> Check my math! I've derived a differential equation for strings starting from Stokes > Theorem to show that energy is conserved along a world-sheet. These diff > eq's involve connection coefficients. And I'm not really sure what it all > means yet. I would appreciate it if some who are more skilled in the art > would take a look at this and comment. The math can be found at: http://www.sirus.com/users/mjake/diffeq.html See link in original post. What does it mean that ? Does this mean that the surface is a geodesic? And what does it mean that ? Does this mean that there is no force in the direction of time? If F is still an arbitrary field, then does this mean that the string must be traveling in a frame where the time component is zero? ==== I just started Complex variables and applications by Churchill, et al. I am flummoxed by something already. In Churchill, they say that an accumulation point is ...a point z0 ... of a set S if each neighborhood if z0 contains at least one point of S distinct from z0. Why is it different from R? To wit, every neighborhood of x0 has an infinite number of points of, say, E. TIA, Lurch ==== They are equivilent. Let me see if I can explain it on the internet. If this helps let me know you can just think of R for simplicity, although I don't think my argument will use this fact. let's say we want to check if x0 is an accumulation point of some set S. I want to show that: Every open neighborhood contains at least one point of S distinct from x0 is equivelent to Every open neighborhood contains infinitely many points of S Suppose every open neighborhood contains at least one point of S distinct from x0. Fix an open neighborhood. (just pick any one) Pick one of the points distinct from x0, call it x1. (again, pick one). Now, clearly there exists an open neighborhood of x0 that doesn't contain x1. For example, if |x0 - x1| = a, then take the a/2 neighborhood of x0 (this is just saying that x1 is some distance away from x0 since it is a different point, so if we take a neighborhood that only goes out half as far, x1 is not in that neighborhood). Now, this smaller neighborhood must contain a point of S. call it x2. now take an even smaller neighborhood that doesn't contain x2. this even smaller neighborhood doesn't contain x1 or x2, but it does contain some point of S distinct from x0. So we can keep repeating this forever. In other words, for any n, however large n is, there is some x_n in S. I hope that made sense. To recap, if you keep taking smaller neighborhoods, you keep finding points. So if every neighborhood contains one point, every neighborhood must contain infinitely many points, since each neighborhood also contains every smaller neighborhood. This might not be very rigorous, but hopefully it will get you started. Obviously, if every neighborhood contains infinitely many points, it contains at least one point. Justin Van Winkle Suppose ----- Original Message ----- > ...a point z0 ... of a set S if each neighborhood if z0 contains at least > one point of S distinct from z0. Why is it different from R? To wit, > every neighborhood of x0 has an infinite number of points of, say, E. TIA, Lurch ==== Lurch > They are equivilent. Let me see if I can explain it on the internet. If > this helps let me know you can just think of R for simplicity, although I don't think my argument > will use this fact. let's say we want to check if x0 is an accumulation point of some set S. I want to show that: > Every open neighborhood contains at least one point of S distinct from x0 > is equivelent to > Every open neighborhood contains infinitely many points of S Suppose every open neighborhood contains at least one point of S distinct > from x0. Fix an open neighborhood. (just pick any one) Pick one of the > points distinct from x0, call it x1. (again, pick one). Now, clearly there > exists an open neighborhood of x0 that doesn't contain x1. For example, if > |x0 - x1| = a, then take the a/2 neighborhood of x0 (this is just saying > that x1 is some distance away from x0 since it is a different point, so if > we take a neighborhood that only goes out half as far, x1 is not in that > neighborhood). Now, this smaller neighborhood must contain a point of S. call it x2. now > take an even smaller neighborhood that doesn't contain x2. this even > smaller neighborhood doesn't contain x1 or x2, but it does contain some > point of S distinct from x0. So we can keep repeating this forever. In > other words, for any n, however large n is, there is some x_n in S. I hope that made sense. To recap, if you keep taking smaller neighborhoods, > you keep finding points. So if every neighborhood contains one point, every > neighborhood must contain infinitely many points, since each neighborhood > also contains every smaller neighborhood. This might not be very rigorous, but hopefully it will get you started. Obviously, if every neighborhood contains infinitely many points, it > contains at least one point. Justin Van Winkle > Suppose > ----- Original Message ----- >I just started Complex variables and applications by Churchill, et al. > I > am flummoxed by something already. In Churchill, they say that an > accumulation point is > ...a point z0 ... of a set S if each neighborhood if z0 contains at least > one point of S distinct from z0. Why is it different from R? To wit, > every neighborhood of x0 has an infinite number of points of, say, E. >TIA, >Lurch > > ==== Charlie Johnson > I just started Complex variables and applications by Churchill, et al. I > am flummoxed by something already. In Churchill, they say that an > accumulation point is > ...a point z0 ... of a set S if each neighborhood if z0 contains at least > one point of S distinct from z0. Why is it different from R? To wit, > every neighborhood of x0 has an infinite number of points of, say, E. It is equivalent. It is always equivalent in a metric space. -- Maxi ==== How is it equivalent? If one has the usual metric in R and C, then according to the books I am reading, they are different. C requiring only one point and R requiring infinite amount of points. (The metric in C is abs(z-z0) < e and the metric in R being the usual: abs(x - x0) < e.) Lurch > Charlie Johnson > I just started Complex variables and applications by Churchill, et al. > I > am flummoxed by something already. In Churchill, they say that an > accumulation point is > ...a point z0 ... of a set S if each neighborhood if z0 contains at least > one point of S distinct from z0. Why is it different from R? To wit, > every neighborhood of x0 has an infinite number of points of, say, E. It is equivalent. > It is always equivalent in a metric space. -- > Maxi ==== > How is it equivalent? If one has the usual metric in R and C, then > according to the books I am reading, they are different. C requiring only > one point and R requiring infinite amount of points. (The metric in C is > abs(z-z0) < e and the metric in R being the usual: abs(x - x0) < e.) If every neighbourhood of z0 contains one point of E distict from z0, then by considering the open balls centered in z0 of radius 1/n you see that there is infinitely many (otherwise there would be none in some of these disks for n sufficiently large). -- Maxi ==== I don't quite get it yet, but I will work on it. Lurch > How is it equivalent? If one has the usual metric in R and C, then > according to the books I am reading, they are different. C requiring only > one point and R requiring infinite amount of points. (The metric in C is > abs(z-z0) < e and the metric in R being the usual: abs(x - x0) < e.) > If every neighbourhood of z0 contains one point of E distict from z0, then > by considering the open balls centered in z0 of radius 1/n you see that > there is infinitely many (otherwise there would be none in some of these > disks for n sufficiently large). -- > Maxi ==== Im having trouble understanding some questions. Some help would be appreciated. I have read the book, but dont follow it. Problem 1 F={B->A,A->C} 1. the non-trivial dependencies are: B->A, A->C, B->C is that right? I was wondering if it was a trick question. 2. Find a non-empty instance of R that satisfied every FD in F, bot not A->B any idea what that question MEANS. Does it want a real example? Im confused. 3. Find an instance of R that satisfies every FD in F, bot not A-> B. again does this mean an example? Im confused. ==== I understand that it is possible to express the number of ways of having k bishops on an n*n board such that they are not attacking each other as a combinatorial formula and i have been trying to derive the formula without any luck, can anyone help me out ? ==== It might be helpful to separate the board into two pieces according to the colors of the squares, then translate the diagonals into rows and columns, so you can use familiar coordinate methods. An inductive proof might work. | I understand that it is possible to express the number of ways of | having k bishops on an n*n board such that they are not attacking each | other as a combinatorial formula and i have been trying to derive the | formula without any luck, can anyone help me out ? ==== > I understand that it is possible to express the number of ways of > having k bishops on an n*n board such that they are not attacking each > other as a combinatorial formula and i have been trying to derive the > formula without any luck, can anyone help me out ? Is this right?.. Each bishop has a coordinate (r,c) (row, column). For all the bishops, the r's are different to each other, and the c's are different to each other. So for k bishops the total amount of possible coordinates is how many different ways you can take k r's out of 8, times how many different ways you can take k c's out of 8. And that should be enough for you.. If it's right..I think so. -- Quaternion ==== > > I understand that it is possible to express the number of ways of > having k bishops on an n*n board such that they are not attacking each > other as a combinatorial formula and i have been trying to derive the > formula without any luck, can anyone help me out ? > > Is this right?.. Each bishop has a coordinate (r,c) (row, column). For all > the bishops, the r's are different to each other, and the c's are different > to each other. No, bishops attack on diagonals, not rows & columns. OP; have you tried starting small? n = 2, n = 3, see whether there are any patterns? Or try searching Sloane's On-Line Encyclopedia of Integer Sequences for the word bishop? -- ==== > I understand that it is possible to express the number of ways of >> having k bishops on an n*n board such that they are not attacking each >> other as a combinatorial formula and i have been trying to derive the >> formula without any luck, can anyone help me out ? Is this right?.. Each bishop has a coordinate (r,c) (row, column). For all >the bishops, the r's are different to each other, and the c's are different >to each other. >So for k bishops the total amount of possible coordinates is how many >different ways you can take k r's out of 8, times how many different ways >you can take k c's out of 8. And that should be enough for you.. If it's >right..I think so. Thats a little too simplistic, as you can see from http://acm.uva.es/p/v102/10237.html with 6 bishops on an 8*8 board there are 5599888 ways and with 5 bishops on a 30*30 board there are 3127859642656 ways. ==== > > I understand that it is possible to express the number of ways of > having k bishops on an n*n board such that they are not attacking each > other as a combinatorial formula and i have been trying to derive the > formula without any luck, can anyone help me out ? >>Is this right?.. Each bishop has a coordinate (r,c) (row, column). For all >>the bishops, the r's are different to each other, and the c's are >>different to each other. >>So for k bishops the total amount of possible coordinates is how many >>different ways you can take k r's out of 8, times how many different ways >>you can take k c's out of 8. And that should be enough for you.. If it's >>right..I think so. > > Thats a little too simplistic, as you can see from > http://acm.uva.es/p/v102/10237.html with 6 bishops on an 8*8 board > there are 5599888 ways and with 5 bishops on a 30*30 board there are > 3127859642656 ways. Yes I thought bishops were towers for some reason.. Different languages -- Quaternion ==== > > I understand that it is possible to express the number of ways of > having k bishops on an n*n board such that they are not attacking each > other Why would bishops attack one another? Oh--over homosexuality perhaps? > as a combinatorial formula and i have been trying to derive the > formula without any luck, can anyone help me out ? -- G.C. ==== It is very good that I heard Andrei Linde speak last night at UC Berkeley as my updated version of http://qedcorp.com/APS/EmergentGravity.doc shows. Also a rather large crowd showed up to hear my talk right in the middle of the talk before mine. Linde explained that he does not believe most of the current textbook presentations of inflationary cosmology. The essence of his talk is in my 2 equations II.9 and II.10 in the new version of http://qedcorp.com/APS/EmergentGravity.doc The key is the friction term in II.9 which is a linearization of my II.8 near the bottom of a local minimum in the effective potential of the vacuum coherence field in the FRW large scale limit assuming homogeneity. The friction term that is the root cause of chaotic inflation with slow descent at large order parameter followed by oscillatory reheating to make the post-inflationary Big Bang is quite simply the effect of the connection field for parallel transport in the second application of the covariant time derivative to the scalar field. Of course Linde has no micro-dynamics for the emergence of the scalar field as I do nor does he derive Einstein's gravity together with the unified exotic vacuum dark energy/matter field in a two-way bootstrap of the never-ending spontaneous self-organizing parallel universes splitting off into baby universes in the sense of Andrei Sakharov's metric elasticity a special case of P.W. Anderson's More is different. Steve Carlip gave a interesting talk on topology change in WKB approx to quantum gravity with a possible explanation of why space is homogeneous on large scale consistent with inflation. ==== Given two nx1 column matrices (vectors) X=(x1 x2 ... xn)^T and Z=(z1 z2 ... zn)^T, all elements real numbers (T=transpose): 1) Is there a simple matrix operation to create the nx1 matrix (x1*z1 x2*z2 ... xn*zn)^T from X and Z? I am referring to a mathematical operation like inner product, rather than an algorithm or computer program. 2) Is there a way using matrix operations to produce the column vector (exp(x1) exp(x2) ... exp(xn))^T (exp=exponential function) from X? 3) Related to (2), is there a way using matrix operations to produce the nxn diagonal matrix D (x1 0 0 ... 0) (0 x2 0 ... 0) ( ... ) (0 0 0 ... xn) from X? And conversely, given D as above, to write X in terms of D using matrix operations? ==== > Given two nx1 column matrices (vectors) X=(x1 x2 ... xn)^T and Z=(z1 z2 > ... zn)^T, all elements real numbers (T=transpose): > > 1) Is there a simple matrix operation to create the nx1 matrix > (x1*z1 x2*z2 ... xn*zn)^T from X and Z? I am referring to a mathematical > operation like inner product, rather than an algorithm or computer program. let Em be the square matrix (eij) such that emm = 1 and eij = 0 otherwise. consider Em.Z.X, m = 1, 2, 3, ..., n > 2) Is there a way using matrix operations to produce the column vector > (exp(x1) exp(x2) ... exp(xn))^T (exp=exponential function) from X? assuming exp(A) = Sum A^n/(n!) over n = 0, 1, 2, ..., infinity you can obtain the jth entry of the sought vector: exp(A)ej, where A is the diagonal matrix aii = xi, aij = 0 for i not equal to j, and ej is the jth elementary vector. > 3) Related to (2), is there a way using matrix operations to produce the > nxn diagonal matrix D > > (x1 0 0 ... 0) > (0 x2 0 ... 0) > ( ... ) > (0 0 0 ... xn) > > from X? And conversely, given D as above, to write X in terms of D using > matrix operations? consider X.ej form a linear combo. ==== theorem to matrix transformations. The textbook that I am using has examples of using the inverse function theorem for ordinary R(n)->R(m) functions but not for matrix transformations such as S(X)=X^3, where X is in Mat(3,3) for example. Here is the problem that I need to solve and my solution. I would greatly appreciate if someone could go over my reasoning and point out any flaws that I have and explain to me how to solve Problem: A = [ 0 1 0 ] [ 0 0 1 ] [ 1 0 0 ] Note that A^3 = I (identity matrix). Is there a cont. differentiable function g such that g(I)=A and (g(A))^3=A in the neighborhood of I? My Solution: Let f: X -> X^3 (X is a matrix) then f(g(x)) = [g(x)]^3 = x if X = I f(g(I)) = I Now use the chain rule for the derivative of f(g(I)): [D(fog)(X)] = [Df(g(X))][Dg(X)] (by definition) Let X = I: [D(fog)(I)] = [Df(g(I))][Dg(I)] = [Df(A)][Dg(I)] (since g(I)=A) Finding [Df(A)] is a bit tricky. I used the general definition of the derivative to show that [Df(A)]H = 3A^2H + 3AH^2 since lim {(1/H) * (f(A+H) - f(A) - (3A^2H + 3AH^2))} = 0. Now I let H = I so that [Df(A)]I = 3A^2 + 3A I then went on to show that the determinant of this matrix (using A from the problem) = 54 (is nonzero). ==== Since we're discussing the axiom of foundation (in the textbooks I've seen, it's called the axiom of regularity), does anyone know what the intuitive justification for this axiom is? I mean, all the other axioms seem pretty natural to me, even the infamous axiom of choice. But where in world did they come up with the axiom of regularity? Have a tolerable existence. Eli ==== >Since we're discussing the axiom of foundation (in the textbooks I've seen, >it's called the axiom of regularity), does anyone know what the intuitive >justification for this axiom is? I mean, all the other axioms seem pretty >natural to me, even the infamous axiom of choice. But where in world did >they come up with the axiom of regularity? It was to avoid having a set x which is its only element, and more complicated versions of this. It is easy to see that both it and its negation are consistent. -- 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 ==== The foundation axiom rules out situations that you might call membership loops. For example, if A1 in A2 in A3 in A1, then {A1,A2,A3} has a nonempty intersection with each of its elements. | Since we're discussing the axiom of foundation (in the textbooks I've seen, | it's called the axiom of regularity), does anyone know what the intuitive | justification for this axiom is? I mean, all the other axioms seem pretty | natural to me, even the infamous axiom of choice. But where in world did | they come up with the axiom of regularity? | | Have a tolerable existence. Eli ==== > Since we're discussing the axiom of foundation (in the textbooks I've seen, > it's called the axiom of regularity), does anyone know what the intuitive > justification for this axiom is? I mean, all the other axioms seem pretty > natural to me, even the infamous axiom of choice. But where in world did > they come up with the axiom of regularity? It is not assumed because of some intuitive justification, but because it doesn't interfere with ordinary mathematics (sets of the kind prohibited by the axiom aren't ever needed) and it simplifies model theory by a jillion times. Thomas ==== |Since we're discussing the axiom of foundation (in the textbooks I've seen, |it's called the axiom of regularity), does anyone know what the intuitive |justification for this axiom is? I mean, all the other axioms seem pretty |natural to me, even the infamous axiom of choice. But where in world did |they come up with the axiom of regularity? i think the answer is that (in the presence of the usual other axioms) it's equivalent to the statement every set belongs to some level of the cumulative hierarchy, which is a powerful statement because it allows you to prove theorems applying to all the sets in the universe, by transfinite induction with respect to the level of the cumulative hierarchy to which they belong (the levels of the hierarchy being indexed by ordinal numbers). the cumulative hierarchy starts out with the empty set as the bottom level, then the power set of the empty set as the next level, and so forth on upwards. at limit ordinals you take the union of the levels below. speaking as a non-specialist in set theory, what most bugged me about the axiom of foundation when i was first trying to learn axiomatic set theory (after already learning first-order logic) was that on the one hand it sounds like it's trying to prevent weird loops and weird infinite regresses of membership chains, while on the other hand the means by which it's trying to prevent it (namely the expressive power of axioms of first-order logic) is notoriously ineffectual at actually preventing the existence of models with infinite regresses. but in retrospect i guess the idea is supposed to be that that shouldn't really bother me any more than the similar fact that first-order peano logic is trying to prevent the same kind of infinite regresses, and essentially fails to do so, for essentially the same reason. in both cases you still end up with a powerful means of proving theorems within the theory by some kind of induction even though some of the models of the theory contain infinite regresses which naively seem to violate the spirit of the inductive principles. -- ==== >Since we're discussing the axiom of foundation (in the textbooks I've seen, >it's called the axiom of regularity), does anyone know what the intuitive >justification for this axiom is? I mean, all the other axioms seem pretty >natural to me, even the infamous axiom of choice. But where in world did >they come up with the axiom of regularity? > The Axiom of Foundation is not really necessary - exactly 99.93% (:-)) of mathematics can be done without it. In fact, it even limits our universe of discourse when we accept it. However, it is there to make things nice. Without it, it is consistent to have pathologies such as a set which is an element of itself, two distinct sets each of which is a member of the other, and a sequence x of distinct sets such that x_(n+1) is an element of x_n. The axiom also allows an ice-cream cone construction of the class of sets. With the axiom, there is a sequence R through the ordinals such that a < b implies R(a) is a subset of R(b) and the universe of sets is the well-founded sets, viz., theunion of all these sets. (In particular, R(0) = 0 and R(a+1) = P(R(a)).) This structure allows certain proofs in set theory; for example, the well-founded sets provide a model for ZF. I get most of this from Kunen. One more question: If we didn't mind invoking the axiom of foundation, wouldn't {a, {a,b}} suffice as a definition of the ordered pair (a,b)? Seems to me the answer is yes. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== > One more question: If we didn't mind invoking the axiom of > foundation, wouldn't {a, {a,b}} suffice as a definition of the > ordered pair (a,b)? Seems to me the answer is yes. Sure, but what makes {a, {a,b}} better than {{a}, {a,b}}? The key is to prove that (a,b) = (c,d) => a=c & b=d. With your proposed definition, the proof is a royal pain, and the resulting thingies aren't really any simpler. Thomas ==== >One more question: If we didn't mind invoking the axiom of >>foundation, wouldn't {a, {a,b}} suffice as a definition of the >>ordered pair (a,b)? Seems to me the answer is yes. >> Sure, but what makes {a, {a,b}} better than {{a}, {a,b}}? The key is >to prove that (a,b) = (c,d) => a=c & b=d. With your proposed >definition, the proof is a royal pain, and the resulting thingies >aren't really any simpler. > Doesn't seem to like the necessary implication is more difficult than with the usual definition.. If {a,{a,b}} = {c,{c,d}}, suppose a = {c,d} and c = {a,b}. Then a is an element of c, which is an element of a. This is not possible with foundation. By contradiction, a = c and {a,b} = {c,d}, whence b = d. I would say that my definition is in a sense simpler, since it requires the construction of one fewer set, or the representation requires two fewer characters. Seems to me that the real objection is that invokes foundation unnecessarily. (Note that Halmos doesn't even mention foundation in NST.) Also, once an ordered pair is well-defined, one needs never refer to the definition again. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== > Doesn't seem to like the necessary implication is more difficult than > with the usual definition.. If {a,{a,b}} = {c,{c,d}}, suppose a = > {c,d} and c = {a,b}. Then a is an element of c, which is an > element of a. This is not possible with foundation. By > contradiction, a = c and {a,b} = {c,d}, whence b = d. I suppose so. (You also need to prove that {a,{a,b}} is necessarily a doubleton, which is also a quick deduction from Foundation.) > I would say that my definition is in a sense simpler, since it > requires the construction of one fewer set, or the representation > requires two fewer characters. Seems to me that the real objection is > that invokes foundation unnecessarily. (Note that Halmos doesn't even > mention foundation in NST.) Ok, I suppose I grant your point then. I'd only add that I think this objection is a very important one. As noted before, Foundation isn't added because of some intuitive confidence, but rather because it is known to be harmless, and it's a big help in model theory. So that means that one must be able to develop ordinary mathematics without using it (or else it wouldn't be so harmless, it would be important), and since you need to show that, once you've done it, it is no longer interesting to show that you could have used it here or there along the way. So invoking Foundation unnecessarily is a bad thing, but in a very different way from (say) invoking Choice unnecessarily. Thomas X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft ==== at 06:35 PM, Elaine Jackson said: >The whole problem is just that you're misquoting the axiom. No he is not. >You say: Every nonempty B contains a y >with (B intersect y) = empty. Not only him. Everybody who uses GBN or ZF says so as well. >I say: Every nonempty B contains a y for which >there is no z with z in y and z in B. In what context do you say it? >My axiom What set theory is your axiom a part of? What are the other axioms? >but it allows for the possibility that there exist >citrus fruits that are not sets. Then it's not part of the same set theory, it is your responsibility to give the complete set of axioms that you are using. >Citrus fruits that have no elements, but aren't the >empty set, are technically called individuals. No. There are sets theories that have individuals without having to abandon extentionality. Again, if you wish to be taken seriously you will need to state what set theory it is that you are using. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== | at 06:35 PM, Elaine Jackson said: | |>The whole problem is just that you're misquoting the axiom. | |No he is not. | |>You say: Every nonempty B contains a y |>with (B intersect y) = empty. | |Not only him. Everybody who uses GBN or ZF says so as well. In a careful presentation, it would be usual to write it out in terms of epsilon and =, and not defined terms like intersect which technically are not part of the language of ZFC. I did a web search, and I see that on Mathworld the axiom is given as a form of the axiom scheme of epsilon-induction, which again does not involve referring directly to intersections. |>I say: Every nonempty B contains a y for which |>there is no z with z in y and z in B. | |In what context do you say it? The point here is that speaking of intersections presupposes that the objects in question are sets, whereas set theories with urelements typically consider epsilon to be a relationship on the whole domain, including both urelements and sets. So saying there doesn't exist a z such that z in y and z in B allows for the possibility that y or B is an urelement (like a piece of fruit). Thus the same statement of the foundation axiom would suffice for a set theory with urelements of this kind. |>My axiom | |What set theory is your axiom a part of? What are the other axioms? ZFU would be a familiar example of this kind of set theory. |>but it allows for the possibility that there exist |>citrus fruits that are not sets. | |Then it's not part of the same set theory, it is your responsibility |to give the complete set of axioms that you are using. Seymour Metz is being overly formal again. |>Citrus fruits that have no elements, but aren't the |>empty set, are technically called individuals. | |No. But they are. |There are sets theories that have individuals without having to |abandon extentionality. She didn't say this was the only sense in which the term was used. True, another way to model urelements/atoms/individuals is to have them bear the epsilon relation to themselves. But that requires either abandoning or adjusting the foundation axiom. |Again, if you wish to be taken seriously you |will need to state what set theory it is that you are using. The context was adequate. Keith Ramsay Ok, I'm trying to work on my homework and am stuck on 4.9 #10 of Vector Analysis by Davis. The question states By means of Stokes' theorem, find S F*dR around the ellipse x^2+y^2=1, z=y, where F=xi+(x+y)j+(x+y+z)k. I got the curl of F and that equalled i-j+k but I'm not really sure how to do the rest of the problem. Any help would be appreciated. I've wasted a lot of time and gotten almost nowhere. ==== I forgot to mention that the answer is in the back of the book: +-2pi depending upon the direction of integration. I just can't figure out how to get that. ==== I'm reading about topological groups and I am having trouble with the definition of such a group. What exactly are the open sets? Any help would be appreciated. X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft ==== at 08:02 PM, Arthur said: >I'm reading about topological groups and I am having trouble with the > definition of such a group. What exactly are the open sets? That's like asking what the open sets are in a topological space. Part of specifying a topological group is specifying a topology; the open sets of a topological group are the open sets of its topology. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== > I'm reading about topological groups and I am having trouble with the > definition of such a group. What exactly are the open sets? > Hey, You can't talk about the open sets since a group is said topological iff it is continuous ( (x,y)-> x+y and x->x^(-1)are continuous); sometimes it is also required to be Hausdorff. But considering a group (G,+) you can put different topologies on G so that G is a topological group (provided that G is continuous (and sometimes, Hausdorff)). We're not definining a special topology here. -- Julien Santini, France. ==== >> I'm reading about topological groups and I am having trouble with the >> definition of such a group. What exactly are the open sets? > > Hey, > > You can't talk about the open sets since a group is said topological > iff it is continuous ( (x,y)-> x+y and x->x^(-1)are continuous); > sometimes it is also required to be Hausdorff. But considering a group > (G,+) you can put different topologies on G so that G is a topological > group (provided that G is continuous (and sometimes, Hausdorff)). > We're not definining a special topology here. > > -- > Julien Santini, > France. > > > I see. Arthur ==== Jack & Jonathan, The *Roots of Consciousness* by Jeffrey Mishlove on the web does NOT include my 40 page appendix paper, Consciousness: a Hyperspace View. Jeffrey wanted to include it but there was some difficulty with the complexity of the figures -- if I remember correctly. Of course, it had never been in pdf form, since the web (& home computers) were in a very primitive state in 1993. This paper has the most detailed account of what I now call ADEX theory -- the application of the A-D-E Coxeter graphs to mathematics, physics, and other fields such as consciousness theory. This appendix paper was written in 1989, and published in 1993. A short summary and update of this paper (with more ADEX examples) was titled, A Mathematical Strategy for a Theory of Consciousness. It was written in 1994 and published in the book, *Toward a Science of Consciousness: the First Tucson Discussions and Debates* (edited by Stuart R. Hameroff, Alfred W. Kaszniak, and Alwyn C. Scott), MIT Press, 1996. People have told me that my papers are hard to read without much more mathematical knowledge than they possess. Jeffrey tells me that Russian scientists have less trouble with the math since their mathematical training is more advanced than that of American psychologists and biologists. Actually much of the mathematics of ADEX theory is of very recent vintage, but the key area of mathematics is quite old -- group theory (both finite groups and Lie groups and relationships between them). Our understanding of these relationships depends on both algebra and geometry -- especially hyperspace geometry -- including differential geometry and algebraic geometry. Physicists who study general relativity know some differential geometry, but they have not studied the more recently developed algebraic geometry -- and very few physicist (or mathematicians) have seen the great utility of the A-D-E Coxeter graphs which have been used to classify more than 20 mathematical objects. The advantage of having these classifications, is that the A-D-E graphs provide the relationships between all these mathematical objects. I will mention a few of these objects -- the study and application of which I call ADEX theory: 1. Finite reflection groups (Coxeter groups also called Weyl groups) 2. Hyperspace polytopes and thus crystallographic lattices 3. Coxeter arrangements (mirrors in reflection space) 4. Lie algebras and Lie groups (& also Kac-Moody Lie algebras) 5. Thom-Arnold catastrophe bundles (useful for Jack's version of Bohm) [BTW: Thom claimed (1975) that it models the mind-body relationship] [Yes, this is the aspect I want to flesh out with you.] 6. McKay groups (finite subgroups of SU(2) -- unit length quaternions) 7. Gravitational Instantons (closely related to Penrose twistors) 8. 2-d Conformal field theories (which live on hyperspace strings) [Jack: It is interesting that O(2) macro-quantum order parameter in ordinary space has string defects e.g. Hagen Kleinert also books on soft condensed matter physics and cosmic strings. Then introduce extra space dimensions including fermi dimensions for supersymmetry to get higher dim branes from the macro-quantum order parameters perhaps with higher O(N) internal symmetry, hyper-complex matrix order parameters over hyper-complex generalized space-time manifolds. My model in http://qedcorp.com/APS/EmergentGravity.doc is only the low energy tail of that. BTW new version shows how to go from my BIT FROM IT Landau-Ginburg eq to Andrei Linde's specific equations for chaotic APS-AAPT. I had discovered the friction term from my equations a year ago not realizing their crucial role in Linde's theory of the continuous creation of the parallel universes.] 9. Error-correcting codes (related to Jack's IT <--> BIT idea) That too. The mind field must have error correcting codes built into it.] 10. Quantizing lattices (analog to digital transforms) As the motto for Plato's academy said, Let no one enter here without geometry. Today, this geometry must include hyperspace geometry -- which dates back to the 19th century. BTW: *Roots of Consciousness* is mentioned at the end of John McKay's very short paper, A Rapid Introduction to ADE Theory. The URL for this paper is: http://math.ucr.edu/home/baez/ADE.html This is on John Baez's very extensive website, and from the above URL you can access 4 much longer (tutorial) papers by John Baez on the ADE related mathematics. Nuff said! Saul-Paul ---------- It's mainly metaphor and not useful given today's advances. ==== >> .... >> This shows that our modern area formula (pi)(r^2) or (pi)(d^2)/4 .... > > Ken , > Is this really a modern formula for circle's area?.... What I meant was modern _notation_ for the formula. I'm sorry if that wasn't clear. Ken Pledger. ==== I noticed that continued fraction expansion for sqrt( (n^2 - n + 2)^2 / 4 - n ) was quite interesting, as soon as n is very great (if n is small, the property is still true, but not easy to see). Take n=2^67; then, you can notice that starting from the 4-th term in the expansion, you will find blocks starting with a great value, and a few very small terms just after. What is interesting is that a formula f(n,i) can be built that returns rationals equal to the rational built from the i-th block. Here is an example: for n=2^67 The 38-th block is [22801,14,1,1,5,15,1,1,3,3,1,2,1,1,1,18,...] (length is 33). Considering this block is the continued fraction of a rational, we have the 38-th rational, being: 238...709 / fibonacci(38+2)^2 Now, I found the formula that gives rational equal to the successive blocks: floor( (n-epsilon) / (fibo(i+1)*fibo(i+2)) ) [Times] ( ((-1)^i [Times] fibo(i+1)[Times](n-fibo(i+2)-1)[Times]fibo(i+2)-1) mod (fibo(i+2)^2)) / (fibo(i+2)^2) Now, just put epsilon=0, and you will see the formula is very good. BUT... The integer part is sometimes greater (diff=1) than the real value; the rest of the block is fine. Thus, I think that an epsilon value should be put in the formula. It looks like epsilon depends on both n and i. It means that floor( n / (fibo(i+1)*fibo(i+2)) ) is almost the right value for the successive big values in the initial continued fraction expansion, but not quite exact. Could someone fix the formula ? PS - may my formula be simplified (for instance, is it possible to remove the (-1)^i ?) Cordially, ==== > Is there a book similar to Berkeley Problems in Mathematics, but > geared more towards the undergraduate, and perhaps more finding > stuff rather than proving stuff? I would hope the problems would be > of the same relative difficulty to an undergraduate as the BPM book > would be to a graduate. You may find some of what you want in D.K. Faddeev & I.S. Sominskii old-fashioned, but has a lot of good problems. Ken Pledger. ==== Some of you may have noticed frenetic activity from posters trying to convince you that there's nothing sinister about mathematicians doing their best to downply my find of a way to count prime numbers by integrating a partial difference equation, but what's the bottom line? Does what I found work or not? It does. End of story, so mathematicians should acknowledge it. If it's not important they can just put it in some math text somewhere, or in some journal and drive on. No big deal. But they're fighting to totally ignore it. Translation: Sinister attempt by academic types to hide something really important. Otherwise, why go to so much effort to fight me, when a simple way to shut me up on the issue is just record it somewhere? And it is a FIRST in human history, so use your common sense. The loser academic world is fighting me over something that works. End of story. These posters trying to convince you otherwise are just insulting your basic intelligence. James Harris My math discoveries, found for profit http://mathforprofit.blogspot.com/ ==== > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about mathematicians doing > their best to downply my find [...] And again you find it perfectly acceptable to hurl insults at millions -- while reserving to play the indignant sensitive little flower when someone hands you a tiny fraction of your insults back. > of a way to count prime numbers by > integrating a partial difference equation, but what's the bottom line? You have yet to present any kind of way to count prime numbers that actually has anything to do with integration at all. > Does what I found work or not? > It has been conclusively proven that it doesn't. I presented the implementation of the exact literal lines you posted here and you yourself could not find anything wrong with it. If you had a quarrel with the implementation, you could even simply have posted your own little fortran or basic or c-routine. No big deal. But of course you can't. It does not work. That is all there is to it. I have given you thebenefit of the doubt long enough to implement exactly what you posted here to see for myself whether you're on to something or not. That's called 'science': I go and examine the evidence myself. And I have seen with my own eyes that you don't have anything here that counts primes. And further *lies* of yours to the contrary will not sway someone who's actually examined the evidence himself. > It does. End of story, so mathematicians should acknowledge it. Ah: you say so and thus it is so. 'Tis a simple world you live in. So why does this go for you but not for everybody else on the planet? Because there's a lot of people out there that say you stuff doesn't work. And contrary to you they have evidence for their claim. > But they're fighting to totally ignore it. Translation: Sinister > attempt by academic types to hide something really important. Ask yourself: how does this line distinguish you from every run-of-the-mill dime-a-dozen psychotic crackpots with a new theory of everything to sell, without a shred of evidence to present and with demonstrated lack of grasp of what they're talking about? > Otherwise, why go to so much effort to fight me, when a simple way to > shut me up on the issue is just record it somewhere? Nobody is going to any particular effort fighting you. Nobody is going to record anything anywhere because there's nothing to record here. > These posters trying to convince you otherwise are just insulting your > basic intelligence. Just to clarify for to odd reader out there: I am not trying to convince you of anything at all. (Contrary to Mr. Harris.) Go and see for yourself, as I did. You'll see for yourself. ==== > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about mathematicians doing > their best to downply my find of a way to count prime numbers by > integrating a partial difference equation, but what's the bottom line? What you are posting is, as usual, complete nonsense. To everyone except you it is obvious that your supposed find of a way to count prime numbers by integrating a partial difference equation is complete and utter garbage. Pointing out that complete and utter garbage is complete and utter garbage is not sinister, it is the obvious thing to do. So the reason why people post that you are wrong is just plain because you are wrong. Nothing sinister about that. ==== > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about mathematicians doing > their best to downply my find of a way to count prime numbers by > integrating a partial difference equation, but what's the bottom line? Your work does *not* involve the integration of a partial difference equation. Integration is used to find anti-derivatives. Difference equations are solved using the sum calculus. Nowhere in the exposition of your find do you ever integrate any equation whatsoever, much less a partial difference equation. -- 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 ==== | |> Some of you may have noticed frenetic activity from posters trying to |> convince you that there's nothing sinister about mathematicians doing |> their best to downply my find of a way to count prime numbers by |> integrating a partial difference equation, but what's the bottom line? | |Your work does *not* involve the integration of a partial difference |equation. Integration is used to find anti-derivatives. Difference |equations are solved using the sum calculus. Nowhere in the exposition of |your find do you ever integrate any equation whatsoever, much less a |partial difference equation. you're lucky herman rubin isn't reading these threads. -- ==== >Some of you may have noticed frenetic activity from posters trying to >convince you that there's nothing sinister about mathematicians doing >their best to downply my find of a way to count prime numbers by >integrating a partial difference equation, but what's the bottom line? Does what I found work or not? It does. End of story, so mathematicians should acknowledge it. If >it's not important they can just put it in some math text somewhere, >or in some journal and drive on. No big deal. But they're fighting to totally ignore it. Translation: Sinister >attempt by academic types to hide something really important. Fascinating. If people were raving about how important it was of course that would prove it was important. In fact people are ignoring it, and curiously that also proves it's important. Do you really think anyone's buying this? >Otherwise, why go to so much effort to fight me, when a simple way to >shut me up on the issue is just record it somewhere? Huh? First, nobody's going to any trouble - what looks to you like people going to a lot of trouble is just people having a bit of good-natured fun with the village idiot. And second, all your stuff _is_ recorded, right there on Google. (You're going to find that fact embarassing if you ever sober up...) > And it is a >FIRST in human history, so use your common sense. The loser academic world is fighting me over something that works. End of story. These posters trying to convince you otherwise are just insulting your >basic intelligence. Uh, right. Our common sense tells us that when people on sci.math, you harass in person _all_ find your work of no interest the only possible explanation is that they all realize it's tremendously important, and every single one of them is quick enough to realize he needs to lie about his opinion before once saying oh my god that's incredible even once. That's not common sense as we know it, Jim. >James Harris My math discoveries, found for profit >http://mathforprofit.blogspot.com/ David C. Ullrich ==== > > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about [snip] > My math discoveries, found for profit [snip] http://www.crank.net/harris.html It's not every braying jackass that gets a whole page at crank.net -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! ==== James Harris > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about mathematicians doing > their best to downply my find of a way to count prime numbers by > integrating a partial difference equation, but what's the bottom line? Does what I found work or not? He's talking about this one: For newbies: www.crank.net/harris.html For harrisologists: mathdb.math.cuhk.edu.hk/forum/e_show.php?msg=705 ==== James I believe that the people responsible for covering up the Roswell incident are the same people responsible for suppressing your world changing ideas about algebraic integers, and your prime counting function. I would be careful about what you say of the government. You may end up speaking with Mulder and Scully. Lurch > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about mathematicians doing > their best to downply my find of a way to count prime numbers by > integrating a partial difference equation, but what's the bottom line? Does what I found work or not? It does. End of story, so mathematicians should acknowledge it. If > it's not important they can just put it in some math text somewhere, > or in some journal and drive on. No big deal. But they're fighting to totally ignore it. Translation: Sinister > attempt by academic types to hide something really important. Otherwise, why go to so much effort to fight me, when a simple way to > shut me up on the issue is just record it somewhere? And it is a > FIRST in human history, so use your common sense. The loser academic world is fighting me over something that works. End of story. These posters trying to convince you otherwise are just insulting your > basic intelligence. > James Harris My math discoveries, found for profit > http://mathforprofit.blogspot.com/ ==== And that are my dogs! Mulder and Scully are two lovely dogs, Amstaffs of course. They love Harris as I do. I and them would like Harris to be honest about his work and commit to failure. Or else!!!! What I really want is that Harris fully gave his work a scrutiny which included all the corrections from the bistanders and helping hands. Then he can conclude and put forward his proof. One last question to Harris: What is the difference between algebraic numbers, algebraic integers, numbers, integers and complex numbers? Karl-Olav Nyberg ==== > One last question to Harris: What is the difference between algebraic numbers, algebraic integers, > numbers, integers and complex numbers? LH ==== What is the difference between algebraic numbers, algebraic integers, > numbers, integers and complex numbers? I believe you might have to wade through JSHs Object Oriented Mathematics and the brilliance that that has yet to shine on all of us losers before you can even ask the self-proclaimed highest ranking number theorist in the world. In fact, Mr. Harris ego is growing at a super-exponential rate and soon no one in sci/math will be able to contain his NPD! ==== > > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about mathematicians doing > their best to downply my find of a way to count prime numbers by > integrating a partial difference equation, but what's the bottom line? > > Does what I found work or not? > > Now that is an interesting question, isn't it! pssst, hey Harris... your stuff doesn't work.... but don't tell anybody... ==== > > Some of you may have noticed frenetic activity from posters trying to > convince you that there's nothing sinister about mathematicians doing > their best to downply my find of a way to count prime numbers by > integrating a partial difference equation, but what's the bottom line? > > Does what I found work or not? > > > Now that is an interesting question, isn't it! > pssst, hey Harris... your stuff doesn't work.... but don't tell anybody... And Sam Wormley, surprise, surprise, is making a false statement as it *does* work, but I'm not terribly surprised by his immature behavior. After all, I found this partial difference equation, which integrates over a certain range to give a count of prime numbers!!! Here's the equation: dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,sqrt(y-1))], Here are the instructions for the integration: S(x,1) = 0. And p(x, y) = floor(x) - S(x, y) - 1, and you get S as the sum of dS from dS(x,2) to dS(x,y). Now for someone like Sam Wormley it probably doesn't seem fair that I'm here posting on Usenet stealing thunder from everyone else, but hey, blame the mathematicians. The bottom line on my work is that it DOES work. Now then, do any of you know *how* it works? Maybe some poster will reply to this post to explain it to you, but how do you trust them? And what about the story--my story--of the discovery? What was I thinking? What motivated me to look in this area? What's the story? If mathematicians hadn't decided to break faith with you and the rest of the world, probably there'd be a book, some popular work, explaining the story. But how can you get that story if mathematicians are playing their academic games? Bottom line: What I have works. So what if I sell my story and get rich. That's how capitalism works. These mathematicians are worse than communists, as how do you explain their behavior? I *am* the American Dream, fighting for what should be mine, having to get past weak-minded academics who are fighting to block my success. But I shall prevail!!! I'm sure some hate that I'm in it for the money. But why screw over the world, why screw *you* over by blocking the story of my success? What's their motivation? James Harris My math discoveries, found for profit http://mathforprofit.blogspot.com/ ==== >If mathematicians hadn't decided to break faith with you and the rest >of the world, probably there'd be a book, some popular work, >explaining the story. But how can you get that story if mathematicians are playing their >academic games? Bottom line: What I have works. So what if I sell my story and get rich. Psst, James, there is a very small market for stories about mathematics. Better find some way to work in spies and the CIA, and pretty girl agents, and such like. And sex. Sex always sells, even when the sex scenes are separated by boring mathematical explanations. People just skip those. -- Wolf Kirchmeir, Blind River ON Canada Nature does not deal in rewards or punishments, but only in consequences. (Robert Ingersoll) ==== > >If mathematicians hadn't decided to break faith with you and the rest >of the world, probably there'd be a book, some popular work, >explaining the story. > >But how can you get that story if mathematicians are playing their >academic games? > >Bottom line: What I have works. > >So what if I sell my story and get rich. > > Psst, James, there is a very small market for stories about mathematics. > Better find some way to work in spies and the CIA, and pretty girl agents, > and such like. And sex. Sex always sells, even when the sex scenes are > separated by boring mathematical explanations. People just skip those. A very small market in today's world can be worth millions of dollars US. The bottom line is that what I have works, people expect mathematicians to report on discoveries, but they are not doing their jobs. It's easy to check using Google. Go search on partial difference equation which can verify for you that they are real. Then search on prime counting or counting primes to see if ANYONE besides me has ever used a partial difference equation to count prime numbers. For those wondering what they might do to help, I think that maybe knows what might happen? James Harris My math discoveries, found for profit http://mathforprofit.blogspot.com/ ==== >A very small market in today's world can be worth millions of dollars >US. IMO, you need to brush up on your arithmetic, too. -- Wolf Kirchmeir, Blind River ON Canada Nature does not deal in rewards or punishments, but only in consequences. (Robert Ingersoll) ==== > For those wondering what they might do to help, I think that maybe > knows what might happen? Dear Time Magazine, Here is what can happen with education gone bad, a person with delusions of grandeur, fame and fortune. This individual believes he is one of the greatest number theorists and analytical researchers of ALL TIME! Can you perhaps run a story on NPD (you have a real-live case here)? Here is a clinical definition of NPD: *** Diagnostic criteria for 301.81 Narcissistic Personality Disorder (cautionary statement) A pervasive pattern of grandiosity (in fantasy or behavior), need for admiration, and lack of empathy, beginning by early adulthood and present in a variety of contexts, as indicated by five (or more) of the following: (1) has a grandiose sense of self-importance (e.g., exaggerates achievements and talents, expects to be recognized as superior without commensurate achievements) (2) is preoccupied with fantasies of unlimited success, power, brilliance, beauty, or ideal love (3) believes that he or she is special and unique and can only be understood by, or should associate with, other special or high-status people (or institutions) (4) requires excessive admiration (5) has a sense of entitlement, i.e., unreasonable expectations of especially favorable treatment or automatic compliance with his or her expectations (6) is interpersonally exploitative, i.e., takes advantage of others to achieve his or her own ends (7) lacks empathy: is unwilling to recognize or identify with the feelings and needs of others (8) is often envious of others or believes that others are envious of him or her (9) shows arrogant, haughty behaviors or attitudes Reprinted with permission from the Diagnostic and Statistical Manual of Mental Disorders, fourth Edition. Copyright 1994 American Psychiatric Association *** ==== ... > After all, I found this partial difference equation, which integrates > over a certain range to give a count of prime numbers!!! > > Here's the equation: > > dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - > p(y-1,sqrt(y-1))], As written this does not look like a partial difference equation to me. perhaps you intend: S(x, y) - S(x, y-1) = the expression on the right, or: S(x, y+1) - S(x, y) = the expression on the right. It is not clear which of the two you intend here. > Here are the instructions for the integration: > S(x,1) = 0. Instruction? Looks more like a boundary condition. > And p(x, y) = floor(x) - S(x, y) - 1, and you get S as the sum of dS > from dS(x,2) to dS(x,y). There must be something wrong. You now instruct how to get S. But I thought in a partial difference equation you had to get S by solving the equation. So how can you now give explicit instructions on how to calculate S? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== In sci.physics, Sam Wormley <3FB7DF7A.13A6732D@mchsi.com>: >> >> Some of you may have noticed frenetic activity from posters trying to >> convince you that there's nothing sinister about mathematicians doing >> their best to downply my find of a way to count prime numbers by >> integrating a partial difference equation, but what's the bottom line? >> >> Does what I found work or not? >> > > Now that is an interesting question, isn't it! > pssst, hey Harris... your stuff doesn't work.... but don't tell anybody... Actually, his stuff worked more or less fine ... but I can't say it was the fastest. http://home.earthlink.net/~ewill3/math/primecounters/index.html was a somewhat tongue-in-cheek contest I sponsored 2 months back that produced a few bizarre results and some interesting algorithms. (However, Christian Bau has a better one anyway, although he didn't submit that particular one for my contest. Perhaps it was because my contest was unworthy thereof. :-) ) James' wasn't the best, especially after a round of memoization which I for one did not foresee. In fact, yours truly beat him with a rather simple Legendrephi entry that was more than twice as fast. However, an even simpler sieving entry (sieve6bool, which did take advantage of a prime being of one of the forms 6k+1 or 6k-1, after the two entries 2 and 3) was 10.5 times as fast, once I got it working -- and the best entry (edgar3), apart from a couple of what were essentially large table lookups by yours truly, was submitted by a hitherto unknown poster James did win a consolation prize for elegance. Of course prize here is a bit of a misnomer. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== > was a somewhat tongue-in-cheek contest I sponsored 2 months back > that produced a few bizarre results and some interesting > algorithms. (However, Christian Bau has a better one anyway, > although he didn't submit that particular one for my contest. > Perhaps it was because my contest was unworthy thereof. :-) ) No, it was not finished at that time, and I have to find some spare time to improve it anyway. What I am quite interested in at the moment is that there seems to be a substantial improvement possible if you want to calculate pi (N) for many different values of N, for example N = k * 10^14 for 1 <= k <= 10000. My implementation should take about O (N^(2/3)) to find pi (N). However, it might be possible to find pi (x) for n different values x <= N in about O (N^(2/3)) * sqrt (n) instead of O (N^(2/3)) * n. ==== In sci.physics, Christian Bau > was a somewhat tongue-in-cheek contest I sponsored 2 months back >> that produced a few bizarre results and some interesting >> algorithms. (However, Christian Bau has a better one anyway, >> although he didn't submit that particular one for my contest. >> Perhaps it was because my contest was unworthy thereof. :-) ) > > No, it was not finished at that time, and I have to find some spare time > to improve it anyway. What I am quite interested in at the moment is > that there seems to be a substantial improvement possible if you want to > calculate pi (N) for many different values of N, for example > > N = k * 10^14 for 1 <= k <= 10000. > > My implementation should take about O (N^(2/3)) to find pi (N). However, > it might be possible to find pi (x) for n different values x <= N in > about O (N^(2/3)) * sqrt (n) instead of O (N^(2/3)) * n. I suppose it might depend in part on the value of max(N_i), where N_i are the numbers fed into pi(N). I really don't know, and haven't researched the issue. Good luck. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== With all the dumb crap in math journals, you people know that my find of a way to count prime numbers by integrating a partial difference equation is worthy of publication somewhere. But you sit by as if there's nothing sinister going on, when all these people are fighting such a choice result. Yet in your ENTIRE CAREERS most of you will never publish anything that's even close in neatness, but you'll fill up journals anyway. That's the math world. If there's anything I really like here it's proving the lie to all of you of pure math, as sure, you may convince others that my work isn't important, but those of you desperate to find something worth publication in your publish or perish world know the reality. Pure math is a fraud, just something you say, when you don't believe in it. It's just a way to get by and pay your bills. Yes, to me you are losers, too weak to handle the truth, so you think you can crap on me, when what you do anyway is fill up journals with junk. I'm better than you, and you know it. James Harris ==== > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. > > But you sit by as if there's nothing sinister going on, when all these > people are fighting such a choice result. > > Yet in your ENTIRE CAREERS most of you will never publish anything > that's even close in neatness, but you'll fill up journals anyway. > > That's the math world. > > If there's anything I really like here it's proving the lie to all of > you of pure math, as sure, you may convince others that my work > isn't important, but those of you desperate to find something worth > publication in your publish or perish world know the reality. > > Pure math is a fraud, just something you say, when you don't believe > in it. > > It's just a way to get by and pay your bills. Yes, to me you are > losers, too weak to handle the truth, so you think you can crap on me, > when what you do anyway is fill up journals with junk. > > I'm better than you, and you know it. > > > James Harris fuffy ==== > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. There are indeed journals that will publish nifty new ways of proving old results if there is some insight to be gained from the new way. In my field for instance SIAM Review has classroom notes where people explain well-known results in ways that are especially easy to grasp. You may want to consider that route if you think that your approach to Legendre's (?) method gives some new perspective on the matter. V. -- homepage: cs utk edu tilde lastname ==== > With all the dumb crap in math journals, How would you know? You've repeatedly refused to attempt answering even the most basic of homework-type questions in algebra. ==== > I'm better than you, and you know it. Do you know that you actually have more than nine of these symptoms? That also deals with numbers, including 9 which is the perfect square 3^2! *** Diagnostic criteria for 301.81 Narcissistic Personality Disorder (cautionary statement) A pervasive pattern of grandiosity (in fantasy or behavior), need for admiration, and lack of empathy, beginning by early adulthood and present in a variety of contexts, as indicated by five (or more) of the following: (1) has a grandiose sense of self-importance (e.g., exaggerates achievements and talents, expects to be recognized as superior without commensurate achievements) (2) is preoccupied with fantasies of unlimited success, power, brilliance, beauty, or ideal love (3) believes that he or she is special and unique and can only be understood by, or should associate with, other special or high-status people (or institutions) (4) requires excessive admiration (5) has a sense of entitlement, i.e., unreasonable expectations of especially favorable treatment or automatic compliance with his or her expectations (6) is interpersonally exploitative, i.e., takes advantage of others to achieve his or her own ends (7) lacks empathy: is unwilling to recognize or identify with the feelings and needs of others (8) is often envious of others or believes that others are envious of him or her (9) shows arrogant, haughty behaviors or attitudes Reprinted with permission from the Diagnostic and Statistical Manual of Mental Disorders, fourth Edition. Copyright 1994 American Psychiatric Association *** ==== > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. You're work does *not* involve the integration of a partial difference equation. Integration is used to find anti-derivatives. Difference equations are solved using the sum calculus. Your prime counting function does not involve integration in any way. If you'd stop misrepresenting your work, maybe someone would pay attention to it. [snip] -- 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 ==== >With all the dumb crap in math journals, you people know that my find >of a way to count prime numbers by integrating a partial difference >equation is worthy of publication somewhere. Why are you telling us this? Do you think that sci.math has something to do with what gets published in math journals? That's not how it works. Send your stuff to a journal. Let us know how it turns out. >But you sit by as if there's nothing sinister going on, when all these >people are fighting such a choice result. Yet in your ENTIRE CAREERS most of you will never publish anything >that's even close in neatness, but you'll fill up journals anyway. That's the math world. If there's anything I really like here it's proving the lie to all of >you of pure math, as sure, you may convince others that my work >isn't important, but those of you desperate to find something worth >publication in your publish or perish world know the reality. Pure math is a fraud, just something you say, when you don't believe >in it. It's just a way to get by and pay your bills. Yes, to me you are >losers, too weak to handle the truth, so you think you can crap on me, >when what you do anyway is fill up journals with junk. I'm better than you, and you know it. >James Harris David C. Ullrich ==== |With all the dumb crap in math journals, you people know that my find |of a way to count prime numbers by integrating a partial difference |equation is worthy of publication somewhere. my comment here is only indirectly related to your comment above. your work on writing computer programs to count prime numbers is so much, much better than your work on trying to prove theorems of any kind (fermat's last theorem, algebraic integer theory, et cetera) that you would have to be a total idiot to continue pursuing the theorem-proving stuff when instead you could be out there trying to do stuff like writing computer programs to win those rsa challenge prizes or whatever they're called. you really, really need something to provide external discipline for you, something to tell you when you've gotten a wrong answer and when you've gotten a right answer. the concept of mathematical proof does provide that kind of external discipline for people who understand that concept, but you're not one of those people! (at least, it would be astonishing if you did understand the concept of mathematical proof, and yet produced the kind of incomprehensible near-gibberish proofs that you regularly produce. i'm willing to bet that you yourself really don't have the slightest idea as to whether or not your proofs are really valid.) if you work on creating algorithms and writing computer programs then you can get concrete answers about whether you've done things correctly simply by running a computer program. i guess if you really have your heart set on pursuing the theorem-proving stuff then it would be ok to pursue it, but only if you first learn what mathematical proof really means. it's not that stupid horseshit about a proof starts with a truth and proceeds by logical steps to a conclusion which then must be true! at least, that stupid slogan is worthless unless you learn what a valid logical step really is. really, it would do you well to learn the simple way in which rigorous (though not necessarily computationally efficient) theorem-checking and theorem-proving computer programs actually work, and maybe even write such a program of your own so you could honestly say that you understand what a mathematical proof is. also, have you ever considered maintaining your opinion of yourself as a million times smarter than everyone else but _not going around telling everyone about it every five minutes_? works for me. also, a lot of the people on this newsgroup who go around attacking you constantly are worthless idiots and when you try to adopt for yourself the same attitudes and the same methods of attacking that they use you just make yourself look like a worthless idiot too, so cut it out. you really need to learn to ignore the rantings of worthles idiots instead of trying to become one of them. -- ==== >also, a lot of the people on this newsgroup who go around attacking >you constantly are worthless idiots It's considered customary to have researched the topic you're speaking about before making an ass of yourself. Doug ==== >if you work on creating >algorithms and writing computer programs then you can get concrete >answers about whether you've done things correctly simply by running a >computer program. > Excellent advice! Few professionals will find an amateur's proof interesting. On the other hand, today's amateur can perform carefully crafted computations that are sometimes of interest to the professional. It's one of the reasons why I believe the 21st century is a great time to be an amateur mathematician. Rich Burge ==== > > |With all the dumb crap in math journals, you people know that my find > |of a way to count prime numbers by integrating a partial difference > |equation is worthy of publication somewhere. > > my comment here is only indirectly related to your comment above. snip > also, a lot of the people on this newsgroup who go around attacking > you constantly are worthless idiots Well, actually, they are not. They probably should find something better to do with their time than pointing out James's deficiencies, but that isn't the same thing. I think that if you look you will find that all of JSH's fiercest critics are helpful and polite to those who are less antisocial than JSH. Mark Atherton ==== |> |> |With all the dumb crap in math journals, you people know that my find |> |of a way to count prime numbers by integrating a partial difference |> |equation is worthy of publication somewhere. |> |> my comment here is only indirectly related to your comment above. | |snip | |> also, a lot of the people on this newsgroup who go around attacking |> you constantly are worthless idiots | |Well, actually, they are not. bullshit. either you have trouble understanding words like a lot, or you're just as worthless as they are. |They probably should find something better to do with their time than |pointing out James's deficiencies, but that isn't the same thing. I |think that if you look you will find that all of JSH's fiercest |critics are helpful and polite to those who are less antisocial than |JSH. if you think that all his fiercest critics are helpful and polite then you're an idiot, since all includes for example the guy that calls himself uncle al. -- ==== > > |> also, a lot of the people on this newsgroup who go around attacking > |> you constantly are worthless idiots > | > |Well, actually, they are not. > > bullshit. either you have trouble understanding words like a lot, > or you're just as worthless as they are. I'm happy for it to be the latter, thanks, since I consider that they are not worthless at all. > > |They probably should find something better to do with their time than > |pointing out James's deficiencies, but that isn't the same thing. I > |think that if you look you will find that all of JSH's fiercest > |critics are helpful and polite to those who are less antisocial than > |JSH. > > if you think that all his fiercest critics are helpful and polite > then you're an idiot, since all includes for example the guy that > calls himself uncle al. That's a fair point, though mostly Uncle Al confines himself to brief insults. By his fiercest critics I meant those who criticise his maths. And if you respond to a polite disagreement by calling me an idiot then you should think twice before commenting on other posters' manners. By the way, why the lower case? ALL UPPER CASE is considered to be rude but normal sentence capitalization is the norm. Have a nice day. :-) Mark Atherton ==== James, I apologize that my previous posts may have been a bit insulting, it wasn't very mature of me. In the following, I'll assume that the things you have been working on are correct. It seems that the prime counting function is generally agreed to be correct, so just consider that if you like. Don't be so bitter. Contemplate on why many people in the math community aren't taking your result seriously. In a legislature bills are presented in a certain way. If someone from the outside were to insist on presenting a bill in another way, many would ignore it. A computer can only compile a program if it is very carefully formatted. If it is not formatted correctly, the computer not only doesn't give the expected output, it will respond with a long list of errors. If you were to approach a chemist with an astounding result, but express your result in the language of alchemy, you would be ignored. Not because the chemist is out to get you, but because they are busy with their work. Mathematicians expect results to be presented a certain way. If a result is handed to them in some other way, they most often will simply not read it. This is because it is much more difficult to read something that isn't presented in the way they expect, and additionally because there are many people who don't know what they are doing. It is a waste of time to spend alot of time working through a paper that is hard to read because it is written in a strange way, only to realize that it is either nonsensical or just completely wrong. There are many such papers. Math departments at universities are inundated with proofs of squared circles and so on, along with less inane things. It would be a generous person indeed who carefully read each letter and carefully responded. Are math journals filled with dumb crap? In the journals I have read I've seen only thoughtful papers. Each has been carefully reviewed, and the author spent alot of time carefully writing it. This doesn't garantee that it will be correct or even especially insightful, but it certainly is the best effort of those involved. It was allowed into the journal on it's own merit, and is almost certainly correct becuase of the time and effort spent checking it, by very smart people who have nothing to gain, and alot to lose, by publishing a false result. If the results aren't very insightful or important, it is probably because those papers are the best that were available, or that one doesn't have the specialized knowlege to understand why they are insightful or important. No one has any interest in preventing your work from being recognized. Quite the opposite. When someone revolutionalizes mathematics, it doesn't discredit every previous result. Instead, it opens new areas of mathematics to exploration. This would only lead to many more papers to be published, and it would certainly add to the prestige of those publishing at this time, since history often remembers those who are working on something in it's early stages. There isn't a lie of 'pure math'. I'm not sure exactly what you mean here. It isn't something to be believed in, or asserted. You are making a category error here. You use 'pure math' as though it is an assertion. 'pure math' is only defined when you also consider 'applied math'. 'pure math' is mathematics with no alterior motive, so to speak. Applied math is using math to solve a specific problem or class of problems in the physical world. You might say that pure math is defined as everything that is not applied math. Beyond this, there really isn't any meaning to the word. I'll end this with an open offer. If you want to learn some things, I would be glad to help however I can. Within reason of course. This newsgroup presents an opportunity for smart people to interact with many other smart people. So if you were to get a copy of Hersteins Topics in Algebra and work through some of the problems, I would be glad to check them over or to make suggestions if you needed it. You can get this book at amazon.com. The result of this would be twofold. First, you would learn the standard way in which mathematical results are presented. Secondly, you would learn alot of very interesting mathematics, and add to your mathematical toolbox so to speak. Sorry the post was so long, I just started taking ritalin again, Justin Van Winkle http://atheism.about.com/library/glossary/general/bldef_categoryerror.htm http://www.amazon.com/exec/obidos/tg/detail/-/0471010901/qid=1069019047/sr=8 -1/ref=sr_8_1/102-4193208-0966565?v=glance&n=507846 Wow, Topics is almost a hundred dollars. Here is an alternative: http://www.amazon.com/exec/obidos/tg/detail/-/0070026556/qid=1069019119/sr=1 -1/ref=sr_1_1/102-4193208-0966565?v=glance&s=books These Schaum's Outlines are very good in my experience. I first learned about them from this book: http://www.amazon.com/exec/obidos/tg/detail/-/0817638660/qid=1069019208/sr=1 -4/ref=sr_1_4/102-4193208-0966565?v=glance&s=books > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. But you sit by as if there's nothing sinister going on, when all these > people are fighting such a choice result. Yet in your ENTIRE CAREERS most of you will never publish anything > that's even close in neatness, but you'll fill up journals anyway. That's the math world. If there's anything I really like here it's proving the lie to all of > you of pure math, as sure, you may convince others that my work > isn't important, but those of you desperate to find something worth > publication in your publish or perish world know the reality. Pure math is a fraud, just something you say, when you don't believe > in it. It's just a way to get by and pay your bills. Yes, to me you are > losers, too weak to handle the truth, so you think you can crap on me, > when what you do anyway is fill up journals with junk. I'm better than you, and you know it. > James Harris ==== > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. But you sit by as if there's nothing sinister going on, when all these > people are fighting such a choice result. Yet in your ENTIRE CAREERS most of you will never publish anything > that's even close in neatness, but you'll fill up journals anyway. That's the math world. If there's anything I really like here it's proving the lie to all of > you of pure math, as sure, you may convince others that my work > isn't important, but those of you desperate to find something worth > publication in your publish or perish world know the reality. Pure math is a fraud, just something you say, when you don't believe > in it. It's just a way to get by and pay your bills. Yes, to me you are > losers, too weak to handle the truth, so you think you can crap on me, > when what you do anyway is fill up journals with junk. I'm better than you, and you know it. > James Harris If you had any common sense, you'd go learn some math so you could know what we're talking about instead of the immature name calling. Yes you ARE Immature, and it shows. David Moran ==== I take it Jimmy that you have given up on the new marketing strategy. Back to the old tricks, eh! Lurch > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. But you sit by as if there's nothing sinister going on, when all these > people are fighting such a choice result. Yet in your ENTIRE CAREERS most of you will never publish anything > that's even close in neatness, but you'll fill up journals anyway. That's the math world. If there's anything I really like here it's proving the lie to all of > you of pure math, as sure, you may convince others that my work > isn't important, but those of you desperate to find something worth > publication in your publish or perish world know the reality. Pure math is a fraud, just something you say, when you don't believe > in it. It's just a way to get by and pay your bills. Yes, to me you are > losers, too weak to handle the truth, so you think you can crap on me, > when what you do anyway is fill up journals with junk. I'm better than you, and you know it. > James Harris ==== > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. But you sit by as if there's nothing sinister going on, when all these > people are fighting such a choice result. Yet in your ENTIRE CAREERS most of you will never publish anything > that's even close in neatness, but you'll fill up journals anyway. That's the math world. If there's anything I really like here it's proving the lie to all of > you of pure math, as sure, you may convince others that my work > isn't important, but those of you desperate to find something worth > publication in your publish or perish world know the reality. Pure math is a fraud, just something you say, when you don't believe > in it. It's just a way to get by and pay your bills. Yes, to me you are > losers, too weak to handle the truth, so you think you can crap on me, > when what you do anyway is fill up journals with junk. I'm better than you, and you know it. James Harris Oh James... I love it when you talk like that. You are so... ... forceful . -- Clive Tooth http://www.clivetooth.dk ==== > > With all the dumb crap in math journals, you people know that my find > of a way to count prime numbers by integrating a partial difference > equation is worthy of publication somewhere. >But you sit by as if there's nothing sinister going on, when all these > people are fighting such a choice result. >Yet in your ENTIRE CAREERS most of you will never publish anything > that's even close in neatness, but you'll fill up journals anyway. >That's the math world. Does the sound like sour grapes? ==== | |> |>|Now it occurs to me that another simple generalization step can bring |>|the concept to a categorical formulation: let A,B be categories and |>|let (A,A) (the monoid of functors A->A) be regarded as a category |>|itself. Then choose a functor sigma:B->(A,A) and define |>| |>|(a2,b2)(a1,b1):=(a2sigma(b2)a1,b2b1) |>| |>|for all a2,b2,a1,b1 s.t. Dom(a2)=sigma(b2)Cod(a1), Dom(b2)=Cod(b1). |> |>actually i'm not sure i understand your notation well enough to see |>whether what you're describing actually works, but i'm a bit skeptical |>about it because you don't seem to be giving the most straightforward |>generalization of the semi-direct product construction to a context |>involving arbitrary categories. | |Well, I didn't know the construction mentioned hereafter, and in this |respect my one is indeed extremely naive. But I don't see how it |could not work, and what is not clear: | |sigma is a functor B->(A,A), so sigma(b) is a functor A->A for all |morphisms b. Its action on objects is obviously trivial. Note that |sigma(e)=1_A (the identical functor) for all identities e of B. | |The notation (a2,b2)(a1,b1):=(a2sigma(b2)a1,b2b1) is a |straightforward generalization of that used for groups/monoids. The |only difference being that the obvious compatibility relation for the |composition of the given morphisms is required: | |a1:X->Y, a2:sigma(b2)Y->Z, |b1:S->T, b2:T->U. | |Verification that both the associative and the identity properties |hold is merely a mechanical task. | |>the most straightforward such generalization is some version of the |>homotopy colimit construction. you start with a category c and a |>functor (or in some versions a pseudo-functor) f from c to the |>category of categories. the homotopy colimit of f is the category |>where an object is a pair (x,y) with x an object in c and y an object |>in f(x), and a morphism from (x,y) to (z,w) is a pair (m,n) with |>m:x->z in c and n:f(m)(y)->w in f(z), with composition of morphisms |>defined in a semi-obvious way. |> |>semi-direct product of monoids is then the special case where c has a |>unique object x and f(x) has a unique object y. | |Then if I'm not mistaken (or misunderstanding), my construction is |the special case when c is arbitrary and f(x) has a unique object A |(in the notation above). In other words it is the same construction |with the difference that the codomain of f is restricted to a |subcategory of the category of categories, namely the category of |functors A->A. certainly the special case where [for any object x in c, f(x) has a unique object] can be considered, but i'm still confused about whether what you tried to describe is really the same thing. if [for any object x in c, f(x) has a unique object], then what we're dealing with is something like a functor from c to (some version of) the category of monoids, but i didn't see anything in your description that really sounded like that. but it's not implausible to me that you might be trying to describe the same thing in different language, because despite what you say i honestly find your notation and terminology confusing, and it makes me wonder whether you might have made what i call a level slip somewhere- getting concepts on the level of objects mixed up with concepts on the level of morphisms, or something like that. -- ==== >certainly the special case where [for any object x in c, f(x) has a >unique object] can be considered, but i'm still confused about whether >what you tried to describe is really the same thing. if [for any >object x in c, f(x) has a unique object], then what we're dealing with >is something like a functor from c to (some version of) the category >of monoids, but i didn't see anything in your description that really >sounded like that. but it's not implausible to me that you might be >trying to describe the same thing in different language, because >despite what you say i honestly find your notation and terminology >confusing, and it makes me wonder whether you might have made what i >call a level slip somewhere- getting concepts on the level of >objects mixed up with concepts on the level of morphisms, or something >like that. I don't think I've made a level slip. To be sure I'm writing the whole lot from scratch: maybe my wording will be more fortunate... (or precise!) Let's start with monoids, say A,B. Aut(A) is a monoid itself, so there may well be something in Hom(B,Aut(A)), and indeed -incidentally- there always is! So, chosen f in Hom(B,Aut(A)), we can define the semidirect product as usual: (*) (a2,b2),(a1,b1)|->(a2 f(b2) a1,b2 b1). Now, as a matter of a fact a Category turns out to be a sort of big monoid with possibly more than one identity and a product not defined for all elements, right? (intendedly loosely speaking!) Now instead of Aut(A) we have the monoid of functors A->A. But a monoid IS a category, with just one object. So we take into account Hom(B,Hom(A,A)): again it is not empty and if we choose f in it we can define a composition *exactly* as in (*) provided that all compositions in it are well defined, i.e. the domain of a2 is the image through f(b2) of the codomain of a1 and the domain of b2 is the codomain of b1. Please do not make me write down extensively (in an ASCII environment) the (trivial) proof that both associativity and identity properties hold! TIA, Michele -- > Comments should say _why_ something is being done. Oh? My comments always say what _really_ should have happened. :) - Tore Aursand on comp.lang.perl.misc ==== I'm having difficulties solving these two limits. I mustn't use L'hospital rule: a) lim(x*(2^(1/x))-x) where x increases to infinite. b) lim(cosh(x)-1)/(x^2) where x approaches 0. ==== > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: > a) lim(x*(2^(1/x))-x) where x increases to infinite. > b) lim(cosh(x)-1)/(x^2) where x approaches 0. ====== SOLUTION for a) in a more general form ====== L:=lim_{x-->infty}F(x) where F:(0,infty)-->R , F(x):= x*(A^{1/x}-1) , with A>0 , A=/=1 . ==== SOLUTION 1 : by supposing as known that lim_{z-->0}(1+z)^{1/z}= e , where ,,e is Napier's constant. Let z:= A^{1/x}-1 . Then x-->infty iff z--->0 .Also x=ln(A)/ln(1+z) . Therefore F(x)= ln(A)/(ln(1+z)^(1/z)) and L=lim_{z-->0} ln(A)/(ln(1+z)^{1/z})= ln(A)/ln(e)=ln(A). === SOLUTION 2 : by supposing known the theory of Riemann integral, and in particular case when x take only integer values. More precisely consider the sequence with general term X_n:= n(A^{1/n}-1) . Let us assume that A>1. On interval [1,A] take the division (D_n) (D_n) 1=x_0(n)infty iff norm ||D_n||:=max_{k=0,1,...,n}(x_{k+1}(n)-x_k(n))= =A*(A^(1/n)-1)-->0 . Take the continuous function f:[1,A]-->R , f(x)=1/x and consider the integral sum S_n(f)=S((D_n), f )= SUM_{k=0 to k=n}(x_(k+1)-x_k(n))f(x_k(n)). We have X_n= S_n(f) . In this manner lim_{n-->infty}X_n = lim_{||D_n||-->0}S_n(f)= =INTEGRAL_{t=1 to t=A}dt/t= ln(A). ================ ==== In sci.math, Roy <7a108ddd.0311161336.5e01fb3e@posting.google.com>: > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: > a) lim(x*(2^(1/x))-x) where x increases to infinite. > b) lim(cosh(x)-1)/(x^2) where x approaches 0. (a) = lim{x->oo} x*(2^(1/x) - 1) = lim{x->oo}x*(exp(ln2/x) - 1) = lim{y->0} (exp(y*ln 2) - 1) / y (y = 1/x) It's now clearly a derivative. Note that we do *not* need to worry about the chain rule here; all we're doing is switching variables within a limit. In fact, using z = ln2/x is instructive; one gets (a) = lim{z->0} (exp(z) - 1) * ln(2) / z which gives the same answer anyway. Or one can write (a) = lim{y->0} (2^y - 1)/y = lim{y->0} {exp(y*ln2) - 1)/y (b) might be doable by setting x^2 = y (y = sqrt(x)); note that one *has* to use the chain rule in this case. A little confusing perhaps but remember that a derivative is lim{d->0} (f(x+d) - f(x))/d; (the traditional notation is delta x, but ASCII is a pain at times :-) ); the value lim{d->0} (f(x+d)-f(x))/d^2 is something else entirely. However, since cosh'(0) = sinh(0) = 0, it works in this case. Note that both are a special case of L^Hopital's rule (note spelling): lim{x->c}f(x)/x = lim{x->c}f'(x)/1 = f'(c) since the derivative of x is the constant 1; be careful how you explain your answer. -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: > a) lim(x*(2^(1/x))-x) where x increases to infinite. We want lim_{x->oo} [2^(1/x)) - 1]/(1/x). As x -> oo, 1/x -> 0. So this is nothing but the derivative of 2^x at x = 0. > b) lim(cosh(x)-1)/(x^2) where x approaches 0. Use Taylor series as others have suggested, or if you're feeling adventurous, show that for any C^2 function f in a neighborhood of 0 with f'(0) = 0, (f(x) - f(0))/x^2 -> f''(0)/2 as x -> 0. ==== > I'm having difficulties solving these two limits. >> I mustn't use L'hospital rule: >> a) lim(x*(2^(1/x))-x) where x increases to infinite. We want lim_{x->oo} [2^(1/x)) - 1]/(1/x). As x -> oo, 1/x -> 0. So this is >nothing but the derivative of 2^x at x = 0. > b) lim(cosh(x)-1)/(x^2) where x approaches 0. Use Taylor series as others have suggested, or if you're feeling >adventurous, show that for any C^2 function f in a neighborhood of 0 with >f'(0) = 0, (f(x) - f(0))/x^2 -> f''(0)/2 as x -> 0. For b), think cosh(x) = sqrt(1 + sinh(x)^2). Rationalize the numerator by multiplying both numerator and denominator by 1 + sqrt(1 + sinh(x)^2) = 1 + cosh(x). Or substitute x = 2*y and use double-angle formulae. sinh(x) is better-behaved in a limit problem since sinh(x) approaches zero as x -> 0. -- After California's recall election, wildfires Schwartz-en-ed the Bush-lands on its geographic right (when we wanted the forests to be Green). pmontgom@cwi.nl Home: San Rafael, California Microsoft Research and CWI ==== > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: The only alternative which springs to mind is to expand in series valid for the limit under consideration. > a) lim(x*(2^(1/x))-x) where x increases to infinite. Write y = 1/x; take limit y -> 0: lim (2^y - 1)/y = lim (exp(y log2) - 1) / y > b) lim(cosh(x)-1)/(x^2) where x approaches 0. > This is an easier exercise in Taylor series expansion. -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. ==== > > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: > > The only alternative which springs to mind is to expand in series valid > for the limit under consideration. > > a) lim(x*(2^(1/x))-x) where x increases to infinite. > > Write y = 1/x; take limit y -> 0: > > lim (2^y - 1)/y = lim (exp(y log2) - 1) / y > > b) lim(cosh(x)-1)/(x^2) where x approaches 0. > > This is an easier exercise in Taylor series expansion. Sorry, but I can't use series development. 10x, Roy ==== Roy escribi.97 en el mensaje > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: > a) lim(x*(2^(1/x))-x) where x increases to infinite. > b) lim(cosh(x)-1)/(x^2) where x approaches 0. Can you use series developments? -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com ==== > Roy escribi.97 en el mensaje > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: > a) lim(x*(2^(1/x))-x) where x increases to infinite. > b) lim(cosh(x)-1)/(x^2) where x approaches 0. > > Can you use series developments? No, I can't. 10x, Roy ==== >> Roy escribiÌÒ en el mensaje >> I'm having difficulties solving these two limits. >> I mustn't use L'hospital rule: >> a) lim(x*(2^(1/x))-x) where x increases to infinite. >> b) lim(cosh(x)-1)/(x^2) where x approaches 0. >> >> Can you use series developments? > No, I can't. What a shame. They make such problems much easier :-( -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== In sci.math, Robin Chapman : > > Roy escribi.97 en el mensaje > I'm having difficulties solving these two limits. > I mustn't use L'hospital rule: > a) lim(x*(2^(1/x))-x) where x increases to infinite. > b) lim(cosh(x)-1)/(x^2) where x approaches 0. > > Can you use series developments? >> No, I can't. > > What a shame. They make such problems much easier :-( > All this one needs is a simple variable change. -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== This is not homework. Could anyone with some numbercrunching power help solve these following diophantine equations (there are infinitely many solutions but I am looking for the one with the smallest A) (and of course, nonnegative integers only please!): A^2 = 729B + 364 A^2 = 2187B + 1093 A^2 = 6561B + 3280 A^2 = 19683B + 9841 The general pattern is A^2 = (3^k)B + SUM( i = 0 to k-1 )3^i k > 5 But help with even just 1 of the equations would be much appreciated. S.A. ==== > This is not homework. Could anyone with some numbercrunching > power help solve these following diophantine equations (there are > infinitely many solutions but I am looking for the one with the > smallest A) (and of course, nonnegative integers only please!): > > A^2 = 729B + 364 > > A^2 = 2187B + 1093 > > A^2 = 6561B + 3280 > > A^2 = 19683B + 9841 > > The general pattern is > A^2 = (3^k)B + SUM( i = 0 to k-1 )3^i k > 5 See also Robert Israel's posts in the thread, Squares that end with four identical digits -- ==== This is not homework. Could anyone with some numbercrunching >power help solve these following diophantine equations (there are >infinitely many solutions but I am looking for the one with the >smallest A) (and of course, nonnegative integers only please!): A^2 = 729B + 364 A^2 = 2187B + 1093 A^2 = 6561B + 3280 A^2 = 19683B + 9841 OK, you're looking at successive approximations to the 3-adic square root of -1/2 So if you ask magma p := pAdicRing(3, 100); Sqrt(p!(-1/2)) you get 14918756739905062250418133874814682552290432142 and if you consider that large number mod 3^n you get the values of A that work at each step. For example, 475^2 = 729 * 309 + 365 1933^2 = 2187 * 1708 + 1093 You just need to look at X, X+3^n and X+2*3^n, modulo 3^(n+1), to see which digit to put on the beginning at each stage. I'm afraid I don't have a good reference for the p-adic numbers; they turn up incidentally in most number-theory books, but you're expected to absorb them instantaneously. Tom ==== > This is not homework. Could anyone with some numbercrunching >power help solve these following diophantine equations (there are >infinitely many solutions but I am looking for the one with the >smallest A) (and of course, nonnegative integers only please!): > >A^2 = 729B + 364 > >A^2 = 2187B + 1093 > >A^2 = 6561B + 3280 > >A^2 = 19683B + 9841 > > OK, you're looking at successive approximations to the 3-adic square root of > -1/2 > > So if you ask magma > > p := pAdicRing(3, 100); > Sqrt(p!(-1/2)) > > you get 14918756739905062250418133874814682552290432142 That's one of the square roots - its negative is the other > and if you consider that large number mod 3^n you get the values of A that > work at each step. But you might not get the smallest A, which OP asked for, as that might sometimes come from the given square root and sometimes from the negative. -- ==== > But help with even just 1 of the equations would be much appreciated A^2 = B*c + d ---------------------- 254^2 = 29*2187 + 1093 254^2 = 88*729 + 364 475^2 = 309*729 + 364 1933^2 = 569*6561 + 3280 983^2 = 1325*729 + 364 1933^2 = 1708*2187 + 1093 1204^2 = 1988*729 + 364 2441^2 = 2724*2187 + 1093 4628^2 = 3264*6561 + 3280 8494^2 = 3665*19683 + 9841 1712^2 = 4020*729 + 364 1933^2 = 5125*729 + 364 11189^2 = 6360*19683 + 9841 4120^2 = 7761*2187 + 1093 2441^2 = 8173*729 + 364 2662^2 = 9720*729 + 364 4628^2 = 9793*2187 + 1093 8494^2 = 10996*6561 + 3280 3170^2 = 13784*729 + 364 3391^2 = 15773*729 + 364 6307^2 = 18188*2187 + 1093 11189^2 = 19081*6561 + 3280 3899^2 = 20853*729 + 364 6815^2 = 21236*2187 + 1093 4120^2 = 23284*729 + 364 4628^2 = 29380*729 + 364 4849^2 = 32253*729 + 364 8494^2 = 32989*2187 + 1093 15055^2 = 34545*6561 + 3280 9002^2 = 37053*2187 + 1093 5357^2 = 39365*729 + 364 28177^2 = 40336*19683 + 9841 5578^2 = 42680*729 + 364 17750^2 = 48020*6561 + 3280 30872^2 = 48421*19683 + 9841 6086^2 = 50808*729 + 364 10681^2 = 52164*2187 + 1093 6307^2 = 54565*729 + 364 11189^2 = 57244*2187 + 1093 6815^2 = 63709*729 + 364 7036^2 = 67908*729 + 364 21616^2 = 71216*6561 + 3280 12868^2 = 75713*2187 + 1093 7544^2 = 78068*729 + 364 13376^2 = 81809*2187 + 1093 7765^2 = 82709*729 + 364 24311^2 = 90081*6561 + 3280 8273^2 = 93885*729 + 364 8494^2 = 98968*729 + 364 15055^2 = 103636*2187 + 1093 15563^2 = 110748*2187 + 1093 9002^2 = 111160*729 + 364 47860^2 = 116373*19683 + 9841 9223^2 = 116685*729 + 364 28177^2 = 121009*6561 + 3280 50555^2 = 129848*19683 + 9841 9731^2 = 129893*729 + 364 9952^2 = 135860*729 + 364 17242^2 = 135933*2187 + 1093 17750^2 = 144061*2187 + 1093 30872^2 = 145264*6561 + 3280 10460^2 = 150084*729 + 364 10681^2 = 156493*729 + 364 11189^2 = 171733*729 + 364 19429^2 = 172604*2187 + 1093 11410^2 = 178584*729 + 364 19937^2 = 181748*2187 + 1093 34738^2 = 183924*6561 + 3280 11918^2 = 194840*729 + 364 12139^2 = 202133*729 + 364 37433^2 = 213569*6561 + 3280 21616^2 = 213649*2187 + 1093 12647^2 = 219405*729 + 364 22124^2 = 223809*2187 + 1093 12868^2 = 227140*729 + 364 67543^2 = 231776*19683 + 9841 13376^2 = 245428*729 + 364 70238^2 = 250641*19683 + 9841 13597^2 = 253605*729 + 364 23803^2 = 259068*2187 + 1093 41299^2 = 259961*6561 + 3280 24311^2 = 270244*2187 + 1093 14105^2 = 272909*729 + 364 14326^2 = 281528*729 + 364 43994^2 = 294996*6561 + 3280 14834^2 = 301848*729 + 364 25990^2 = 308861*2187 + 1093 15055^2 = 310909*729 + 364 26498^2 = 321053*2187 + 1093 15563^2 = 332245*729 + 364 15784^2 = 341748*729 + 364 47860^2 = 349120*6561 + 3280 28177^2 = 363028*2187 + 1093 16292^2 = 364100*729 + 364 16513^2 = 374045*729 + 364 28685^2 = 376236*2187 + 1093 87226^2 = 386545*19683 + 9841 50555^2 = 389545*6561 + 3280 17021^2 = 397413*729 + 364 17242^2 = 407800*729 + 364 89921^2 = 410800*19683 + 9841 30364^2 = 421569*2187 + 1093 17750^2 = 432184*729 + 364 30872^2 = 435793*2187 + 1093 17971^2 = 443013*729 + 364 54421^2 = 451401*6561 + 3280 18479^2 = 468413*729 + 364 18700^2 = 479684*729 + 364 32551^2 = 484484*2187 + 1093 57116^2 = 497216*6561 + 3280 33059^2 = 499724*2187 + 1093 19208^2 = 506100*729 + 364 19429^2 = 517813*729 + 364 19937^2 = 545245*729 + 364 34738^2 = 551773*2187 + 1093 20158^2 = 557400*729 + 364 60982^2 = 566804*6561 + 3280 35246^2 = 568029*2187 + 1093 106909^2 = 580680*19683 + 9841 20666^2 = 585848*729 + 364 20887^2 = 598445*729 + 364 109604^2 = 610325*19683 + 9841 63677^2 = 618009*6561 + 3280 36925^2 = 623436*2187 + 1093 21395^2 = 627909*729 + 364 37433^2 = 640708*2187 + 1093 21616^2 = 640948*729 + 364 22124^2 = 671428*729 + 364 22345^2 = 684909*729 + 364 67543^2 = 695329*6561 + 3280 39112^2 = 699473*2187 + 1093 22853^2 = 716405*729 + 364 39620^2 = 717761*2187 + 1093 23074^2 = 730328*729 + 364 70238^2 = 751924*6561 + 3280 23582^2 = 762840*729 + 364 23803^2 = 777205*729 + 364 41299^2 = 779884*2187 + 1093 41807^2 = 799188*2187 + 1093 24311^2 = 810733*729 + 364 126592^2 = 814181*19683 + 9841 24532^2 = 825540*729 + 364 74104^2 = 836976*6561 + 3280 129287^2 = 849216*19683 + 9841 25040^2 = 860084*729 + 364 43486^2 = 864669*2187 + 1093 25261^2 = 875333*729 + 364 43994^2 = 884989*2187 + 1093 76799^2 = 898961*6561 + 3280 25769^2 = 910893*729 + 364 25990^2 = 926584*729 + 364 45673^2 = 953828*2187 + 1093 26498^2 = 963160*729 + 364 46181^2 = 975164*2187 + 1093 26719^2 = 979293*729 + 364 80665^2 = 991745*6561 + 3280 for 1 <= B <= 1,000,000 courtesy of GP-Pari and about 4 minutes time. ==== > This is not homework. Could anyone with some numbercrunching > power help solve these following diophantine equations (there are > infinitely many solutions but I am looking for the one with the > smallest A) (and of course, nonnegative integers only please!): > > A^2 = 729B + 364 A = 254 (you can work out the B value) > A^2 = 2187B + 1093 254 > A^2 = 6561B + 3280 1933 > A^2 = 19683B + 9841 8494 > The general pattern is > A^2 = (3^k)B + SUM( i = 0 to k-1 )3^i k > 5 You're after the 3-adic expansion of sqrt(-1/2). I wouldn't expect much of a pattern in the answers. -- ==== Dear Michelle, I'm really glad that you didn't forget this discussion, only don't understand, why did you take just this place of it. Someone likes trapezoid, someone other - picket fence, this is of no importance for integral. Of course, if you integrate numerically (even with a very small step), there will be the difference, and the more step the more difference. Numerical integration generally is better in parabolic curves, and even in curves of higher order. When you pass to infinitesimals, all these differences disappear, because ALWAYS only first term of expansion remains - just as in differentiating. The fact that expansion in Fourier series in irreversible, especially after operations with this series - this also is a well-known difficulty which one can solve only finding some class of analytical functions which would generalise all existing functions and would continuously transform from one analytic function into another with the parameters variation. I'm afraid, this task in general case is some alike seeking the philosophic stone. But in particular problems it is possible. I don't remember, have I told it in that discussion, but we dealt with such functions when studied the models of resistant elastic lines. The shape of one such function you can see in Fig. 2, page 22 of our paper Some features of vibrations in homogeneous 1d resistant elastic lumped line, http://angelfire.lycos.com/la3/selftrans/v2_1/resist22/resist22.html This function is described by the algebraic expression (16) in page 20 of that paper, http://angelfire.lycos.com/la3/selftrans/v2_1/resist20/resist20.html As you can see, the function varies with varied r from the asymptotic line to a line with a bend, retaining always analytical and algebraic! Besides, in the limiting case r = 0, this curve is described by a system of two functions - linear and asymptotically descending, with a bend at critical regime (see (25) in the page 22)! With it the analytical function approaches to a broken curve WITH ANY ACCURACY. Anyway, lest to build a separate plot in the diagram in Fig. 2, we built the limiting curve with the use of analytical function. Yes, this is a complicated function even in the complex form. Its complicacy grows even more in passing to real variables. Its integration is out of question. But we needn't to integrate, as this is just the solution of problem. If we have another elastic line, there will be simply another solution. ;-) Have you ever thought in this direction? ;-) I can show you a function that changes the number of resonance peaks with changing parameter, and many other. This means, transforming functions exist, just as exist the functions that combine several analytical functions. This is just what differs our solutions from conventional techniques. We yield these complicated functions in the end of solving, not trying to pluck the solution out of integral or to rape the Green function. This is why we have not problems which make a trouble to others. Though we have other problems, indeed. ;-) However this pragmatism is not utilitarianism. Our solutions are exact, analytic, not replicable by other techniques, they can fine work also in combination with numerical techniques, providing the accuracy and effectiveness of these last. But the main, our solutions are in full agreement with experiments. This is why I cannot understand, what for have you raised from archive that thought from our dialogue with Archimedes. Could you explain your meaning better? Sergey. PS: And there exists, of course, relation between the geometric representation of second derivative and acceleration in physics. ;-) > >> just this Archimedes Plutonium meant when said of a slope of one of >> rectangle sides. To simplify a trapezium to a rectangle is simpler >> than to resume the lost information of the slope. So in most cases we >> are unable to integrate. The same, we can easily expand a function >> into Fourier series, but to reconstruct the original by the Fourier >> series - it's practically unsolvable problem. Or, if a function has > >better than picketfence. > >Can you think of a term for a rectangle that has an end-sides of just a >mere point rather than a true rectangle whose four sides are more than >just a point? > >In the derivative there are no strange objects but in the integral there >is this >strange object of a rectangle whose end-sides are mere one point. > >So in differentiation, there appears to be no real strange objects for a >set of trapezoids is normal geometric objects but in integration we have >this strange set of objects of rectangles whose end sides are one point. > >And thus, I see geometrically the inverse relationship between derivative >and integral as that between trapezoids to one-point-sided rectangles. When >differentiating one makes trapezoids and when one integrates they collapse >the trapezoids into these strange rectangles. > >Sergey, can you comment on the issue that derivative has normal geometric >objects of trapezoids (what I call picketfences) yet the integral relies >upon these >strange abnormal geometric objects of a collapsed rectangle whose end-sides >consist of one mere point. >i think trapezoids are stupid!!!! > >So can we say that the essence of the Inverse relation between >derivative and >integral is the geometric idea that one is to expand strange rectangles into >trapezoids and the other is to collapse trapezoids into one-point end sided >rectangles. > >P.S. I do not know if I should bother with a geometric explanation of the >2nd derivative that is acceleration in physics. > >whole entire Universe is just one big atom where dots >of the electron-dot-cloud are galaxies ==== is there a way to compute sqrt(-1) mod p easily (for a large prime p, say 7-digits)? i know that -1 has a square mod p by the law of quadratic reciprocity, but that doesnt help constructing an x such that x^2+1=0 (mod p). i also know that the problem is equivalent to find numbers of order 4 (mod p), since order of sqrt(-1) is 4, but yet again that seems like a difficult problem. i am trying to do this without checking every number between 1 and p. anyone know about this? thanks ==== Henri Cohen's book A Course in Computational Algebraic Number Theory(Graduate Texts in Mathematics, Vol 138), Springer-Verlag has a nice explanation of efficient ways to compute square roots modulo p in general, although as one poster pointed out, there are may well be quicker ways to compute the square root of -1. JoeS > is there a way to compute sqrt(-1) mod p easily (for a large prime p, > say 7-digits)? i know that -1 has a square mod p by the law of > quadratic reciprocity, but that doesnt help constructing an x such > that x^2+1=0 (mod p). i also know that the problem is equivalent to > find numbers of order 4 (mod p), since order of sqrt(-1) is 4, but yet > again that seems like a difficult problem. > > i am trying to do this without checking every number between 1 and p. > > anyone know about this? > > thanks ==== |>is there a way to compute sqrt(-1) mod p easily (for a large prime p, |>say 7-digits)? i know that -1 has a square mod p by the law of |>quadratic reciprocity, but that doesnt help constructing an x such |>that x^2+1=0 (mod p). i also know that the problem is equivalent to |>find numbers of order 4 (mod p), since order of sqrt(-1) is 4, but yet |>again that seems like a difficult problem. In Maple 9, for example: > msolve(x^2=-1, 1299709); {x = 329008}, {x = 970701} takes very little time, even on my rather slow computer. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== tungsteneer3@yahoo.com (billy d.) asked: > is there a way to compute sqrt(-1) mod p > easily (for a large prime p, > say 7-digits) You can use the continued fraction expansion of sqrt(p) to solve x^2 - py^2 = -1 and then x = sqrt(-1) (mod p). For an efficient way to do this, see Solving x^2 - Dy^2 = +-1 in Solving the generalized Pell equation at http://hometown.aol.com/jpr2718 Other than that, Henri Cohen, A Course in Computational Algebraic Number Theory, Section 1.5 might help. John Robertson ==== > tungsteneer3@yahoo.com (billy d.) asked: > is there a way to compute sqrt(-1) mod p >> easily (for a large prime p, >> say 7-digits) You can use the continued fraction >expansion of sqrt(p) to solve x^2 - py^2 = -1 and then x = sqrt(-1) (mod p). For an >efficient way to do this, see Are you sure this is wise? For p of the range indicated by the OP, one expects to find continued fractions with periods up to the thousands. Even though this is not overwhelmingly large in terms of space or time complexity, I'm not sure I see the point of doing it this way. -- Erick ==== >is there a way to compute sqrt(-1) mod p easily (for a large prime p, >say 7-digits)? i know that -1 has a square mod p by the law of >quadratic reciprocity, but that doesnt help constructing an x such >that x^2+1=0 (mod p). i also know that the problem is equivalent to >find numbers of order 4 (mod p), since order of sqrt(-1) is 4, but yet >again that seems like a difficult problem. It's not difficult at all (unless you demand a deterministic algorithm). For any b, b^[(p-1)/4] has order at most 4 (mod p), so if you pick b randomly there is a 50% chance that it will have order exactly 4 (the only other possibilities are order 1 and order 2 which are +1 and -1 respectively). -- Erick ==== >is there a way to compute sqrt(-1) mod p easily (for a large prime p, >say 7-digits)? i know that -1 has a square mod p by the law of >quadratic reciprocity, You mean, you know that -1 has a square mod p when p is congruent to 1 mod 4; this does not use quadratic reciprocity per se (which is about two odd primes), but rather the Legendre symbol and similar results. > but that doesnt help constructing an x such >that x^2+1=0 (mod p). i also know that the problem is equivalent to >find numbers of order 4 (mod p), since order of sqrt(-1) is 4, but yet >again that seems like a difficult problem. i am trying to do this without checking every number between 1 and p. Usually, one proves that x^2 = -1 (mod p) has a solution if and only if p=2 or p=1 (mod 4) by ->constructing<- a solution. Namely, take (p-1)/2, and compute its factorial mod p. That's a solution, a consequence of Wilson's Theorem. ====================================================================== 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 ==== >is there a way to compute sqrt(-1) mod p easily (for a large prime p, >say 7-digits)? > Usually, one proves that x^2 = -1 (mod p) has a solution if and only > if p=2 or p=1 (mod 4) by ->constructing<- a solution. Namely, take > (p-1)/2, and compute its factorial mod p. That's a solution, a > consequence of Wilson's Theorem. Yes, but if you actually want to compute a solution, finding (p - 1) / 2 factorial mod p is a really bad way to do it, at any rate for the 7-digit p that OP has. A couple of algorithms are given in Crandall & Pomerance, Prime Numbers, Section 2.3.2 -- ==== I find it difficult to find the homology group of S^n x S^m. Can anyone give me some ideas? I am thinking of using the MV sequence (exact): ... -->H_n(A and B)-->H_n(A) + H_n(B)--> H_n(A union B)-->H_(n-1) (A and B) --> ... ==== > I find it difficult to find the homology group of >S^n x S^m. Can anyone give me some ideas? I am thinking of using the MV >sequence (exact): ... -->H_n(A and B)-->H_n(A) + H_n(B)--> H_n(A union B)-->H_(n-1) (A and >B) --> ... I'm not sure that's the best approach but you could make it work. What will you choose for A and B? You might try decomposing S^m = C union D and then take A = S^n x C, B = S^n x D. You'll need to do the calculation for all m this way so that you can use induction. Don't forget to use the naturality of the MV sequence so that you can identify the maps between the groups. By the way, Mr V just died about a year ago -- at the time he was the oldest living person in Austria (110.8 yrs). dave ==== > I find it difficult to find the homology group of > S^n x S^m. Can anyone give me some ideas? Apply the Eilenberg-Zilber and Kunneth theorems. Let S(X) denote the singular complex of a space X. The Eilenberg-Zilber theorem states that S(X x Y) is chain homotopy equivalent to the tensor product of S(X) and S(Y). The Kunneth theorem expresses the homology of the tensor product of two complexes in terms of the homology of each one. Here life is simple since the homology of S^m and of S^n is torsion-free so we can forget the Tor factor in the Kunneth theorem and conclude that the total homology of S^m x S^n is 4-dimensional with generators in dimensions 0, m, n and m+n. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== Looking for a good lie algebra book at the introductory level. thanks. ==== > Looking for a good lie algebra book at the introductory level. Varadarajan's Lie groups, Lie algebras, and their representations and Serre's Lie Algebras and Lie Groups are good choices. Jose Carlos Santos ==== > Looking for a good lie algebra book at the introductory level. Humphreys, James E. Introduction to Lie algebras and representation theory. (2.ed) Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978. should still serve as a good introduction. It requires not muuch more that linear algebra. Marc ==== re: http://qedcorp.com/APS/EmergentGravity.doc http://qedcorp.com/APS/StarGate1.mov Commentary 1 The fiber bundle as an idea has 4 parts. 1. A structure symmetry group G. 2. The total hyperspace H or, in some applications Wheeler's BIT. 3. The projection map P. 4. The base space M or, in some applications. Wheeler's IT. The hyperspace H consists of fibers f(x) that are either copies of or representations of the symmetry group G. The projection map P collapses a fiber f(x) in the hyperspace H to a point x in the base space M. All of these objects are continuum differential manifolds depending on the continuum of real numbers which its associated issues of Cantor's infinity of infinities of Cabalistic Aleph's in an ascending Jacob's Ladder. This is not a discrete combinatoric mathematics although such a skeletal structure is associated with it as in Herman Weyl's Theory of Groups and Quantum Mechanics and as in Saul-Paul Sirag's presentation of V.I. Arnold's A-D-E mathematics of everything. The base space is covered by an atlas of local coordinate patches with all important overlap transition functions sewing the patches together like a quilt. M is space-time in local micro-quantum field theory of point The extra-dimensions of hyperspace form the Calabi-Yau space of vibrations of the superstring beyond space-time. The connection on the total hyperspace H is the potential of a local gauge force. Examples of connections is the 4 potential Au(x) in Maxwell's electromagnetism with G as U(1). There are similar connections for the Yang-Mills weak force with G = SU(2) and the strong force with G = SU(3). Classical general relativity, as distinct from local micro-quantum field theory, has the torsion-free symmetric three-index non-tensor Levi-Civita connection with G as the Diff(4) group. The latter comes from locally gauging the 4 parameter translation subgroup (generated by the 4-momentum Pu of globally flat special relativity ) of the 15 parameter conformal group of Roger Penrose's massless twistors. Bottom -> Up: Given base space M and symmetry group G construct the hyperspace H as a quilt patchwork. Top -> Down: Given hyperspace H and symmetry group G construct the base space M as the non-overlapping partition of hyperspace into G-orbits called the quotient space of H mod G in the principal bundle. Micro-quantum source renormalizable local fields of spin 1/2 lepto-quarks are associated vector bundles. Micro-quantum force renormalizable local fields of spin 1 gauge force bosons (electro-weak and strong) are from the principal bundle. There is no renormalizable quantum gravity in this precise sense. This is because classical Einstein gravity is a More is different (P.W. Anderson) emergent collective effect as in Andrei Sakharov's metric elasticity of an instability in the globally flat false vacuum of the interacting lepto-quark source/electroweak-strong force. Einstein's gravity + unified exotic vacuum dark energy/matter with Andrei Linde's chaotic inflationary cosmology are the result of the continual phase transitions from globally flat false high entropy micro-quantum vacua to locally curved macro-quantum low entropy metastable vacua. to be continued: ==== Mr. Sarfatti Sir How are you..how do you feel? dave ==== [snip] Nothing, as usual, and was prolix about it. Hey Jacko - are you going to present at the 2004 American Physical Society national meeting in Denver next year? Uncle Al knows a fellow who was in your audience last year. He said you had everybody rolling on the floor clutching their tummies - some from laughter, others puking, and the remainder collecting lint for tinder for igniting your faggots. -- Uncle Al http://www.mazepath.com/uncleal/qz.pdf http://www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) ==== Tiz a frightened man whom scoffs! > [snip] > Nothing, as usual, and was prolix about it. Hey Jacko - are you going to present at the 2004 American Physical > Society national meeting in Denver next year? Uncle Al knows a fellow > who was in your audience last year. He said you had everybody rolling > on the floor clutching their tummies - some from laughter, others > puking, and the remainder collecting lint for tinder for igniting your > faggots. -- > Uncle Al > http://www.mazepath.com/uncleal/qz.pdf > http://www.mazepath.com/uncleal/eotvos.htm > (Do something naughty to physics) ==== > God isn't responsible for the bad things that happen, Satan is. Read > your damn Bible, specifically, the Book of Job. >Lol and god created Satan knowing *FULL* well what would happen > (unless hes not omniscient). > So then the question is, if God knew what would happen, then perhaps Satan > isn't such a bad guy after all? Consider: he obeyed God's commands > concerning how much he was allowed to afflict Job. Sounds like an obedient > son of God to me... > Hehe maybe satan isn't? Or maybe god just isn't as powerful or as loving as everyone says he is (if he exists at all)? I don't know what you mean by an obedient son of god but Job did get a bit drilled by the almighty. ==== Ok, I'm trying to work on my homework and am stuck on 4.9 #10 of Vector >Analysis by Davis. The question states By means of Stokes' theorem, find S F*dR around the >ellipse x^2+y^2=1, z=y, where F=xi+(x+y)j+(x+y+z)k. z=y ? what? I got the curl of F and that equalled i-j+k but I'm not really sure how to do >the rest of the problem. Any help would be appreciated. I've wasted a lot of >time and gotten almost nowhere. > write down an expresion for the (vector) dA for your circle... I think that you should use (vector) dA = (scalar) dA k then curl F dot dA is just (scalar) dA and a circle of radius one has area 2 Pi adam ==== >>Ok, I'm trying to work on my homework and am stuck on 4.9 #10 of Vector >>Analysis by Davis. >>The question states By means of Stokes' theorem, find S F*dR around the >>ellipse x^2+y^2=1, z=y, where F=xi+(x+y)j+(x+y+z)k. z=y ? what? >I got the curl of F and that equalled i-j+k but I'm not really sure how to do >>the rest of the problem. Any help would be appreciated. I've wasted a lot of >>time and gotten almost nowhere. write down an expresion for the (vector) dA for your circle... I think that you should use (vector) dA = (scalar) dA k then curl F dot dA is just (scalar) dA and a circle of radius one has area 2 Pi adam uhhh. duh no a circle of radius one has area Pi. hmmm, maybe the problem here is that I'm not sure what the z=y condition means... sorry adam