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What is a word for "non-arbitrarily close" as in "We can find the equation of a function f if we know its degree to be n and n+1 "non-arbitrarily close" points on f"? 68.76.159.51 ( talk) 00:12, 24 August 2010 (UTC)
Certainly there is a unique nth degree polynomial that fits prescribed values at n + 1 distinct points in the domain. I suspect that is what is meant. I don't know what "non-arbitrarily close" means, unless maybe it's a clumsy way of saying "distinct". Michael Hardy ( talk) 01:52, 25 August 2010 (UTC)
I was looking at Dice's coefficient and Jaccard index. In some papers, it is claimed that Dice's coefficient is twice the Jaccard index. In others, it is claimed that Dice's coefficient cannot be translated to Jaccard index. In the articles here, it claims that the relationship between Dice's coefficient (D) and Jaccard index (J) is D=2J/(1+J). It appears that it is converting |X|+|Y| = |X∪Y|+|X∩Y|. Is any of this correct? If I could translate Dice's coefficient directly to Jaccard index, it would be helpful. -- kainaw ™ 12:08, 24 August 2010 (UTC)
If I have a quaternion , which I know is equal to , where the overdot denoted differentiation with respect to time, how is it possible to find ? I know is a unit quaternion if that helps.-- Leon ( talk) 18:13, 24 August 2010 (UTC)
Could the expression have emerged from a chain rule? Did you mean that q varies or that Ω varies? Are you saying Ω is a known unit-quaternion-valued function of time, or maybe that Ω is a fixed (constant) and known unit quaternion? Michael Hardy ( talk) 01:29, 25 August 2010 (UTC)
If
where a, b, c, d are real, then q is a unit quaternion precisely if
and the derivative of that with respect to time is
If I look at and expand it, I get
and the sum of the first four terms vanishes as above. Now is the numerator in an application of the quotient rule, as is , etc. I don't know where this will take us. So just some preliminary scratchwork. Michael Hardy ( talk) 02:02, 25 August 2010 (UTC)
The equation is equivalent to on account of |q| = 1. If it were complex numbers rather than quaternions, then the solutions to this equation would be , where c is any constant (with real part 0 to ensure |q| = 1). I don't know whether one can do such thing with quaternions, but since the complex numbers are a special case (where the j and k components of q and Ω are 0), the solution will have to involve some quaternion generalization of the exponential.— Emil J. 11:43, 25 August 2010 (UTC)
Mathematics desk | ||
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< August 23 | << Jul | August | Sep >> | August 25 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
What is a word for "non-arbitrarily close" as in "We can find the equation of a function f if we know its degree to be n and n+1 "non-arbitrarily close" points on f"? 68.76.159.51 ( talk) 00:12, 24 August 2010 (UTC)
Certainly there is a unique nth degree polynomial that fits prescribed values at n + 1 distinct points in the domain. I suspect that is what is meant. I don't know what "non-arbitrarily close" means, unless maybe it's a clumsy way of saying "distinct". Michael Hardy ( talk) 01:52, 25 August 2010 (UTC)
I was looking at Dice's coefficient and Jaccard index. In some papers, it is claimed that Dice's coefficient is twice the Jaccard index. In others, it is claimed that Dice's coefficient cannot be translated to Jaccard index. In the articles here, it claims that the relationship between Dice's coefficient (D) and Jaccard index (J) is D=2J/(1+J). It appears that it is converting |X|+|Y| = |X∪Y|+|X∩Y|. Is any of this correct? If I could translate Dice's coefficient directly to Jaccard index, it would be helpful. -- kainaw ™ 12:08, 24 August 2010 (UTC)
If I have a quaternion , which I know is equal to , where the overdot denoted differentiation with respect to time, how is it possible to find ? I know is a unit quaternion if that helps.-- Leon ( talk) 18:13, 24 August 2010 (UTC)
Could the expression have emerged from a chain rule? Did you mean that q varies or that Ω varies? Are you saying Ω is a known unit-quaternion-valued function of time, or maybe that Ω is a fixed (constant) and known unit quaternion? Michael Hardy ( talk) 01:29, 25 August 2010 (UTC)
If
where a, b, c, d are real, then q is a unit quaternion precisely if
and the derivative of that with respect to time is
If I look at and expand it, I get
and the sum of the first four terms vanishes as above. Now is the numerator in an application of the quotient rule, as is , etc. I don't know where this will take us. So just some preliminary scratchwork. Michael Hardy ( talk) 02:02, 25 August 2010 (UTC)
The equation is equivalent to on account of |q| = 1. If it were complex numbers rather than quaternions, then the solutions to this equation would be , where c is any constant (with real part 0 to ensure |q| = 1). I don't know whether one can do such thing with quaternions, but since the complex numbers are a special case (where the j and k components of q and Ω are 0), the solution will have to involve some quaternion generalization of the exponential.— Emil J. 11:43, 25 August 2010 (UTC)