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HallsofIvy

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Everything posted by HallsofIvy

  1. In order to combine rotation and translation you need to use "homogeneous coordinates". That is, we represent the point (x, y, z) by the vector (x, y, z, 1) while identifying the vector (ax, ay, az, a), for any non-zero a, with (x, y, z, 1). Then the rotation around the z- axis, through angle [math]\theta[/math] together with translation by (dx, dy, dz), is given by the 4 by 4 matrix [itex]\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 & 0 \\ sin(\theta) & cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ dx & dy & dz & 1\end{bmatrix}[/itex]. [itex]\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 & 0 \\ sin(\theta) & cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ dx & dy & dz & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ 1\end{bmatrix}= \begin{bmatrix}xcos(\theta)- y sin(\theta)+ dx \\ xsin(\theta)+ ycos(\theta)+ dy \\ z+ dz \\ 1\end{bmatrix}[/itex].
  2. With my eyes, that probably wouldn't help! But thanks for the correction.
  3. Actually, the book specifically tells you what omega is in problem 2! There the equation is d^2x/dt^2= omega x. In problem 3 the equation is d^2x/dt^2= -x. Comparing that to problem 2 you should immediately see that problem 3 is the same as problem 2 with omega= 1.
  4. Before you- or anyone- can say which apples are "doing better", you need to decide what you mean by "better". That's part of the set up of the experiment.
  5. Would "a field of daisys" constitute a herd?
  6. It would help if you posted the definitions of "[math]\phi[/math]" and "[math]\sigma_0[/math]"
  7. I would say that the answer is clearly "10". As far as an explanation is concerned my first thought would be to note that the numbers in the left column are decreasing by "1" each time, that 54 is 9 times 6, that 40 is 8 times 5, and that 28 is 7 times 4 so following that pattern, the "next row" would have 3 on the left, and 6 times 3= 18 but that row is missing so the next row has "2" on the left and then 2 times 5= 10. But you don't like that explanation because, for some reason, you don't like the appeal to a "missing row", So, instead, I would say that each number on the left is decreasing by "`1" until we get to the last, "2", which is 2 less than 4. I would also note that each number on the right is the number on the left multiplied by 9, 8, 7, also decreasing by 1. but since for the last step the left number decreased by 2, I would also decrease the multiplier by 2 to 7- 2= 5 and so the number on the right is 2*5= 10. That is the same explanation but without appealing to a "missing row".
  8. I would say that molecules NEVER "evaporate". It is the body of the water that evaporates as molecules leave the body of the water. Of course when you heat water to its boiling point, bubbles will appear in the water. Do you want to count that as "evaporation"?
  9. I always thought that was the point or "Shrodinger's cat"- that quantum actions, a nucleus ejecting a subatomic particle, can have macro consequences, a cat dying or living. As for the human body, or any large collection of quantum objects, the quantum effects will tend to average out.
  10. I am suddenly starting to feel irregular.
  11. " In what sense are functions curves? I think you are talking about the graphs of functions. Do you know the definition of "function"? It follows immediately from that definition that the graph of any function, elementary or not, is a simple curve. In fact, the graph of $x^2+ y^2= 1$ is a circle, so a "simple curve" even though y is not a function of x nor is x a function of y.
  12. I don't know where you got this idea! The last ice age, "when most of the planet was covered in ice" ended about 3,000,000 years ago (perhaps you just dropped ",000"). There was a period, sometimes called "the little ice age" of lower than normal tempartures but no ice sheets that only ended about 200 years ago (that's why the painting "Washington crossing the Delaware" shows ice in the river).
  13. So how do I move there?
  14. HallsofIvy

    Question?

    It's hard to believe you are serious! From your other posts I had thought that you were past third or fourth grade arithmetic- which is basically what this is. I would hope that you know that 1x1= 1, 2x1= 2, and 1- 1= 0. And if 1- ?= ? the, adding "?" to both sides 1= 2?. What number, multiplied by 2, gives1?
  15. "Vast organisms"? Blue whales, today, are much larger than dinosaurs were!
  16. That's not what "consistency" means. Saying the Peano axiom are consistent means that the Peano axioms cannot be used to prove both statement "p" and "not p". Saying that the Peano axioms cannot be used to prove its own consistency means that the axioms cannot be used to prove that statement.
  17. This isn't really relevant to the main point here, but most people don't die because of "events". They die because their body wears out. And that would happen in any universe.
  18. If I remember correctly, a region in $R^2$ is "regular" if its boundary is a simple closed curve. (And a curve is "simple" if it does not cross itself.) Yes, the rectangular region $a\le x\le b$, $c\le y \le d$ is a "regular region". As to the integral, $\int_{u(x)}^{v(x)} f(x,y)dy$, Assuming that f is an "integrable" function of y, then $\int_{u(x)}^{v(x)} f(x,y)dy$ is a function of x, F(x). I might think of the "x" as a "parameter" in the integral but as a variable in f(x,y) and in F(x).
  19. The only thing "The Architect" has definitely shown is that he is NOT a mathematician! (And I am inclined to hope he is also NOT an architect- I would hate to have to live on an upper floor or a building he designed.)
  20. Do you understand that this primarily an exercise in READING COMPEHENSION? (Which is an extremely important skill in programming or software engineering). There is little mathematics or programming involved.
  21. How does that contradict what I said? A person who is on the dole still needs money- staying alive is not living in a satisfying way!
  22. If they had offered "Effing Science" when I was in school, I certainly would have taken it! That is, if "Effing" means what I think it does.
  23. I remember when I was very young thinking about this. Suppose we just let people go into a store and TAKE whatever they want! Since everyone can have whatever they want, no one would complain. Then as I got older I discovered a very important, very disturbing fact- work is hard! Most people would not want to work if they did not have to have money! What if we could just go into a store and take whatever we want, but discovered that there was not anything in the store? Who would plant, grow, and harvest the vegetables? That's hard work! If farmers did not need the money they wouldn't do it! If tailor's did not need the money they wouldn't make shirts, slacks, suits!
  24. "I am a layman trying to understand above theorems. This could be a stupid question." There is no such thing as a stupid question! "Does these theorems imply that we actually cannot prove that 2+2 = 4???" Gosh, I may have to reconsider! No, Godel's theorem say that, given any set of axioms large enough to encompass the properties of the non-negative integers there must exist some theorem that can neither be prove nor disproved. It does not say that a specific theorem cannot be proved. In fact, if we were able to identify a specific theorem that can not be proved nor disproved, we can always extend the axioms, perhaps by adding that theorem itself as an axiom, so that theorem can be proved. Of course, there would then be still another theorem that cannot be proved nor disproved.
  25. So you are arguing that NO ONE can criticize your paper because YOU did not include a "working roadmap" or "outline"?
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