -- not just, they are used in > electronics, or physics, or whatever, but rather, something like: M is the matrix which describes such-and-such physical property or > transformation or process, its eigenvectors V correspond to such and > such property, and the eigenvalues of V and M indicate such-and-such. Steve O. *be* an idiot, but even idiots have feelings. A common use of eigenvalues and eigenvectors is in the analysis of dynamic mechanical systems. Given an undamped mechanical system described by the differential equations [M](d^2x)/(dt^2)+[K]u=0 where M is mass, K is spring stiffness (d^2x)/(dt^2) is acceleration and x is position, the eigenvalues of the system notes the squared ressonant frequencies of the system and the eigenvectors are the decomposed patterns of motion. -- ---------------------------- Christopher Grinde Ph.D student Mobile:+47 91137588 Tlph: +47 33037717 Web:http://cg.ans.hive.no ----------------------------- Vestfold University College Institute of microsystem technology. http://ri.hive.no/imst ==== > Math is a hobby for me. I've been reading up on Eigenvectors and > Eigenvalues. It get the manipulations involved, but can't imagine the > applications -- and the books I have don't help. Can people provide a > few examples? Specific examples, if possible -- not just, they are used in > electronics, or physics, or whatever, but rather, something like: M is the matrix which describes such-and-such physical property or > transformation or process, its eigenvectors V correspond to such and > such property, and the eigenvalues of V and M indicate such-and-such. The inertia tensor of 3d objects can be written as a symmetric positive definite matrix. It is used to calculate angular momentum out of angular velocity. If the axis of rotation of a (free) body is along one of the eigenvectors, angular momentum will have the same direction as angular velocity and the axis of rotation will remain the same. There are three such axes, perpendicular to each other. (Symmetric matrices are diagonalizable in an orthogonal basis.) > Steve O. > ==== I got some great news for you all. Yesterday a friend called to tell me > about the new project she's starting. > I have some great news for you. I just saved a ton of money on my auto insurance by switching to Geico. ==== http://www22.pair.com/csdc/car/carfre64.htm#OPENSETS He defines the sets in a topology T of a topological space X as closed sets. I've read many definitions that define an open set to be any subset U of X that is in T : Definition: A topology T1(closed) on a set X is a collection or class of subsets that obeys the following axioms: a.. A1(closed): X and the null set 0 are elements of the collection. b.. A2(closed): The arbitrary intersection of any number of elements of the collection belongs to the collection. c.. A3(closed): The arbitrary union of any pair of elements of thecollection belongs to the collection. The elements of the collection, T1(closed), are defined to be closed sets. The compliments of the closed sets are defined as open sets. l8r, Mike N. Christoff ==== > http://www22.pair.com/csdc/car/carfre64.htm#OPENSETS He defines the sets in a topology T of a topological space X as closed sets. > I've read many definitions that define an open set to be any subset U of X > that is in T : subsets that obeys the following axioms: > a.. A1(closed): X and the null set 0 are elements of the collection. > b.. A2(closed): The arbitrary intersection of any number of elements of > the collection belongs to the collection. > c.. A3(closed): The arbitrary union of any pair of elements of > thecollection belongs to the collection. The elements of the collection, T1(closed), are defined to be closed sets. > The compliments of the closed sets are defined as open sets. > [...] I didn't realize how close the definition of open set and properties of >closed sets were. They seem almost identical Let X be a topological space. A subset of X is closed if its complement is >open. The closed sets satisfy the following conditions. >(i) The empty set ; and the set X are closed. >(ii) Any finite union of closed sets is closed. >(iii) Any intersection of closed sets is closed. except b) refers to arbitrary union / ii) finite union ; c) pair >intersection / iii) arbitrary intersection. This would also mean X and {} >are both open and closed. Is a set closed iff it has the above properties with respect to some >topology? Or can it have the above, yet not be closed (ie: not be the >complement of an open set)? > A set is closed iff its complement is open. Your question is actually not welll-posed: How can a *single* set have properties (i)-(iii)? As you and others have noted, most authors define a topology as the collection of open sets. Actually, the sequence is usually to define topology (closed under arbtirary union, finite intersection), define a set to be open iff it is in the topology, and define a set to be closed iff its complement is open. But it is not unheard of to define a topology by its closed sets (as you do above), and the two approaches are equivalent. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== [...] I didn't realize how close the definition of open set and properties of >closed sets were. They seem almost identical Let X be a topological space. A subset of X is closed if its complement is >open. The closed sets satisfy the following conditions. >(i) The empty set ; and the set X are closed. >(ii) Any finite union of closed sets is closed. >(iii) Any intersection of closed sets is closed. except b) refers to arbitrary union / ii) finite union ; c) pair >intersection / iii) arbitrary intersection. This would also mean X and {} >are both open and closed. Is a set closed iff it has the above properties with respect to some >topology? Or can it have the above, yet not be closed (ie: not be the >complement of an open set)? A set is closed iff its complement is open. Your question is actually > not welll-posed: How can a *single* set have properties (i)-(iii)? > rephrased: Let X be a topological space and T be a collection of sets with the following properties: (i) The empty set ; and the set X are in T (ii) Any finite union of sets in T is in T. (iii) Any intersection of sets in T sets is in T. Let U be an arbitrary set in T. Is U neccessarily closed? l8r, Mike N. Christoff ==== >Let X be a topological space and T be a collection of sets with the >following properties: >(i) The empty set ; and the set X are in T >(ii) Any finite union of sets in T is in T. >(iii) Any intersection of sets in T sets is in T. Let U be an arbitrary set in T. Is U neccessarily closed? > Unless X is discrete, no: Let A be a set which is not closed. Then T = {{}, A, X} satisfies the three properties. Also, P(X) (the power set) satisfies the three properties. Any such collection T indicates a topology (viz., the one where the sets of T are the closed sets). If X has more than one element, there are multiple topologies on X. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== rephrased: Let X be a topological space and T be a collection of sets with the > following properties: > (i) The empty set ; and the set X are in T > (ii) Any finite union of sets in T is in T. > (iii) Any intersection of sets in T sets is in T. Let U be an arbitrary set in T. Is U neccessarily closed? l8r, Mike N. Christoff Let U be any not closed subset in the topology of X. Let T be {empty set,U,X} Will that do? In general, given a topology on X, just define another topology it in your favourite way (discrete, cofinite whatever) and its closed sets will satisfy those requirements, and there should be a closed set in the new topology that wasn't in the old one. ==== Let X be a topological space and T be a collection of sets with the sets? or subsets of X? > following properties: > (i) The empty set ; and the set X are in T > (ii) Any finite union of sets in T is in T. > (iii) Any intersection of sets in T sets is in T. Let U be an arbitrary set in T. Is U neccessarily closed? closed? closed in what? If you mean closed in the topological space X then the answer is not necessarily . The sets in T may have no relation with the given topology on X. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > http://www22.pair.com/csdc/car/carfre64.htm#OPENSETS He defines the sets in a topology T of a topological space X as closed sets. > I've read many definitions that define an open set to be any subset U of X > that is in T : subsets that obeys the following axioms: > a.. A1(closed): X and the null set 0 are elements of the collection. > b.. A2(closed): The arbitrary intersection of any number of elements of > the collection belongs to the collection. > c.. A3(closed): The arbitrary union of any pair of elements of > thecollection belongs to the collection. The elements of the collection, T1(closed), are defined to be closed sets. > The compliments of the closed sets are defined as open sets. > This looks OK. You can define a topology by describing either > open or closed sets. or convergent nets, or neighborhoods, or convergent filter-bases, or non-standard monads, or... -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ ==== This looks OK. You can define a topology by describing either > open or closed sets. or convergent nets, or neighborhoods, or convergent filter-bases, or > non-standard monads, or... or a closure operator. That's a nice one. There is a whole book done that way... (some eastern European I believe, can't think of which now). ==== --------------------------------- <^> <(áÀá)> <^> ----------------------------------- > The mathematician comes home early from a conference > and finds his wife...eh, you know the routine. > Off all the...With a traveling salesman? > Well, at least he was P-space hard. That was NF Herc ==== I need help on the following problem. Let X,Y be Banach spaces and B(X,Y) the space of continuous linear operators. Let S:(Y*,sigma)-->(X*,tau) is linear continuous mapping ,where sigma and tau are the coresponding weak* topologies on X*,Y*.Prove that S=T* where T is in B(X,Y).And * denotes the dual operator or dual space corespondingly. Greetings Mladen ==== I need help on the following problem. Let X,Y be Banach spaces and B(X,Y) the space of continuous linear > operators. > Let S:(Y*,sigma)-->(X*,tau) is linear continuous mapping ,where sigma > and tau are the coresponding weak* topologies on X*,Y*.Prove that S=T* > where T is in B(X,Y).And * denotes the dual operator or dual space > corespondingly. > T should be the restriction of T** to (the cononical subpace identified with) X. So you only have to show that T** maps X into Y. How can you identify an element of Y** as belonging to Y? That has something to do with the weak* topology on Y. ==== I need help on the following problem. Let X,Y be Banach spaces and B(X,Y) the space of continuous linear >operators. >Let S:(Y*,sigma)-->(X*,tau) is linear continuous mapping ,where sigma >and tau are the coresponding weak* topologies on X*,Y*.Prove that S=T* > where T is in B(X,Y).And * denotes the dual operator or dual space >corespondingly. A hint to get started: Note that one can talk about dual operators (or adjoint operators) of operators mapping topological vector spaces to topological vector spaces; the notion is not restricted to operators on Banach sapces. Now given S:(Y*,sigma)-->(X*,tau) it follows that we have S*:(X*,tau)* -->(Y*,sigma)*. What is (X*,tau)* ? It probably says somewhere in the book what (X*,tau)* is. If not: Suppose that L is in (X*,tau)*. It follows from the definition of tau that there exist L_1, ... L_n in X such that |L| <= |L_1| + ... |L_n| pointwise on X*. And hence... Greetings Mladen ************************ David C. Ullrich ==== I remember many years at school, a math teacher proving this. Has anyone got a math example of how this is possible? ==== > I remember many years at school, a math teacher proving this. > Has anyone got a math example of how this is possible? Not strictly related, but I remember the proof that girls are evil from approximately the same era: (1) girl = time * money (2) money = sqrt(evil) (root of all evil) (3) time = money so, girl = time * money <=> girl = money * money , (by (3)) <=> girl = (sqrt(evil)) ^ 2 , (by (2)) <=> girl = evil This was a consolation in times of trial. Richard ==== >I remember many years at school, a math teacher proving this. >Has anyone got a math example of how this is possible? > 1 + 1 = 3 for sufficiently large values of 1. --Lynn ==== >I remember many years at school, a math teacher proving this. >Has anyone got a math example of how this is possible? Your teacher was of course incorrect. *Everyone* knows 1 + 1 = 10: -4 = 2 * (-2) = Sqrt(4) * Sqrt(4) = Sqrt(4 * 4) = Sqrt(16) = 4 and -4 = 4 <=> -4 + 5 = 4 + 5 <=> 1 = 9 <=> 1 + 1 = 10 ==== >>I remember many years at school, a math teacher proving this. >>Has anyone got a math example of how this is possible? > Your teacher was of course incorrect. *Everyone* knows 1 + 1 = 10: -4 = 2 * (-2) = Sqrt(4) * Sqrt(4) = Sqrt(4 * 4) = Sqrt(16) = 4 and -4 = 4 <=> -4 + 5 = 4 + 5 <=> 1 = 9 <=> 1 + 1 = 10 That's not right, Sqrt(4) is always 2 (not -2). But if you use complex numbers it all becomes clear: -4 = 2i * 2i = 2 * sqrt(-1) * 2 * sqrt(-1) = 2 * 2 * sqrt(-1) * sqrt(-1) = 4 * sqrt((-1)*(-1)) = 4 * sqrt(1) = 4 The rest of the proof is the same. ==== > I remember many years at school, a math teacher proving this. > Has anyone got a math example of how this is possible? The most popular method is probably to divide both sides of a valid equation by zero, like so: 0*(1 + 1) = 0*3 <=> 1 + 1 = 3. That was not hard, was it? But if multiplication and division are too difficult, here is a simple proof that uses mere addition: add one to both sides of a valid equation, like so: 1 + 1 = 2 <=> 1 + 1 = 3. Or you can start from a proof that 0 = 1. Just take a square root on both sides of 1 = 1 to get -1 = 1, add 1 to get 0 = 2 and divide that by 2 to get 0 = 1. Add the same number to both sides of 1 + 1 = 2: 1 + 1 = 2 <=> 1 + 1 = 3. Or just state that 1 + 1 = 3 by definition. There really is no end to the errors that you can choose to make. Obfuscation helps, too. Write something incomprehensible and then state that the result follows. I will not attempt this here. -- ==== The Barcelona conjecture: Let c=(x+y+z)^p/(pxyz2^p) for integer c,x,y,z and p prime greater than or equal to 5, the Barcelona conjecture is that no solutions exist with gcd(c,xyz)=1 (no c exist that shares no factor with x or y or z). While this is a very interesting problem in itself (the heart of it is the relation between the factors of a sum versus the product of the addends - ie (x+y+z)^p/(xyz)), and may at first seem to be easy to resolve, it is in fact intimately related to Fermat's Last Theorem. In fact, proving the Barcelona conjecture also proves FLT (no integer ABC that satisfy A^p+B^p=C^p for p prime greater than 2) for prime exponents greater than 3. [note that letting c=1 for p=3 solves this exponent for FLT] There are integer solutions to the above, but all found to date have a factor in common for c and xyz. ==== I am forwarding this post from synergeo group. Some here might be interested. Dick Fischbeck I've just created a sourceforge project for the various programs I've been writing - http://sourceforge.net/projects/packinon The associated website is here - http://packinon.sourceforge.net/ Not too much there at the moment. I've started with the Python as I get round to completing their next versions. Adrian Rossiter ==== I would be interested in solutions to the following problem. Elementary solutions are especially welcome. In an ABC triangle, D is a point on BC from which we construct segments that form equal angles at the sides AB, AC, at the points I, K respectively. Which is the position of D so that the length of IK will be minimum? It is assumed that the angles BID=DKC in question are constant as the point D moves on BC. You may assume that the result is known when BID=DKC=right anlge. ==== Dear Mathematicians, I plan on attending graduate school next year in Mathematics to begin my PhD, and was hoping to receive some insight or advice from the board as it consists of such a wide variety of mathematicians. If someone/some people can reply, I would highly appreciate it. (As background, I am completing a Master of Arts at University of Pennsylvania) My question is as follows : To those who have completed a PhD in math in Algebra (or really this applies to any subfield of math), as you look back at your graduate school days, if you were to go back and restart graduate school, what would you have done differently? How would you go about graduate school in general? Moshe madrianATmath.upenn.edu) ==== Dear all, I was wondering : I feel that I am very strong in Algebra, but not as much in Analysis, and I would say Topology is in the middle (I haven't studied Differential Geometry yet, that is next semester). Is it me, or is thinking in an Analysis type way different than in an Algebraic type way? I just feel that they are very different. I feel that Algebra is much more visual than Analysis. For me I can just see Linear Algebra, representation theory (not too advanced, but advanced enough), introductory Field theory, etc etc. It is much harder for me to see Analysis. Does anyone have a recommendation for me as to this problem? I feel I should just pick up Rudin's Analysis textbook and read through it and do all of the problems, or maybe a different textbook, and maybe that would help me see Analysis better. Moshe ==== > .... It is much harder for me to see Analysis. Does anyone have a > recommendation for me as to this problem? I feel I should just pick > up Rudin's Analysis textbook and read through it and do all of the > problems, or maybe a different textbook, and maybe that would help me > see Analysis better.... If you're after a really good basic feeling for analysis rather than advanced modern technical facts, then don't forget the classic by G. H. Hardy, A Course of Pure Mathematics. Just read it for pleasure! :-) It was written when analysis was neglected in Cambridge, and with an emphasis and enthusiasm which seem rather ridiculous now. If I were to rewrite it now I should not write (to use Prof. Littlewood's simile) like 'a missionary talking to cannibals', but with decent terseness and restraint .... Luckily he didn't, and his missionary-talking-to-cannibals style has been of great value to many of us since. Ken Pledger. ==== Rudin seems to be a book that one would read after having mastered the subject. However , if complex analysis is a topic you are interested in you might like Visual Complex Analysis by Tristan Needham. > Dear all, I was wondering : I feel that I am very strong in Algebra, but not as much > in Analysis, and I would say Topology is in the middle (I haven't studied > Differential Geometry yet, that is next semester). Is it me, or is thinking > in an Analysis type way different than in an Algebraic type way? I just > feel that they are very different. I feel that Algebra is much more visual > than Analysis. For me I can just see Linear Algebra, representation theory > (not too advanced, but advanced enough), introductory Field theory, etc etc. > It is much harder for me to see Analysis. Does anyone have a recommendation > for me as to this problem? I feel I should just pick up Rudin's Analysis > textbook and read through it and do all of the problems, or maybe a > different textbook, and maybe that would help me see Analysis better. Moshe > ==== I wonder if someone can help me with the following probability problem, and show me the method by which it can be solved. If there is an event with three possible outcomes, and each outcome has an equal probability of occurring, what are the odds that over 80 trials, one of the three possible outcomes only results on two occasions (while the other two possible outcomes are successful on the remaining 78 trials). Also, is there a significance test that can be applied to this type of probability problem, such that it is possible to declare that the results are statistically significant? Anon ==== I wonder if someone can help me with the following probability > problem, and show me the method by which it can be solved. If there > is an event with three possible outcomes, and each outcome has an > equal probability of occurring, what are the odds that over 80 trials, > one of the three possible outcomes only results on two occasions > (while the other two possible outcomes are successful on the remaining > 78 trials). Also, is there a significance test that can be applied to > this type of probability problem, such that it is possible to declare > that the results are statistically significant? > Anon As far as I know, given the way the question is asked, you can do no more than calculating the probability (I know I'll be corrected if I'm wrong): P(one outcome occurs twice in 80) = bin(80,2)*(1/3)^2*(2/3)^78 = 6.46e-12 Jeroen ==== >> I wonder if someone can help me with the following probability >> problem, and show me the method by which it can be solved. If there >> is an event with three possible outcomes, and each outcome has an >> equal probability of occurring, what are the odds that over 80 trials, >> one of the three possible outcomes only results on two occasions >> (while the other two possible outcomes are successful on the remaining >> 78 trials). Also, is there a significance test that can be applied to >> this type of probability problem, such that it is possible to declare >> that the results are statistically significant? >> Anon >As far as I know, given the way the question is asked, you can do no >more than calculating the probability (I know I'll be corrected if I'm >wrong): > P(one outcome occurs twice in 80) = bin(80,2)*(1/3)^2*(2/3)^78 = >6.46e-12 This is the probability that a particular outcome occurs twice; there are three such outcomes, so multiplying this by 3 gives a not quite correct answer. The reason it is not quite correct is that two oucomes could each occur twice. The probability of this is for a particular ordering of the outcomes is [80!/2!2!76!]*(1/3)^80, and 3 times this needs to be subtracted. This is MUCH smaller. There are standard significance tests, such as the chi-squared test, and exact tests. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University ==== > Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. I'm curious about the mental processes that allow *some* of you to > claim that 49 divides off as a *variable* dependent on x, so I'm > giving another opportunity for you to speak your minds. To my knowledge, in the history of mathematics, no one has ever > presented such a proposition, so it is a unique one, and I must say > that I'm intrigued. Speak your minds. > James Harris Why does this not count as presenting such a proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). then > g1(x)*g2(x) = 3(5-x) but neither g1 nor g2 is divisible by 3 for all x. - William Hughes OOPS! I just replied to this post with my own mistake, as I had 1 + (1-sqrt(1+3x))/3 being an integer if 1+3x is a square, when x=40 refutes that notion. James Harris ==== > OOPS! I just replied to this post with my own mistake, as I had 1 + (1-sqrt(1+3x))/3 being an integer if 1+3x is a square, when x=40 refutes that notion. James Harris Your OOPS! count just went up by one. Funny how *your* mistakes are simple, easily corrected faults, but other posters mistakes are signs of ignorance, incompetence or dishonesty. Get a grip, James. You have been thoroughly and conclusively refuted. Your *research* is a worthless pile of pseudo-scientific, incomprehensible and error-ridden junk -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== OOPS! I just replied to this post with my own mistake, as I had 1 + (1-sqrt(1+3x))/3 being an integer if 1+3x is a square, when x=40 refutes that notion. James Harris Your OOPS! count just went up by one. Funny how *your* mistakes are simple, easily corrected faults, but other > posters mistakes are signs of ignorance, incompetence or dishonesty. Get a grip, James. You have been thoroughly and conclusively refuted. Your *research* is a worthless pile of > pseudo-scientific, incomprehensible and error-ridden junk -- > There are two things you must never attempt to prove: the unprovable -- and the obvious. > -- > Democracy: The triumph of popularity over principle. > -- > http://www.crbond.com Heh, did you see that message a few days ago from someone who asked a question and James told him if he had to ask, he should read something else? I thought that was funny considering James doesn't seem to know what he's talking about. -- David Moran Chief Meteorologist Oklahoma Storm Team ==== > Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. I'm curious about the mental processes that allow *some* of you to > claim that 49 divides off as a *variable* dependent on x, so I'm > giving another opportunity for you to speak your minds. To my knowledge, in the history of mathematics, no one has ever > presented such a proposition, so it is a unique one, and I must say > that I'm intrigued. Speak your minds. > James Harris Why does this not count as presenting such a proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). then > g1(x)*g2(x) = 3(5-x) I noted there's a *sign* ambiguity in the sqrt() operator, which sparked a lot of debate. One thing I found interesting is that posters ignored that if you divide both sides by 3, with the convention that you're taking the positive of the sqrt() operator for *integer* results, you have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer (remember x is an integer and remember 1+3x is a square). > but neither g1 nor g2 is divisible by 3 for all x. - William Hughes Prove it. Readers should note that this poster presented a *later* post claiming a result that covers integer results of the square root operator but made a rather simple mistake. In my reply to that post I noted the sign ambiguity in the square root operator. However sqrt(x), where x is a square can be taken to be the positive result since you can *give* the result, but the ambiguity remains if you see sqrt(x) without a given value. That is, for instance, sqrt(4) is 2 *or* -2, but by convention, it's *usually* taken as 2, though if you do enough analysis you will run into situations where you need the negative!!! Mathematics is *logical* and consistent, which is something that many people seem to have a problem with, as they try to twist it to their own needs and interests. Note lack of proper explication from other sci.math posters. James Harris ==== > Given, where x is in the ring of algebraic integers, I've shown the > factorization > > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = > > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > > where b_3(x) = a_3(x) - 3 and the a's are roots of > > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > > so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. > > I'm curious about the mental processes that allow *some* of you to > claim that 49 divides off as a *variable* dependent on x, so I'm > giving another opportunity for you to speak your minds. > > To my knowledge, in the history of mathematics, no one has ever > presented such a proposition, so it is a unique one, and I must say > that I'm intrigued. > > Speak your minds. > > James Harris Why does this not count as presenting such a proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). then > g1(x)*g2(x) = 3(5-x) I noted there's a *sign* ambiguity in the sqrt() operator, which > sparked a lot of debate. One thing I found interesting is that posters ignored that if you > divide both sides by 3, with the convention that you're taking the > positive of the sqrt() operator for *integer* results, you have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer (remember x is an integer and > remember 1+3x is a square). but neither g1 nor g2 is divisible by 3 for all x. - William Hughes Prove it. Readers should note that this poster presented a *later* > post claiming a result that covers integer results of the square root > operator but made a rather simple mistake. In my reply to that post I > noted the sign ambiguity in the square root operator. However sqrt(x), where x is a square can be taken to be the positive > result since you can *give* the result, but the ambiguity remains if > you see sqrt(x) without a given value. That is, for instance, sqrt(4) is 2 *or* -2, but by convention, it's > *usually* taken as 2, though if you do enough analysis you will run > into situations where you need the negative!!! > Well, if you dislike the function sqrt(x) so much, note this alternate presentation [1] Let g1(x) and g2(x) be the two roots of the quadratic g^2 - 8g + 3(5-x) For x=0, the roots are 3 and 5. Choose g1 and g2 such that g1(0)=3 and g2(0) = 5 We have (the product of the roots is the constant term) g1(x) g2(x) = 3(5-x) Clearly g1(x)/3 and g2(x)/3 are both roots of 3g^2 - 8g + (5-x) but for x=2 this is primitive, irreducible and non-monic. Hence neither g1(2) nor g2(2) is divisible by 3. -William Hughes [1] My thanks to Rick Decker whose post I adapted ==== > Given, where x is in the ring of algebraic integers, I've shown the > factorization > > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = > > 49(300125 x^3 - 18375 x^2 - 360 x + 22) > > where b_3(x) = a_3(x) - 3 and the a's are roots of > > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) > > so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. > > I'm curious about the mental processes that allow *some* of you to > claim that 49 divides off as a *variable* dependent on x, so I'm > giving another opportunity for you to speak your minds. > > To my knowledge, in the history of mathematics, no one has ever > presented such a proposition, so it is a unique one, and I must say > that I'm intrigued. > > Speak your minds. > > James Harris Why does this not count as presenting such a proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). then > g1(x)*g2(x) = 3(5-x) I noted there's a *sign* ambiguity in the sqrt() operator, which > sparked a lot of debate. One thing I found interesting is that posters ignored that if you > divide both sides by 3, with the convention that you're taking the > positive of the sqrt() operator for *integer* results, you have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer (remember x is an integer and > remember 1+3x is a square). but neither g1 nor g2 is divisible by 3 for all x. - William Hughes Prove it. Consider x =2: g1(2)/3 = (4-sqrt(7))/3 is a root of the primitive, irreducible, non-monic polynomial P(x)=3x^2 - 8x + 3. Thus g1(2)/3 is not an algebraic integer and g1(2) is not divisible by 3. Now James has made the bizzare claim that sqrt(x) and hence g1(x) must be considered multi-valued, and furthmore if either value of g1(x) is divisible by 3 then g1(x) is divisible by 3. So let's try the other square root of 7. g1(2)/ = (4+sqrt(7))/3 is a root of the primitive, irreducible, non-monic polynomial P(x)=3x^2 - 8x + 3. Thus g1(2)/3 is not an algebraic integer and g1(2) is not divisible by 3. So, even if we accept the ambiguity argument, g1(2) is not divisible by 3. Similarly, g2(2) is not divisible by 3. Thus neither g1 nor g2 is divisible by 3 for all x. -William Hughes ==== > I noted there's a *sign* ambiguity in the sqrt() operator, which > sparked a lot of debate. There was no debate. Everyone except the village idiot agrees how the sqrt(x) function is defined. The village idiot is screaming as usual, but nobody is bothered too much. ==== > >Why does this not count as presenting such a proposition ? >>Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). >>then >> g1(x)*g2(x) = 3(5-x) >> I noted there's a *sign* ambiguity in the sqrt() operator, which > sparked a lot of debate. One thing I found interesting is that posters ignored that if you > divide both sides by 3, with the convention that you're taking the > positive of the sqrt() operator for *integer* results, you have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer (remember x is an integer and > remember 1+3x is a square). >>but neither g1 nor g2 is divisible by 3 for all x. >> - William Hughes >> Prove it. Clearly, the g's satisfy g^2 - 8g + 3(5 - x). If 3 were to divide (in the algebraic integers) one of the g's, then one or both of g1(x)/3 and g2(x)/3 would be algebraic integers. Letting h1(x) = g1(x)/3, h2(x) = g2(x)/3 we find that the h's satisfy 3h^2 - 8h + (5 - x). This is obviously a primitive polynomial and is irreducible over Z[x] if and only if 1 + 3x isn't a square. In particular, when x = 3 we see that neither of 4 + sqrt(10) and 4 - sqrt(10) is divisible by 3, though their product, 6, is. Note, BTW, that even if you're bumfuzzled about the ambiguity of the sqrt function, there should be no confusion here, since we're using both of the second roots of 10. Rick ==== > Why does this not count as presenting such a proposition ? > Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). > then >> g1(x)*g2(x) = 3(5-x) I noted there's a *sign* ambiguity in the sqrt() operator, which > sparked a lot of debate. One thing I found interesting is that posters ignored that if you > divide both sides by 3, with the convention that you're taking the > positive of the sqrt() operator for *integer* results, you have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer (remember x is an integer and > remember 1+3x is a square). Why must 1+3x be a square? > but neither g1 nor g2 is divisible by 3 for all x. >> erm, how about x =1 for g1, cos then 2/3 wold be an algebraic integer. and zero for g2, cos then 5/2 would be an algebraic integer. Of course in your core error world then you may think differently. And I may be wrong, if I am I hope someone can point it out. >> - William Hughes Prove it. Readers should note that this poster presented a *later* > post claiming a result that covers integer results of the square root > operator but made a rather simple mistake. In my reply to that post I > noted the sign ambiguity in the square root operator. However sqrt(x), where x is a square can be taken to be the positive > result since you can *give* the result, but the ambiguity remains if > you see sqrt(x) without a given value. That is, for instance, sqrt(4) is 2 *or* -2, but by convention, it's > *usually* taken as 2, though if you do enough analysis you will run > into situations where you need the negative!!! Mathematics is *logical* and consistent, which is something that many > people seem to have a problem with, as they try to twist it to their > own needs and interests. Note lack of proper explication from other sci.math posters. > James Harris ==== In sci.math, James Harris <3c65f87.0312151404.50d586a2@posting.google.com>: >> Okay, so you're dividing both sides by 49. Point taken. > :> In other words, are you >> :> claiming that (5 a_1(x) + 7) and (5 a_2(x) + 7) are divisible by 7 for >> :> all x? If so, could you please explain why the fact that it's true for >> :> x=0 implies it's true for other x also? > If you are not claiming this, then please explain what you are claiming. > Many thanks, >> Justin Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now you can divide both sides by 49. I could also divide them by 2198/401, add 5.23e11 to both sides, take the cosh of the natural log, and even use partial differentiation on the variable b, which I declare as an independent variable for both a = a(b) and x = x(b). :-) You'll have to be a little more specific than that here as to what you expect to happen. Some posters have claimed that when you divide by 49, what results > varies depending on what value x has. The result *does* depend on x, since x is part of both sides. In the declaration, say, f(x) = x^3 - 3x + 1 f(x) definitely depends on x. Your functions aren't all that different. Do you understand? No. Now I shall answer your question. It is clear that the quantities (5 a_1(x) + 7) and (5 a_2(x) + 7) are not divisible by 7 for all x (counterexamples: a_1(x) = 1, a_2(x) = 1). You will probably want to clarify your domain here. Is x: [1] an integer? [2] a positive integer? [3] an algebraic integer? [4] a rational number? [5] pi? [6] the square root of the sum of the squares of a series? [7] something else? If you want me to actually produce an x such that a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 has a root 1, I'll have to do a bit of work. The simplest method might be to set a = 1, then solve for x: 1^3 + 3(-1 + 49x)1^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 which ultimately generates the equation -117649*x^3 + 7203*x^2 - 2 = 0 This equation has 3 real roots, which are approximately: -0.01493981239936484272504992533 0.02040816326530612244897959183 0.05575613892997708762300910901 These are obviously not algebraic integers, as the equation of which they are roots is not of the requisite type. Substituting x = any one of these will result in your product becoming either 12 ( 5 a_2(x) + 7) (5 b_3(x) + 22) or (5 a_1(x) + 7) (5 a_2(x) + 7) 12 What conclusions can be drawn from all this, I for one do not know. [rest snipped] -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== > ... > Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). > > then > g1(x)*g2(x) = 3(5-x) > > but neither g1 nor g2 is divisible by 3 for all x. Ah, but the constant term of g1(x) = 3, and the constant term of g2(x) = 5, > so g1(x) should be divisible by 3. A core error in mathematics ;-). It's actually not that simple. It's an interesting case to highlight > your ineptitude with basic mathematics though. Consider that the constant term of g1(x) is 3 *or* 5 because the > sqrt() operator is ambiguous. I've explained that before in replying > to Arturo Magidin, but mathematics is a difficult discipline for some, > so repetition is necessary, and still often not enough. > You say the sqrt() operator is ambiguous. Well, that makes YOUR OWN claims ambiguous. You probably dont realize it, but the a_1, a_2 and a_3 in your core error CONTAIN SQUARE ROOTS! Just use the Cardano(sp?) formulas to find the a_n's and you will see the square roots. > You see Dik Winter, actually *knowing* mathematics versus talking as > if you know it can be two different things. And you clearly by posts like yours here do not actually know > mathematics. You do post a lot though living in a fantasy world. > James Harris ==== > Consider that the constant term of g1(x) is 3 *or* 5 because the > sqrt() operator is ambiguous. > You say the sqrt() operator is ambiguous. Well, that makes YOUR OWN > claims ambiguous. You probably dont realize it, but the a_1, a_2 and > a_3 in your core error CONTAIN SQUARE ROOTS! Just use the > Cardano(sp?) formulas to find the a_n's and you will see the square > roots. No, a_1(x), a_2(x) and a_3(x) do not CONTAIN SQUARE ROOTS! . They are functions defined in terms of the roots of a polynomial with coefficients depending on x (actually, James leaves out a couple of steps needed to completely specify the functions). While one can certainly use the Cardano formulae to find expressions for the a_i, and these formulae contain square roots, there are ways to express the a_i that do not involve the symbol sqrt. One should be careful to distinguish between the existence of the underlying function and the notation used to express the function. - William Hughes ==== > Consider that the constant term of g1(x) is 3 *or* 5 because the > sqrt() operator is ambiguous. You say the sqrt() operator is ambiguous. Well, that makes YOUR OWN > claims ambiguous. You probably dont realize it, but the a_1, a_2 and > a_3 in your core error CONTAIN SQUARE ROOTS! Just use the > Cardano(sp?) formulas to find the a_n's and you will see the square > roots. No, a_1(x), a_2(x) and a_3(x) do not CONTAIN SQUARE ROOTS! . They are > functions defined in terms of the roots of a polynomial with > coefficients depending on x (actually, James leaves out a couple of > steps needed to completely specify the functions). > While one can certainly use the Cardano formulae to > find expressions for the a_i, and these formulae contain square roots, > there are ways to express the a_i that do not involve the > symbol sqrt. One should be careful to distinguish between the > existence of the underlying function and the notation used to > express the function. - William Hughes Given JHS poly: JHS(x) = a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) taking roots of the equation: JHS(x) = 0 we have, a_1 = 1 - 49*x + (2^(1/3)*(-1 + 49*x)^2)/ (2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 - 847425747*x^6])^(1/3) + (2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 - 847425747*x^6])^(1/3)/2^(1/3) a_2 = 1 - 49*x - ((1 + I*Sqrt[3])*(-1 + 49*x)^2)/ (2^(2/3)*(2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 - 847425747*x^6])^(1/3)) - ((1 - I*Sqrt[3])*(2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 - 847425747*x^6])^(1/3))/(2*2^(1/3)) a_3 = 1 - 49*x - ((1 - I*Sqrt[3])*(-1 + 49*x)^2)/ (2^(2/3)*(2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 - 847425747*x^6])^(1/3)) - ((1 + I*Sqrt[3])*(2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 - 847425747*x^6])^(1/3))/(2*2^(1/3)) where the root assignments can be permuted. Doubtless other expressions can be found by suitable transforms. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== > ... > Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). > > then > g1(x)*g2(x) = 3(5-x) > > but neither g1 nor g2 is divisible by 3 for all x. > > Ah, but the constant term of g1(x) = 3, and the constant term of g2(x) = 5, > so g1(x) should be divisible by 3. A core error in mathematics ;-). > > It's actually not that simple. It's an interesting case to highlight > your ineptitude with basic mathematics though. > > Consider that the constant term of g1(x) is 3 *or* 5 because the > sqrt() operator is ambiguous. I've explained that before in replying > to Arturo Magidin, but mathematics is a difficult discipline for some, > so repetition is necessary, and still often not enough. Yup, it appears so. In mathematics the sqrt when applied to reals and > delivering a real is defined to give the positive result. This has > been said before, but it is apparently to difficult for you to understand. That doesn't change the *inherent* ambiguity in the sqrt() operator. That's easy to show as consider sqrt(4), and you wish to say it's *defined* to be 2, but what about -2? Does your definition take away -2 as a solution? And besides, you lied, it's by *convention* that the positive is taken, not by definition. > > You see Dik Winter, actually *knowing* mathematics versus talking as > if you know it can be two different things. It appears so, yes. Since people working with square roots *usually* want the positive root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2 as a solution, as is easily proven: sqrt(4) = -2, square both sides, 4 = 4. QED Now then, can you *prove* that -2 is not a solution to sqrt(4)? James Harris ==== ... > Yup, it appears so. In mathematics the sqrt when applied to reals and > delivering a real is defined to give the positive result. This has > been said before, but it is apparently to difficult for you to understand. ... > And besides, you lied, it's by *convention* that the positive is > taken, not by definition. What is the distinction? > That's necessary because, like sqrt(4) has *either* 2 or -2 as a > solution, as is easily proven: > > sqrt(4) = -2, square both sides, 4 = 4. QED 1 = -1, square both sides, 1 = 1, QED > Now then, can you *prove* that -2 is not a solution to sqrt(4)? I see no equation, so how can you talk about a solution? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== > ... > Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). > > then > g1(x)*g2(x) = 3(5-x) > > but neither g1 nor g2 is divisible by 3 for all x. > > Ah, but the constant term of g1(x) = 3, and the constant term of g2(x) = 5, > so g1(x) should be divisible by 3. A core error in mathematics ;-). > > It's actually not that simple. It's an interesting case to highlight > your ineptitude with basic mathematics though. > > Consider that the constant term of g1(x) is 3 *or* 5 because the > sqrt() operator is ambiguous. I've explained that before in replying > to Arturo Magidin, but mathematics is a difficult discipline for some, > so repetition is necessary, and still often not enough. Yup, it appears so. In mathematics the sqrt when applied to reals and > delivering a real is defined to give the positive result. This has > been said before, but it is apparently to difficult for you to understand. That doesn't change the *inherent* ambiguity in the sqrt() operator. That's easy to show as consider sqrt(4), and you wish to say it's > *defined* to be 2, but what about -2? Well, as sqrt(4) *is* defined to be to 2 I think Dik may get his wish. Does your definition take away -2 as a solution? > Solution to what? -2 is a solution to x^2=4, x=sqrt(4) is not an equation. > And besides, you lied, it's by *convention* that the positive is > taken, not by definition. > In this context there is no difference between convention and definition. > You see Dik Winter, actually *knowing* mathematics versus talking as > if you know it can be two different things. It appears so, yes. Since people working with square roots *usually* want the positive > root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2 as a > solution, as is easily proven: > To say that sqrt(4) has a solution is nonsensical. > sqrt(4) = -2, square both sides, 4 = 4. QED The same logic shows that -x = x for any x The whole discussion is silly in any case. There certainly exits a single valued function, g(z) defined on the complex plane such that for all complex z: g(z)*g(z) = z, either real(g(z)) > 0 or (real(g(z)) = 0 and imag(g(z))>=0) If you have a deep psycho-sexual need not to call g(z) sqrt(z), then call it sqrt_pb(z) or harris(z) or hughes(z) or whatever else you want [1]. Define g_1(x) = 4 - g(1+3x), g_2(x) = 4 + g(1+3x) (note that both g_1 and g_2 are single valued) Then g_1(x) g_2(x) = 3(5-x) g_1(0) = 3, g_2(0) = 5 g_1(1) = 2 is not divisible by 3 - William Hughes [1] But be aware that others will still use sqrt(z). ==== ... > The whole discussion is silly in any case. There certainly > exits a single valued function, g(z) defined on the complex plane > such that for all complex z: > > g(z)*g(z) = z, > either real(g(z)) > 0 or (real(g(z)) = 0 and imag(g(z))>=0) > > If you have a deep psycho-sexual need not to call g(z) sqrt(z), > then call it sqrt_pb(z) or harris(z) or hughes(z) or whatever > else you want [1]. Define > > g_1(x) = 4 - g(1+3x), g_2(x) = 4 + g(1+3x) > > (note that both g_1 and g_2 are single valued) > > Then g_1(x) g_2(x) = 3(5-x) > > g_1(0) = 3, g_2(0) = 5 > > g_1(1) = 2 is not divisible by 3 More interesting, neither g_1(2) nor g_2(2) is divisible by 3, while g_1(2)*g_2(2) = 9. Moreover, g_1(2) and g_2(2) are coprime: (12 + sqrt(7)).g_1(2) - (24 - 8.sqrt(7)).g_2(2) = 1. moreover: gcd(g_1(2), 3) = (sqrt(2) - sqrt(14))/2, gcd(g_2(2), 3) = (sqrt(2) + sqrt(14))/2, gcd(g_1(2), 9) = gcd(g_1(2), 3)^2 and gcd(g_2(2), 9) = gcd(g_2(2), 3)^2. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ <3c65f87.0312151213.7bbfb1b7@posting.google.com> <3c65f87.0312160631.117923f5@posting.google.com> <4d5e4663.0312161059.154f9ecc@posting.google.com> ==== > x=sqrt(4) is not an equation. Surely, you didn't mean to say that. -- Jesse F. Hughes What you call reasonable is suspect since you've proven yourself to be an enemy of mathematics. -- James S. Harris defends the cause. ==== x=sqrt(4) is not an equation. Surely, you didn't mean to say that. Um, right. This should read sqrt(4) is not an equation -William Hughes ==== [...] > Since people working with square roots *usually* want the positive > root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2 as a > solution, as is easily proven: sqrt(4) has no solution as it isn't an equation, sqrt(x) is a function wich gives you by *definition* your positive result. [...] > Now then, can you *prove* that -2 is not a solution to sqrt(4)? -2 is a solution of the equation x ^ 2 = 4, not of sqrt(4). the word solution doesn't apply to sqrt(4), see above. -- Edgar ==== > ... > Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). > > then > g1(x)*g2(x) = 3(5-x) > > but neither g1 nor g2 is divisible by 3 for all x. > > Ah, but the constant term of g1(x) = 3, and the constant term of g2(x) = 5, > so g1(x) should be divisible by 3. A core error in mathematics ;-). > > It's actually not that simple. It's an interesting case to highlight > your ineptitude with basic mathematics though. > > Consider that the constant term of g1(x) is 3 *or* 5 because the > sqrt() operator is ambiguous. I've explained that before in replying > to Arturo Magidin, but mathematics is a difficult discipline for some, > so repetition is necessary, and still often not enough. Yup, it appears so. In mathematics the sqrt when applied to reals and > delivering a real is defined to give the positive result. This has > been said before, but it is apparently to difficult for you to understand. That doesn't change the *inherent* ambiguity in the sqrt() operator. That's easy to show as consider sqrt(4), and you wish to say it's > *defined* to be 2, but what about -2? Does your definition take away -2 as a solution? And besides, you lied, it's by *convention* that the positive is > taken, not by definition. Quit saying people lie when they make honest mistakes, there is a BIG difference. The sqrt of 4 is 2 OR -2; if you dispute this, go back to algebra I in high school. You apparently have gaps in your algebra. > You see Dik Winter, actually *knowing* mathematics versus talking as > if you know it can be two different things. It appears so, yes. Since people working with square roots *usually* want the positive > root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2 as a > solution, as is easily proven: sqrt(4) = -2, square both sides, 4 = 4. QED Now then, can you *prove* that -2 is not a solution to sqrt(4)? James Harris -- David Moran Chief Meteorologist Oklahoma Storm Team ==== >> ... >>[...] >> That's easy to show as consider sqrt(4), and you wish to say it's >> *defined* to be 2, but what about -2? >> Does your definition take away -2 as a solution? >> And besides, you lied, it's by *convention* that the positive is >> taken, not by definition. Quit saying people lie when they make honest mistakes, there is a BIG >difference. Excellent point, but irrelevant here, because there was no mistake in what Dik said, honest or otherwise. >The sqrt of 4 is 2 OR -2; if you dispute this, go back to >algebra I in high school. You apparently have gaps in your algebra. No. _The_ square root of 4 is 2. >> You see Dik Winter, actually *knowing* mathematics versus talking as >> if you know it can be two different things. >> It appears so, yes. >> Since people working with square roots *usually* want the positive >> root, by convention the positive is taken, not by definition. >> That's necessary because, like sqrt(4) has *either* 2 or -2 as a >> solution, as is easily proven: >> sqrt(4) = -2, square both sides, 4 = 4. QED >> Now then, can you *prove* that -2 is not a solution to sqrt(4)? >> James Harris ************************ David C. Ullrich ==== [.snip.] >Quit saying people lie when they make honest mistakes, there is a BIG >difference. The sqrt of 4 is 2 OR -2; Ehr, no. Both 2 and -2 are square roots of 4 (that is, both of them, when squared, give 4); but the sqrt of 4 is defined to be the principal branch, and therefore it is unabiguously equal to 2. That's why sqrt(a^2) = |a| for real numbers, for example. -- ====================================================================== It's not denial. I'm just very selective about what I accept as reality. --- Calvin ( Calvin and Hobbes ) ====================================================================== Arturo Magidin magidin@math.berkeley.edu ==== [.snip.] Quit saying people lie when they make honest mistakes, there is a BIG >difference. The sqrt of 4 is 2 OR -2; Ehr, no. Both 2 and -2 are square roots of 4 (that is, both of them, > when squared, give 4); but the sqrt of 4 is defined to be the > principal branch, and therefore it is unabiguously equal to 2. That's > why sqrt(a^2) = |a| for real numbers, for example. -- > ====================================================================== > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin ( Calvin and Hobbes ) > ====================================================================== Arturo Magidin > magidin@math.berkeley.edu > Arturo, thanks for the correction. I stand corrected. -- David Moran Chief Meteorologist Oklahoma Storm Team ==== Since people working with square roots *usually* want the positive > root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2 as a > solution, as is easily proven: sqrt(4) = -2, square both sides, 4 = 4. QED Nice logic! -1 is a solution of 1 (whatever that means), since squaring both sides gives 1=1. QED Wilbert ==== Since people working with square roots *usually* want the positive > root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2 as a > solution, as is easily proven: sqrt(4) = -2, square both sides, 4 = 4. QED Nice logic! -1 is a solution of 1 (whatever that means), since > squaring both sides gives 1=1. QED Wilbert -1 IS a solution for sqrt(1). It's interesting that is a point of debate, but not surprising for the sci.math newsgroup! James Harris ==== > > Since people working with square roots *usually* want the positive > root, by convention the positive is taken, not by definition. > > That's necessary because, like sqrt(4) has *either* 2 or -2 as a > solution, as is easily proven: > > sqrt(4) = -2, square both sides, 4 = 4. QED Nice logic! -1 is a solution of 1 (whatever that means), since > squaring both sides gives 1=1. QED Wilbert -1 IS a solution for sqrt(1). It's interesting that is a point of debate, but not surprising for the > sci.math newsgroup! Not surprising that you don't understand something simple like that. sqrt(x) is a function defined for real numbers x >= 0: sqrt(x) is by definition the positive solution of y^2 = x. y = -1 is one of the two solutions of y^2 = 1, but -1 is not the square root of 1. ==== > Since people working with square roots *usually* want the positive >> root, by convention the positive is taken, not by definition. > That's necessary because, like sqrt(4) has *either* 2 or -2 as a >> solution, as is easily proven: > sqrt(4) = -2, square both sides, 4 = 4. QED > Nice logic! -1 is a solution of 1 (whatever that means), since >> squaring both sides gives 1=1. QED > Wilbert -1 IS a solution for sqrt(1). It's interesting that is a point of debate, but not surprising for the >sci.math newsgroup! Yup. Just like you thought it was interesting that the fact that integers are irrational was a point of debate... >James Harris David C. Ullrich ************************** As far as I'm concerend you're trying to wait until I die, so I figure maybe you should die instead. How about that, eh? Wouldn't that be a better twist? You refuse to follow the math, so the great Powers that control reality and *speak* in mathematics decide to kill you instead of me. So what do you think about that, eh? Oh, can't hear Them talking? Well, I guess that's because you don't really understand Mathematics, the true language, which is THE language. They're talking about you now, and They agree with my assessment, and will not penalize me as They allowed the others like Galois and Abel to be penalized. They will kill you instead. James Harris speaking on Weird factorization, genius ==== >> ... >> Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). >> > then >> g1(x)*g2(x) = 3(5-x) >> > but neither g1 nor g2 is divisible by 3 for all x. >> > Ah, but the constant term of g1(x) = 3, and the constant term of g2(x) = 5, >> so g1(x) should be divisible by 3. A core error in mathematics ;-). >> > It's actually not that simple. It's an interesting case to highlight >> your ineptitude with basic mathematics though. >> > Consider that the constant term of g1(x) is 3 *or* 5 because the >> sqrt() operator is ambiguous. I've explained that before in replying >> to Arturo Magidin, but mathematics is a difficult discipline for some, >> so repetition is necessary, and still often not enough. > Yup, it appears so. In mathematics the sqrt when applied to reals and >> delivering a real is defined to give the positive result. This has >> been said before, but it is apparently to difficult for you to understand. That doesn't change the *inherent* ambiguity in the sqrt() operator. Uh, yes it does. >That's easy to show as consider sqrt(4), and you wish to say it's >*defined* to be 2, but what about -2? Does your definition take away -2 as a solution? A solution to what? It doesn't take away -2 as a solution to the equation x^2 = 4. In fact -2 is _a_ square root of 4. But by definition the square root of 4, aka sqrt(4) , is 2. >And besides, you lied, it's by *convention* that the positive is >taken, not by definition. My god you can be an idiot. Saying sqrt(4) = 2 by convention is the same as saying sqrt(4) = 2 by definition . A definition is precisely a convention regarding the meaning of a word (or phrase or operator.) You should probably try to avoid calling people liars when your reason for saying they lie is based on your profound ignorance of the meaning of simple English words. >> > You see Dik Winter, actually *knowing* mathematics versus talking as >> if you know it can be two different things. > It appears so, yes. Since people working with square roots *usually* want the positive >root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2 as a >solution, as is easily proven: sqrt(4) = -2, square both sides, 4 = 4. QED Now then, can you *prove* that -2 is not a solution to sqrt(4)? _equations_ have solutions. Sqrt(4) is not an equation, so saying that -2 is a solution to sqrt(4) makes no sense, just as saying that 2 is a solution to sqrt(4) makes no sense. In fact 2 and -2 are both solutions to the equation x^2 = 4; nobody has said otherwise. >James Harris David C. Ullrich ************************** As far as I'm concerend you're trying to wait until I die, so I figure maybe you should die instead. How about that, eh? Wouldn't that be a better twist? You refuse to follow the math, so the great Powers that control reality and *speak* in mathematics decide to kill you instead of me. So what do you think about that, eh? Oh, can't hear Them talking? Well, I guess that's because you don't really understand Mathematics, the true language, which is THE language. They're talking about you now, and They agree with my assessment, and will not penalize me as They allowed the others like Galois and Abel to be penalized. They will kill you instead. James Harris speaking on Weird factorization, genius ==== Consider that the constant term of g1(x) is 3 *or* 5 because the > sqrt() operator is ambiguous. I've explained that before in replying > to Arturo Magidin, but mathematics is a difficult discipline for some, > so repetition is necessary, and still often not enough. > Damn, James, does this mean your prime counter's wrong cos it had an ambiguous sqrt in it? ==== > > : Why does this not count as presenting such a proposition ? > : Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x). > : then > : g1(x)*g2(x) = 3(5-x) > : but neither g1 nor g2 is divisible by 3 for all x. > > the product *are* divisible by 7 for a *specific* value of x, then they > are divisible by 7 for *all* values of x. In your example neither is ever > divisible by 3. > > Justin Consider any integer a such that 1+3a is a perfect square. Then > g1(a) and g2(a) are both integers. Since their product is divisible > by 3 one of them must be divisible by 3. Note that posters have gone off on a tangent with a made up example, > and they're even getting *it* wrong! Fascinating behavior, but not > surprising for the sci.math newsgroup. > Examples g1(0)=3, g2(0)=5 That is incorrect as the sqrt() operator is ambiguous > No, by sqrt() I mean the unambiguous principle branch. - William Hughes ==== > Okay, so you're dividing both sides by 49. Point taken. :> In other words, are you > :> claiming that (5 a_1(x) + 7) and (5 a_2(x) + 7) are divisible by 7 for > :> all x? If so, could you please explain why the fact that it's true for > :> x=0 implies it's true for other x also? If you are not claiming this, then please explain what you are claiming. Many thanks, > Justin Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now you can divide both sides by 49. Some posters have claimed that when you divide by 49, what results > varies depending on what value x has. > Okay, so you're dividing both sides by 49. Point taken. :> In other words, are you > :> claiming that (5 a_1(x) + 7) and (5 a_2(x) + 7) are divisible by 7 for > :> all x? If so, could you please explain why the fact that it's true for > :> x=0 implies it's true for other x also? If you are not claiming this, then please explain what you are claiming. Many thanks, > Justin Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now you can divide both sides by 49. Some posters have claimed that when you divide by 49, what results > varies depending on what value x has. For Q(x) = (5a_1(x) + 7)(5a_2(x) + 7)(5b_3(x) + 22): x Q(x) JSH 'a' ----------------------------- 0 1078 0 0 3 1 13789188 25.5404... -138.21... -31.330... 2 114013298 51.6126... -279.30... -63.312... 3 388910158 77.6850... -420.39... -95.295... -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== One of those hard lessons to learn in mathematics is operator ambiguity. For instance the square root and cuberoot operators are ambiguous, and there's nothing you can do about it. Given (1)^{1/3} there are *three* solutions, and not one, which is the operator ambiguity that gave me fits for a while back last year when I posted and posted trying to find some trick around it. The square root operator has ambiguity in that it gives *two* solutions, even if you only want one. I managed to get myself in trouble yet again today trying yet again to escape operator ambiguity by making an earlier post trying to go with the sign convention of taking the positive solution of the square root operator. That doesn't work. It bothers me that I keep fighting operator ambiguity and trying to find ways around it, as if some part of me just can't accept that if you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which refuse to go away, no matter how hard you wish. James Harris ==== : For instance the square root and cuberoot operators are ambiguous, and : there's nothing you can do about it. Pray tell, what's your definition of operator ? The sqrt(c) *function* is defined to be the positive real number whose square is c . There is no *solution* here because this is not an *equation*. There is no ambiguity. The *equation* x^2-c=0 has *two solutions*, one is sqrt(c) and the other is -sqrt(c). How do *you* define operator ? Justin ==== > One of those hard lessons to learn in mathematics is operator > ambiguity. For instance the square root and cuberoot operators are ambiguous, and > there's nothing you can do about it. Given (1)^{1/3} there are *three* solutions, and not one, which is the > operator ambiguity that gave me fits for a while back last year when I > posted and posted trying to find some trick around it. The square root operator has ambiguity in that it gives *two* > solutions, even if you only want one. I managed to get myself in trouble yet again today trying yet again to > escape operator ambiguity by making an earlier post trying to go with > the sign convention of taking the positive solution of the square > root operator. That doesn't work. It bothers me that I keep fighting operator ambiguity and trying to > find ways around it, as if some part of me just can't accept that if > you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which > refuse to go away, no matter how hard you wish. > James Harris I, for one, find it odd that JSH can be so concerned about the action of the Galois group in this case (of the equation x^n - c = 0), when to date he has been a Galois Luddite, and proud of it. Dale ==== > One of those hard lessons to learn in mathematics is operator > ambiguity. For instance the square root and cuberoot operators are ambiguous, and > there's nothing you can do about it. Given (1)^{1/3} there are *three* solutions, and not one, which is the > operator ambiguity that gave me fits for a while back last year when I > posted and posted trying to find some trick around it. The square root operator has ambiguity in that it gives *two* > solutions, even if you only want one. I managed to get myself in trouble yet again today trying yet again to > escape operator ambiguity by making an earlier post trying to go with > the sign convention of taking the positive solution of the square > root operator. That doesn't work. It bothers me that I keep fighting operator ambiguity and trying to > find ways around it, as if some part of me just can't accept that if > you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which > refuse to go away, no matter how hard you wish. > James Harris James, your argument is equivalent to the following: Roads have two lanes. Therefore roads are ambiguous and there is nothing you can do about it. Therfore you can't drive on a road because there is no way for you to know what lane you should be in. However, we know from common experience that people do drive on roads. The point is that there is an unabiguous way for people to decide which is the right lane [1]. The same holds with ambiguous operators . The fact that every complex number has two square roots does not stop us defining a single valued square root function, as long as we can find an unambiguous way to select between them. If we are only interested in the positive real axis, it is enough to select the positive square root (it is easy to show that if x is a positive real then x has two real square roots, on positive and one negative). If we need to deal with all posible complex numbers z, then we choose the square root that has positive real part (positive imaginary part if z is negative real number). For cube roots things get a bit trickier (though not much). Look up principal branch for more details. -William Hughes [1] Indeed it's the right lane. And to deal with the square root you take the values from the right side of the plane. And to deal with the cube root you take ... Ok, eventualy this breaks down. ==== > [1] Indeed it's the right lane. Except where the left lane is the right lane. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== > [1] Indeed it's the right lane. Except where the left lane is the right lane. But suppose you don't drive in the left lane: then right is left, right? Jim Burns ==== > However, we know from common experience that people do drive on > roads. The point is that there is an unabiguous way for > people to decide which is the right lane [1]. [1] Indeed it's the right lane. And to deal with the square root > you take the values from the right side of the plane. And to deal > with the cube root you take ... Ok, eventualy this breaks down. That is more like taking the square root of -1. Of course the square root of -1 is i, but it could just as well have been j where j = -i, and nobody would be any the wiser. Just like you should always drive on the right lane, but in some countries the left lane is the right line to use. ==== [...] > It bothers me that I keep fighting operator ambiguity and trying to > find ways around it, as if some part of me just can't accept that if > you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which > refuse to go away, no matter how hard you wish. James you've spent so many years of your life on maths, and you're still such an ignorant. Impressive... This operator ambiguity bothers you because you still don't understand what an operator is. I'll try to help with the following question addressed to you: What is an operator? -- Edgar P.S. If you can't answer it, then everybody will know you've no idea what you're talking about. ==== > One of those hard lessons to learn in mathematics is operator > ambiguity. For instance the square root and cuberoot operators are ambiguous, and > there's nothing you can do about it. Given (1)^{1/3} there are *three* solutions, and not one, which is the > operator ambiguity that gave me fits for a while back last year when I > posted and posted trying to find some trick around it. The square root operator has ambiguity in that it gives *two* > solutions, even if you only want one. I managed to get myself in trouble yet again today trying yet again to > escape operator ambiguity by making an earlier post trying to go with > the sign convention of taking the positive solution of the square > root operator. That doesn't work. It bothers me that I keep fighting operator ambiguity and trying to > find ways around it, as if some part of me just can't accept that if > you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which > refuse to go away, no matter how hard you wish. James Harris By your scholarly exposition of the *alleged* operator ambiguity in the sqrt function, you have completely invalidated your own research results. Time to step up to the bar and admit that, BY YOUR OWN CRITERIA, your work is ambiguous, misleading and outright false! Next time you try to box someone else into a corner, make sure you aren't painting yourself into one. Hahahahahaha............ -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== > One of those hard lessons to learn in mathematics is operator > ambiguity. For instance the square root and cuberoot operators are ambiguous, and > there's nothing you can do about it. Given (1)^{1/3} there are *three* solutions, and not one, which is the > operator ambiguity that gave me fits for a while back last year when I > posted and posted trying to find some trick around it. The square root operator has ambiguity in that it gives *two* > solutions, even if you only want one. I managed to get myself in trouble yet again today trying yet again to > escape operator ambiguity by making an earlier post trying to go with > the sign convention of taking the positive solution of the square > root operator. That doesn't work. It bothers me that I keep fighting operator ambiguity and trying to > find ways around it, as if some part of me just can't accept that if > you have sqrt(x), or x^{1/3}, you have *multiple* solutions, which > refuse to go away, no matter how hard you wish. > James Harris You say that sqrt(x) has two solutions, but a function doesn't have solutions, only an equation. sqrt(x) is a function, and as such, has only values. The equation x^2 = 5 has two solutions, namely sqrt(5) and -sqrt(5), but that's just a result of solving the equation. It's not inherent in the functions which make up the solution. For example, the equation 2x = 2sqrt(5) has exactly one solution, namely sqrt(5). In the same way, while x^3 = 1 has three solutions, 1^y for any y is exactly 1. ==== >excuse my ignorance...but is there any explicit result for the following Tr(AAAAA....A) in terms of the componants a[i,j] of the matrix A where the product is taken >N (finite) times? (where i=1...n, j=1...m) Diagonalize: if r_1, ..., r_n are the entries on the diagonal of the Jordan canonical form of A (in an extension field of the field of definition of A) then the quantity T_N you want is the sum of the N-th powers of the r_i. That sum can be expressed as a polynomial in the elementary symmetric functions on the r_i, which are the coefficients of det(A + X I) (essentially the characteristic polynomial of A). You can compute the T_N easily from the previous T_k's using Newton's identities. If n or N is small you can write this out explicitly in symbolic form. (Of course you must have n=m for the product AA to be defined.) dave ==== >excuse my ignorance...but is there any explicit result for the following >Tr(AAAAA....A) >in terms of the componants a[i,j] of the matrix A where the product is taken >N (finite) times? (where i=1...n, j=1...m) If A is n x m and you want A^N to make sense, you'd better have n=m. Tr(A^N) = sum_{i_1=1}^n ... sum_{i_n=1}^n product_{j=1}^n a[i_j, i_{j+1}] where i_{N+1} = i_1. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== excuse my ignorance...but is there any explicit result for the following Tr(AAAAA....A) in terms of the componants a[i,j] of the matrix A where the product is taken >N (finite) times? (where i=1...n, j=1...m) If A is n x m and you want A^N to make sense, you'd better have n=m. Tr(A^N) = sum_{i_1=1}^n ... sum_{i_n=1}^n product_{j=1}^n a[i_j, i_{j+1}] > where i_{N+1} = i_1. thank you. cheers moth ==== For those of you who have been following the Peter Lynds' controversy, you may be interested on knowing that a new scandal has happened recently in the field of Maths. You can check here (and leave your opinion in the forum if you like): http://www.thequantummachine.com Cesar Sirvent ==== > For those of you who have been following the Peter Lynds' controversy, > you may be interested on knowing that a new scandal has happened > recently in the field of Maths. > You can check here (and leave your opinion in the forum if you like): Looks like the same old silliness to me. ==== > Suppose p : E --> B is a covering space and f : X --> B is a continuous > map where X is connected and B is Hausdorff. Show that if g and h > are two lifts of f such that there exists an a in X such that > g(a) = h(a), then g(x) = h(x) for all x in X. (Should we show > that {x in X : g(x) = h(x) is both open and closed}?) > That would probably be fine. What I would do is to consider the path- > component of a , and look at lifts via g and h, respectively, of any > path from a to an arbitrary other point in this path-component. Since covering spaces have unique path lifting, then g and h have to > agree on each path component. By continuity, they have to agree on > the closures of path components. Eventually, you can exhaust the > space X by progressing from point a to its path-component, to > the closure of that, to the path-components occupied by that closure, > to the closure of that set, and so on (there's probably something non- > trivial to prove here, and that's most likely the core of your > show that {...} is open and closed in X argument). > No offense, but ... that seems _way_ too complicated. To the OP: showing that A(g,h) = { x in X : g(x) = h(x) } is both open and closed _is_ the way to go. (1) Since B is Hausdorff and p: E --> B is a covering map, it follows that E is Hausdorff. (Proof ??) (2) Given _any_ two maps X --> E, the subset of X on which they agree is closed (since E is Hausdorff). So A(g,h) is closed. (3) For any x in A(g,h), you can find an open neighborhood U of g(a) = h(a) in E such that p|U is a homeomorphism. Then define V = g^{-1}(U) intersect h^{-1}(U). Show that V is contained in A(g,h), which proves that A(g,h) is open. > Suppose that G is a finite group which acts freely on a Hausdorff space > X. Show that the action must be properly discontinuous. (Can we use > the above problem for this?) > About your parenthetical question: it seems unlikely. If you _knew_ that the quotient map q: X --> X/G is a covering map, then you might have a chance. But ... one usually proves that q is a covering map as a consequence of the fact that G acts properly discontinuously ... (and that's proved in the manner that Dale indicates below). Think on these three conditions: G is finite > X is Hausdorff. > The action is free. Recall that a properly discontinuous action is one for which, > for each point x in X, there is a neighborhood U_x of x in X, > such that all non-identity elements of G map U to a set > disjoint from U. Here, you can take the orbit Gx of your > favorite x in X, and form disjoint open sets U_gx of gx. > By a suitable song and dance, you can force the union > to be G-invariant (hint: G is finite, and a finite > intersection of open sets is open), and then take the > patch containing your original x. I'm not sure how to apply the first problem to address > the second one. Steven Dale ==== >>Suppose p : E --> B is a covering space and f : X --> B is a continuous >map where X is connected and B is Hausdorff. Show that if g and h >are two lifts of f such that there exists an a in X such that >g(a) = h(a), then g(x) = h(x) for all x in X. (Should we show >that {x in X : g(x) = h(x) is both open and closed}?) >That would probably be fine. What I would do is to consider the path- >>component of a , and look at lifts via g and h, respectively, of any >>path from a to an arbitrary other point in this path-component. >>Since covering spaces have unique path lifting, then g and h have to >>agree on each path component. By continuity, they have to agree on >>the closures of path components. Eventually, you can exhaust the >>space X by progressing from point a to its path-component, to >>the closure of that, to the path-components occupied by that closure, >>to the closure of that set, and so on (there's probably something non- >>trivial to prove here, and that's most likely the core of your >> show that {...} is open and closed in X argument). >> > No offense, but ... that seems _way_ too complicated. To the OP: > showing that A(g,h) = { x in X : g(x) = h(x) } is both open and > closed _is_ the way to go. (1) Since B is Hausdorff and p: E --> B is a covering map, it > follows that E is Hausdorff. (Proof ??) (2) Given _any_ two maps X --> E, the subset of X on which they > agree is closed (since E is Hausdorff). So A(g,h) is closed. (3) For any x in A(g,h), you can find an open neighborhood U of > g(a) = h(a) in E such that p|U is a homeomorphism. Then define > V = g^{-1}(U) intersect h^{-1}(U). Show that V is contained > in A(g,h), which proves that A(g,h) is open. > I agree. That proof is far simpler. ... stuff deleted ... >>Dale > Dale ==== I came up with a question regarding limits today, let n be real, x = lim { n/(n!)^(1/) } n->inf How can I solve this problem? I tried L'Hosptal's rule but I dont know the answer of d(x!) ------ dx Could somebody please help me to answer these two questions? thanks a lot in advance, /lucas ==== > let n be real, > x = lim { n/(n!)^(1/) } n->inf How can I solve this problem? I tried > L'Hosptal's rule but I dont know > the answer of d(x!) > ------ > dx One way to do it is to use Stirling's approximation to the factorial function n! ~~ sqrt(2 pi) n^(n + 1/2) exp( -n ) ==== > I came up with a question regarding limits today, let n be real, > x = lim { n/(n!)^(1/) } n->inf You probably mean the limit of n/(n!)^(1/n) as n -> oo. Take the log to get sum_(j=1,n) (1/n)*ln(n/j), which is a Riemann sum for int_[0,1] ln(1/x) dx = 1. So the answer is e. ==== I came up with a question regarding limits today, let n be real, >x = lim { n/(n!)^(1/) } n->inf How can I solve this problem? If you mean x = lim(n/(n!)^(1/n)), n->oo then taking the logarithm of x gives: log(x) = lim[ log n - 1/n * log(n!) ] and using Stirling's approximation log(n!) ~= nlog n - n (for large n) we get: log(x) = lim[ log n - 1/n * (n * log n - n) ] = lim[ log n - log n + 1 ] = 1 So that x = e^1 = e. ==== > [snipped] >>let n be real, >>x = lim { n/(n!)^(1/) } n->inf >>How can I solve this problem? If you mean x = lim(n/(n!)^(1/n)), n->oo then taking the logarithm of > x gives: > That's it! Sorry, my mistake. again, x = lim(n/(n!)^(1/n)) n->infinity. > log(x) = lim[ log n - 1/n * log(n!) ] and using Stirling's approximation log(n!) ~= nlog n - n (for large n) > we get: log(x) = lim[ log n - 1/n * (n * log n - n) ] > = lim[ log n - log n + 1 ] > = 1 So that x = e^1 = e. This is the kind of trick that I didnt see in the engineering course. Its perfectly clear for me now, I appreciate that, thanks a lot everybody that answerd my question. Again, what is the answer of d(x!)/dx ? thank you, /lucas ==== [snip] > Again, what is the answer of d(x!)/dx ? [I assume, of course, that you're thinking about x! = Gamma(x+1), rather than having x! defined only for nonnegative integer x.] See (16) at . David ==== > [snip] >> Again, what is the answer of d(x!)/dx ? [I assume, of course, that you're thinking about x! = Gamma(x+1), rather > than having x! defined only for nonnegative integer x.] See (16) at . > Not really. n! = n * (n - 1) * (n - 2) * (n - 3) * ... * 2 * 1 Am I missing something? This is my definition of `n!', and what I wanted to calculate in the above problem. thanks, /lucas ==== > [snip] >> Again, what is the answer of d(x!)/dx ? [I assume, of course, that you're thinking about x! = Gamma(x+1), rather > than having x! defined only for nonnegative integer x.] See (16) at . Not really. > n! = n * (n - 1) * (n - 2) * (n - 3) * ... * 2 * 1 > Am I missing something? This is my definition of `n!', and what > I wanted to calculate in the above problem. > Well, then, what you are missing is the definition of derivative. It requires a function defined on an interval. Not merely at discrete points like 1,2,3,... So for THAT factorial function, your question >> Again, what is the answer of d(x!)/dx ? is nonsense. ==== > Not really. >> n! = n * (n - 1) * (n - 2) * (n - 3) * ... * 2 * 1 >> Am I missing something? This is my definition of `n!', and what >> I wanted to calculate in the above problem. > > Well, then, what you are missing is the definition of derivative. It > requires a function defined on an interval. Not merely at discrete > points like 1,2,3,... So for THAT factorial function, your question > I interpreted the above statement as a polynomial expression. Such as I could distribute the values and somehow apply a reduction to that expression being able to derivate it. For example, i can derivate `ax^2 + bx + c' -> `2ax + b'. I wonder if I can do the same with this infinity expression. I thought i could express `n!' as a polynomial, p(n) of n'th degree. > Again, what is the answer of d(x!)/dx ? is nonsense. Also, I was working on another solution to the same limit problem and I blocked myself on this: y = lim(x/[(x!)^(1/x)]) y = lim([x^x/x!]^1/x) i know that x^x > x!, because, x^x = x * x * x * x * x * ... * x and x! = x * (x - 1) * (x - 2) ... * 1 so, lim(x^x/x!) = oo (infinity) i cant manage to solve (oo)^(1/x). thanks for everything, /lucas ==== I came up with a question regarding limits today, let n be real, > x = lim { n/(n!)^(1/) } n->inf How can I solve this problem? I tried > L'Hosptal's rule but I dont know AAAAAAAAAAAAAARRRRRRRRRRGGGGGGGGGGGGGGGGGHHHHHHHHHHHHHHHHHHHHH!!!!!!!!! Forget the Hospital. I presume you mean n/(n!)^(1/n). Use Stirling's formula. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > I came up with a question regarding limits today, let n be real, > x = lim { n/(n!)^(1/) } n->inf > It seems something is missing here... After correcting that... Did you see what Stirling's formula tells you? > How can I solve this problem? I tried > L'Hosptal's rule but I dont know > the answer of d(x!) > ------ > dx Could somebody please help me to answer these > two questions? thanks a lot in advance, /lucas ==== I know there are a number of methods available readily for approximating the complete gamma function (e.g., the Lanczos approximation). However, what I need is a formula/method for getting approximate values for the LOWER INCOMPLETE GAMMA FUNCTION; i.e., where zero is the lower limit of integration and the upper limit is specified as one of the parameters of the function. Does anyone know ==== >I know there are a number of methods available readily for >approximating the complete gamma function (e.g., the Lanczos >approximation). However, what I need is a formula/method for getting >approximate values for the LOWER INCOMPLETE GAMMA FUNCTION; i.e., >where zero is the lower limit of integration and the upper limit is >specified as one of the parameters of the function. Does anyone know If G(a,z) = int_0^z t^(a-1) exp(-t) dt is your function, then for small z you can use G(a,z) = sum_{k=0}^infinity (-1)^k z^(a+k)/(k! (a+k)) Or for positive integers a, G(a,z) = Gamma(a) (1 - exp(-z) sum_{k=0}^{a-1} z^k/k!) If a is not an integer, this becomes an asymptotic series G(a,z) = Gamma(a) (1 - exp(-z) sum_{j=0}^infinity z^(a-1-j)/Gamma(a-j)) Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== >I know there are a number of methods available readily for >approximating the complete gamma function (e.g., the Lanczos >approximation). However, what I need is a formula/method for getting >approximate values for the LOWER INCOMPLETE GAMMA FUNCTION; i.e., >where zero is the lower limit of integration and the upper limit is >specified as one of the parameters of the function. Does anyone know There are lots of these as well; look in numerical analysis texts dealing with such functions. Which ones I would use depends on the actual parameters. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University ==== Given, where x is in the ring of algebraic integers, I've shown the factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now consider the factorization shown again, but with the 49 multiplied through: (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 and since a_1(0)= a_2(0) = b_3(0) = 0, it's not surprising that the values thus shown to be constant in the factors on the left side i.e. 7, 7 and 22 are in fact factors of what's constant on the right side i.e. 1078. Now if I divide both sides by 49, I end up with a change where now I have constant factors 1, 1, and 22 on the left which are still factors of 22 on the right. Does that fact tell you that *two* of the factors on the left, the ones that have 7 as a constant factor were each divided by 7, or does it tell you *nothing* at all? Test is of the ability to understand constant factors as independent of variables, with a check of ability to convince large groups. Note sci.logic is included to emphasize logical thinking and see if it matters to readers. That is, the test is of self-doubt. James Harris ==== Is there any rationality test that JSH has not yet failed? ==== I am going to try to follow the convention of addressing the assembly as a whole, not just the poster to whom I am replying. I think this practice improves the ratio of light to heat in a discussion. > Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of > [Polynomial P] > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. It took me two tries, even with Mathematica, but this computation checks out OK. Now consider the factorization shown again, but with the 49 multiplied > through: (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 and since a_1(0)= a_2(0) = b_3(0) = 0, it's not surprising that the > values thus shown to be constant in the factors on the left side i.e. > 7, 7 and 22 are in fact factors of what's constant on the right side > i.e. 1078. Sustitute 0 for x and the equation reduces to 7 * 7 * 22 = 1078. Yes. Now if I divide both sides by 49, I end up with a change where now I > have constant factors 1, 1, and 22 on the left which are still factors > of 22 on the right. Well, let's see if I understand. The left side could just be written (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22)/49. But one can say, Well, let's merge the factor of 1/49 into the other factors. The way that looks nicest is to multiply each of the first two factors by 1/7. Then we have this: [Equation A] ((5/7) a_1(x) + 1)((5/7)a_2(x) + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 Does that fact tell you that *two* of the factors on the left, the > ones that have 7 as a constant factor were each divided by 7, or does > it tell you *nothing* at all? It tells me a little bit, but I don't think it tells me as much as Mr. Harris wants. I am not sure, but I think he wants to infer that the three factors in Equation A will be algebraic integers when x is a non-zero algebraic integer. Hmm. Are they? When x = 0, Polynomial P has roots 0, with multiplicity 2, and 1 with multiplicity 1. It seems clear that Mr. Harris intends that a_1(x) and a_2(x) specialize to 0 when x = 0, whereas a_3(x) specializes to 1. OK, I think. Suppose we let x = y^2, where y is an algebraic integer. This does not lose any generality. And suppose, when y != 0, we let a_i = 7 y c_i. We can substitute into Polynomial A and divide out by a factor of 49y^2. When the dust settles, we find that the c_i are roots of this polynomial: 7 y c^3 -3(1 - 49 y^2)c^2 -(3 - 147 y^2 + 2401 y^4) I think Mr. Harris would claim that two of the roots of this polynomial are algebraic integers, when y is a non-zero algebraic integer. I doubt this, but I would be glad to learn more. I am going to take another look at the paper by Messrs. Magidin and MacKinnon on Gauss's Lemma , available at http://www.math.umt.edu/magidin/preprints/preprints.html I recommend it to all interested readers of this post. -- Chris Henrich ==== > Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now consider the factorization shown again, but with the 49 multiplied > through: (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 and since a_1(0)= a_2(0) = b_3(0) = 0, it's not surprising that the > values thus shown to be constant in the factors on the left side i.e. > 7, 7 and 22 are in fact factors of what's constant on the right side > i.e. 1078. Now if I divide both sides by 49, I end up with a change where now I > have constant factors 1, 1, and 22 on the left which are still factors > of 22 on the right. Does that fact tell you that *two* of the factors on the left, the > ones that have 7 as a constant factor were each divided by 7, or does > it tell you *nothing* at all? It provides a special case in which the expressions are evaluated at 'x' = 0. It does *not* provide insight into the general case. For Q(x) = (5a_1(x)+7)(5a_2(x)+7)(5b_3(x)+22): x Q(x) JSH 'a' (b_3(x) = a_3(x) - 3) --------------------------------------------------- 0 1078 0 0 3 1 13789188 25.5404 -138.21 -31.3299 2 114013298 51.6126 -279.3 -63.3123 3 388910158 77.685 -420.39 -95.2947 > Test is of the ability to understand constant factors as independent > of variables, with a check of ability to convince large groups. Note > sci.logic is included to emphasize logical thinking and see if it > matters to readers. rather in 'alt. crackpot.theories'. Give up, James! You are *toast*! -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== >[...] Test is of the ability to understand constant factors as independent >of variables, with a check of ability to convince large groups. Note >sci.logic is included to emphasize logical thinking So today you decided to add what's the point to sci.logic? to the list of things you're clueless about. Congratulations. >and see if it >matters to readers. That is, the test is of self-doubt. James Harris David C. Ullrich ************************** As far as I'm concerend you're trying to wait until I die, so I figure maybe you should die instead. How about that, eh? Wouldn't that be a better twist? You refuse to follow the math, so the great Powers that control reality and *speak* in mathematics decide to kill you instead of me. So what do you think about that, eh? Oh, can't hear Them talking? Well, I guess that's because you don't really understand Mathematics, the true language, which is THE language. They're talking about you now, and They agree with my assessment, and will not penalize me as They allowed the others like Galois and Abel to be penalized. They will kill you instead. James Harris speaking on Weird factorization, genius ==== > Given, where x is in the ring of algebraic integers, I've shown the > factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x) - 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0. Now consider the factorization shown again, but with the 49 multiplied > through: (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 and since a_1(0)= a_2(0) = b_3(0) = 0, it's not surprising that the > values thus shown to be constant in the factors on the left side i.e. > 7, 7 and 22 are in fact factors of what's constant on the right side > i.e. 1078. Now if I divide both sides by 49, I end up with a change where now I > have constant factors 1, 1, and 22 on the left which are still factors > of 22 on the right. Does that fact tell you that *two* of the factors on the left, the > ones that have 7 as a constant factor were each divided by 7, or does > it tell you *nothing* at all? Test is of the ability to understand constant factors as independent > of variables, with a check of ability to convince large groups. Note > sci.logic is included to emphasize logical thinking and see if it > matters to readers. That is, the test is of self-doubt. James Harris James, You are a total moron. You act like we are nothing except for your experiment. We are not your experiment; we are all humans. You are totally out of line by claiming to use us as part of your experiment . -- David Moran Chief Meteorologist Oklahoma Storm Team ==== [...snip JSH nonsense...] James, You are a total moron. You act like we are nothing except for your > experiment. We are not your experiment; we are all humans. You are totally > out of line by claiming to use us as part of your experiment . No explanation needed. All that is required is: James, You are a total moron. The rest is just filler. :):):) Bye, Jay ==== [snip] > best wishes > George After having some rest, I am back. Scrolling the threads in sci.physics, I have noticed that nobody remembers that Wright brothers took first flight in history of mankind All Aeroplanes I see here in Bombay flying in sky is because of them. 17 years ago, when I left my small village at the age of 17, I had never seen any aeroplane flying right over my head so closely. And because of those Aeroplanes flying in sky, somehow I began to dream that I want to be Pilot, not Engineer. And I just left this place despite stiff opposition from my family. For 17 years, bouncing from this career to that career, somehow I have this action device which we can use to hang the things in air, make small bikes, cars etc. and any one, even small kids can become pilot to fly them in blue sky. I pay my tributes to Wright Brothers who gave me dreams to fly 17 years ago. But Everything That Has A Beginning Has An End... http://www.geocities.com/actiondevice -Abhi. ==== > [snip] > best wishes > George After having some rest, I am back. I hope you are feeling better. Now instead of snipping what I said, how about reading it. It shows why it is not possible to build a machine such as you describe regardless of how you construct it: > I have once again changed the webpage. Perhaps these magnets facing > each other will explain the working of this action device in better > way. Abhi, you said that logic was a tool you could use so > consider this explanation of basic mechanics. 1) Two magnets pull towards each other: ==== F--> <--F ==== The forces are equal and opposite. 2) A stretched spring pulls the ends towards each other: ---VVVVVVVVVVVVVV--- > F--> <--F The forces are equal and opposite. 3) A compressed rod pushes back against the compression. >|=============|< > <--F F-- The forces are equal and opposite. 4) Try to bend a rod: F F > | | > v v > |=============| > ^ > | > 2*F The total forces are equal and opposite. (Forces in this case are two of 'F' each downwards > and one of twice as much upwards.) Now here's the logic: Since the forces created by every rod, spring > or pair of magnets are always equal and opposite, > the total of all forces in your device must also > be equal and opposite. Adding equal but opposite > forces always totals zero, therefore your device > will always produce zero total force no matter > how many springs, rods and magnets you introduce. George ==== > For 17 years, bouncing from this career to that career, somehow I have > this action device which we can use to hang the things in air, make > small bikes, cars etc. and any one, even small kids can become pilot > to fly them in blue sky. WHAT??!! YOU SPEND 17 YEARS ON THAT THING AND NEVER BUILD IT?? Whyever that? Greetings! Volker ==== Can someone please explain how to permute n objects in a circle without going too much into the mathematics of it ==== > Can someone please explain how to permute n objects in a circle without > going too much into the mathematics of it Nail one to the ground. ==== Can someone please explain how to permute n objects in a circle without > going too much into the mathematics of it > Nail one to the ground. Good one :-) ... or throw one away and permute the remaining ones in a line. Dirk Vdm ==== does anybody know about properties concerning the zeros of the derivatives of smmoth symmetric functions f(t)=f(1-t) with compact support? I was wondering if he zeros of the derivatives of a function of the form f(t)=exp(-1/(t*(1-t))), t in (0,1) Thomas ==== does anybody know about properties concerning the zeros of the > derivatives of smmoth symmetric functions f(t)=f(1-t) with compact > support? I was wondering if he zeros of the derivatives of a function > of the form f(t)=exp(-1/(t*(1-t))), t in (0,1) > Thomas The derivative is going to be a rational function in t times an exponential and therefore cannot have more than finitely many. ==== >does anybody know about properties concerning the zeros of the >derivatives of smmoth symmetric functions f(t)=f(1-t) with compact >support? I was wondering if he zeros of the derivatives of a function >of the form >f(t)=exp(-1/(t*(1-t))), t in (0,1) For this particular one, I don't know, but it's certainly not true in general. For example, there are functions of this type that coincide with cos(t-1/2) on a neighbourhood of 1/2, and the derivatives of those have no zeros in that neighbourhood except at 1/2. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== We'd like to call your attention to a proposal for a new usenet newsgroup in the sci.physics.* hierarchy with the proposed name being sci.physics.strings . This new group is supposed to provide a place for intensive discussion of string theory and related topics by people working in the field as well as interested laymen. In order to create such a new usenet group the proposed group's charter has to go through a discussion period at news.groups , which can be accessed for instance via http://groups.google.de/groups?&lr=&ie=UTF-8&group=news.groups . You can find and discuss the proposed charter of sci.physics.strings as well as the rationale for its creation at http://groups.google.de/groups?selm=1071590601.19921%40isc.org After the discussion period (which will last presumably about three to six weeks) there will be a general poll where everybody is asked to vote in favour or against the creation of sci.physics.strings on usenet. We'd like to kindly invite everybody with an interest in string theory or string theory related topics to have a look at the above proposal and to participate in the voting. Since a majority of 2/3 of the valid votes must be in favor of the creation of sci.physics.strings, we ask everybody who would enjoy seeing interesting online discussion of string theory to participate in the voting. P.S. General information on the nature of usenet discussion fora is given at http://www.faqs.org/faqs/usenet/what-is/part1/ . More information on the general procedure for creating new usenet newsgroups can be found at http://www.faqs.org/faqs/usenet/creating-newsgroups/part1/ . ==== >please don't take too much comfort from my post. My reference, if taken as implying that you might be sympathetic to my position, seems not to be welcome. Clarification accepted. However,back to the bone of contention. There may be others besides myself who are happier with the older way of defining Riemann surfaces for use in analysis so let me briefly go into it. Having introduced cuts to branch-points and taken copies of the cut plane (sheets) I parametrize the Riemann surface with two variables (n,z) where n is the sheet and z is the point on the sheet. Then one comes to the problem of boundary identifications. I found a flaw in the way this is treated in the literature and have noted it elsewhere. I also found a remedy but have kept this to myself. Others might find it for themselves. It does not need great ability, just a degree of openmindedness. ==== >>please don't take too much comfort from my post. My reference, if taken as implying that you might be sympathetic to >my position, seems not to be welcome. Clarification accepted. >However,back to the bone of contention. There may be others besides myself who are happier with the older >way of defining Riemann surfaces for use in analysis so let me briefly >go into it. Having introduced cuts to branch-points and taken copies >of the cut plane (sheets) I parametrize the Riemann surface with two >variables (n,z) where n is the sheet and z is the point on the sheet. >Then one comes to the problem of boundary identifications. I found a >flaw in the way this is treated in the literature and have noted it >elsewhere. I also found a remedy but have kept this to myself. Others >might find it for themselves. It does not need great ability, just a >degree of openmindedness. This was _not_ the bone of contention. If you'd stated that you're happier with branch cuts than with the definition in terms of continuations along paths nobody would have argued. The bone of contention was your statement (in a post which as far as I can see has been cancelled or something, although it's still available at google) that there is necessarily some arbitrariness in the construction of the Riemann surface for a function. That statement is simply false. ************************ David C. Ullrich ==== My Commentary on Lenny's lecture in the context of my dialogues with Paul Zielinski and Harold Puthoff on Metric Engineering: Making Star Trek Real from my third book in the Space-Time and Beyond series now being written with full color illustrations and cartoons. I examine Lenny's ideas to establish where mainstream cutting edge physics is these days in 2004 as a proper context to evaluate Hal's and my own fringe ideas relating to the UFO controversy. For the record I suggested the problem for Lenny's first published physics paper at Cornell in 1963 on the the problem of lack of a Hermitian operator for both time and wave phase in quantum theory. I had been working on that problem with George Parrent Jr, a student of Emil Wolf's, at Tech/Ops associated with Mitre on Route 2 near Boston. I also brought Johnny Glogower to Cornell with Phil Morrison's help. Johnny was part of Walter Breen's Super Kids group from Columbia University in a project allegedly funded by Eugene McDermott a co-founder of Texas Instruments. We were all rebels . Lenny was a high school dropout. Johnny was a Quiz Kid, Westinghouse Finalist who flunked out of Brandeis. Excerpts from: Physics World Superstrings by Leonard Susskind String theory is either a theory of everything - which automatically unites gravity with the other three forces in nature - or a theory of nothing, but finding the correct form of the theory is like searching for a needle in a stupendous haystack rather like trying to summarize the history of the world in 10 pages. It is just too large a subject, with too many lines of thought and too many threads to weave together. In the 34 years since it began, string theory has developed into an enormous body of knowledge that touches on every aspect of theoretical physics. String theory is a theory of composite hadrons, quantum theory of gravity, and a framework for understanding black holes. It is also a powerful technical tool for taming strongly interacting quantum field theories and, perhaps, a basis for formulating a fundamental theory of the universe. It even touches on problems in condensed-matter physics, and has also provided a whole new world of mathematical problems and tools. ... String theory is considered to be a branch of high-energy or the 1950s, 1960s or 1970s would be surprised to read a recent string-theory paper and find not a single Feynman diagram, literature. What the reader would find are black-hole metrics, Einstein equations, Kaluza-Klein theories and plenty of fancy geometry and topology. The energy scales of interest are not MeV, GeV or even TeV, but energies at the Planck scale - the scale at which the classical concepts of space and time break down. The Planck energy is equal to h-bar5/G, where h-bar is Planck's constant divided by 2!, c is the speed of light and G is the gravitational constant, and it corresponds to masses that are some 19 orders of magnitude larger than the proton mass. There is actually a typo editor's error here, not Lenny's. The Planck energy is hbar c/Lp = hbar c/(hbarG/c^3)^1/2 = (hbar c^5/G)^1/2 Back to Lenny's talk: This is the energy of the universe when it was just 10-43s old, and it will understand physics at the Planck scale we need a quantum theory of gravity. In the days when my career was beginning, a typical colloquium on high-energy physics would often begin by stating that there are four forces in nature - electromagnetic, weak, strong and gravitational - followed by a statement that the gravitational force is much too from now on. That has all changed. Today the other three forces are described by the gauge theories of quantum chromodynamics (QCD) and quantum electrodynamics physics. These quantum field theories describe the fundamental quanta: the photon for the electromagnetic force, the W and Z bosons for the weak force, and the gluon for the strong force. In the string-theory community, however, the electromagnetic, strong and weak forces are generally considered to be manifestations of certain compactifications of space from 10 or 11 dimensions to the four familiar dimensions of space-time. ... Why quantum gravity? charge, colour, parity and hypercharge - to be truly elementary. scale. Protons and mesons reveal their parts at the modestly small distance of about 10-15 m, but quarks, leptons and photons hide their structure much more effectively. Indeed, no experiment has ever seen direct evidence of size or structure for any of these JS Comment: This point-like structure may be from a huge space-warp effect depending on the momentum transfer from scattering probe to target from an exotic vacuum dark matter core of the spatially extended lepto-quarks where the effective gravity coupling at short range is 40 powers of ten greater than Newton's. Back to Lenny: ... coupling constants are not really constants at all - they vary with energy. If the known couplings are extrapolated they all intersect the predictions of the unified theory at roughly the same scale. JS Comment: This GUT scale is ~ Lp/(alpha) where Lp^2 ~ hG/c^3 and alpha ~ e^2/hc ~ 1/137. Back to Lenny: Moreover, this scale is close to the Planck scale. The implication of this was clear: the scale of the internal machinery of elementary appears in the definition of the Planck energy, to many of us this inevitably meant that gravitation must play an essential role in The earliest attempts to reconcile gravity and quantum mechanics - notably by Richard Feynman, Paul Dirac and Bryce DeWitt, who is now at the University of Texas at Austin - were based on trying to fit Einstein's general theory of relativity into a quantum field theory like the hugely successful QED. The goal was to find a set of rules for calculating scattering amplitudes in which the photons of QED are replaced by the quanta of the gravitational field: gravitons. But gravitational forces become increasingly strong as the energy of the participating quanta increases, and the theory proved to be wildly simply gave rise to far too many degrees of freedom at short distances. In a sense the failure of this quantum gravity theory was a good sign. The theory itself gave no insight into the internal machinery of forces of nature. At best it was more of the same: an effective (but not very) description of gravitation with no deeper insight into the nonsense. Strings as hadrons We all know that science is full of surprising twists, but the discovery of string theory was particularly serendipitous. The theory grew out of attempts in the 1960s to describe the interactions of neutron. This was a problem that had nothing to do with gravity. Gabriele Veneziano, now at CERN, and others had written down a simple mathematical expression for scattering amplitudes that had certain properties that were fashionable at that time. It was soon discovered by Yoichiro Nambu of the University of Chicago and myself, and in a slightly different form by Holger Bech Nielsen at the Niels Bohr Institute, that these amplitudes were the solution of a definite physical system that consists of extended 1D elastic strings. ... Fermionic versions of string theory were soon discovered and, moreover, they turned out to have a surprising symmetry called supersymmetry that is now totally pervasive in high-energy physics. In supersymmetric theories all bosons have a fermionic superpartner and vice versa. ... Another apparently serious problem with the string theory of hadrons concerned dimensions. Although the original assumptions in string theory were simple enough, the mathematics proved internally inconsistent, at least if the number of dimensions of space-time was four. The source of this problem was quite deep, but, strangely, if space-time has 10 dimensions it contrives to cancel out. ... A mathematical string can vibrate in many patterns, angular momentum ( spin-two ). There are certainly spin-two hadrons, but none that have anything like zero mass. Despite all made massive. ... A massless spin-two field might not be good for hadronic physics, but it is just what was needed for quantum gravity, albeit in 10D. This is because just as the photon is the quantum of the electromagnetic field, the graviton is the quantum of the gravitational field. But the gravitational field is a symmetric tensor rather than a vector, and this means the graviton is spin-two, rather than spin-one like the photon. This difference in spin is the principal reason why early attempts to quantize gravity based on QED did not work. A theory of everything, ... either all matter is strings, or string theory is wrong. This is one of the most exciting features of the theory. But what about the problem of dimensions? Here again, a sow's ear was turned into a silk purse. The basic idea goes back to Theodor Kaluza in 1919, who tried to unify Einstein's gravitational theory with electrodynamics by introducing a compact space-like fifth dimension. Kaluza discovered the beautiful fact that the extra components of the gravitational field tensor in 5 dimensions behaved exactly like the electromagnetic field plus one additional scalar field. Somewhat later, in 1938, Oskar Klein and then Wolfgang Pauli generalized Kaluza's work so that the single compact dimension was replaced by a 2D space. If the 2D space is the surface of a sphere then a remarkable thing happens when Kaluza's procedure is followed. Instead of electrodynamics, Klein and Pauli discovered the first non-Abelian gauge theory, which was later rediscovered by Chen Ning Yang and Robert Mills. This is exactly the same class of theories that is so successful in describing the strong and electromagnetic interactions in the Standard Model. JS Comment: A 2D Kaluza-Klein space has group structure of a 2D sphere embedded in flat 3D space with 3 rotation charge generators, i.e. SU(2) group for the weak force with 3 charges. In general we have N^2 - 1 charges for the SU(N) internal symmetry gauge forcegroup at a fixed space-time point where the minimal coupling local independence of phase rotations introduces the compensating spin 1 gauge force fields to restore the broken global symmetry. This force generator idea is re-expressed in the geometrodynamics of hyperspace. Here's Lenny: that appears to be standing still in our usual 3D space have velocity or momentum components in the compact dimensions? The answer is yes, and the corresponding components of momentum define new conserved quantities. What is more, these quantities are quantized in discrete units. In short, they are charges similar to electric charge, isospin and all the other the problem of dimensions in string theory is obvious: six of the 10 dimensions should be wrapped up into some very small compact space, and the corresponding quantized components of momenta determines their quantum numbers. Life in six dimensions Much of the development of string theory is therefore concerned with 6D spaces. These spaces, which can be thought of as generalized Kaluza-Klein compactification spaces, were originally studied by mathematicians and are known as Calabi-Yau spaces. They are tremendously complicated and are not completely understood. But in the process of studying how strings move on them, physicists have created an unexpected revolution in the study of Calabi-Yau spaces. JS Comment: Recall that the classical gravity radius is proportional to M and the quantum radius is proportional to 1/M. That is Rg = GM/c^2 Rq = h/Mc Therefore RgRq = Gh/c^3 = Lp^2 = 1 Bekenstein BIT. We have a germ of a duality between black holes and quantum momenta Rg = Lp^2/Rq Note also the Blackett empirical relation e = G*^1/2M where for an electron G* ~ 10^40G The quantum momenta p in the compactified extra-dimensions are charges Q (sources of the spin 1 gauge forces) where by the Blackett relation Q/G*^1/2 = M = h/cR Q = G*^1/2h/cR R is a compactification scale. G* = e^phiG(Newton) Back to Lenny In particular, it was discovered that a compactification radius of size R is completely equivalent to a space with size 1/R from the point of view of string theory. This connection, which is known as T-duality, has a mathematically profound generalization called mirror symmetry, which states that there is an equivalence between small and large spaces ... . Mirror symmetry of Calabi-Yau spaces - which are not only of different sizes but have completely different topologies - was completely unsuspected before physicists began studying quantum strings moving on them. I wish it was possible to draw a Calabi-Yau space but they are tremendously complicated. They are six-dimensional, which is three more than I can visualize, and they have very complicated topologies, including holes, tunnels and handles. Furthermore, there are thousands of them, each with a different topology. And even when their topology is fixed there are hundreds of parameters called moduli that determine the shape and size of the various dimensions. Indeed, it is the complexity of Calabi-Yau geometry that makes string theory so intimidating to an outsider. However, we can abstract a few useful things from the mathematics, one of them being the idea of moduli. The simplest example of a modulus is just the compactification radius, R, when there is only a single compact dimension. In more complicated cases, the moduli determine the sizes and shapes of the various features of the geometry. The moduli are not constants but depend on the geometry of the space itself, in the same way that the radius of the universe changes with time in a manner that is controlled by dynamical equations of motion. Since the compact dimensions are too small to see, the moduli can simply be thought of as fields in space that determine the local conditions. Electric and magnetic fields are examples of such fields but the moduli are even simpler: they are scalar fields (i.e. they have only one component), rather than vector fields. String theory always has lots of scalar-field moduli and these can potentially play important roles in All of this raises an interesting question: what determines the compactification moduli in the real world of experience? Is there some principle that selects a special value of the moduli of a particular Calabi-Yau space and therefore determines the coupling constants of the forces, and so on? The answer seems to be no: all values of the moduli apparently give rise to mathematically consistent theories. Whether or not this is a good thing, it is certainly surprising. Ordinarily we might expect the vacuum or ground state of the world to be the state of lowest energy. Furthermore, in the absence of very special symmetries, the energy of a region of space will depend non-trivially on the values of the fields in that region. Finding the true vacuum is then merely an exercise in computing the energy for a given field configuration and minimizing it. This is, to be sure, a difficult task, but it is possible in principle. In string theory, however, we know from the beginning that the potential energy stored in a given configuration has no dependence on the moduli fields. The reason that the field potential is exactly zero for every value of the moduli is that string theory is supersymmetric. Supersymmetry has both desirable and undesirable consequences. Its most obvious drawback is the requirement that for every fermion there is a boson with exactly the same mass, which is clearly not a property of our world. A more subtle difficulty involves the aforementioned fact that the vacuum energy is independent of the moduli. As well as telling us that we cannot determine the moduli by minimizing the energy, supersymmetry also tells us that the quanta of the moduli fields are exactly massless. No such massless fields are known in nature and, furthermore, such fields are very dangerous. Indeed, massless moduli would probably lead to long-range forces that would compete with gravity and violate the equivalence principle - the cornerstone of general relativity - at an observable level. On the plus side, the vanishing vacuum energy that is implied by supersymmetry ensures that the cosmological constant vanishes. If it were not for supersymmetry, the vacuum would have a huge zero-point energy density that would make the radius of curvature of space-time not much bigger than the Planck scale - a most undesirable situation. JS Comment: I have a different much simpler explanation for the smallness of the cosmological constant in http://qedcorp.com/APS/EmergentGravity.pdf Also the observational fact of dark energy with FRW Omega ~ 0.7 means that the cosmological constant is not exactly zero, which is a problem for the physics Lenny is talking about. Back to Lenny: Supersymmetry also stabilizes the vacuum against various hypothetical instabilities, and it allows us to make exact mathematical conclusions. Indeed, T-duality and mirror symmetry are examples of those exact consequences. Black holes Throughout the 1980s and early 1990s progress in string theory largely consisted of working out the detailed rules of perturbation theory for the five known versions of the theory, which would allow theorists to arrive at actual solutions (figure 2). These perturbative rules were generalizations of the Feynman diagrams of QED and QCD in which the world lines of point moving strings. The study of world-sheet physics created a huge body of knowledge about 2D quantum field theory, but it did not offer much insight into the inner workings of quantum gravity. At best, string theory provided an especially consistent way to introduce a small distance scale and thereby regulate the divergences that had plagued the older attempts at quantizing gravity. Personally I found the whole enterprise dry, overly technical and, above all, disappointing. I felt that a quantum theory of gravity should profoundly affect our views of space-time, quantum mechanics, the origin of the universe, and the mysteries of black holes. But string theory was largely silent about all these matters. Then in 1993 all this began to change, and the catalyst was the awakening interest in Stephen Hawking's earlier speculations about black holes. The starting point for Hawking's speculations was the thermal behaviour of black holes, which built on earlier work by Jacob Bekenstein of the Hebrew University in Israel. Rather than the cold, dead objects that they were originally thought to be, black holes turned out to have a heat content and to glow like black bodies. Because they glow they lose energy and evaporate, and because they have a temperature and an energy, they also have an entropy. This entropy, S, is defined by the Bekenstein-Hawking equation: S = AkBc3/4h-barG, where A is the surface area of the horizon and kBis Boltzmann's constant. After realizing that black holes must evaporate by the emission of black-body radiation, Hawking raised an extremely profound question: what happens to all the detailed information that falls into a black hole? Once it falls through the horizon it cannot subsequently reappear on the outside without violating causality. That is the meaning of a horizon. But the black hole will eventually evaporate, leaving only photons, gravitons and other elementary information must ultimately be lost to our world. But one of the fundamental principles of quantum mechanics is that information is never lost, because the information in the initial state of a quantum system is permanently imprinted in the quantum state. Hawking's view was that conventional quantum mechanics must be violated during the formation and evaporation of the black hole. He rightly understood that if this is true, the rules of quantum mechanics must be drastically modified as the Planck scale is for unified theories, should have been obvious. But initially Hawking's idea generated little interest among high-energy theorists, apart from myself and Gerard 't Hooft at the University of Utrecht. We were convinced that by modifying the rules of quantum mechanics in the way advocated by Hawking, all hell would break loose, such as causing empty space to quickly heat up to stupendous temperatures and energy densities. We were sure that Hawking was wrong. By the early 1990s, however, the issue was becoming critical, especially to string theorists. String theory by its very definition is based on the conventional rules of quantum mechanics and if Hawking was right, the entire foundation of the theory would be destroyed. JS Comment: I tend to disagree with Lenny and t'Hooft that the unitarity of micro-quantum theory is an absolute. P.W. Anderson's More is different suggests otherwise. Think of the relation between special relativity and general relativity -- similarly with micro-quantum theory and MACRO-quantum theory of superfluids. 4D space-time is a non-dynamical absolute in special relativity. It acts on mass-energy without any direct reaction of mass-energy back on it. This is because the string tension is infinite in that limit. Special relativity is action without reaction. General relativity corrects that giving a finite value to string tension? How? Because general relativity emerges from a macro-quantum theory as shown in http://qedcorp.com/APS/EmergentGravity.pdf The finite string tension in Ed Witten's sense of alpha' is actually a quantum h effect added to G and c. & 14.6, micro-quantum theory is like special relativity because the quantum BIT pilot wave is a non-dynamical absolute. It acts on the IT extra variable without any direct reaction of IT back on its quantum BIT. That is, in Wheeler's terms, micro-quantum theory is IT FROM BIT In contrast, MACRO-quantum theory adds to that BIT FROM IT. Micro-quantum theory is linear and nonlocal in configuration space for entangled composite systems with unitary time evolution and a probability interpretation in Lenny's sense but with signal locality in a detente passion at a distance (A. Shimony) with retarded causality. In contrast, MACRO-quantum theory with superfluid signal generalized phase rigidity (e.g. string tension) is nonlinear (Landau-Ginzburg eq.) and local in ordinary (hyper) space with non-unitary time evolution and a complete breakdown of the Born probability interpretation. Also it allows signal nonlocality violating retarded causality. Micro-quantum theory still works for the normal fluid noisy component. Back to Lenny: Over the last decade the apparent clash between standard quantum principles and black-hole evaporation has been resolved, favouring, I should add, the views of 't Hooft and myself. The formation and evaporation of a black hole is similar to many other process in and chaotic spectrum of intermediate states. In the case of a black hole, the collisions are between the original protons, neutrons and electrons in a collapsing star. Roughly speaking a black hole is nothing but a very excited string with a total length that is proportional to the area of its horizon. JS Comment. Already in 1973 I published a paper in Herbert Frohlich's Collective Phenomena that the Regge string hadronic trajectories showed that the hadronic resonances were tiny black holes in Abdus Salam's strong short-range f-gravity with G* ~ 10^40 G(Newton). Spin ~ G*E^2/hc^5 + intercept G*/hc^5 ~ (String Tension)^-1 G*/hc^5 ~ (1 Gev)^-2 UNIVERSAL SLOPE (micro-geometrodynamics) The decay of the hadrons would be like Hawking radiation. Abdus Salam invited me to work with him at Trieste because of this paper. We now see that this idea I had before its time was essentially on the right path. Back to Lenny During the collision or collapse process, all the energy of the initial state goes into forming a single long, tangled string, and the entropy of the configuration is the logarithm of the number of configurations of a random-walking quantum string. The correspondence between string configurations and black-hole entropy was checked for all of the various kinds of charged and neutral black holes that occur in compactifications of string theory. In most of the cases the entropy of the string configuration could be estimated and it agreed with the Bekenstein-Hawking entropy to within a factor of order unity. But string theorists wanted to do better. The Bekenstein-Hawking formula for the entropy of a black hole is very precise: the entropy is one quarter of the horizon area, measured in Planck units, for every kind of black hole, be it static, rotating, charged or even higher-dimensional. Surely the universal factor of a quarter should be computable in string theory? The key to a precise calculation was obvious. Certain black holes called extremal black holes - which are the ground states of charged black holes that carry electric and magnetic charges - are especially tractable in a supersymmetric theory. The only problem was that in 1993 no-one knew how to build an extremal black hole out of the right type out of strings. This had to wait a couple of years for the discovery of entities called D-branes. Brane world In 1995 Joe Polchinski of the University of California in Santa Barbara electrified the string-theory community with a major discovery that has subsequently impacted every field of physics. As we have seen, T-duality is the strange symmetry that interchanges the Kaluza-Klein momenta and winding numbers of a closed string. But what happens to an open string? Obviously the idea of a winding number does not make sense for such a string. What actually happens to open strings under T-duality is that the free ends become fixed on surfaces called D-branes. D-branes come in various dimensions; 2D branes, for example, can also be called membranes. They have an energy or mass per unit surface area and are localized physical objects in their own right. In a sense they seem to be no less fundamental than the strings themselves. To an outsider, D-branes may seem to be arbitrary additions to the theory. They are not. Their existence is absolutely essential to the mathematical consistency of the theory. In addition to allowing T-duality to act on an open string in Type I string theory, they are necessary for implementing the deep dualities that link the five different kinds of string theory together. But from the point of view of black holes, the importance of D-branes is that you can build extremal black holes from them. In fact, just by placing a large number of D-branes at the same location you can build an extremal supersymmetric black hole. And because of the special properties of supersymmetric systems, the statistical entropy of that black hole can be precisely computed. The result, which was first derived by Andrew Strominger and Cumrun Vafa at Harvard in 1996, is that the entropy is equal to exactly one quarter of the horizon area in Planck units! This suggested that the microscopic degrees of freedom implied by the Bekenstein-Hawking entropy are the degrees of freedom describing strings, and was a major boost for the superstring community. At about the same time as D-branes were discovered, another very important development took place. As I mentioned, the coupling constant of string theory is not really a constant at all, and in many respects it is very similar to the compactification moduli. String theorists took a surprisingly long time to make the connection, but it turns out that 10D string theory is itself a Kaluza-Klein compactification of an 11D theory that became known as M-theory . M-theory appears to underlie all string theories. The five different versions of string theory are just different ways of compactifying its 11 dimensions. But M-theory is not itself a string theory. It has membranes but no strings, and the strings only appear when the 11th dimension is compactified. Furthermore, the momentum in the compact 11th direction (the Kaluza-Klein momentum) is identified as the number of D0-branes - i.e. zero-dimensional branes, or points - in a particular type of string theory. This connection between Kaluza-Klein momentum and D0-branes led to another breakthrough. In 1996 myself, Tom Banks and Steve Shenker (at Rutgers University), and Willy Fischler (at the University of Texas) realized that M-theory could be cast in a form no more complicated than the quantum mechanics of a system of is called Matrix theory, is an exact and complete quantum theory that describes the microscopic degrees of freedom of M-theory. As such it is the first precise formulation of a quantum theory of gravity. Duality Matrix theory was just one example of how D-branes can be used to formulate a theory of quantum gravity. Soon after its discovery, Juan Maldacena, who is now at the Institute for Advanced Study (IAS) in Princeton, came up with a new direction to explore. Ed Witten of the IAS and others had previously shown that D-branes have their own dynamics. But it turned out that the fluctuations and motions of a D-brane can be quantized in the form of a gauge theory that is restricted to the D-brane itself. The theory that lives on a coincident collection of D3-branes, for example, is a supersymmetric non-Abelian gauge theory. In other words, it is a supersymmetric version of QCD - the theory describing quarks and gluons. In a sense, string theory is returning to its roots as a pp35-38). Maldacena realized that in an appropriate limit the theory of D3-branes should be a complete description of string theory - not just on the branes, but in the entire geometry in which the branes are embedded. A gauge theory would therefore also be a description of quantum gravity in a particular background space-time. This space-time is called anti-de Sitter space, which, roughly speaking, is a universe inside a cavity. The walls of the cavity behave like reflecting surfaces so that nothing escapes it (figure 4). This duality between quantum field theory and gravity is an exact realization of what is called the holographic principle. This strange principle, formulated by 't Hooft and myself, grew from our debate with Hawking regarding the validity of quantum mechanics in the formation and evaporation of black holes. According to the holographic principle, everything that ever falls into a black hole can be described by degrees of freedom that reside in a thin layer just above the horizon. In other words, the full 3D world inside the horizon can be described by the 2D degrees of freedom on its surface. Even more generally, it should be possible to describe the physics of any region of space in terms of holographic degrees of freedom that reside on the boundary of that region. This leads to a drastic reduction of the number of degrees of freedom in a field theory, and most theorists found it very hard to swallow until Maldacena's work came along. Maldacena's duality replaces a gravitational theory in anti-de Sitter space by a field theory that lives on its boundary in a very precise way. In other words, the 3 + 1-dimensional boundary field theory is a holographic description of the interior of 4 + 1-dimensional anti-de Sitter space. The D-brane revolution has been very far reaching. Matrix theory and the Maldacena duality are both formulations of quantum gravity that conform to the standard rules of quantum mechanics, and should therefore lay to rest any further questions about black holes violating these rules. Googles of possibilities I would like to end by discussing the future of string theory, not as cosmology. The final evaluation of string theory will rest on its ability to explain the facts of nature, not on its own internal beauty and consistency. String theory is well into its fourth decade, but so convincing explanation of any cosmological observation. Many of the models that are based on specific methods of compactifying either 10D string theory or 11D M-theory have a good deal in common with the real world. They have bosons and fermions, for example, and gauge theories that are similar to those in the Standard Model. Furthermore, unlike any other theory, they inevitably include gravity. But the devil is in the details, and so far the details have eluded string theorists. It is, of course, possible that string theory is the wrong theory, but I believe that would be a very premature judgement and probably incorrect. The problem does not seem to be a lack of richness, but rather the opposite. String theory contains too many possibilities. For most physicists, the ideal physical theory is one that is unique and perfect, in that it determines all that can be determined and that it could not logically be any other way. In other words, it is not only a theory of everything but it is the only theory of everything. To the orthodox string theorist, the goal is to discover the one true consistent version of the theory and then to demonstrate that the solution manifests the known laws of nature, such as the Standard But the more we learn about string theory the more non-unique it seems to be. There are probably millions of Calabi-Yau spaces on which to compactify string theory. Each space has hundreds of moduli and hundreds of subspaces on which branes can be wrapped, fluxes imposed upon and so on. A conservative estimate of the number of distinct vacua of the theory is in the googles - that is, more than 10100. The space of possibilities is called the Landscape, and it is huge. To mix metaphors, it is a stupendous haystack that contains googles of straws and only one needle. Worse still, the theory itself gives us no hint about how to pick among the possibilities (see The string-theory landscape ). This enormous variety may, however, be exactly what cosmology is looking for. A common theme among cosmologists is that the observed universe may merely be a minuscule part of a vastly bigger universe that contains many local environments, or what Alan Guth at MIT calls pocket universes . According to this view, so many pocket universes formed during the early inflationary epoch - each of which with its own vacuum structure - that the entire landscape of possibilities is represented. The reasons for this view are not just idle speculation but are rooted in the many accidental fine-tunings that are necessary for a universe that supports life. Thus it may be that the enormous number of possible vacuum the doctor ordered for cosmology. Further information T-duality In a single compact dimension there are two kinds of quantum numbers: momentum in the compact direction and the winding number. Both of these are quantized in integer multiples of a basic unit, and each has a certain energy associated with it. In the case of momentum, for example, the energy is just the kinetic energy of units of compact momentum is equal to n/R, where R is the circumference of the compact direction. Note that the energy grows as the size of the compact space gets smaller. On the other hand, the winding modes also have energy, which is the potential energy needed to stretch the string around the compact co-ordinate. If we call the winding number m, then the winding energy is equal to mR. In this case the energy decreases as the size of the compact direction decreases. The surprising thing is that the spectrum of energies is unchanged if we change the compactification radius from R to 1/R, and at the same time interchange the Kaluza-Klein momentum and winding modes. In other words, just by looking at the spectrum of energies you could never tell the difference between a theory that is compactified on a space of size R or on one of size 1/R. As you try to make the compactification scale smaller than the natural string scale - i.e. the size of a vibrating string - the theory begins to behave as if the compactification radius was getting bigger. Physically, the smallest compactification value of R is the string scale. But from a mathematical viewpoint, two different spaces - one large, the other small - are completely equivalent. This equivalence is called T-duality. Author Leonard Susskind is in the Department of Physics, Stanford susskind@stanford.edu Further reading J Maldacena 1999 The large N limit of superconformal field theories and supergravity Int. J. Theor. Phys. 38 1113-1133 J Polchinski 1995 Dirichlet-branes and Ramond-Ramond charges Phys. Rev. Lett. 75 4724-4727 J Polchinski 1998 String Theory (volume 2): Superstring Theory and Beyond (Cambridge University Press) J H Schwarz et al. 1981 Superstring Theory (volume 1): Introduction (Cambridge University Press) A Strominger and C Vafa 1996 Microscopic origin of the Bekenstein-Hawking entropy Phys. Lett. B 379 99-104 The official string theory website: superstringtheory.com/ Note above my new Blackett formula for charge Q in the Calabi-Yau space, i.e. Q = G*^1/2h/cR R is a compactification scale modulus in the extra-dimensional generalized Kaluza-Klein space. G* = e^phiG(Newton) See Saul-Paul Sirag's Nature paper on Blackett effect in astrophysics. ==== My Commentary on Lenny's lecture in the context of my dialogues with Paul Zielinski and Harold Puthoff > on Metric Engineering: Making Star Trek Real from my third book > in the Space-Time and Beyond series now being written with full > color illustrations and cartoons. I examine Lenny's ideas to establish where mainstream cutting edge > physics is these days in 2004 as a proper context to evaluate Hal's and > my own fringe ideas relating to the UFO controversy. For the record I > suggested the problem for Lenny's first published physics paper at > Cornell in 1963 on the the problem of lack of a Hermitian operator for > both time and wave phase in quantum theory. I had been working on that > problem > with George Parrent Jr, a student of Emil Wolf's, at Tech/Ops associated > with Mitre on Route 2 near Boston. I also brought Johnny Glogower to > Cornell with Phil Morrison's help. Johnny was part of Walter Breen's > Super Kids group from Columbia University in a project allegedly > funded by Eugene McDermott a co-founder of Texas Instruments. We were > all rebels . Lenny was a high school dropout. Johnny was a Quiz Kid, > Westinghouse Finalist who flunked out of Brandeis. Excerpts from: > Physics World > Superstrings by Leonard Susskind String theory is either a theory of everything - which > automatically unites gravity with the other three forces in > nature - or a theory of nothing, but finding the correct form > of the theory is like searching for a needle in a stupendous > haystack rather like trying to summarize the history of the world in 10 pages. > It is just too large a subject, with too many lines of thought and too > many threads to weave together. In the 34 years since it began, > string theory has developed into an enormous body of knowledge > that touches on every aspect of theoretical physics. > String theory is a theory of composite hadrons, > quantum theory of gravity, and a framework for > understanding black holes. It is also a powerful > technical tool for taming strongly interacting > quantum field theories and, perhaps, a basis for > formulating a fundamental theory of the > universe. It even touches on problems in > condensed-matter physics, and has also > provided a whole new world of mathematical > problems and tools. ... String theory is considered to be a branch of high-energy or > the 1950s, 1960s or 1970s would be surprised to read a recent > string-theory paper and find not a single Feynman diagram, > literature. What the reader would find are black-hole metrics, > Einstein equations, Kaluza-Klein theories and plenty of fancy > geometry and topology. The energy scales of interest are not MeV, > GeV or even TeV, but energies at the Planck scale - the scale at > which the classical concepts of space and time break down. > The Planck energy is equal to h-bar5/G, where h-bar is Planck's > constant divided by 2!, c is the speed of light and G is the > gravitational constant, and it corresponds to masses that are some > 19 orders of magnitude larger than the proton mass. There is actually a typo editor's error here, not Lenny's. The Planck energy is hbar c/Lp = hbar c/(hbarG/c^3)^1/2 = (hbar c^5/G)^1/2 Back to Lenny's talk: > This is the energy of the universe when it was just 10-43s old, and it will > understand physics at the Planck scale we need a quantum theory > of gravity. In the days when my career was beginning, a typical colloquium on > high-energy physics would often begin by stating that there are four > forces in nature - electromagnetic, weak, strong and gravitational - > followed by a statement that the gravitational force is much too > from now on. That has all changed. Today the other three forces are described by the gauge theories of > quantum chromodynamics (QCD) and quantum electrodynamics > physics. These quantum field theories describe the fundamental > quanta: the photon for the electromagnetic force, the W and Z > bosons for the weak force, and the gluon for the strong force. In > the string-theory community, however, the electromagnetic, strong > and weak forces are generally considered to be manifestations of > certain compactifications of space from 10 or 11 dimensions to > the four familiar dimensions of space-time. ... Why quantum gravity? > charge, colour, parity and hypercharge - to be truly elementary. > scale. Protons and mesons reveal their parts at the modestly small > distance of about 10-15 m, but quarks, leptons and photons hide > their structure much more effectively. Indeed, no experiment has > ever seen direct evidence of size or structure for any of these JS Comment: This point-like structure may be from a huge space-warp effect > depending on the momentum transfer from scattering probe to target > from an exotic vacuum dark matter core of the spatially extended > lepto-quarks where the effective gravity coupling at short range is > 40 powers of ten greater than Newton's. Back to Lenny: > ... coupling constants are not really constants at all - they vary with > energy. If the known couplings are extrapolated they all intersect > the predictions of the unified theory at roughly the same scale. JS Comment: This GUT scale is ~ Lp/(alpha) where > Lp^2 ~ hG/c^3 and alpha ~ e^2/hc ~ 1/137. Back to Lenny: > Moreover, this scale is close to the Planck scale. The implication of > this was clear: the scale of the internal machinery of elementary > appears in the definition of the Planck energy, to many of us this > inevitably meant that gravitation must play an essential role in The earliest attempts to reconcile gravity and quantum mechanics - > notably by Richard Feynman, Paul Dirac and Bryce DeWitt, who is > now at the University of Texas at Austin - were based on trying to > fit Einstein's general theory of relativity into a quantum field theory > like the hugely successful QED. The goal was to find a set of rules > for calculating scattering amplitudes in which the photons of QED > are replaced by the quanta of the gravitational field: gravitons. But > gravitational forces become increasingly strong as the energy of the > participating quanta increases, and the theory proved to be wildly > simply gave rise to far too many degrees of freedom at short > distances. In a sense the failure of this quantum gravity theory was a good > sign. The theory itself gave no insight into the internal machinery of > forces of nature. At best it was more of the same: an effective (but > not very) description of gravitation with no deeper insight into the > nonsense. Strings as hadrons > We all know that science is full of surprising twists, but the > discovery of string theory was particularly serendipitous. The theory > grew out of attempts in the 1960s to describe the interactions of > neutron. This was a problem that had nothing to do with gravity. > Gabriele Veneziano, now at CERN, and others had written down a > simple mathematical expression for scattering amplitudes that had > certain properties that were fashionable at that time. It was soon > discovered by Yoichiro Nambu of the University of Chicago and > myself, and in a slightly different form by Holger Bech Nielsen at > the Niels Bohr Institute, that these amplitudes were the solution of > a definite physical system that consists of extended 1D elastic > strings. > ... Fermionic versions of string theory were soon discovered and, > moreover, they turned out to have a surprising symmetry called > supersymmetry that is now totally pervasive in high-energy physics. > In supersymmetric theories all bosons have a fermionic > superpartner and vice versa. ... Another apparently serious problem with the string theory of > hadrons concerned dimensions. Although the original assumptions in > string theory were simple enough, the mathematics proved > internally inconsistent, at least if the number of dimensions of > space-time was four. The source of this problem was quite deep, > but, strangely, if space-time has 10 dimensions it contrives to > cancel out. ... A mathematical string can vibrate in many patterns, > angular momentum ( spin-two ). There are certainly spin-two > hadrons, but none that have anything like zero mass. Despite all > made massive. > ... > A massless spin-two field might not be good for hadronic physics, but > it is just what was needed for quantum gravity, albeit in 10D. This is > because just as the photon is the quantum of the electromagnetic > field, the graviton is the quantum of the gravitational field. But the > gravitational field is a symmetric tensor rather than a vector, and > this means the graviton is spin-two, rather than spin-one like the > photon. This difference in spin is the principal reason why early > attempts to quantize gravity based on QED did not work. A theory of everything, > ... either all matter is strings, or string > theory is wrong. This is one of the most exciting features of the > theory. But what about the problem of dimensions? Here again, a sow's ear > was turned into a silk purse. The basic idea goes back to Theodor > Kaluza in 1919, who tried to unify Einstein's gravitational theory > with electrodynamics by introducing a compact space-like fifth > dimension. Kaluza discovered the beautiful fact that the extra > components of the gravitational field tensor in 5 dimensions > behaved exactly like the electromagnetic field plus one additional > scalar field. Somewhat later, in 1938, Oskar Klein and then > Wolfgang Pauli generalized Kaluza's work so that the single compact > dimension was replaced by a 2D space. If the 2D space is the > surface of a sphere then a remarkable thing happens when Kaluza's > procedure is followed. Instead of electrodynamics, Klein and Pauli > discovered the first non-Abelian gauge theory, which was later > rediscovered by Chen Ning Yang and Robert Mills. This is exactly the > same class of theories that is so successful in describing the strong > and electromagnetic interactions in the Standard Model. JS Comment: A 2D Kaluza-Klein space has group structure > of a 2D sphere embedded in flat 3D space with 3 rotation charge > generators, > i.e. SU(2) group for the weak force with 3 charges. > In general we have N^2 - 1 charges for the SU(N) internal symmetry > gauge forcegroup > at a fixed space-time point where the minimal coupling local > independence of phase > rotations introduces the compensating spin 1 gauge force fields to > restore the broken > global symmetry. This force generator idea is re-expressed in the > geometrodynamics of hyperspace. Here's Lenny: > that appears to be standing still in our usual 3D > space have velocity or momentum components > in the compact dimensions? The answer is yes, > and the corresponding components of > momentum define new conserved quantities. What is > more, these quantities are quantized in discrete units. In short, they > are charges similar to electric charge, isospin and all the other > the problem of dimensions in string theory is obvious: six of the 10 > dimensions should be wrapped up into some very small compact > space, and the corresponding quantized components of momenta > determines their quantum numbers. Life in six dimensions > Much of the development of string theory is therefore concerned > with 6D spaces. These spaces, which can be thought of as > generalized Kaluza-Klein compactification spaces, were originally > studied by mathematicians and are known as Calabi-Yau spaces. > They are tremendously complicated and are not completely > understood. But in the process of studying how strings move on > them, physicists have created an unexpected revolution in the study > of Calabi-Yau spaces. JS Comment: Recall that the classical gravity radius is proportional to > M and the > quantum radius is proportional to 1/M. That is Rg = GM/c^2 Rq = h/Mc Therefore RgRq = Gh/c^3 = Lp^2 = 1 Bekenstein BIT. We have a germ of a duality between black holes and quantum momenta Rg = Lp^2/Rq Note also the Blackett empirical relation e = G*^1/2M where for an electron G* ~ 10^40G The quantum momenta p in the compactified extra-dimensions are > charges Q (sources of the spin 1 gauge forces) where by the Blackett > relation Q/G*^1/2 = M = h/cR Q = G*^1/2h/cR R is a compactification scale. G* = e^phiG(Newton) Back to Lenny > In particular, it was discovered that a compactification radius of size > R is completely equivalent to a space with size 1/R from the point > of view of string theory. This connection, which is known as > T-duality, has a mathematically profound generalization called > mirror symmetry, which states that there is an equivalence between > small and large spaces ... . Mirror symmetry of > Calabi-Yau spaces - which are not only of different sizes but have > completely different topologies - was completely unsuspected > before physicists began studying quantum strings moving on them. > I wish it was possible to draw a Calabi-Yau space but they are > tremendously complicated. They are six-dimensional, which is three > more than I can visualize, and they have very complicated > topologies, including holes, tunnels and handles. Furthermore, there > are thousands of them, each with a different topology. And even > when their topology is fixed there are hundreds of parameters > called moduli that determine the shape and size of the various > dimensions. Indeed, it is the complexity of Calabi-Yau geometry > that makes string theory so intimidating to an outsider. However, > we can abstract a few useful things from the mathematics, one of > them being the idea of moduli. The simplest example of a modulus is just the compactification > radius, R, when there is only a single compact dimension. In more > complicated cases, the moduli determine the sizes and shapes of > the various features of the geometry. The moduli are not constants > but depend on the geometry of the space itself, in the same way > that the radius of the universe changes with time in a manner that > is controlled by dynamical equations of motion. Since the compact > dimensions are too small to see, the moduli can simply be thought > of as fields in space that determine the local conditions. Electric and > magnetic fields are examples of such fields but the moduli are even > simpler: they are scalar fields (i.e. they have only one component), > rather than vector fields. String theory always has lots of > scalar-field moduli and these can potentially play important roles in All of this raises an interesting question: what determines the > compactification moduli in the real world of experience? Is there > some principle that selects a special value of the moduli of a > particular Calabi-Yau space and therefore determines the > coupling constants of the forces, and so on? The answer seems to > be no: all values of the moduli apparently give rise to > mathematically consistent theories. Whether or not this is a good > thing, it is certainly surprising. Ordinarily we might expect the vacuum or ground state of the world > to be the state of lowest energy. Furthermore, in the absence of > very special symmetries, the energy of a region of space will > depend non-trivially on the values of the fields in that region. > Finding the true vacuum is then merely an exercise in computing > the energy for a given field configuration and minimizing it. This is, > to be sure, a difficult task, but it is possible in principle. In string > theory, however, we know from the beginning that the potential > energy stored in a given configuration has no dependence on the > moduli fields. The reason that the field potential is exactly zero for every value of > the moduli is that string theory is supersymmetric. Supersymmetry > has both desirable and undesirable consequences. Its most obvious > drawback is the requirement that for every fermion there is a boson > with exactly the same mass, which is clearly not a property of our > world. A more subtle difficulty involves the aforementioned fact that the > vacuum energy is independent of the moduli. As well as telling us > that we cannot determine the moduli by minimizing the energy, > supersymmetry also tells us that the quanta of the moduli fields are > exactly massless. No such massless fields are known in nature and, > furthermore, such fields are very dangerous. Indeed, massless > moduli would probably lead to long-range forces that would > compete with gravity and violate the equivalence principle - the > cornerstone of general relativity - at an observable level. On the plus side, the vanishing vacuum energy that is implied by > supersymmetry ensures that the cosmological constant vanishes. If > it were not for supersymmetry, the vacuum would have a huge > zero-point energy density that would make the radius of curvature > of space-time not much bigger than the Planck scale - a most > undesirable situation. JS Comment: I have a different much simpler explanation for the > smallness of the cosmological constant in > http://qedcorp.com/APS/EmergentGravity.pdf Also the observational fact of dark energy with FRW Omega ~ 0.7 > means that the cosmological constant is not exactly zero, which is > a problem for the physics Lenny is talking about. Back to Lenny: > Supersymmetry also stabilizes the vacuum > against various hypothetical instabilities, and it allows us to make > exact mathematical conclusions. Indeed, T-duality and mirror > symmetry are examples of those exact consequences. Black holes > Throughout the 1980s and early 1990s progress > in string theory largely consisted of working out > the detailed rules of perturbation theory for the > five known versions of the theory, which would > allow theorists to arrive at actual solutions > (figure 2). These perturbative rules were > generalizations of the Feynman diagrams of QED > and QCD in which the world lines of point > moving strings. The study of world-sheet physics created a huge > body of knowledge about 2D quantum field theory, but it did not > offer much insight into the inner workings of quantum gravity. At > best, string theory provided an especially consistent way to > introduce a small distance scale and thereby regulate the > divergences that had plagued the older attempts at quantizing > gravity. Personally I found the whole enterprise dry, overly technical and, > above all, disappointing. I felt that a quantum theory of gravity > should profoundly affect our views of space-time, quantum > mechanics, the origin of the universe, and the mysteries of black > holes. But string theory was largely silent about all these matters. > Then in 1993 all this began to change, and the catalyst was the > awakening interest in Stephen Hawking's earlier speculations about > black holes. The starting point for Hawking's speculations was the thermal > behaviour of black holes, which built on earlier work by Jacob > Bekenstein of the Hebrew University in Israel. Rather than the cold, > dead objects that they were originally thought to be, black holes > turned out to have a heat content and to glow like black bodies. > Because they glow they lose energy and evaporate, and because > they have a temperature and an energy, they also have an entropy. > This entropy, S, is defined by the Bekenstein-Hawking equation: S > = AkBc3/4h-barG, where A is the surface area of the horizon and > kBis Boltzmann's constant. After realizing that black holes must evaporate by the emission of > black-body radiation, Hawking raised an extremely profound > question: what happens to all the detailed information that falls into > a black hole? Once it falls through the horizon it cannot > subsequently reappear on the outside without violating causality. > That is the meaning of a horizon. But the black hole will eventually > evaporate, leaving only photons, gravitons and other elementary > information must ultimately be lost to our world. But one of the > fundamental principles of quantum mechanics is that information is > never lost, because the information in the initial state of a quantum > system is permanently imprinted in the quantum state. Hawking's view was that conventional quantum mechanics must be > violated during the formation and evaporation of the black hole. He > rightly understood that if this is true, the rules of quantum > mechanics must be drastically modified as the Planck scale is > for unified theories, should have been obvious. But initially > Hawking's idea generated little interest among high-energy > theorists, apart from myself and Gerard 't Hooft at the University of > Utrecht. We were convinced that by modifying the rules of quantum > mechanics in the way advocated by Hawking, all hell would break > loose, such as causing empty space to quickly heat up to > stupendous temperatures and energy densities. We were sure that > Hawking was wrong. By the early 1990s, however, the issue was > becoming critical, especially to string theorists. String theory by its > very definition is based on the conventional rules of quantum > mechanics and if Hawking was right, the entire foundation of the > theory would be destroyed. JS Comment: I tend to disagree with Lenny and t'Hooft that the unitarity > of micro-quantum theory is an absolute. P.W. Anderson's More is > different suggests otherwise. Think of the relation between special > relativity and general relativity -- similarly with micro-quantum theory > and MACRO-quantum theory of superfluids. 4D space-time is a non-dynamical absolute in special relativity. It acts > on mass-energy without any direct reaction of mass-energy back on it. > This is because the string tension is infinite in that limit. Special > relativity is action without reaction. General relativity corrects that > giving a finite value to string tension? How? Because general relativity > emerges from a macro-quantum theory as shown in > http://qedcorp.com/APS/EmergentGravity.pdf The finite string tension in > Ed Witten's sense of alpha' is actually a quantum h effect added to G > and c. & 14.6, micro-quantum theory is like special relativity because the > quantum BIT pilot wave is a non-dynamical absolute. It acts on the IT > extra variable without any direct reaction of IT back on its quantum BIT. That is, in Wheeler's terms, micro-quantum theory is IT FROM BIT In contrast, MACRO-quantum theory adds to that BIT FROM IT. Micro-quantum theory is linear and nonlocal in configuration space for > entangled composite systems with unitary time evolution and a > probability interpretation in Lenny's sense but with signal locality > in a detente passion at a distance (A. Shimony) with retarded causality. In contrast, MACRO-quantum theory with superfluid signal generalized > phase rigidity (e.g. string tension) is nonlinear (Landau-Ginzburg eq.) > and local in ordinary (hyper) space with non-unitary time evolution and > a complete breakdown of the Born probability interpretation. Also it > allows signal nonlocality violating retarded causality. Micro-quantum > theory still works for the normal fluid noisy component. Back to Lenny: > Over the last decade the apparent clash between standard quantum > principles and black-hole evaporation has been resolved, favouring, > I should add, the views of 't Hooft and myself. The formation and > evaporation of a black hole is similar to many other process in > and chaotic spectrum of intermediate states. In the case of a black > hole, the collisions are between the original protons, neutrons and > electrons in a collapsing star. Roughly speaking a black hole is > nothing but a very excited string with a total length that is > proportional to the area of its horizon. JS Comment. Already in 1973 I published a paper in Herbert Frohlich's > Collective Phenomena that the Regge string hadronic trajectories > showed that the hadronic resonances were tiny black holes in > Abdus Salam's strong short-range f-gravity with G* ~ 10^40 G(Newton). Spin ~ G*E^2/hc^5 + intercept G*/hc^5 ~ (String Tension)^-1 G*/hc^5 ~ (1 Gev)^-2 UNIVERSAL SLOPE (micro-geometrodynamics) The decay of the hadrons would be like > Hawking radiation. Abdus Salam invited me to work with him at > Trieste because of this paper. We now see that this idea I had before > its time > was essentially on the right path. Back to Lenny > During the collision or > collapse process, all the energy of the initial state goes into forming > a single long, tangled string, and the entropy of the configuration is > the logarithm of the number of configurations of a random-walking > quantum string. The correspondence between string configurations and black-hole > entropy was checked for all of the various kinds of charged and > neutral black holes that occur in compactifications of string theory. > In most of the cases the entropy of the string configuration could be > estimated and it agreed with the Bekenstein-Hawking entropy to > within a factor of order unity. But string theorists wanted to do better. The Bekenstein-Hawking > formula for the entropy of a black hole is very precise: the entropy > is one quarter of the horizon area, measured in Planck units, for > every kind of black hole, be it static, rotating, charged or even > higher-dimensional. Surely the universal factor of a quarter should > be computable in string theory? The key to a precise calculation was > obvious. Certain black holes called extremal black holes - which are > the ground states of charged black holes that carry electric and > magnetic charges - are especially tractable in a supersymmetric > theory. The only problem was that in 1993 no-one knew how to > build an extremal black hole out of the right type out of strings. > This had to wait a couple of years for the discovery of entities > called D-branes. Brane world > In 1995 Joe Polchinski of the University of California in Santa > Barbara electrified the string-theory community with a major > discovery that has subsequently impacted every field of physics. As > we have seen, T-duality is the strange symmetry that interchanges > the Kaluza-Klein momenta and winding numbers of a closed string. > But what happens to an open string? Obviously the > idea of a winding number does not make sense for such a string. > What actually happens to open strings under T-duality is that the > free ends become fixed on surfaces called D-branes. > D-branes come in various dimensions; 2D > branes, for example, can also be called > membranes. They have an energy or > mass per unit surface area and are localized > physical objects in their own right. In a sense > they seem to be no less fundamental than the > strings themselves. To an outsider, D-branes may seem to be > arbitrary additions to the theory. They are not. Their existence is > absolutely essential to the mathematical consistency of the theory. > In addition to allowing T-duality to act on an open string in Type I > string theory, they are necessary for implementing the deep > dualities that link the five different kinds of string theory together. > But from the point of view of black holes, the importance of > D-branes is that you can build extremal black holes from them. In > fact, just by placing a large number of D-branes at the same > location you can build an extremal supersymmetric black hole. And > because of the special properties of supersymmetric systems, the > statistical entropy of that black hole can be precisely computed. The > result, which was first derived by Andrew Strominger and Cumrun > Vafa at Harvard in 1996, is that the entropy is equal to exactly one > quarter of the horizon area in Planck units! This suggested that the > microscopic degrees of freedom implied by the Bekenstein-Hawking > entropy are the degrees of freedom describing strings, and was a > major boost for the superstring community. At about the same time as D-branes were discovered, another very > important development took place. As I mentioned, the coupling > constant of string theory is not really a constant at all, and in many > respects it is very similar to the compactification moduli. String > theorists took a surprisingly long time to make the connection, but > it turns out that 10D string theory is itself a Kaluza-Klein > compactification of an 11D theory that became known as > M-theory . M-theory appears to underlie all string theories. The five > different versions of string theory are just different ways of > compactifying its 11 dimensions. But M-theory is not itself a string > theory. It has membranes but no strings, and the strings only > appear when the 11th dimension is compactified. Furthermore, the > momentum in the compact 11th direction (the Kaluza-Klein > momentum) is identified as the number of D0-branes - i.e. > zero-dimensional branes, or points - in a particular type of string > theory. This connection between Kaluza-Klein momentum and D0-branes > led to another breakthrough. In 1996 myself, Tom Banks and Steve > Shenker (at Rutgers University), and Willy Fischler (at the University > of Texas) realized that M-theory could be cast in a form no more > complicated than the quantum mechanics of a system of > is called Matrix theory, is an exact and complete quantum theory > that describes the microscopic degrees of freedom of M-theory. As > such it is the first precise formulation of a quantum theory of > gravity. Duality > Matrix theory was just one example of how D-branes can be used to > formulate a theory of quantum gravity. Soon after its discovery, > Juan Maldacena, who is now at the Institute for Advanced Study > (IAS) in Princeton, came up with a new direction to explore. Ed > Witten of the IAS and others had previously shown that D-branes > have their own dynamics. But it turned out that the fluctuations and > motions of a D-brane can be quantized in the form of a gauge > theory that is restricted to the D-brane itself. The theory that lives > on a coincident collection of D3-branes, for example, is a > supersymmetric non-Abelian gauge theory. In other words, it is a > supersymmetric version of QCD - the theory describing quarks and > gluons. In a sense, string theory is returning to its roots as a > pp35-38). Maldacena realized that in an appropriate limit > the theory of D3-branes should be a complete > description of string theory - not just on the > branes, but in the entire geometry in which the > branes are embedded. A gauge theory would > therefore also be a description of quantum > gravity in a particular background space-time. > This space-time is called anti-de Sitter space, > which, roughly speaking, is a universe inside a > cavity. The walls of the cavity behave like > reflecting surfaces so that nothing escapes it (figure 4). > This duality between quantum field theory and gravity is an exact > realization of what is called the holographic principle. This strange > principle, formulated by 't Hooft and myself, grew from our debate > with Hawking regarding the validity of quantum mechanics in the > formation and evaporation of black holes. According to the holographic principle, everything that ever falls into > a black hole can be described by degrees of freedom that reside in > a thin layer just above the horizon. In other words, the full 3D world > inside the horizon can be described by the 2D degrees of freedom > on its surface. Even more generally, it should be possible to > describe the physics of any region of space in terms of holographic > degrees of freedom that reside on the boundary of that region. This > leads to a drastic reduction of the number of degrees of freedom in > a field theory, and most theorists found it very hard to swallow until > Maldacena's work came along. Maldacena's duality replaces a > gravitational theory in anti-de Sitter space by a field theory that > lives on its boundary in a very precise way. In other words, the 3 + > 1-dimensional boundary field theory is a holographic description of > the interior of 4 + 1-dimensional anti-de Sitter space. The D-brane revolution has been very far reaching. Matrix theory > and the Maldacena duality are both formulations of quantum gravity > that conform to the standard rules of quantum mechanics, and > should therefore lay to rest any further questions about black holes > violating these rules. Googles of possibilities > I would like to end by discussing the future of string theory, not as > cosmology. The final evaluation of string theory will rest on its > ability to explain the facts of nature, not on its own internal beauty > and consistency. String theory is well into its fourth decade, but so > convincing explanation of any cosmological observation. Many of the > models that are based on specific methods of compactifying either > 10D string theory or 11D M-theory have a good deal in common > with the real world. They have bosons and fermions, for example, > and gauge theories that are similar to those in the Standard Model. > Furthermore, unlike any other theory, they inevitably include > gravity. But the devil is in the details, and so far the details have > eluded string theorists. It is, of course, possible that string theory is the wrong theory, but I > believe that would be a very premature judgement and probably > incorrect. The problem does not seem to be a lack of richness, but > rather the opposite. String theory contains too many possibilities. > For most physicists, the ideal physical theory is one that is unique > and perfect, in that it determines all that can be determined and > that it could not logically be any other way. In other words, it is not > only a theory of everything but it is the only theory of everything. > To the orthodox string theorist, the goal is to discover the one true > consistent version of the theory and then to demonstrate that the > solution manifests the known laws of nature, such as the Standard > But the more we learn about string theory the more non-unique it > seems to be. There are probably millions of Calabi-Yau spaces on > which to compactify string theory. Each space has hundreds of > moduli and hundreds of subspaces on which branes can be > wrapped, fluxes imposed upon and so on. A conservative estimate > of the number of distinct vacua of the theory is in the googles - that > is, more than 10100. The space of possibilities is called the > Landscape, and it is huge. To mix metaphors, it is a stupendous > haystack that contains googles of straws and only one needle. > Worse still, the theory itself gives us no hint about how to pick > among the possibilities (see The string-theory landscape ). This enormous variety may, however, be exactly what cosmology is > looking for. A common theme among cosmologists is that the > observed universe may merely be a minuscule part of a vastly > bigger universe that contains many local environments, or what > Alan Guth at MIT calls pocket universes . According to this view, so > many pocket universes formed during the early inflationary epoch - > each of which with its own vacuum structure - that the entire > landscape of possibilities is represented. The reasons for this view > are not just idle speculation but are rooted in the many accidental > fine-tunings that are necessary for a universe that supports life. > Thus it may be that the enormous number of possible vacuum > the doctor ordered for cosmology. Further information > T-duality > In a single compact dimension there are two kinds of quantum > numbers: momentum in the compact direction and the winding > number. Both of these are quantized in integer multiples of a basic > unit, and each has a certain energy associated with it. In the case > of momentum, for example, the energy is just the kinetic energy of > units of compact momentum is equal to n/R, where R is the > circumference of the compact direction. Note that the energy grows > as the size of the compact space gets smaller. On the other hand, > the winding modes also have energy, which is the potential energy > needed to stretch the string around the compact co-ordinate. If we > call the winding number m, then the winding energy is equal to mR. > In this case the energy decreases as the size of the compact > direction decreases. The surprising thing is that the spectrum of energies is unchanged if > we change the compactification radius from R to 1/R, and at the > same time interchange the Kaluza-Klein momentum and winding > modes. In other words, just by looking at the spectrum of energies > you could never tell the difference between a theory that is > compactified on a space of size R or on one of size 1/R. As you try > to make the compactification scale smaller than the natural string > scale - i.e. the size of a vibrating string - the theory begins to > behave as if the compactification radius was getting bigger. > Physically, the smallest compactification value of R is the string > scale. But from a mathematical viewpoint, two different spaces - > one large, the other small - are completely equivalent. This > equivalence is called T-duality. > Author > Leonard Susskind is in the Department of Physics, Stanford > susskind@stanford.edu > Further reading > J Maldacena 1999 The large N limit of superconformal field theories > and supergravity Int. J. Theor. Phys. 38 1113-1133 > J Polchinski 1995 Dirichlet-branes and Ramond-Ramond charges > Phys. Rev. Lett. 75 4724-4727 > J Polchinski 1998 String Theory (volume 2): Superstring Theory and > Beyond (Cambridge University Press) > J H Schwarz et al. 1981 Superstring Theory (volume 1): Introduction > (Cambridge University Press) > A Strominger and C Vafa 1996 Microscopic origin of the > Bekenstein-Hawking entropy Phys. Lett. B 379 99-104 > The official string theory website: superstringtheory.com/ Note above my new Blackett formula for charge Q in the Calabi-Yau > space, i.e. Q = G*^1/2h/cR R is a compactification scale modulus in the extra-dimensional > generalized Kaluza-Klein space. G* = e^phiG(Newton) See Saul-Paul Sirag's Nature paper on Blackett effect in astrophysics. wow... - Arthur ==== Is there any free software like Maple for linux? thank you, /lucas ==== > Is there any free software like Maple for linux? thank you, /lucas Maxima and Axiom (http://www.sciface.com/download.shtml) are free software. http://www.sciface.com/download.shtml They are all fine! Raymond ==== Is there any free software like Maple for linux? Take a look perhaps at http://maxima.sourceforge.net/ ==== > [snip] >Surely if every transaction takes a finite time there is no end. > simple paradoxes of Zeno! Believe it or not, an infinite series of positive terms (times) can have a > finite sum. For example, 1/2 sec + 1/4 sec + 1/8 sec +...+ 1/2^N sec +... = 1 sec. Why do some people leap to conclusions? I knew about Zeno while you were learning how to count. I thought it was obvious that I meant a finite _constant_ time (the same time for each transaction - why should they get faster?) Gib ==== [snip] >Surely if every transaction takes a finite time there is no end. simple paradoxes of Zeno! Believe it or not, an infinite series of positive terms (times) can > have a finite sum. For example, 1/2 sec + 1/4 sec + 1/8 sec +...+ 1/2^N sec +... = 1 sec. Why do some people leap to conclusions? way in this case that I could reasonably have guessed that you actually had intended something else.) > I knew about Zeno while you were learning how to count. Glad to know that. > I thought it was obvious that I meant a finite _constant_ time (the same > time for each transaction It certainly wasn't obvious to me. > - why should they get faster?) Response 0: Why shouldn't they? Response 1: So that all transactions can be completed in a finite time. David ==== > If it is not in at the end, and being not in is true of every ball, > which (allegedly large number of) balls are in the bucket at the end? Only the finitely labeled balls are absent. The infinite number of balls remaining all have infinite labels. You got a problem with that ? Lew Mammel, Jr. ==== If it is not in at the end, and being not in is true of every ball, > which (allegedly large number of) balls are in the bucket at the end? Only the finitely labeled balls are absent. The infinite number of > balls remaining all have infinite labels. You got a problem with that ? Lew Mammel, Jr. Since initially all balls have finite labels and no new labels are introduced, where do those infinite labels come from? ==== > Only the finitely labeled balls are absent. The infinite number of > balls remaining all have infinite labels. You got a problem with that ? Yes, as there is no natural number n such that 10*n is infinite. At each stage, n is finite, and therefore only label with finite values are ever created. There is no statement in the procedure that says, at noon, give them all infinite labels . Since all label management happens BEFORE noon, n is always finite. Jonathan Hoyle Gene Codes Corporation ==== > A transaction consists of adding ten balls to a bucket and removing 1. > (Obviously a transaction is a net increase of nine balls.) Assume that > infinitely many transactions somehow occur. Some people here think > there will be no balls in the bucket afterward! There are some people who think that there will only be zero balls > left if we label the balls in a certain way, otherwise there will be > more than zero! Yes, people who are even more stupid than you are. ==== A transaction consists of adding ten balls to a bucket and removing 1. > (Obviously a transaction is a net increase of nine balls.) Assume that > infinitely many transactions somehow occur. Some people here think > there will be no balls in the bucket afterward! There are some people who think that there will only be zero balls > left if we label the balls in a certain way, otherwise there will be > more than zero! Yes, people who are even more stupid than you are. Too bad that doesn't exclude yourself. ==== In a few months I will be teaching a course to undergraduate math majors called Mathematical Structures . The catalog description is: A rigorous study of the mathematical structures which form the foundation of higher mathematics. Set theory, logic, formal development of the number systems from the natural numbers through the complex numbers, basic algebraic structures (groups, rings, and fields), and elementary topological concepts. The course is also supposed to introduce the students to the practice of good mathematical writing and the construction of proofs. It will be a ten week course (which is supposed to be the equivalent of a normal 15 week semester course). A student will take this course after completing the calculus sequence, linear algebra, and a couple of other courses, but before taking abstract algebra or analysis. Can anyone recommend a good text or two for my students? I realize that one most likely cannot find a book covering set theory and logic and also algebra and topology. I am most intersested in recommendations for the set theory, logic, and proof writing parts of the course (which will probably take up 80% of the course). If necessary I can teach the course without a text (I have extensive notes on set theory and logic and the construction of the number systems from various sources). Any thoughts on that? ==== In a few months I will be teaching a course to undergraduate math majors >called Mathematical Structures . The catalog description is: A rigorous study of the mathematical structures which form the foundation >of higher mathematics. Set theory, logic, formal development of the number >systems from the natural numbers through the complex numbers, basic >algebraic structures (groups, rings, and fields), and elementary topological >concepts. That's quite a lot for one course! >The course is also supposed to introduce the students to the practice of >good mathematical writing and the construction of proofs. And each of those two by itself is quite a lot. > It will be a >ten week course (which is supposed to be the equivalent of a normal >15 week semester course). Hey, why not. Since there's already impossibly too much stuff in it for a 15 week course, there's little additional harm in having impossibly*150% too much stuff. I expect that's why you say supposed to a couple of times. Now seriously, what do you ACTUALLY expect to accomplish in the available time with the available students? It is IMO very important for your own stress management to have realistic(ish) expectations. It sounds like this is a new course. If so, that increases all risk factors markedly. If not, you would definitely benefit from talking with the previous instructors of this course about what went well and what went badly. >A student will take this course after completing the calculus sequence, >linear algebra, and a couple of other courses, but before taking abstract >algebra or analysis. Can anyone recommend a good text or two for my students? I realize that >one most likely cannot find a book covering set theory and logic and also >algebra and topology. I am most intersested in recommendations for >the set theory, logic, and proof writing parts of the course (which will >probably take up 80% of the course). And the other topics will take up the other 80%... >If necessary I can teach the course without a text (I have extensive notes >on set theory and logic and the construction of the number systems from >various sources). Any thoughts on that? In my experience, having a text provides a security blanket for many students (expecially if the instructor may have to go on medical stress leave half way through...) Gun, Will Travel TV series. -- --------------------------- | B B aa rrr b | | BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb | ----------------------------- Cancel-Lock: sha1:6eqNfLB8izgvwsh3H8fXWRMM+ik= ==== Ok, I am getting closer and closer... > [ Feller shows S_n/(n*log2(n)) -> 1 where log2 is log base 2. You have to show S_n/(n*log(n)) -> log(2) where log is log base e. In both cases, -> is convergence in probability. ] Well, you should be able to express log2 in terms of log. And you should be able to use the properties of convergence in probability to decide whether or not the two statements are equivalent. As far as I can tell, they *aren't* equivalent. So, Feller's problem is slightly different from yours, you misread has made a mistake. -- Kevin ==== > Is there an algorithm that maps positive integers 1-1 to primes that > is substantially more efficient than the function that maps n to the > n-th prime? > A lot of work has been done on this. Fermat thought 2^(2^n) + 1 would > always give you a prime number, but he was wrong. I haven't counted up the operations, but calculating 2^(2^n) (in decimal, say) may not be more efficient than finding the n-th prime. Dale ==== bcc list JS: The name of the game is to control the local vacuum coherence. I would like to see how Hal Puthoff uses his PV model for this problem. HP: You want me to send you the blueprint already?! (Let's see, where did I put it?) :-) JS: You already have mine. Seriously, I am much further along than you on all this. HP: Check with Table 2 in my JBIS paper. (The Table you don't like, that's not even wrong! ) Time runs very fast for the drivers (i.e., the rest of the universe is in slow motion relative to them) when they fire up to reduce K (i.e., manipulate the vacuum polarizability to make it very stiff, i.e., to resist polarizing). Ahh, how to do that? I think you're telling me you already have the knob with <0|e+(x)e-(x)|0> mobilzing / zpf, right? JS: Yes, it's the Josephson effect where the space-time stiffness plays no role. I could be mistaken, but it's an idea how to get to the next step, which is more than you have as far as I can see and you have revealed. You should come to London March 8-12. HP: So what are you asking me for? Hal JS: Numero Uno, come clean and fess up where you stand of the reality of the flying saucers and how that influences your theoretical work both with Ibison on PV and with Haisch et-al on zero point energy. Numero Due, make a public statement about all these claims of existing zero point energy machines like Bearden's and all the Tesla maniacs etc. You know like what we were doing at ISSO. All these nuts are citing you as their Guru. On your K-control. Looks like you have no way to do it. Even if you could it is not clear to me that it would help. Your K in your simple model is goo = K^-1 You have K = e^2GM/c^2r with maybe some EM corrections. Please write down the K extended for EM. Also you have no rotation no gravimagnetism which must play an important role. OK for the flying saucer, M is the mass of the saucer correct? The saucer must make its own timelike geodesic, correct? Are we on the same page here or not? If so, we can go maybe to next step. OK dT = goo^1/2dt dR = grr^1/2dr So how do you get saucer to fly starting from there? You want to reduce K why? Reducing K increases goo and decreases grr. Now you have c' = c/K and if you have K < 1 your c' > c but that's for a null geodesic. The saucer of mass M is on a timelike self-made geodesic so how does K < 1 at the saucer help assuming I can help you figure out how to MAKE IT SO with my detailed QED vacuum coherence model. At least I have h in my equations, which you do not have. I also have a detailed formula for guv in terms of h and even the Au from EM. I got a lot more than you got Hal. :-) We also have the clue from Ray Chiao's gravity radio that the metric cross terms of rotation (gravimagnetism) are important. I want to look at Chiao's insight for the near field. He is only thinking far field . If we can convert near induction EM fields to near guv fields we have done it! Do you get what I am saying here? ==== >Prove or disprove: if two matrices commutes (AB=BA) then they can be >expressed with polynomials in terms of the same matrix P ( A=f(P), >B=g(P) ). What would P, f, and g be if A,B = [0 1 0 0] [0 0 1 0] > [ ] [ ] > [0 0 0 0] [0 0 0 1] > [ ], [ ] > [0 0 0 1] [0 0 0 0] > [ ] [ ] > [0 0 0 0] [0 0 0 0] By considering the potential Jordan canonical forms for P, and noting > that A^2 = B^2 = 0, I think one can show there is no possible f and g > no matter what P is. For example, if P were diagonalizable, f(P) and > g(P) would be too (but A and B are not). We have a 4-dimensional commutative algebra spanned by I, A, B, and AB. If A and B are polynomials in P, then so is AB. But in 4 by 4 matrices, the polynomials in P can form an algebra of at most dimension 4. Hence P itself must be a combination of I, A, B, and AB. Polynomials in P can all be written as polynomials in P - cI for any c, so we can assume P is just rA + sB + tAB. But this P has P^2 = 2rAB and P^3 = 0, so it does not produce a sufficiently large algebra. William C. Waterhouse Penn State ==== In many mathematics papers and books, you see the comment We will need this in the sequel or We adopt this notation throughout the sequel. What exactly does the sequel and in the sequel mean here, precisely? Does it mean in what follows? ==== >In many mathematics papers and books, you see the comment We will need this >in the sequel or We adopt this notation throughout the sequel. What >exactly does the sequel and in the sequel mean here, precisely? Does it >mean in what follows? Yes. Sequel comes from the Latin sequella , meaning to follow . In papers, sometimes below is used instead of in the sequel , but if you are refering to something which is some way ahead (e.g., in books), below is not normally used. -- ====================================================================== It's not denial. I'm just very selective about what I accept as reality. --- Calvin ( Calvin and Hobbes ) ====================================================================== Arturo Magidin magidin@math.berkeley.edu ==== >In many mathematics papers and books, you see the comment We will need this >in the sequel or We adopt this notation throughout the sequel. What >exactly does the sequel and in the sequel mean here, precisely? Does it >mean in what follows? Yes. Lee Rudolph ==== I have a question about linear, least-squares (LLS) data fitting: What is the current best algorithm for solving LLS problems? I am considering straightforward data-fitting with no bounds, or any other conditions. At this point, I plan to fit polynomials to the data (does the fact I am using a polynomial function affect the choice of algorithm to use?). After reading some material, I was lead to believe that Singular Value Decomposition (SVD) is a better algorithm than a QR factorization (more stable, but much more expensive in terms of computations). Upon visiting the netlib site to see what they have to offer for LLS code, I noticed they have two SVD algorithms available: DGELSS (SVD via QR) DGELSD (SVD via Divide and Conquer) Could somebody please let me know what the strengths and weaknesses of these two approaches are. Which one is better for my application (fitting data to a polynomial)? My purpose does not mind if the computational cost is high, but the results must be consistent and accurate. David I have been told the following fact by someone who didn't know where they knew it from: Every periodic sequence can be expressed as the sum of a constant sequence and an irreducible sequence; where an irreducible sequence is one such that if you take successive differences iteratively, you eventually end up with the sequence you started with; a constant sequence is one whose terms are all equal; and the sum of two sequences is the sequence whose nth term is the sum of the nth terms of the two summand sequences. In other words, and stated even less rigorously than the above: every periodic sequence is offset by some constant amount from an irreducible sequence. Does this seem obvious? Does it even seem true? Is it a result that you've seen published anywhere? I find it surprising but my intuition is not to be trusted. him on to branches of math so he can see what's out there. He likes quirky number theory stuff, he loved Conway's Book of Numbers, and he loved the book Knots and Surfaces by Farmer and Stanford, which we went through in a blitz - he did almost all of the exercises. I was wondering whether someone knows of an introduction to group theory that is intuitive enough that he can get a feeling for essential concepts without too much formalism. I feel like something that had a good presentation of why there are only a few groups of size smaller than some limit, and what those groups are, for example, could capture his imagination. (He likes to know what all the possibilities are in various contexts.) The texts I know are too terse for a kid that age, who will have time (and patience) to do the formal stuff later, but I think it would be good for him to discover as soon as possible that he loves math in a broad way - if indeed he does love math in a broad way. He's fine on simple set stuff and linear algebra. ==== > wondering whether someone knows of an introduction to group theory that > is intuitive enough that he can get a feeling for essential concepts > without too much formalism.... Brief but beautiful: W.W. Sawyer, Prelude to Mathematics, pp.97-102 and 201-214. Actually, he would very probably enjoy this whole little book. A bit longer, but also very well done: Vol. 3, Chapter XX (pp.263 ff.) Ken Pledger. ==== -- www.StealthHostiing.com You rule Truman. http://tinyurl.com/iky4 Hey Trueman...love the show. YOU ARE the Truman I heard him. Very spooky! >Is the truman living in Townsville? I've been hearing stuff, yeah. Webmasters help the TRUEman by joining www.theBanner.net Current:1 Goal:1000 ---------------------------------------------------------------------------- ------ > As can be seen by the number of posts in this thread, > and the references to his web site in thousands of other posts, > a computer programmer, who took some data processing classes > at a third rate California college, has become a highly regarded expert > in math, physics, and other science disciplines, and > many people, who pretend to be rational, intelligent, open-minded > scientists (Or at least, pretend to have a scientific mind.), > frequently use this programmer as a major reference. > > What third rate California college? Who rated it? What criteria? Hey Wormley, > as you use this programmer's web site as your primary rederence, > it seems to me that you should know what college your resident expert > attended. If you want to know how this college rates, > I suggest that you learn how to use Google. I'll give you some hints. > Caltech and Stanford and first rate California colleges. > The college that Baez teaches at is a second rate college. > Your expert took some data processing classes at a third rate college. Most scientist are computer programmers... are you knocking us Potter? Wormley, > why do you always try to identify yourself with some group? Does identifying yourself with a group make you feel more secure, > or do you think [sic] that it lends strength to your position? Do you have the courage to express any independent ideas you have > (Assuming that you have an independent idea.), > or the knowledge to address the point of a dichotomy, > rather than try to position an opponents point > against some group that you identify with? In other words Wormley, > are you a man or a mouse? -- > Tom Potter http://tompotter.us You show 2 mutually exclusive beliefs here : One must belong to a named college One must act independently Which is it? Will you proudly tell us what University *you* attended? Or do you proudly stand as an idividual with your own exploits? Herc