text stringlengths 1 2.12k | source dict |
|---|---|
You can do it with a negative sign too.
You want to find the minimum value of (distance from -6) - (distance from 1)
Make a number line with -6 and 1 on it.
(-6)..........................(1)
Think of a point in the center of -6 and 1. Its distance from -6 is equal to distance from 1 and hence (distance from -6) - (distance from 1) = 0 .
What if instead, the point x is at -6? Distance from -6 is 0 and distance from 1 is 7 so (distance from -6) - (distance from 1) = 0 - 7 = -7
If you keep moving to the left, (distance from -6) - (distance from 1) will remain -7 so the minimum value is -7.
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog
Get started with Veritas Prep GMAT On Demand for \$199
Veritas Prep Reviews
Re: Question of the Day - II [#permalink] 20 Mar 2012, 21:17
Go to page 1 2 3 4 Next [ 68 posts ]
Similar topics Replies Last post
Similar
Topics:
3 If 3^x = 81, then 3^(x+3)*4^(x+1) = 4 03 Jun 2017, 19:10
15 If x<0 What is the value of [(x-3)^4]^1/4 + (-x|x|)^1/2 A. -3 B. x+3 16 23 Jun 2017, 23:28
2 If f(x)=4x−1 and g(x)=2x+3 for all integers, which of the 2 04 Feb 2016, 21:14
4 If x = -1, then -(x^4 + x^3 + x^2 + x) = 7 08 Feb 2014, 03:16
1 If x = -1, then (x^4 - x^3 + x^2)/(x - 1) = 3 28 Oct 2015, 07:51
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 1,
"lm_q1q2_score": 0.8539127548105611,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 1845.2245836373818,
"openwebmath_score": 0.7273833751678467,
"tags": null,
"url": "https://gmatclub.com/forum/if-f-x-4x-1-x-3-x-1-question-of-the-day-ii-103935.html"
} |
# Computing $\int\limits_0^\infty x \left \lfloor{\frac1x}\right \rfloor \, dx$
This is an integral I computed but can't find the result online or on wolfram. So here's a proof sketch, please indulge this sanity check:
$$\int_0^\infty x \left \lfloor{\frac1x}\right \rfloor \ dx = \int_0^1 x \left \lfloor{\frac1x}\right \rfloor \ dx$$ $$= \sum_{n=1}^\infty \int_{1/(n+1)}^{1/n} nx \ dx =\sum_{n=1}^\infty\frac n2 \left(\frac{1}{n^2} - \frac{1}{(n+1)^2}\right)$$ $$= \sum_{n=1}^\infty\frac n2 \left(\frac{2n+1}{n^2(n+1)^2}\right)$$ $$= \sum_{n=1}^\infty\frac{1}{(n+1)^2} + \frac12 \sum_{n=1}^\infty \frac{1}{n(n+1)^2}$$ $$= \frac{\pi^2}{6} -1 + \frac12\left(\sum_{n=1}^\infty \frac1n - \frac{1}{n+1} - \frac{1}{(n+1)^2}\right)$$ $$=\frac{\pi^2}{6} -1 + \frac12\left(\sum_{n=1}^\infty \frac1n - \frac{1}{n+1}\right) -\frac12\left(\sum_{n=1}^\infty\frac{1}{(n+1)^2}\right)$$ $$= \left(\frac{\pi^2}{6} -1\right) + \left(\frac12\cdot 1\right) - \frac12\left(\frac{\pi^2}{6} -1\right)$$ $$= \frac{\pi^2}{12}.$$
Basically, I used the Basel sum several times, and the fifth line follows from a partial sum decomposition. The seventh follows from the known result for the Basel sum, as well as the fact that the first series in the 6th line telescopes.
I hope this is all correct.
• @XanderHenderson For $x>1$, floor of $\frac{1}{x}$ is $0$ – BallBoy Dec 28 '17 at 3:34
• @XanderHenderson What you wrote is certainly not true, because you forgot the $x$ term. However, what Y. Forman says is correct, and I should've been more explicit there. – David Bowman Dec 28 '17 at 3:37
• Oi... derp. Sorry for being dyslexic. I missed the $x$. – Xander Henderson Dec 28 '17 at 3:39
• $$\int_0^1 x \lfloor 1/x \rfloor dx = \int_1^\infty \frac{1}{t} \lfloor t \rfloor \frac{dt}{t^2}= \sum_{n=1}^\infty \int_n^\infty t^{-3}dt = \sum_{n=1}^\infty \frac{n^{-2}}{2} = \frac{\zeta(2)}{2}$$ – reuns Dec 28 '17 at 3:43
• @reuns Nice, better post it as answer! – samjoe Dec 28 '17 at 4:40
Use | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.989013056721729,
"lm_q1q2_score": 0.853905582280985,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 997.5000517186052,
"openwebmath_score": 0.9329135417938232,
"tags": null,
"url": "https://math.stackexchange.com/questions/2582606/computing-int-limits-0-infty-x-left-lfloor-frac1x-right-rfloor-dx"
} |
Use
$$n\left(\frac1{n^2}-\frac1{(n+1)^2}\right)=\frac n{n^2}-\frac{n+1-1}{(n+1)^2}=\frac1n-\frac1{n+1}+\frac1{(n+1)^2}.$$
The first two terms do telescope and the Basel series remains.
• @stressedout: of course, but much simpler. And also simpler than yours, which is also essentially the OP's. – Yves Daoust Sep 5 '18 at 6:28
• @stressedout: your solution isn't different, just longer. A one-liner is simpler. – Yves Daoust Sep 5 '18 at 6:38
Alternatively, one may follow the same line of thought and use summation by parts formula to calculate the infinite sum. Similarly, $$\int_0^{+\infty} x\lfloor \frac{1}{x}\rfloor dx =\sum_{n=1}^{\infty}\frac{n}{2}\left(\frac{1}{n^2}-\frac{1}{(n+1)^2}\right)=-\frac{1}{2}\sum_{n=1}^{\infty}f_n(g_{n+1}-g_n)$$
where $f_n = n$ and $g_n = 1/n^2$. Now, summation by parts for the last expression gives:
$$-\frac{1}{2}\sum_{n=1}^{\infty}f_n(g_{n+1}-g_n) = -\frac{1}{2}\left(\ \lim_{n\to\infty}\frac{n}{(n+1)^2} -1 -\sum_{n=2}^{\infty}\frac{1}{n^2}\right)$$
But it is well-known that $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$
Hence,
$$-\frac{1}{2}\sum_{n=1}^{\infty}f_n(g_{n+1}-g_n) = -\frac{1}{2}\left(\ \lim_{n\to\infty}\frac{n}{(n+1)^2} -1 - (\frac{\pi^2}{6}-1)\right) = \frac{\pi^2}{12}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.989013056721729,
"lm_q1q2_score": 0.853905582280985,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 997.5000517186052,
"openwebmath_score": 0.9329135417938232,
"tags": null,
"url": "https://math.stackexchange.com/questions/2582606/computing-int-limits-0-infty-x-left-lfloor-frac1x-right-rfloor-dx"
} |
# Prove that each integer n ≥ 12 is a sum of 4's and 5's using strong induction
So I've been given the following problem: Prove that each integer n ≥ 12 is a sum of 4's and 5's What I have so far: (Basis):
n ≥ 12 Therefore,
12 ≤ 4(x) + 5(y)
x = 3 | y = 0
12 ≤ 4(3) + 5(0)
12 ≤ 12 = Correct
However, what I don't understand is how would I use the x & y variable in induction step. Where exactly would I place these? and how would I go on to solve this?
The trick to these sorts of problems is to realize that if we can find four consecutive integers ($$4$$ is the smallest of our numbers we're taking a combination of) that can be represented as a non-negative sum of the fours and fives.
For example $$12 = 4(3) + 5(0)$$
$$13 = 4(2) + 5(1)$$ $$14 = 4(1)+ 5(2)$$ $$15 = 4(0) + 5(3).$$
With this information is it clear how you could represent 16? Just take the solution from the 12 case, and add 4! (so increase $$x$$ by $$1$$.)
Here's the formal argument. Let $$P(n)$$ be the open sentence "$$n$$ can be written as a non-negative combination of $$4$$ and $$5$$". By what we've shown above, we know that $$P(k)$$ is true for $$k = 12,13,14,15.$$ We wish to prove that $$P(k+1)$$ is true. Our strong inductive hypothesis is to suppose that for some $$k \in \mathbb{Z}$$ that for every $$i$$ with $$12\leq i\leq k$$ that $$P(k)$$ is true, and we need to prove that $$P(k+1)$$ is true.
If $$k = 12,13$$ or $$14$$, we've already seen that $$p(k+1)$$ is also true, so suppose that $$k \geq 15$$. Then we know that $$12 \leq k-3\leq k$$ so then by our strong inductive hypothesis, there exists $$x,y \in \mathbb{Z}$$ such that $$k-3 = 4(x) + 5(y)$$. Then adding $$4$$ to both sides gives that $$k+1 = 4(x+1)+5(y)$$ so that $$P(k+1)$$ is true. Thus by the principle of mathematical induction, $$P(n)$$ must be true for all $$n \geq 12$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232950125849,
"lm_q1q2_score": 0.8539032067010474,
"lm_q2_score": 0.8688267881258485,
"openwebmath_perplexity": 89.83641628509169,
"openwebmath_score": 0.828149139881134,
"tags": null,
"url": "https://math.stackexchange.com/questions/3113510/prove-that-each-integer-n-%E2%89%A5-12-is-a-sum-of-4s-and-5s-using-strong-induction"
} |
• Wow this is very detailed and helped me understand it more, thank you! – MathNoob Feb 15 '19 at 4:15
• No problem! I'm glad this helped! – JonHales Feb 15 '19 at 4:17
$$4x+5y=\underbrace{4(x-1)+5y}_{n-4}+4$$
So, if $$n-4$$ is expessible so will be $$n$$
$$12=4\cdot3$$
$$13=2\cdot4+5$$
$$14=4+2\cdot5$$
$$15=3\cdot5$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232950125849,
"lm_q1q2_score": 0.8539032067010474,
"lm_q2_score": 0.8688267881258485,
"openwebmath_perplexity": 89.83641628509169,
"openwebmath_score": 0.828149139881134,
"tags": null,
"url": "https://math.stackexchange.com/questions/3113510/prove-that-each-integer-n-%E2%89%A5-12-is-a-sum-of-4s-and-5s-using-strong-induction"
} |
# Finding dimension of a kernel from a linear transformation.
I'm trying to solve an exercise where I'm asked to find this linear transformation's kernel and its dimension:
$$f(x_1,x_2,x_3,x_4)=(x_1-x_2,2x_3+x_4)$$
So, from the definition of kernel and converting to parametric equations, I found $x_1=x_2=\lambda$, $x_3=-\frac{1}{2}\mu$ and $x_4=\mu$. Therefore, I assumed: $$\ker(f)=\{(\lambda,\lambda,-\frac{1}{2}\mu,\mu): \lambda,\mu \in \mathbb{R}\}$$
And since the kernel can be described with only one vector (with two parameters), I thought $\dim(\ker(f))=1$, but my textbook says it's actually equal to $2$. Where is my reasoning flawed?
• A basis for the kernel is $\{(1,1,0,0), (0,0,-\frac{1}{2}, 1) \}$ – leibnewtz Jan 9 '18 at 23:20
• @leibnewtz Hm, so that means I can only 'fix' one parameter at once? I thought it'd be possible to establish something like $\lambda=1$, $\mu=2$ so that a basis for the kernel is $\{(1,1,-1,2)\}$. Why isn't that possible? – Manuel Jan 9 '18 at 23:22
• Of course you can do what you say...but that way you'll only get one vector, and you already know you need two lin. ind. vectors to have a basis... – DonAntonio Jan 9 '18 at 23:30
• Since $f:\mathbb R^4\to\mathbb R^2$, its image is at most two-dimensional, so you know that its kernel must be at least two-dimensional. – amd Jan 9 '18 at 23:35
• @Manuel Please, if you are ok, you can accept the answer and set it as solved. Thanks! – user Feb 4 '18 at 0:09 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232924970204,
"lm_q1q2_score": 0.8539031961701196,
"lm_q2_score": 0.8688267796346598,
"openwebmath_perplexity": 250.62895218820742,
"openwebmath_score": 0.883671224117279,
"tags": null,
"url": "https://math.stackexchange.com/questions/2598977/finding-dimension-of-a-kernel-from-a-linear-transformation"
} |
It is all correct, the dimension of the ker(f) is 2 because you have 2 free parameters that define it $(\lambda,\mu)$.
• @Manuel They are equivalent concepts. As noted in the comments you can set $\lambda=1$ and $\mu=0$ and define a basis vector, then $\lambda=0$ and $\mu=1$ and define a second vector linearly independent from the first, thus the dimension is 2. This is true for any number of free parameter (EG a line or a plane in $\mathbb{R^3}$). – user Jan 9 '18 at 23:25
• @Manuel There are two vectors in your description of the kernel: $\left(\lambda,\lambda,-\frac12\mu,\mu\right)=\lambda(1,1,0,0)+\mu\left(0,0,-\frac12,1\right)$; every vector in the kernel is a linear combination of these two linearly-independent vectors. – amd Jan 9 '18 at 23:36 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232924970204,
"lm_q1q2_score": 0.8539031961701196,
"lm_q2_score": 0.8688267796346598,
"openwebmath_perplexity": 250.62895218820742,
"openwebmath_score": 0.883671224117279,
"tags": null,
"url": "https://math.stackexchange.com/questions/2598977/finding-dimension-of-a-kernel-from-a-linear-transformation"
} |
Is every group an ideal class group of a number field?
The inverse Galois problem asks whether every finite group appears as the Galois group of some finite extension of $\mathbb Q$. I was wondering to what extent the analogous problem for ideal class groups has been investigated. More precisely, consider the following question:
Is every finite abelian group the ideal class group of some number field (finite extension of $\mathbb Q$)?
I'd be interested to hear about any partial results, as I suppose this question is still open. I'd be also interested in any results about a weaker problem:
Is every positive integer the ideal class number of some number field?
Again, any reference, even to a partial result, will be appreciated.
• Not an answer to your question, but are you aware of Claborn's theorem that says every finite abelian group is the class group of a Dedekind domain (though not necessarily, as far as one can tell from Claborn's work, of a ring of integers)? – Steven Landsburg Oct 16 '16 at 19:52
• @StevenLandsburg I was not aware of that result, since, to be honest, I am not so interested in general Dedekind's domain. I will definitely take a look at this result though. – Wojowu Oct 16 '16 at 19:55
• duplicate: math.stackexchange.com/questions/10949/… – Franz Lemmermeyer Oct 16 '16 at 20:36
• @FranzLemmermeyer Thanks, I haven't seen that question when searching about the topic. I suppose at this point my question can be closed. – Wojowu Oct 16 '16 at 20:39
• The system does not allow closing a question on one site as a duplicate of a question on a different site. One option, Wojowu, is for you to post an answer here, summarizing what's over there, and linking to it, and then accept your answer. – Gerry Myerson Oct 16 '16 at 22:04 | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232930001334,
"lm_q1q2_score": 0.853903189930967,
"lm_q2_score": 0.8688267728417087,
"openwebmath_perplexity": 267.0133536510818,
"openwebmath_score": 0.7085285782814026,
"tags": null,
"url": "https://mathoverflow.net/questions/252313/is-every-group-an-ideal-class-group-of-a-number-field/299691"
} |
The recent paper by Homlin, Jones, Kurlberg, McLeman, and Petersen (Experimental Math., to appear) is devoted to these questions especially in the context of imaginary quadratic fields. One should expect that every natural number arises as the class number of an imaginary quadratic field. Refining an earlier conjecture of Soundararajan, in this paper a precise asymptotic is formulated for the number of imaginary quadratic fields with a given class number. They also formulate conjectures on what kind of groups can arise as class groups of imaginary quadratic fields -- for example, one expects that for any odd prime $p$, $({\Bbb Z}/p{\Bbb Z})^3$ is not the class group of any imaginary quadratic field. The paper gives much data on such questions together with many related references.
• I think $(\dfrac{\Bbb Z}{2\Bbb Z})^3$ can occur as a class group for $D=-420$, and maybe it is better to restrict $p$ to the odd primes, i.e: $p\neq2$. – Davood KHAJEHPOUR May 8 '18 at 5:51
• Thanks for the reference! I will look into it as soon as I can. – Wojowu May 8 '18 at 7:26
Just for the sake of completeness, here I am posting the answer of Pete Clark which was posted in the math.SE question. I am making this post CW so as not to create the unseemly impression that I am gaining any undeserved reputation from this. Here it is:
Virtually nothing is known about the question of which abelian groups can be the ideal class group of (the full ring of integers of) some number field. So far as I know, it is a plausible conjecture that all finite abelian groups (up to isomorphism, of course) occur in this way. Conjectures and heuristics in this vein have been made, but unfortunately for me I'm not so familiar with them. | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232930001334,
"lm_q1q2_score": 0.853903189930967,
"lm_q2_score": 0.8688267728417087,
"openwebmath_perplexity": 267.0133536510818,
"openwebmath_score": 0.7085285782814026,
"tags": null,
"url": "https://mathoverflow.net/questions/252313/is-every-group-an-ideal-class-group-of-a-number-field/299691"
} |
The situation for imaginary quadratic fields is different. Here there is an absolute bound on the size of an integer $$k$$ such that the class group of an imaginary quadratic field can be isomorphic to $$(\mathbb{Z}/2\mathbb{Z})^k$$. Conditionally on the Generalized Riemann Hypothesis, the largest such $$k$$ is $$4$$. This has do to with idoneal numbers, of which the following paper provides a very fine survey:
http://www.mast.queensu.ca/~kani/papers/idoneal-f.pdf
Actually the truth is slightly stronger: let $$H_D$$ be the class group of the imaginary quadratic field $$\mathbb{Q}(\sqrt{-D})$$. Then, as $$D$$ tends to negative infinity through squarefree numbers, the size of $$2H_D$$ (the image of multiplication by $$2$$) tends to infinity. See for instance
http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.0358v2.pdf
for some recent explicit bounds on this.
For $S$-class groups: Perret, Marc On the ideal class group problem for global fields. Zbl 0933.11053 J. Number Theory 77, No. 1, 27-35 (1999). https://zbmath.org/?any=&au=perret&ti=On+the+Ideal+Class+Group+Problem+for+Global+Fields&so=&ab=&cc=&ut=&an=&la=&py=&rv=&sw=&dm= | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232930001334,
"lm_q1q2_score": 0.853903189930967,
"lm_q2_score": 0.8688267728417087,
"openwebmath_perplexity": 267.0133536510818,
"openwebmath_score": 0.7085285782814026,
"tags": null,
"url": "https://mathoverflow.net/questions/252313/is-every-group-an-ideal-class-group-of-a-number-field/299691"
} |
Prove $\forall n > 2, \ \exists p\in \Bbb{P} : n < p < n!$
I need to prove that:
$$(1) \ \forall n\in\Bbb{N}_{\ge2}, \ \exists p\in \Bbb{P} : n < p < n!$$
I already know how to prove the $n < p$ part; it directly follows from the proof that there is no largest prime. However, I am stumped on the $p < n!$ part.
One idea I had for showing this is as follows: we know that $(2) \ \forall n \in \Bbb{N}_{\ge2},\ \exists m\in\Bbb{N} : n!=2m$. By testing a few values, I conjectured that $(3) \ \forall n\in\Bbb{N}_{\ge2}, \ \exists p\in \Bbb{P} : n < p < 2n$. If (3) could be shown, it would be simple to prove (1), but there does not seem to be no easy way to prove (3), if it's even correct.
• "that there is always a prime number between $n$ and $2n$" The names Bertrand and Chebyshev spring to mind. – Daniel Fischer Jul 26 '13 at 14:32
• Oh, wow! I didn't know that! I was just trying to find a 'lazy' solution. But it's nice to know. :) – dotslash Jul 26 '13 at 14:33
• There's also an easy direct way. Do you know Euclid's proof that there are infinitely many primes? – Daniel Fischer Jul 26 '13 at 14:35
• Yes, I do. But not sure how to use it here. – dotslash Jul 26 '13 at 14:36
• I was thinking of what Goos answered. That relieves of the burden of proving that the product of primes $< n$ is $> n$ (although that's not very hard). – Daniel Fischer Jul 26 '13 at 14:40
Just consider $n! - 1$. Clearly this is between $n$ and $n!$ as long as there is anything between $n$ and $n!$, and it isn't divisible by any prime $p \le n$. Thus it contains a prime factor bigger than $n$ and smaller than $n!$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232909876816,
"lm_q1q2_score": 0.8539031881824951,
"lm_q2_score": 0.8688267728417087,
"openwebmath_perplexity": 104.26105333668434,
"openwebmath_score": 0.8525522351264954,
"tags": null,
"url": "https://math.stackexchange.com/questions/452806/prove-forall-n-2-exists-p-in-bbbp-n-p-n/452816"
} |
• Correct me if I'm wrong, but $n! -1$ isn't divisible by any prime less than $n$ because all the primes are already included in $n!$ ? – dotslash Jul 26 '13 at 14:53
• @dotslash For any prime $p \le n$, $p$ is contained in the product $n!$, so $n! - 1$ is one less than a multiple of $p$. – 6005 Jul 26 '13 at 14:55
• Precisely my thinking! Thank you for the concise and beautiful answer! – dotslash Jul 26 '13 at 14:59
You can obviously proceed by stating the Bertrand's postulate. However, I think the question is asking you to solve it a bit differently.
Let us solve it using a method similar to that used by Euclid when he tried to prove that there are infinite number of primes. Let $p_{1},p_{2},\cdots ,p_{k}$ be all the primes less than or equal to $n$. Obviously, $k<n$.Let, $P=p_{1}\times p_{2}\times\cdots \times p_{k}+1$.
Notice that $P$ is not divisible by any of the given primes. Hence $P$ is either a prime or is divisible by a prime $> n$.
It can be easily seen that $$P<n!$$ Furthermore, $P$ needs to be greater than $n$, otherwise we would find another prime (other than $p_{1},p_{2},\cdots,p_{k}$) which would lead to a contradiction.
• I think you mean that $p_1, p_2, \cdots , p_k$ are less than or equal to $n$. You also have to be careful when claiming that $P < n!$; this isn't true in the case $n = 3$. – 6005 Jul 26 '13 at 14:42
• You need it to not be divisible by $n$ if $n$ is prime, so you need to include primes less than or equal to $n$ in the product. Thus you get $3*2 + 1 = 7$, not less than $3! = 6$. – 6005 Jul 26 '13 at 14:47
• Sorry!My bad,I guess the case n=3 has to be checked manually.Other than that I think there are no exceptions. – Shaswata Jul 26 '13 at 14:49 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232909876816,
"lm_q1q2_score": 0.8539031881824951,
"lm_q2_score": 0.8688267728417087,
"openwebmath_perplexity": 104.26105333668434,
"openwebmath_score": 0.8525522351264954,
"tags": null,
"url": "https://math.stackexchange.com/questions/452806/prove-forall-n-2-exists-p-in-bbbp-n-p-n/452816"
} |
What you speculate on is correct, there is always a prime between $n$ and $2n$. This is known as Bertrand's postulate and is really a theorem. But for between $n$ and $n!$ you can modify the Euclid proof for a much easier approach. It is true for $3, 3 \lt 5 \lt 6$. For $n \gt 3,$ take the product of all primes less than $n$ and add $1$. This is less than $n!$ because it has only one factor of $2$, not the three coming from $2$ and $4$. Then it is either prime or composite ...
• Great! But your explanation of "only one factor of $2$" confuses me. Isn't it simpler to say that $n!$ contains all the integers between $2$ and $n$, including those that are prime. Hence, $n! > p$? – dotslash Jul 26 '13 at 14:46
• But we need a prime greater than $n$ and $n!$ doesn't contain that as a factor. I wanted to justify that the product of all the primes less than $n$ is strictly less than $n!$ so that I could add one and still be less. Identifying a factor of $n!$ that is not part of the product of primes was my way of doing that. That is also why I did $3$ separately-it is not true in that case as you can see. If we just take all the primes less than or equal to $3$, the product plus $1$ is $7$ and we have not found a prime between $3$ and $3!$ – Ross Millikan Jul 26 '13 at 15:10
• I follow you now. Thanks for the answer! – dotslash Jul 26 '13 at 15:42 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232909876816,
"lm_q1q2_score": 0.8539031881824951,
"lm_q2_score": 0.8688267728417087,
"openwebmath_perplexity": 104.26105333668434,
"openwebmath_score": 0.8525522351264954,
"tags": null,
"url": "https://math.stackexchange.com/questions/452806/prove-forall-n-2-exists-p-in-bbbp-n-p-n/452816"
} |
# Why is $\gcd(a,b)=\gcd(b,r)$ when $a = qb + r$?
Given: $a = qb + r$
Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?
Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math.stackexchange.com/a/4110/53259 and averting a duplicate)
What's the intuition behind this result? I only recognise the proof and examples solely due to algebraic properties and formal definitions; I'd like to apprehend the result naturally.
-
$a$, $b$, $r$ are integers? (If yes, tagging the question elementary-number-theory would be suitable.) Or are you looking for some greater generality, e.g. they are elements of some type of commutative ring? If yes, you should say so and you should also mention, what are the assumptions about this ring. – Martin Sleziak Jan 2 '12 at 13:12
If you're asking about integers, then this question is very similar to yours: math.stackexchange.com/questions/59147/… – Martin Sleziak Jan 2 '12 at 13:16
see page 13 here www-groups.dcs.st-and.ac.uk/~martyn/teaching/1003/… – Bhargav Jan 2 '12 at 13:26
@Martin The result does not depend upon the underlying ring (assuming $\rm\:gcd(a,b)\:$ exists). – Bill Dubuque Jan 2 '12 at 18:19
If $d$ is a divisor of $a$ and of $b$, then \begin{align} a & = dn, \\ b & = dm. \end{align} So $$a-b= dn-dm=d(n-m)= (d\cdot\text{something}).$$ So $d$ is a divisor of $a-b$.
Thus: All divisors that $a$ and $b$ have in common are divisors of $a-b$.
If $d$ is a divisor of $a$ and of $a-b$, then \begin{align} a & = dn, \\ a-b & = d\ell. \end{align} So $$b=a-(a-b)=dn-d\ell=(d\cdot\text{something}).$$ So $d$ is a divisor of $b$.
Thus: All divisors that $a$ and $a-b$ have in common are divisors of $b$.
Therefore, the set of all common divisors of $a$ and $b$ is the same as the set of all common divisors of $a$ and $a-b$.
Subtracting one member of a pair from the other never alters the set of all common divisors; therefore it never alters the $\gcd$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8539031843278371,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 242.20783499029983,
"openwebmath_score": 0.9606935381889343,
"tags": null,
"url": "http://math.stackexchange.com/questions/95799/why-is-gcda-b-gcdb-r-when-a-qb-r"
} |
-
Very clear answer! Little remark: $a-b \neq r$, but $a-qb = r$, but all your explanation will work using that :). – Kevin Jan 2 '12 at 21:17
@Kevin Beware that this fails if the ring is not $\rm\:\mathbb Z\:$ since then the remainder $\rm\:r\:$ generally cannot be obtained by repeatedly subtracting $\rm\:b\:$ from $\rm\ q\ b + r\:,\:$ as it can when $\rm\:q\in\mathbb N\:.\:$ In other words, in general Euclidean rings, e.g. the polynomial ring $\rm\:F[x]\:$ over a field $\rm\:F\:,\:$ the Euclidean algorithm generally requires division with remainder, not simply iterated subtraction, in order to effect descent to a "smaller" remainder. So this ad-hoc special-case doesn't generally reveal the essence of the matter. – Bill Dubuque Jan 2 '12 at 22:37
@Kevin : Thanks; I've fixed the typo. – Michael Hardy Jan 3 '12 at 0:52
To express Bill Dubuque's point in other way: Euclid's algorithm works for polynomials, but the argument I give in my answer doesn't work for polynomials unless you do some adaptations. – Michael Hardy Jan 3 '12 at 0:53
HINT $\rm\ \$ If $\rm\ d\ |\ b\$ then $\rm\ d\ |\ q\ b + r\ \iff\ d\ |\ r\:.\$ Therefore $\rm\ \{b\:,\:q\ b+r\}\$ and $\rm\ \{b\:,\: r\}\$ have the same set of common divisors $\rm\:d\:,\:$ hence they have the same greatest common divisor.
Modly: $\$ if $\rm\ b\equiv 0\$ then $\rm\ q\ b+r\equiv 0\: \iff\: r\equiv 0\ \pmod{d}$
NOTE $\$ The result holds true because $\rm\:\mathbb Z\:$ forms a subring of its fraction field $\rm\:\mathbb Q\:.\:$ More generally, given any subring $\rm\:Z\:$ of a field $\rm\:F\:$ we define divisibility relative to $\rm\ Z\$ by $\rm\ x\ |\ y\ \iff\ y/x\in Z\:.\:$ Then the above proof still works, since if $\rm\ q,\ b/d\ \in Z\$ then $\rm\ q\:(b/d) + r/d\in Z\ \iff\ r/d\in Z\:.\:$ In other words, the usual divisibility laws follow from the fact that rings are closed under the operations of subtraction and multiplication; being so closed, $\rm\:Z\:$ serves as a ring of "integers" for divisibility tests. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8539031843278371,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 242.20783499029983,
"openwebmath_score": 0.9606935381889343,
"tags": null,
"url": "http://math.stackexchange.com/questions/95799/why-is-gcda-b-gcdb-r-when-a-qb-r"
} |
For example, to focus on the prime $2$ we can ignore all odd primes and define a divisibility relation so that $\rm\ m\ |\ n\$ if the power of $2$ in $\rm\:m\:$ is $\le$ that in $\rm\:n\:$ or, equivalently if $\rm\ n/m\$ has odd denominator in lowest terms. The set of all such fractions forms a ring $\rm\:Z\:$ of $2$-integral fractions. Moreover, this ring enjoys parity, so arguments based upon even/odd arithmetic go through. Similar ideas lead to powerful local-global techniques of reducing divisibility problems from complicated "global" rings to simpler "local" rings, where divisibility is decided by simply comparing powers of a prime.
-
You can show that for any integer $d$, we have $d\; |\; a$ and $d\; |\; b$ if and only if $d\; |\; b$ and $d\; |\; r$. In other words, $a$ and $b$ have exactly the same common divisors as $b$ and $r$. Thus $\gcd(a,b)$ is the same as $\gcd(b,r)$.
-
excellent answer, +1 for it. – ncmathsadist Jan 2 '12 at 13:44
Thanks, but why do they have the same common divisors? – Kevin Jan 2 '12 at 21:15
@Kevin: If $d$ divides $a$ and $b$, then $d$ divides $-qb$ and thus divides the sum $a - qb = r$. This shows that any common divisor of $a$ and $b$ is a common divisor of $b$ and $r$. If $d$ divides $b$ and $r$, then $d$ divides $qb$ and thus divides the sum $qb + r$. This shows that any common divisor of $b$ and $r$ is a common divisor of $a$ and $b$. – Mikko Korhonen Jan 2 '12 at 21:52
Since set of common divisors of $a-b$ and $b$ coincides with the set of common divisors of $a$ and $b$ then $\operatorname{gcd}(a,b)=\operatorname{gcd}(a-b,b)$. If $a=qb+r$, where $b>0$ and $0\leq r<b$, you can apply this equality $q$ times and obtain $\operatorname{gcd}(a,b)=\operatorname{gcd}(r,b)$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8539031843278371,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 242.20783499029983,
"openwebmath_score": 0.9606935381889343,
"tags": null,
"url": "http://math.stackexchange.com/questions/95799/why-is-gcda-b-gcdb-r-when-a-qb-r"
} |
-
But $a-b$ and $b$ don't have the same set of divisors. They have the same set of common divisors with $a$. – Peter Taylor Jan 2 '12 at 14:29
Ok common divisors. I always miss some details. – no identity Jan 2 '12 at 14:40
@Norbert : what you must have meant is that $\{a,b\}$ and $\{a-b,b\}$ have the same set of common divisors. – Michael Hardy Jan 2 '12 at 18:01
Yes, this is what I meant. – no identity Jan 2 '12 at 18:17
Beware that this "repeated subtraction" implementation of division with remainder does not generally yield the Euclidean algorithm in other domains, e.g. for polynomials. See my comment to Hardy's answer. – Bill Dubuque Jan 2 '12 at 22:58
Let $A$ be a commutative ring. For any $a_1,\dots,a_n$ in $A$ let $(a_1,\dots,a_n)$ the ideal generated by the $a_i$.
Then, for any $q,b,r$ in $A$, we have $$(qb+r,b)=(b,r).$$ Indeed, $qb+r$ is in $(b,r)$, and $r$ is in $(qb+r,b)$.
EDIT. Dear Kevin: Your question, I think, would be better understood if put in a wider context, involving rings and ideals. The most basic fact behind the question is, I believe, the fact that, in any commutative ring, the elements $qb+r$ and $b$ generate the same ideal as the elements $b$ and $r$. If you make additional hypothesis, this fact can be interpreted in terms of divisibility. (See Bill's comment.) The simplest is to assume that your ring is a principal ideal domain.
I could try to explain this in greater details, but many mathematicians much better than I have already done that. So, my advice would be to take a look at at least one of the many Algebra textbooks written by great mathematicians. Here are some of these books: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8539031843278371,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 242.20783499029983,
"openwebmath_score": 0.9606935381889343,
"tags": null,
"url": "http://math.stackexchange.com/questions/95799/why-is-gcda-b-gcdb-r-when-a-qb-r"
} |
-
This seems to implicitly assume some relation between gcds and ideals, e.g. for Bezout domains $\rm\ (a,b) = (\gcd(a,b))\:.$ – Bill Dubuque Jan 2 '12 at 22:28
Dear @Bill: Thanks for your comment. I edited the answer. (Of course, I agree with you.) – Pierre-Yves Gaillard Jan 3 '12 at 5:50
Divisor theory gives one nice way to better understand the relations between ideals and gcds. For a brief overview see Friedemann Lucius, Rings with a theory of greatest common divisors and for a longer exposition see Olaf Neumann, Was sollen und was sind Divisoren? (What are divisors and what are they good for?), Math. Semesterber, 48, 2, 139-192 (2001). – Bill Dubuque Jan 3 '12 at 6:38
I'm going to use the notation $(a,b)$ for the GCD of $a$ and $b$.
If $d|a$ and $d | b$ then $d|(a,b)$, by the definition of GCD. (Well, by one common definition...if that's not the definition you learned, then you probably learned it as a theorem).
Since $(a,b)|a$ and $(a,b)|b$, by the definition of $(a,b)$, it divides $a-qb$, so we have $(a,b)|r$. This gives us $(a,b)|b$ and $(a,b)|r$, hence $(a,b)|(b,r)$.
Now let's go the other way. $(b,r)|b$ and $(b,r)|b$, both by definition, so it also divides $r+qb$, giving us $(b,r)|a$. That gives is $(b,r)|(a,b)$.
From $(a,b)|(b,r)$ and $(b,r)|(a,b)$, we get $(a,b)=(b,r)$ or $(a,b)=-(b,r)$. The latter can be eliminated because GCD is by definition greater than 0.
-
a,b q ,r are integers how can we say that HCF(a,b) = HCF (b,r) we can say common divisor of a and b = common divisor of b and r. for example hcf of 4 and 2 = 2 4= 2x2 + o but here HCF of 4 and2 =2 but HCF of 2 and 0 =1 so how can we say that HCF(a,b) = HCF (b,r)
-
Theorem: Let $a > b > 0$ with $a = bq + r$, $0\leq{}r<b$ then $\gcd(a;b) = \gcd(b;r)$.
Proof: Need to show that $C(a;b) = C(b;r)$ for then the result will hold. To show that the two sets are equal requires showing that $C(a;b) \subseteq C(b;r)$ and that $C(b;r) \subseteq C(a;b)$: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8539031843278371,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 242.20783499029983,
"openwebmath_score": 0.9606935381889343,
"tags": null,
"url": "http://math.stackexchange.com/questions/95799/why-is-gcda-b-gcdb-r-when-a-qb-r"
} |
Let $y \in C(a;b)$ thus $y|a$ and $y|b$,
then $y|[a+(-q)b]$,
and so $y|r$, since $r=a-bq$,
but since $y|b$ is also true, we now have that $y \in C(b;r)$, finally this means that $C(a;b) \subseteq C(b;r)$.
Now let $y \in C(b;r)$ thus $y|b$ and $y|r$,
then $y|[(q)b+r]$,
and so $y|a$, since $a=bq+r$,
but since $y|b$ is also true, we now have that $y \in C(a;b)$, finally this means that $C(b;r) \subseteq C(a;b)$.
Therefore the required results have been proven.
Note: Where $C(a;b)$ denotes the set of common divisors/factors of $a$ and $b$, that is: $C(a;b)=\{y: y|a \land y|b\}$
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9828232884721166,
"lm_q1q2_score": 0.8539031843278371,
"lm_q2_score": 0.8688267711434708,
"openwebmath_perplexity": 242.20783499029983,
"openwebmath_score": 0.9606935381889343,
"tags": null,
"url": "http://math.stackexchange.com/questions/95799/why-is-gcda-b-gcdb-r-when-a-qb-r"
} |
Here is a very simple table showing data lined up in columns. Example. We will look at how to do this using an example and then also look at how to answer questions about a data set using a two way table. 3 Some Table Examples Example-1: A table with combined columns is given below. Practice your multiplication tables. Syntax of this Tableau ABS Function is: ABS(number) To demonstrate these Tableau math functions, we use Calculated Fields. Add rows, call Compute and Merge, and set PrimaryKey. Tree Diagrams in Math: Definition & Examples 4:43 Truth Table: Definition, Rules & Examples 6:08 6:52 Notice that I include the table in a center'' environment to display it properly. The table results can usually be used to plot results on a graph. Illustrated definition of Table: Information (such as numbers and descriptions) arranged in rows and columns. Size of the preallocated table, specified as a two-element numeric vector. According to the survey, what is the probability that a male likes rap music? The game element in the times tables games make it even more fun learn. A truth table is a handy little logical device that shows up not only in mathematics but also in Computer Science and Philosophy, making it an awesome interdisciplinary tool. \multicolumn{num}{col}{text} command is used to combine the following num columns into a single column with their total width. What is. The results, separated by gender, are displayed above. This is an example of a blank math lesson plan where you can fill out the pertinent information in the blank space provided, such as the lesson title, unit title, lesson number, class level, duration of the lessons, number of classes, teacher, lesson objectives, standards, essential questions, and evidence of learning. Example Example The table shows the math achievement test scores for a random sample of n = 10 college freshmen, along with their final calculus grades. Notes. Table Data - Basic Example A total of 72 people participated in a | {
"domain": "specialsports.info",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9658995723244552,
"lm_q1q2_score": 0.8538931567790912,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 1302.6804645855257,
"openwebmath_score": 0.4584738612174988,
"tags": null,
"url": "https://specialsports.info/benzamide-molecular-fbulsmr/viewtopic.php?af9d97=table-math-example"
} |
final calculus grades. Notes. Table Data - Basic Example A total of 72 people participated in a survey about their music preferences. Answer: 0.9332 To find the answer using the Z-table, find where the row for 1.5 intersects with the column for 0.00; this value is 0.9332.The Z-table shows only “less than” probabilities so it gives you exactly what you need for this question. Sample Questions Header Block Open sample questions menu Math . Cap table math is confusing. Use these sample z-score math problems to help you learn the z-score formula. Loved by kids and parent worldwide. The idea is to convert the word-statement to a symbolic statement, then use logical equivalences as we did in the last example. There are four variables in the JavaScript block. 22 Examples of Mathematics in Everyday Life. The following table summarizes them: As math requires special environments, there are naturally the appropriate environment names you can use in the standard way. Example example the table shows the math achievement. Times Tables. You read it right; basic mathematical concepts are followed all the time. The title is created simply as another paragraph in the center environment, rather than as part of the table itself. Hope this helps! So now how many people can we fit? Logic. The following HTML example contains the HTML, CSS, and JavaScript necessary to build a dynamic HTML table. Learn the multiplication tables in an interactive way with the free math multiplication learning games for 2rd, 3th, 4th and 5th grade. \begin{center} Numbers of Computers on Earth Sciences Network, By Type. Yes! Some of the times tables are illustrated below: Times Table … Consumer Math. For example, in the table shown below, Jacket is listed twice in column A. To complete a two way table for a set of data, you need to determine the variables of interest, their possible values, and then finally, the frequencies. But, maths is the universal language which is applied in almost every aspect of life. | {
"domain": "specialsports.info",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9658995723244552,
"lm_q1q2_score": 0.8538931567790912,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 1302.6804645855257,
"openwebmath_score": 0.4584738612174988,
"tags": null,
"url": "https://specialsports.info/benzamide-molecular-fbulsmr/viewtopic.php?af9d97=table-math-example"
} |
frequencies. But, maths is the universal language which is applied in almost every aspect of life. Go to First Question . Aligned to Common Core. The answer obtained in every step is a multiple of 2 and is known as multiplication fact. The opposite of a tautology is a contradiction or a fallacy, which is "always false". Tableau Math Functions. Skip to main content. Try the free Mathway calculator and problem solver below to practice various math topics. View sample math questions and directions students will encounter on SAT test day. The argument col contains one of the position symbols, l, r, or c. The argument text contains the content of the column. The following examples will show you the list of Math Functions in Tableau. Because complex Boolean statements can get tricky to think about, we can create a truth table to keep track of what truth values for the simple statements make the complex statement true and false. Example. Tally Table - Definition with Examples. There's now 3 parts to the tutorial with two extra example videos at the end. Trusted by teachers across schools. Possible values will be from 1-10, instead of 0-9. Uploaded By BrigadierIronJay8680. Example. Truth Table. Determining the post-money cap table for an equity round with an option pool refresh and one or more convertible notes converting to equity can be overwhelming. Here you can find additional information about practicing multiplication tables at primary school. Comprehensive Curriculum. According to some people, maths is just the use of complicated formulas and calculations which won’t be ever applied in real life. There are two disadvantages of writing tables by hand as described in this tutorial. Most of the time the data will be collected in form of a spreadsheet and we don't want to enter the data twice. Tables. Example: table([1:3]',{'one';'two';'three'},categorical({'A';'B';'C'})) creates a table from variables with three rows, but different data types. Mathematics for the | {
"domain": "specialsports.info",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9658995723244552,
"lm_q1q2_score": 0.8538931567790912,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 1302.6804645855257,
"openwebmath_score": 0.4584738612174988,
"tags": null,
"url": "https://specialsports.info/benzamide-molecular-fbulsmr/viewtopic.php?af9d97=table-math-example"
} |
creates a table from variables with three rows, but different data types. Mathematics for the Liberal Arts. Example. A math function table is a table used to plot possible outcomes of a function, which is a kind of rule. Reading over the table; Exercising using the Math Trainer; But here are some "tips" to help you even more: Tip 1: Order Does Not Matter. Go to next Question. Back to overview Forward to Shortdocumentation 1 Examples of Latex Here an example of a very small Latex document \documentclass{article} \begin{document} example for a very \tiny{tiny} \normalsize \LaTeX \ document \end{document} Truth Tables . For example, in the example given below, 7 … A tautology in math (and logic) is a compound statement (premise and conclusion) that always produces truth. Roman numerals.. my mind ~ my math .. Tables/Graphs/Charts BEST CHART EVER !!! Unlike most other environments, however, there are some handy shorthands for declaring your formulas. Two way tables, also known as contingency tables, show frequencies (counts) as they relate to two variables. Some math sections allow the use of a calculator, others do not. By Grades. Mathematical tables are lists of numbers showing the results of a calculation with varying arguments.Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy and celestial navigation.They continued to be widely used until electronic calculators became cheap and plentiful, in order to simplify and drastically speed up computation. Construct a truth table for the formula . The Tableau ABS function is used to return the absolute positive value. This tutorial provides you with commonly used SQL math functions that allow you to perform business and engineering calculations. This is more typical of what you'll need to do in mathematics. 50,000 Schools . Another Example: 2×9=18, and 9×2=18. The Complete K-5 Math Learning Program Built for Your Child. For instance, if we want to work out the times table for | {
"domain": "specialsports.info",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9658995723244552,
"lm_q1q2_score": 0.8538931567790912,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 1302.6804645855257,
"openwebmath_score": 0.4584738612174988,
"tags": null,
"url": "https://specialsports.info/benzamide-molecular-fbulsmr/viewtopic.php?af9d97=table-math-example"
} |
Math Learning Program Built for Your Child. For instance, if we want to work out the times table for 2, we start with 2 and then add 2 in each step. A function table in math is a table that describes a function by displaying inputs and corresponding outputs in tabular form. HTML tables allow web developers to arrange data into rows and columns. About the book: I am honored to see my latest book, Table Talk Math, published by Dave Burgess Consulting and available on Amazon.In it, math educators and experts from around the world offer ideas and share stories about ways in which parents can engage their children with a math-based conversation in an easy-to-read format and plenty of examples. In some tables, there might not be unique values any column in the lookup table. The notation may vary… 30 Million Kids. To fully understand function tables and their purpose, you need to understand functions, and how they relate to variables. However, there is only one record for each jacket and size combination -- Jacket Medium in row 4 and Jacket Large in row 5. Extras. No matter what the individual parts are, the result is a true statement; a tautology is always true. C# DataTable Examples Store data in memory with a DataTable. SAT Suite of Assessments Sample Questions. We can fit one, two, three, four, five. (See Commutative Property.) Tables, graphs, and charts are an easy way to clearly show your data. A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. Let's see. sz — Size of preallocated table two-element numeric vector. A times table is a list of multiples of a number. Be sure to consider how to best show your results with appropriate graph forms. School St. John's University; Course Title MTH 1250C; Type. | {
"domain": "specialsports.info",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9658995723244552,
"lm_q1q2_score": 0.8538931567790912,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 1302.6804645855257,
"openwebmath_score": 0.4584738612174988,
"tags": null,
"url": "https://specialsports.info/benzamide-molecular-fbulsmr/viewtopic.php?af9d97=table-math-example"
} |
results with appropriate graph forms. School St. John's University; Course Title MTH 1250C; Type. First, I list all the alternatives for P and Q. Search for: Truth Tables and Analyzing Arguments: Examples. Tableau ABS Function. And then on this table, which is identical, you could fit six, seven, eight, nine. Z Score Table Sample Problems. When we multiply two numbers, it does not matter which is first or second, the answer is always the same. Parents, Sign Up for Free Teachers, Sign Up for Free. Logic Symbols in Math; Truth Table; Tautology Math Examples; Tautology Definition. Pages 47. Example: 3×5=15, and 5×3=15. Math.floor(Math.random() * (max - min + 1)) + min // We can introduce a min variable as well so we can have a range. While it works for small tables similar to the one in our example, it can take a long time to enter a large amount of data by hand. This preview shows page 11 - 23 out of 47 pages. So here we have one table, and it's going to touch ends with this table right over here. And because it touches ends right over here-- we're making it one big continuous table-- you can't fit someone here anymore. | {
"domain": "specialsports.info",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9658995723244552,
"lm_q1q2_score": 0.8538931567790912,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 1302.6804645855257,
"openwebmath_score": 0.4584738612174988,
"tags": null,
"url": "https://specialsports.info/benzamide-molecular-fbulsmr/viewtopic.php?af9d97=table-math-example"
} |
## table math example
Zapp's Chips Careers, Enchanted Sword Seed, Emmer Wheat Vs Wheat, Momeni Koi Area Rugs, Best Garlic Mayonnaise, 1965 Impala 4 Door For Sale, Deer Resistant Honeysuckle, Steakhouse Blue Cheese Dressing Recipe, Drama Genre Examples, | {
"domain": "specialsports.info",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9658995723244552,
"lm_q1q2_score": 0.8538931567790912,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 1302.6804645855257,
"openwebmath_score": 0.4584738612174988,
"tags": null,
"url": "https://specialsports.info/benzamide-molecular-fbulsmr/viewtopic.php?af9d97=table-math-example"
} |
# How can one show that $\sum_{n=0}^\infty\frac{n}{n!}=e$? [duplicate]
How can one show that $$\sum_{n=0}^\infty\frac{n}{n!}=e$$?
I understand that $$\sum_{n=0}^\infty\frac{x^n}{n!}=e^x$$ and that letting $$x=1$$would give $$\sum_{n=0}^\infty\frac{1}{n!}=e$$
But why does the sum $$\sum_{n=0}^\infty\frac{n}{n!}$$ give an answer of $$e$$ also?
• In the picture on the bottom sum the numerator should be n, apologies. – user677704 Jun 1 '19 at 2:51
• Are you familiar with derivatives ? – DanielV Jun 1 '19 at 2:53
• Your question is rather unclear. Do you want to prove: $\sum _{n=0}^{\infty }\:\frac{x}{n!}=e?$ – NoChance Jun 1 '19 at 3:12
• ‘No chance’ yes that was my question more or less, I just didn’t understand why different sums all add to e – user677704 Jun 1 '19 at 3:13
• It can be proven that $\sum _{n=0}^{\infty }\:\frac{x^n}{n!}=e^x$. However, in your expression, you don't raise x to the power n. – NoChance Jun 1 '19 at 3:15
Note that $$n/n!=1/(n-1)!$$, which is not meaningfully distinct in the context of the infinite sum from the terms $$1/n!$$.
In fact $$\sum_{n\geq 0} \frac{n}{n!}\stackrel{(1)}{=}\sum_{n\geq 1} \frac{n}{n!}= \sum_{n\geq 1} \frac{1}{(n-1)!}\stackrel{(2)}{=}\sum_{n\geq 0} \frac{1}{n!}=e^1=e$$ (1) comes from the fact that the first term is zero, (2) is a shift of indices by one.
You can repeat this shifting procedure ad infinitum and obtain, for each fixed $$k$$ $$\sum_{n\geq 0} \frac{n(n-1)\ldots (n-k)}{n!}=e$$ because your sum 'starts' at index $$k+1$$.
• This is what I meant in my answer. Nice presentation. +1 – The Count Jun 1 '19 at 3:14
• So I, in fact, unfolded your idea ? (+1) – Duchamp Gérard H. E. Jun 1 '19 at 3:15
• Teamwork! I like it. – The Count Jun 1 '19 at 3:16 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109516378093,
"lm_q1q2_score": 0.8538663134065706,
"lm_q2_score": 0.8670357615200475,
"openwebmath_perplexity": 530.2218829793999,
"openwebmath_score": 0.921451210975647,
"tags": null,
"url": "https://math.stackexchange.com/questions/3247145/how-can-one-show-that-sum-n-0-infty-fracnn-e"
} |
$$\begin{array} {rcl} % \displaystyle e^x & = & \displaystyle \sum_{n = 0}^{\infty} \frac{x^n}{n!} \\ % \displaystyle\frac{d}{dx}e^x & = & \displaystyle \frac{d}{dx}\sum_{n = 0}^{\infty} \frac{x^n}{n!} \\ % e^x & = & \displaystyle \sum_{n = 0}^{\infty} \frac{nx^{n-1}}{n!} \\ % e^1 & = & \displaystyle \sum_{n = 0}^{\infty} \frac{n1^{n-1}}{n!} \\ \end{array}$$
• Thanks for your response it’s really clear and perfectly understood, I appreciate it. So would I be right in assuming you can repeat this process ad infinitum and acquire many different expressions which all sum to e? – user677704 Jun 1 '19 at 3:15
• Yes, in this case the multipliers will be $$n,n(n-1),n(n-1)(n-2),\ldots$$ and so on ... – Duchamp Gérard H. E. Jun 1 '19 at 3:18
• That’s incredibly interesting, aside from showing this with basic differentiation would you know where to find a geometric or more intuitive proof. The logic is perfectly reasonable but I prefer to see why. – user677704 Jun 1 '19 at 3:25
• @User3457884334 I added repetition of this process ad infinitum in my answer. – Duchamp Gérard H. E. Jun 1 '19 at 3:26
• I understand that but wondered if there was another proof to show it geometrically. – user677704 Jun 1 '19 at 3:28
Suppose $$e^x=a_0+a_1x+a_2x^2+a_3x^3+...$$ try to find the constants by evaluating $$e^0$$ and then differentiating to eliminate the current $$0$$th degree term.
Here are the first three terms : $$e^0=a_0+a_1*0+...=1\Leftrightarrow a_0=1$$ $$\frac{d}{dx}(e^x)=e^x=\frac{d}{dx}(a_0+a_1x+a_2x^2+...)=a_1+2a_2x+3a_3x^2+...$$ $$e^0=a_1+2a_2*0+3a_3*0^2+...=1\Leftrightarrow a_1=1$$ $$\frac{d}{dx}(e^x)=e^x=\frac{d}{dx}(a_1+2a_2x+3a_3x^2+...)=2a_2+(3*2)a_3x+...$$ $$e^0=2a_2+(3*2)a_3*0\,+...=1\Leftrightarrow a_1=\frac{1}{2}$$ You'll find that $$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$$ switch $$1$$ for $$x$$ and you get your result (refer to The Count's answer). | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109516378093,
"lm_q1q2_score": 0.8538663134065706,
"lm_q2_score": 0.8670357615200475,
"openwebmath_perplexity": 530.2218829793999,
"openwebmath_score": 0.921451210975647,
"tags": null,
"url": "https://math.stackexchange.com/questions/3247145/how-can-one-show-that-sum-n-0-infty-fracnn-e"
} |
Differentiate the series $$e^x=\sum_{n=0}^{\infty}\dfrac {x^n}{n!}$$ term by term. Get $$\sum_{n=0}^{\infty}\dfrac {nx^{n-1}}{n!}$$. Then set $$x=1$$. Note that $$(e^x)'=e^x$$ (for$$\sum_{n=0}^{\infty}\dfrac {nx^{n-1}}{n!}=\sum_{n=0}^{\infty}\dfrac {x^n}{n!}$$) . | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109516378093,
"lm_q1q2_score": 0.8538663134065706,
"lm_q2_score": 0.8670357615200475,
"openwebmath_perplexity": 530.2218829793999,
"openwebmath_score": 0.921451210975647,
"tags": null,
"url": "https://math.stackexchange.com/questions/3247145/how-can-one-show-that-sum-n-0-infty-fracnn-e"
} |
# Find the number of distributions of seven distinct balls into three distinct boxes if at least two balls must go into each box.
My Answer: I chose to place three balls into one box and two balls in the other two boxes. This can be done three times: $$\binom73\binom42\binom22=\binom{7}{3,2,2}\\ \binom72\binom53\binom22=\binom{7}{2,3,2} \\ \binom72\binom52\binom33=\binom{7}{2,2,3}$$ I'm not sure if this is correct, but we have not covered Stirling numbers yet, so I cannot use that as my explanation. Please help and thank you!
An alternative method, noting that 2/2/3 is the only distribution of seven balls to three boxes with at least two balls in each box:
• 3 choices for the box with three balls
• $\binom73=35$ ways to put three balls into that box
• $\binom42=6$ ways to put two balls into one of the remaining boxes; the third box is then forced to contain the last two balls
This yields $3\cdot35\cdot6=630$ possibilities, and your approach yields the same result but through a longer process.
• Of course the solution appear to be the same once one recognises that the three numbers in the OP's solution must be the same by symmetry. – Carsten S Jul 15 '17 at 11:38
Your answer is correct. Here is a variation based upon exponential generating functions.
• A selection of at least two distinct balls can be encoded as \begin{align*} \frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=e^x-1-x \end{align*}
• Since we have three boxes we consider \begin{align*} (e^x-1-x)^3\tag{1} \end{align*}
We denote with $[x^n]$ the coefficient of $x^n$ in a series.
Since we are looking for the number of placing $7$ distinct balls we consider the coefficient of $x^7$ in (1) and obtain with some help of Wolfram Alpha
\begin{align*} 7^3=\color{blue}{630} \end{align*} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984810949854636,
"lm_q1q2_score": 0.8538663118604956,
"lm_q2_score": 0.8670357615200474,
"openwebmath_perplexity": 204.07628654221293,
"openwebmath_score": 0.9417623281478882,
"tags": null,
"url": "https://math.stackexchange.com/questions/2359335/find-the-number-of-distributions-of-seven-distinct-balls-into-three-distinct-box"
} |
\begin{align*} 7^3=\color{blue}{630} \end{align*}
• Let's say we have a red box, a blue box, and a green box (and the observer is not colorblind). There are $\binom{7}{3}$ ways of placing three balls in the red box, $\binom{4}{2}$ ways of placing two of the remaining four balls in the blue box, and $\binom{2}{2}$ ways of placing the remaining two balls in the green box. Hence, there are $$\binom{7}{3}\binom{4}{2}\binom{2}{2} = \binom{7}{3, 2, 2}$$ ways to distribute the balls so that three are placed in the red box, two are placed in the blue box, and two are placed in the blue box. By symmetry, the answer is $$3\binom{7}{3, 2, 2}$$ – N. F. Taussig Jul 19 '17 at 12:31
• @N.F.Taussig: Thanks for your hint! – Markus Scheuer Jul 19 '17 at 12:38
• @N.F.Taussig: I was misleading by myself. Original answer restored. – Markus Scheuer Jul 19 '17 at 16:58
• I have written a solution based on your idea of first placing the seven distinguishable ways in three indistinguishable boxes, then arranging the boxes. – N. F. Taussig Jul 19 '17 at 19:16
This solution is a modification of an attempted solution that Markus Scheuer deleted.
First, we count the number of ways seven distinguishable balls can be placed in three indistinguishable boxes if at least two balls are placed in each box. Then we will arrange the boxes.
The only permissible way to distribute the balls is to place three balls in one bag and two each in the other bags.
Method 1: There are $\binom{7}{3}$ ways to select three balls to be placed in one of the boxes. Line up the remaining balls in a row. Take the ball at the left end of the row and place it in an empty box. There are three ways of selecting one of the other three balls to be placed in the same box as that ball. The remaining two balls must be placed in the remaining empty box. Hence, there are $$3\binom{7}{3}$$ ways to place seven distinguishable balls in three indistinguishable boxes. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984810949854636,
"lm_q1q2_score": 0.8538663118604956,
"lm_q2_score": 0.8670357615200474,
"openwebmath_perplexity": 204.07628654221293,
"openwebmath_score": 0.9417623281478882,
"tags": null,
"url": "https://math.stackexchange.com/questions/2359335/find-the-number-of-distributions-of-seven-distinct-balls-into-three-distinct-box"
} |
Method 2: There are $\binom{7}{3}$ ways to place three of the balls in one box. That leaves $\binom{4}{2}$ ways to place two of the remaining four balls in another box. The remaining two balls are place in the third box. That suggests the answer is $$\binom{7}{3}\binom{4}{2}$$ However, since the boxes are indistinguishable, we cannot distinguish between the two boxes that received two balls. Thus, we must multiply the above result by $1/2$. Hence, there are $$\frac{1}{2}\binom{7}{3}\binom{4}{2}$$ ways of distributing seven distinguishable balls to three indistinguishable boxes.
To make the boxes distinguishable, we paint one box blue, one box green, and one box red. There are $3!$ ways to do this. Hence, there are $$3! \cdot 3\binom{7}{3} = 630$$ ways to place seven distinguishable balls in three distinguishable boxes if at least two balls must be placed in each box.
• @NFTaussig: Clear and extensive elaboration. Very nice! (+1) – Markus Scheuer Jul 19 '17 at 19:38 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984810949854636,
"lm_q1q2_score": 0.8538663118604956,
"lm_q2_score": 0.8670357615200474,
"openwebmath_perplexity": 204.07628654221293,
"openwebmath_score": 0.9417623281478882,
"tags": null,
"url": "https://math.stackexchange.com/questions/2359335/find-the-number-of-distributions-of-seven-distinct-balls-into-three-distinct-box"
} |
# Probability of picking matching socks, after partitioning the drawer.
Apologies if this is a duplicate. I searched and didn't find anything quite like it.
Suppose I have a drawer with an equal number of N black socks and N white socks. They're all mixed up. So, my chances of picking a matching pair in the first two selections is (N-1)/(2N-1), right? Well, what if, before I pick the first sock, I randomly (so I don't know the colors of the socks I'm moving) partition the drawer so that there are N socks on each side, and I draw one sock from each side. Do the chances of drawing a matching pair change?
On the one hand, we can see that selection from one side doesn't change the composition of socks on the other side of the partition. However, whichever color I choose from the "first" side, it's likely that there are more of that color on that side. On other words, if I draw a black sock from one side, it's more likely that that side had N-1 blacks and 1 white than it is that that side had 1 black and N-1 whites.
My suspicion is that I need to do some kind of hypothesis testing, where I consider the chances of every possible partitioning, but that's way above my skill level.
• Shouldn't the probability in the simple case be $\frac{N-1}{2N-1}$, because there are no longer $2N$ socks in the drawer? Feb 19 '18 at 20:17
• with replacement? Feb 19 '18 at 20:18
• @GTonyJacobs Yep $P=\frac {N-1}{2N-1}$ Feb 19 '18 at 20:27
• You certainly don't need hypothesis testing. We're not making inferences from a sample about an otherwise unknowable population. This is just a probability question with all of the parameters known. Kind of a tricky one, maybe, but there might be a good symmetry argument for why there's no difference. Feb 19 '18 at 20:37
• @GTonyJacobs Correct. I'll see if I can edit the post. Feb 20 '18 at 21:34
It doesn't matter if the partitioning happens before or after the first draw. Suppose it happens after. Suppose also that the first sock drawn was black. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109480714625,
"lm_q1q2_score": 0.8538663035472853,
"lm_q2_score": 0.8670357546485407,
"openwebmath_perplexity": 340.77304883709394,
"openwebmath_score": 0.7827708125114441,
"tags": null,
"url": "https://math.stackexchange.com/questions/2657571/probability-of-picking-matching-socks-after-partitioning-the-drawer"
} |
Now, we partition off $N$ from the remaining $2N-1$, to obtain our pool for the second draw. On average, the composition of this pool is $\frac{N-1}{2N-1}$ black and $\frac{N}{2N-1}$ white. Drawing from it, we have a matching pair if we draw from the portion of it that is black, i.e., $\frac{N-1}{2N-1}$.
The trick to simplifying the work is to work with expected values for the number of socks of each color in the second part of the partition, not actual values.
No, the chances don't change. If you worked out all the conditional probabilities based on how the socks split and summed appropriately you'd get the same answer: $(N-1)/(2N -1)$. The partition gives you no new information.
Here's the brute force calculation for $N=2$. There are two possible partitions, WW|BB and WB|WB. The first of these occurs with probability $1/3$. the second with probability $2/3$. In the first case you fail for sure. In the second you succeed half the time. On average you succeed with probability $$0 \times \frac{1}{3} + \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} = \frac{2-1}{4-1}.$$
(There's probably a really elegant argument from symmetry that I don't see.)
Of course it shouldn't change!
Let's see if the math works out for a simple example, say $N=3$
First, the 'normal' calculation says that the chances of getting a match $M$ is:
$$P(M)= \frac{N-1}{2N-1}=\frac{2}{5}$$
Now let's see what happens when you randomly partition them.
Let's say you first pick from the left side. After randomly splitting the socks into two groups, there can be $0$, $1$, $2$, or $3$ socks there, with respective probabilities of:
$$P(0)= \frac{{3 \choose 0}\cdot{3 \choose 3}}{6 \choose 3} = \frac{1}{20}$$
$$P(1)= \frac{{3 \choose 1}\cdot{3 \choose 2}}{6 \choose 3} = \frac{9}{20}$$
$$P(2)= \frac{{3 \choose 2}\cdot{3 \choose 1}}{6 \choose 3} = \frac{9}{20}$$
$$P(3)= \frac{{3 \choose 3}\cdot{3 \choose 0}}{6 \choose 3} = \frac{1}{20}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109480714625,
"lm_q1q2_score": 0.8538663035472853,
"lm_q2_score": 0.8670357546485407,
"openwebmath_perplexity": 340.77304883709394,
"openwebmath_score": 0.7827708125114441,
"tags": null,
"url": "https://math.stackexchange.com/questions/2657571/probability-of-picking-matching-socks-after-partitioning-the-drawer"
} |
$$P(3)= \frac{{3 \choose 3}\cdot{3 \choose 0}}{6 \choose 3} = \frac{1}{20}$$
Now, getting a match $M$ in the first and last situation is impossible, so $$P(M|0)=P(M|3)=0$$
When there is $1$ white sock on the left, the chances of getting a match are the chance of getting that white sock times the chances of getting one of the two socks on the right side, plus the chances of getting one of the two black ones on the left and the one black one on the right. And, with $2$ white socks on the left it's all symmetrical. So:
$$P(M|1)=P(M|2)=\frac{1}{3}\cdot \frac{2}{3} + \frac{2}{3} \cdot \frac{1}{3} = \frac{4}{9}$$
In sum:
$$P(M)=P(0)\cdot P(M|0) + P(1)\cdot P(M|1) +P(2)\cdot P(M|2) +P(3)\cdot P(M|3) =$$
$$\frac{1}{20} \cdot 0 + \frac{9}{20} \cdot \frac{4}{9} + \frac{9}{20} \cdot \frac{4}{9} + \frac{1}{20} \cdot 0 = \frac{8}{20} = \frac{2}{5}$$
OK, so yes, same chance!
Now, I'm sure you can generalize this for any $N$ ... thus evaluating the series:
$$P(M)=\sum_{i=0}^N \frac{{N \choose i} \cdot {N \choose {N-i}}}{2N \choose N} \cdot 2 \cdot \frac{i}{N} \cdot \frac{N-i}{N}$$
[... Insert some math that's over my head ....]
... and you'll see that this ends up being $$\frac{N-1}{2N-1}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109480714625,
"lm_q1q2_score": 0.8538663035472853,
"lm_q2_score": 0.8670357546485407,
"openwebmath_perplexity": 340.77304883709394,
"openwebmath_score": 0.7827708125114441,
"tags": null,
"url": "https://math.stackexchange.com/questions/2657571/probability-of-picking-matching-socks-after-partitioning-the-drawer"
} |
# Proving that a number is non-negative?
The numbers $a$,$b$ and $c$ are real. Prove that at least one of the three numbers $$(a+b+c)^2 -9bc \hspace{1cm} (a+b+c)^2 -9ca \hspace{1cm} (a+b+c)^2-9ab$$ is non-negative.
Any hints would be appreciated too.
• Have you considered adding them up and seeing if they are a square ? – Belgi Aug 16 '14 at 17:42
Hint:
If all three numbers are negative, then:
$$ab > \left(\frac{a+b+c}{3}\right)^2 \hspace{1cm} ac > \left(\frac{a+b+c}{3}\right)^2 \hspace{1cm} bc > \left(\frac{a+b+c}{3}\right)^2 \hspace{1cm}$$
Therefore, if we multiply the three inequalities:
$$a^2b^2c^2 > \left(\frac{a+b+c}{3}\right)^6$$
Or equivalently:
$$\left(\sqrt[3]{abc}\right)^6 > \left(\frac{a+b+c}{3}\right)^6$$
Do you know any inequality you can use here do disprove this?
• @Sudhanshu Look for the Arithmetic-Geometric Median Inequality en.wikipedia.org/wiki/… It says that: $$\frac{a+b+c}{3} \geq \sqrt[3]{abc}$$ – Darth Geek Aug 16 '14 at 17:50
• Oh, how can I forget that...Thanks btw. – Sudhanshu Aug 16 '14 at 17:54
• But since we assume these numbers to be negative how can we apply A.M > G.M ? – Sudhanshu Aug 16 '14 at 18:04
• If we assume those three numbers are negative, then they contradict the "AM>GM" inequality wich we know to be true. Therefore our assumption was incorrect, i.e. at least one of those numbers was not negative. This sort of proof is called "proof by contradiction" or "reductio ad absurdum". You start with a hypothesis and you end up in a contradiction, wich means that the hypothesis was false. – Darth Geek Aug 16 '14 at 18:14
• But I think the A.M > G.M rule is applicable only for positive numbers – Sudhanshu Aug 16 '14 at 18:15
$$\sum [(a+b+c)^2-9bc]=3\sum[a^2+b^2+c^2-ab-bc-ca]=\frac32\sum (a-b)^2\ge0$$
If each $(a+b+c)^2-3bc<0,$ $$\sum [(a+b+c)^2-3bc]<0$$
• You can also observe that $\sum[a^2+b^2+c^2-ab-bc-ca]\ge0$ is just Cauchy -Schwarz. – N. S. Aug 16 '14 at 18:38 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109540896723,
"lm_q1q2_score": 0.853866300306369,
"lm_q2_score": 0.8670357460591568,
"openwebmath_perplexity": 935.3657109660286,
"openwebmath_score": 0.7888849377632141,
"tags": null,
"url": "https://math.stackexchange.com/questions/900315/proving-that-a-number-is-non-negative"
} |
# interior point in metric space | {
"domain": "narrativmedicin.se",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109480714625,
"lm_q1q2_score": 0.8538663001637177,
"lm_q2_score": 0.8670357512127872,
"openwebmath_perplexity": 280.91345862380405,
"openwebmath_score": 0.9142648577690125,
"tags": null,
"url": "https://narrativmedicin.se/gzh7rm/fa9583-interior-point-in-metric-space"
} |
FACTS A point is interior if and only if it has an open ball that is a subset of the set x 2intA , 9">0;B "(x) ˆA A point is in the closure if and only if any open ball around it intersects the set x 2A , 8">0;B "(x) \A 6= ? Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. Definition: We say that x is an interior point of A iff there is an such that: . Browse other questions tagged metric-spaces or ask your own question. Defn Suppose (X,d) is a metric space and A is a subset of X. True. A point x ∈ E is said to be an interior point of E if E contains an open ball centered at x. Since you can construct a ball around 3, where all the points in the ball is in the metric space. (d) Describe the possible forms that an open ball can take in X = (Q ∩ [0; 3]; dE). Proposition A set O in a metric space is open if and only if each of its points are interior points. When we encounter topological spaces, we will generalize this definition of open. $\begingroup$ As an addendum, singletons are open if and only if the metric space is induced by a discrete metric, so there's only one case in which you have a nonempty interior of singleton sets. - the exterior of . We do not develop their theory in detail, and we leave the verifications and proofs as an exercise. In most cases, the proofs Let E be a subset of a metric space X. However, this definition of open in metric spaces is the same as that as if we regard our metric space as a topological space. - the boundary of Examples. $\endgroup$ – Alan Apr 18 '15 at 8:32 METRIC SPACES 77 where 1˜2 denotes the positive square root and equality holds if and only if there is a real number r, with 0 n r n 1, such that yj rxj 1 r zj for each j, 1 n j n N. Remark 3.1.9 Again, it is useful to view the triangular inequalities on “familiar \begin{align} \quad \mathrm{int} \left ( \bigcup_{S \in \mathcal F} S\right ) \supseteq \bigcup_{S \in \mathcal F} \mathrm{int} (S) \quad | {
"domain": "narrativmedicin.se",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109480714625,
"lm_q1q2_score": 0.8538663001637177,
"lm_q2_score": 0.8670357512127872,
"openwebmath_perplexity": 280.91345862380405,
"openwebmath_score": 0.9142648577690125,
"tags": null,
"url": "https://narrativmedicin.se/gzh7rm/fa9583-interior-point-in-metric-space"
} |
( \bigcup_{S \in \mathcal F} S\right ) \supseteq \bigcup_{S \in \mathcal F} \mathrm{int} (S) \quad \blacksquare \end{align} If is the real line with usual metric, , then A point x is called an isolated point of A if x belongs to A but is not a limit point of A. Interior Point Not Interior Points Definition: ... A set is said to be open in a metric space if it equals its interior (= ()). This intuitively means, that x is really 'inside' A - because it is contained in a ball inside A - it is not near the boundary of A. Let (X;d) be a metric space and A ˆX. Let be a metric space, Define: - the interior of . Let be a metric space. (c) The point 3 is an interior point of the subset C of X where C = {x ∈ Q | 2 < x ≤ 3}? The interior of the set E is the set of all its interior points. A point is exterior … An open ball of radius centered at is defined as Definition. Metric space: Interior Point METRIC SPACE: Interior Point: Definitions. This set is denoted by intE. 1. In these examples, all sets under consideration are subsets of the metric space R. Example 2.7. Appendix A. If has discrete metric, 2. Metric Spaces, Topological Spaces, and Compactness 253 Given Sˆ X;p2 X, we say pis an accumulation point of Sif and only if, for each ">0, there exists q2 S\ B"(p); q6= p.It follows that pis an The purpose of this chapter is to introduce metric spaces and give some definitions and examples. Proposition A set C in a metric space is closed if and only if it contains all its limit points. 3.2. Featured on Meta “Question closed” notifications experiment results and graduation | {
"domain": "narrativmedicin.se",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9848109480714625,
"lm_q1q2_score": 0.8538663001637177,
"lm_q2_score": 0.8670357512127872,
"openwebmath_perplexity": 280.91345862380405,
"openwebmath_score": 0.9142648577690125,
"tags": null,
"url": "https://narrativmedicin.se/gzh7rm/fa9583-interior-point-in-metric-space"
} |
# Are there any alternatives for NIntegrate to calculate the area for which $f(x,y)<0$?
I have this two-variable function $$f(x,y)= (8 \cos (x+y)+7)\cos \left(\frac{x}{2}\right)+\cos \frac{x-2 y}{2}+2 \cos \left(\frac{3 x}{2}\right)$$ where $$0. I want to calculate numerically the area for which the function is negative $$f(x,y)<0$$. I use this code
NIntegrate[
Boole[2 Cos[(3 x)/2] + Cos[1/2 (x - 2 y)] +
Cos[x/2] (7 + 8 Cos[x + y]) < 0], {x, 0, Pi}, {y, 0, Pi}]
and it gives the answer $$3.49458$$, but Mathematica gives the following warnings. Are there any other ways to calculate this value that is more reliable and more accurate than this method?
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 3.494581434480605 and 0.002397336775896384 for the integral and error estimates. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.951142217223021,
"lm_q1q2_score": 0.8538358840174605,
"lm_q2_score": 0.8976952852648487,
"openwebmath_perplexity": 4621.230410626539,
"openwebmath_score": 0.46566471457481384,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/249789/are-there-any-alternatives-for-nintegrate-to-calculate-the-area-for-which-fx-y"
} |
• Your f[x, y] is a constant rather than a function of x and y. Jun 17, 2021 at 19:03
• @BobHanlon Thanks, it was a typo mistake.
– user80186
Jun 17, 2021 at 23:21
• There is an exact solution to your integral $-2 \text{Li}_2\left(-\frac{1}{2}\right)+\text{Li}_2\left(\frac{1}{4} \left(i \sqrt{3}+1\right)\right)+\text{Li}_2\left(\frac{1}{4} \left(1-i \sqrt{3}\right)\right)+\frac{2 \pi ^2}{9}$ Jun 18, 2021 at 13:12
• @yarchik I think you should add that and its derivation as an answer.
– Kiro
Jun 18, 2021 at 14:03
• @SaraChem @Kiro I was simply a bit more patient than @eyorble. I used u = Integrate[upperCurve, {x, 0, Pi}] and v = Integrate[lowerCurve, {x, 0, 2 Pi/3}], followed by u-v. The lowerCurve and the upperCurve are defined in eyorble's post. Jun 18, 2021 at 19:46
Let us name the function of interest f[x,y]:
f[x_, y_] := (8 Cos[x + y] + 7) Cos[x/2] + Cos[(x - 2 y)/2] + 2 Cos[3 x/2]
Attempt to ContourPlot to find the zero lines:
ContourPlot[f[x, y] == 0, {x, 0, Pi}, {y, 0, Pi}]
Simple numerical evaluation determines that the negative region is the inside of these two curves. Notice that the ContourPlot is quite rapid and has very clean lines. Interesting coincidence. Perhaps there exists an analytical solution to these curve lines?
sol = Solve[{f[x, y] == 0}]
This returns a list of 4 possible curves, while also stating that some solutions may be missing. By manual inspection (such as by using Plot), we can find that the 2nd and 4th solutions are of interest to us, so we shall label them:
upperCurve = y /. sol[[2]];
lowerCurve = y /. sol[[4]];
Plot[{upperCurve, lowerCurve}, {x, 0, Pi}, PlotRange -> {0, Pi}]
Checking the curves manually by plotting them against the original ContourPlot, we see that upperCurve matches the upper line for the whole domain, and that lowerCurve matches the lower line up until it reaches its minimum.
Find the minimum of the lowerCurve:
FindMinimum[{lowerCurve, 0 < x < Pi}, x, WorkingPrecision -> 25] | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.951142217223021,
"lm_q1q2_score": 0.8538358840174605,
"lm_q2_score": 0.8976952852648487,
"openwebmath_perplexity": 4621.230410626539,
"openwebmath_score": 0.46566471457481384,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/249789/are-there-any-alternatives-for-nintegrate-to-calculate-the-area-for-which-fx-y"
} |
FindMinimum[{lowerCurve, 0 < x < Pi}, x, WorkingPrecision -> 25]
{1.872299341324760554288429*10^-8, {x -> 2.094395111754692173633430}}
The warning about a small imaginary part is of little concern here, but you can increase the WorkingPrecision and PrecisionGoal if you would like more digits.
Michael Seifert also pointed out that an exact form can be found for this solution by applying TrigFactor to f[x,y]:
TrigFactor[f[x,y]]
2 (Cos[x/2 - y/2] + 2 Cos[x/2 + y/2]) (2 Cos[x + y/2] + Cos[y/2]) == 0
The lower line happens to correspond to the second variable factor in this expression, and its minimum is found when y is set to 0 and solved.
Solve[{(2 Cos[x + y/2] + Cos[y/2]) == 0 /. y -> 0, 0 < x < Pi}, x]
{{ x -> 2 Pi/ 3 }}
Integrate the area below the 2nd curve over the whole domain minus the area under the 4th curve for 0 through 2.094...
NIntegrate[upperCurve, {x, 0, Pi}, WorkingPrecision -> 25] -
NIntegrate[lowerCurve, {x, 0, 2.0943951117546921736334297747816890478125.},
WorkingPrecision -> 25]
3.49805583366099845069196
Or with the exact form, we can see a slightly different answer:
NIntegrate[upperCurve, {x, 0, Pi}, WorkingPrecision -> 25] -
NIntegrate[lowerCurve, {x, 0, 2 Pi/3}, WorkingPrecision -> 25]
3.49805583366099836305434
While this method is not universally applicable, it does work for this function and is much faster than Area or direct application of NIntegrate and Boole for high precisions. As yarchik notes, you can swap NIntegrate for Integrate here to acquire an exact solution, though it takes a bit longer to evaluate. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.951142217223021,
"lm_q1q2_score": 0.8538358840174605,
"lm_q2_score": 0.8976952852648487,
"openwebmath_perplexity": 4621.230410626539,
"openwebmath_score": 0.46566471457481384,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/249789/are-there-any-alternatives-for-nintegrate-to-calculate-the-area-for-which-fx-y"
} |
• Using TrigFactor on f[x,y] shows that the two curves are $\cos((x-y)/2) + 2 \cos((x+y)/2) = 0$ (upper) and $2 \cos (x+y/2)+ \cos(y/2) = 0$ (lower). I don't know if that really helps, though. Jun 17, 2021 at 20:10
• If nothing else, it shows that the intersection of the lower curve with the $x$-axis is when $2 \cos x + 1 = 0$, or $x = 2 \pi/3$. Jun 17, 2021 at 20:12
• There are a lot of inconsistencies in your answer. What is y that you are integrating? Jun 18, 2021 at 11:51
• @yarchik The y in the integration should be read as y /. sol[[2]] (which is the upper curve from the contour plot) or y /. sol[[4]] (which is the lower curve from the contour plot). These are just explicit forms (in x) of the curves where f[x,y]==0, selected to match the curves we are interested in. Jun 18, 2021 at 12:02
• @yarchik I have modified the answer to hopefully clarify what's being integrated here. Jun 18, 2021 at 12:11
Clear["Global*"]
RegionPlot[
2 Cos[(3 x)/2] + Cos[1/2 (x - 2 y)] + Cos[x/2] (7 + 8 Cos[x + y]) < 0,
{x, 0, Pi}, {y, 0, Pi},
Frame -> True]
rgn = ImplicitRegion[{2 Cos[(3 x)/2] + Cos[1/2 (x - 2 y)] +
Cos[x/2] (7 + 8 Cos[x + y]) < 0 && 0 < x < Pi &&
0 < y < Pi}, {x, y}];
Area[rgn, WorkingPrecision -> MachinePrecision]
(* 3.49805 *)
Area[rgn, WorkingPrecision -> 15]
(* 3.49805 *)
A similar way is
reg=ImplicitRegion[
2 Cos[(3 x)/2] + Cos[1/2 (x - 2 y)] + Cos[x/2] (7 + 8 Cos[x + y]) <
0, {{x, 0, Pi}, {y, 0, Pi}}];
NIntegrate[1, {x, y} ∈ reg]
3.49485
• Thanks. Does WorkingPrecission matter in this method? I see a small difference in the different answers here obtained by different methods others suggested; if I need the most accurate one, which one is more reliable?
– user80186
Jun 18, 2021 at 12:54
Method ->"LocalAdaptive" evaluates without errormessage
NIntegrate[Boole[2 Cos[(3 x)/2] + Cos[1/2 (x - 2 y)] +Cos[x/2] (7 + 8 Cos[x + y]) < 0], {x, 0, Pi}, {y,0, Pi}, Method -> "LocalAdaptive"]
(*3.49818*) | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.951142217223021,
"lm_q1q2_score": 0.8538358840174605,
"lm_q2_score": 0.8976952852648487,
"openwebmath_perplexity": 4621.230410626539,
"openwebmath_score": 0.46566471457481384,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/249789/are-there-any-alternatives-for-nintegrate-to-calculate-the-area-for-which-fx-y"
} |
• Thanks. Does WorkingPrecission matter in this method? I see a small difference in the different answers here obtained by different methods others suggested; if I need the most accurate one, which one is more reliable?
– user80186
Jun 18, 2021 at 12:55
• @SaraChem Sorry, I was offline some days. If you add the option , IntegrationMonitor :> ((errors = Through[#1@"Error"]) &) inside NIntegrate , it's possible to evaluate the integrationerror Total@errors afterwards. Might be helpful. Jun 20, 2021 at 14:27 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.951142217223021,
"lm_q1q2_score": 0.8538358840174605,
"lm_q2_score": 0.8976952852648487,
"openwebmath_perplexity": 4621.230410626539,
"openwebmath_score": 0.46566471457481384,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/249789/are-there-any-alternatives-for-nintegrate-to-calculate-the-area-for-which-fx-y"
} |
A few words about PWW #33: subset permutations
My previous post showed four rows of diagrams, where the $n$th row (counting from zero) has diagrams with $n+2$ dots. The diagrams in the $n$th row depict all possible paths that
1. start at the top left dot,
2. end at the top right dot, and
3. never visit any dot more than once.
For a given diagram, we can choose which dots to visit (representing a subset of the dots), and we can visit the chosen dots in any order (a permutation). I will call the resulting things “subset permutations”, although I don’t know if there is some other more accepted term for them.
It’s not necessary to draw subset permutations as paths; it’s just one nice visual representation. If we number the dots from $1 \dots n$ (excluding the top two, which are uninteresting), each path corresponds precisely to a permutation of some subset of $\{1 \dots n\}$. Like this:
I thought of this idea the other day—of counting the number of “subset permutations”—and wondered how many there are of each size. I half expected it to turn out to be a sequence of numbers that I already knew well, like the Catalan numbers or certain binomial coefficients or something like that. So I was surprised when it turned out to be a sequence of numbers I don’t think I have ever encountered before (although it has certainly been studied by others).
So, how many subset permutations are there of size $n$? Let’s call this number $\mathit{SP}_n$. From my previous post we can see that starting with $\mathit{SP}_0 = 1$, the first few values are $1, 2, 5, 16$. (When I saw 1, 2, 5, I thought sure it was going to be the Catalan numbers—but then the next number was 16 instead of 14…) Can we compute these more generally without just listing them all and counting? | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
Well, to choose a particular subset permutation of size $n$, we can first choose how big we want the subset to be, i.e. how many dots to visit, which could be zero, or $n$, or anything in between. Once we choose to visit $k$ out of $n$ dots, then we have $n$ choices for which dot to visit first, then $(n-1)$ choices for which dot to visit second, all the way down to $(n-k+1)$ choices for the last dot (you should convince yourself that $(n-k+1)$, not $(n-k)$, is correct!). This is often notated
$\displaystyle {}_n P_k = n (n-1) (n-2) \dots (n-k+1) = \frac{n!}{(n-k)!}.$
So, in total then, $\mathit{SP}_n$ is the sum of ${}_n P_k$ over all possible $k$, that is,
$\displaystyle \mathit{SP}_n = \sum_{0 \leq k \leq n} \frac{n!}{(n-k)!}.$
This is already better than just listing all the subset permutations and counting. For example, we can compute that
$\mathit{SP}_4 = 4!/4! + 4!/3! + 4!/2! + 4!/1! + 4!/0! = 1 + 4 + 12 + 24 + 24 = 65.$
And sure enough, if we list them all, we get 65 (you’ll just have to take my word that this is all of them, I suppose!):
However, with a little algebra, we can do even better! First, let’s write out the sum for $\mathit{SP}_n$ explicitly:
$\mathit{SP}_n = 1 + n + n(n-1) + n(n-1)(n-2) + \dots + n!$
If we factor $n$ out of all the terms after the initial $1$, we get
$\mathit{SP}_n = 1 + n[1 + (n-1) + (n-1)(n-2) + \dots + (n-1)!] = 1 + n \mathit{SP}_{n-1}$
so the subset permutation numbers actually follow a very simple recurrence! To get the $n$th subset permutation number, we just multiply the previous one by $n$ and add one.
$\begin{array}{rcl} \mathit{SP}_0 &=& 1 \\ \mathit{SP}_1 &=& 1 \cdot 1 + 1 = 2 \\ \mathit{SP}_2 &=& 2 \cdot 2 + 1 = 5 \\ \mathit{SP}_3 &=& 3 \cdot 5 + 1 = 16 \end{array}$
and sure enough, $\mathit{SP}_4 = 4 \cdot 16 + 1 = 65$. We can now easily calculate the next few subset permutation numbers:
$1,2,5,16,65,326,1957,13700,109601,986410, \dots$ | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
$1,2,5,16,65,326,1957,13700,109601,986410, \dots$
A commenter also wondered about the particular order in which I listed subset permutations in my previous post. The short, somewhat disappointing answer is “whatever order came out of the standard library functions I used”. However, there are definitely more interesting things to say about the ordering, and I think I’ll probably write about that in another post.
Posted in combinatorics, posts without words | Tagged , , , | 6 Comments
Post without words #33
Posted in combinatorics, posts without words | Tagged , , | 4 Comments
Happy tau day!
Happy Tau Day! The Tau Manifesto by Michael Hartl is now available in print, if you’re in the market for a particularly nerdy coffee table book and conversation starter:
I also wrote about Tau Day ten years ago; see that post for more links! Using $\tau$ we can make the most beautiful equation in the world even more beautiful:
$e^{i \tau} = 1 + 0i$
Please enjoy eating two pi(e)s today. I’ll be back soon with some solutions to the parallelogram area challenge!
Posted in famous numbers, links | Tagged , , | 1 Comment
Challenge: area of a parallelogram
And now for something completely different!1
Suppose we have a parallelogram with one corner at the origin, and two adjacent corners at coordinates $(a,b)$ and $(c,d)$. What is the area of the parallelogram?
1. …or is it?
Posted in challenges, geometry | Tagged , | 18 Comments
Post without words #32
Posted in posts without words | Tagged , , , | 8 Comments
The Natural Number Game
Hello everyone! It has been quite a while since I have written anything here—my last post was in March 2020, and since then I have been overwhelmed dealing with online and hybrid teaching, plus a newborn (who is now almost 5 months old and very cute): | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
But the spring semester is finally over, and I have a sabbatical in the fall, so I plan to do a bunch more writing! I have several ideas of things to come (feel free to send me suggestions as well), but to start things out for today I just have a fun link:
The idea is to use a computer proof assistant to formally prove a lot of basic facts about arithmetic. That is, you get to build proofs using a special precise notation, and the computer will automatically check whether your proof is correct. But the whole thing is organized like a game, with a tutorial, levels that build on each other and introduce new ideas and techniques just before you need them, etc. It’s a lot of fun and honestly kind of addicting.
There is very little background needed other than some capability for abstract thinking. Things start out quite easy and progress slowly, though things become quite challenging by the time you make it all the way to the end.
Posted in challenges, computation, proof | Tagged , , , , , | 6 Comments
An exploration of forward differences for bored elementary school students
Last week I made a mathematics worksheet for my 8-year-old son, whose school is closed due to the coronavirus pandemic. I’m republishing it here so others can use it for similar purposes.
Figurate numbers and forward differences
There are lots of further directions this could be taken but I’ll leave that to you and your kids. I tried to create something that was conducive to open-ended exploration rather than something that had a single particular goal in mind.
• For the curious: Babbage Difference Engine
• For the intrepid: Concrete Mathematics by Graham, Knuth, and Patashnik, section 2.6 (“Finite and Infinite Calculus”) | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
Seeing as how we’ve got at least four more weeks of (effectively) homeschooling ahead of us, and probably more than that, in all likelihood I will be making more of these, and I will certainly continue to share them here! If you use any of these with your kids I’d love to hear about your experiences.
Posted in arithmetic, teaching | | 1 Comment
Ways to prove a bijection
You have a function $f : A \to B$ and want to prove it is a bijection. What can you do?
By the book
A bijection is defined as a function which is both one-to-one and onto. So prove that $f$ is one-to-one, and prove that it is onto.
This is straightforward, and it’s what I would expect the students in my Discrete Math class to do, but in my experience it’s actually not used all that much. One of the following methods usually ends up being easier in practice.
By size
If $A$ and $B$ are finite and have the same size, it’s enough to prove either that $f$ is one-to-one, or that $f$ is onto. A one-to-one function between two finite sets of the same size must also be onto, and vice versa. (Of course, if $A$ and $B$ don’t have the same size, then there can’t possibly be a bijection between them in the first place.)
Intuitively, this makes sense: on the one hand, in order for $f$ to be onto, it “can’t afford” to send multiple elements of $A$ to the same element of $B$, because then it won’t have enough to cover every element of $B$. So it must be one-to-one. Likewise, in order to be one-to-one, it can’t afford to miss any elements of $B$, because then the elements of $A$ have to “squeeze” into fewer elements of $B$, and some of them are bound to end up mapping to the same element of $B$. So it must be onto. | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
However, this is actually kind of tricky to formally prove! Note that the definition of “$A$ and $B$ have the same size” is that there exists some bijection $g : A \to B$. A proof has to start with a one-to-one (or onto) function $f$, and some completely unrelated bijection $g$, and somehow prove that $f$ is onto (or one-to-one). Also, a valid proof must somehow account for the fact that this becomes false when $A$ and $B$ are infinite: a one-to-one function between two infinite sets of the same size need not be onto, or vice versa; we saw several examples in my previous post, such as $f : \mathbb{N} \to \mathbb{N}$ defined by $f(n) = 2n$. Although tricky to come up with, the proof is cute and not too hard to understand once you see it; I think I may write about it in another post!
Note that we can even relax the condition on sizes a bit further: for example, it’s enough to prove that $f$ is one-to-one, and the finite size of $A$ is greater than or equal to the finite size of $B$. The point is that $f$ being a one-to-one function implies that the size of $A$ is less than or equal to the size of $B$, so in fact they have equal sizes.
By inverse
One can also prove that $f : A \to B$ is a bijection by showing that it has an inverse: a function $g : B \to A$ such that $g(f(a)) = a$ and $f(g(b)) = b$ for all $a \in A$ and $b \in B$. As we saw in my last post, these facts imply that $f$ is one-to-one and onto, and hence a bijection. And it really is necessary to prove both $g(f(a)) = a$ and $f(g(b)) = b$: if only one of these hold then $g$ is called a left or right inverse, respectively (more generally, a one-sided inverse), but $f$ needs to have a full-fledged two-sided inverse in order to be a bijection. | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
…unless $A$ and $B$ are of the same finite size! In that case, it’s enough to show the existence of a one-sided inverse—say, a function $g$ such that $g(f(a)) = a$. Then $f$ is (say) a one-to-one function between finite equal-sized sets, hence it is also onto (and hence $g$ is actually a two-sided inverse).
We must be careful, however: sometimes the reason for constructing a bijection in the first place is in order to show that $A$ and $B$ have the same size! This kind of thing is common in combinatorics. In that case one really must show a two-sided inverse, even when $A$ and $B$ are finite; otherwise you end up assuming what you are trying to prove.
By mutual injection?
I’ll leave you with one more to ponder. Suppose $f : A \to B$ is one-to-one, and there is another function $g : B \to A$ which is also one-to-one. We don’t assume anything in particular about the relationship between $f$ and $g$. Are $f$ and $g$ necessarily bijections?
Posted in logic, proof | | 7 Comments
One-sided inverses, surjections, and injections
Several commenters correctly answered the question from my previous post: if we have a function $f : A \to B$ and $g : B \to A$ such that $g(f(a)) = a$ for every $a \in A$, then $f$ is not necessarily invertible. Here are a few counterexamples:
• Commenter Buddha Buck came up with probably the simplest counterexample: let $A$ be a set with a single element, and $B$ a set with two elements. It does not even matter what the elements are! There’s only one possible function $g : B \to A$, which sends both elements of $B$ to the single element of $A$. No matter what $f$ does on that single element $a \in A$ (there are two choices, of course), $g(f(a)) = a$. But clearly $f$ is not a bijection. | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
• Another counterexample is from commenter designerspaces: let $f : \mathbb{N} \to \mathbb{Z}$ be the function that includes the natural numbers in the integers—that is, it acts as the identity on all the natural numbers (i.e. nonnegative integers) and is undefined on negative integers. $g : \mathbb{Z} \to \mathbb{N}$ can be the absolute value function. Then $g(f(n)) = |n| = n$ whenever $n$ is a natural number, but $f$ is not a bijection, since it doesn’t match up the negative integers with anything.
Unlike the previous example, in this case it is actually possible to make a bijection between $\mathbb{N}$ and $\mathbb{Z}$, for example, the function that sends even $n$ to $n/2$ and odd $n$ to $-(n+1)/2$.
• Another simple example would be the function $f : \mathbb{N} \to \mathbb{N}$ defined by $f(n) = 2n$. Then the function $g(n) = \lfloor n/2 \rfloor$ satisfies the condition, but $f$ is not a bijection, again, because it leaves out a bunch of elements.
• Can you come up with an example $f : \mathbb{R} \to \mathbb{R}$ defined on the real numbers $\mathbb{R}$ (along with a corresponding $g$)? Bonus points if your example function is continuous.
All these examples have something in common, namely, one or more elements of the codomain that are not “hit” by $f$. Michael Paul Goldenberg noted this phenomenon in general. And in fact we can make this intuition precise.
Theorem. If $f : A \to B$ and $g : B \to A$ such that $g(f(a)) = a$ for all $a \in A$, then $f$ is injective (one-to-one).
Proof. Suppose for some $a_1, a_2 \in A$ we have $f(a_1) = f(a_2)$. Then applying $g$ to both sides of this equation yields $g(f(a_1)) = g(f(a_2))$, but because $g(f(a)) = a$ for all $a \in A$, this in turn means that $a_1 = a_2$. Hence $f$ is injective.
Corollary. since bijections are exactly those functions which are both injective (one-to-one) and surjective (onto), any such function $f : A \to B$ which is not a bijection must not be surjective. | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
And what about the opposite case, when there are functions $f : A \to B$ and $g : B \to A$ such that $f(g(b)) = b$ for all $b \in B$? As you might guess, such functions are guaranteed to be surjective—can you see why?
Posted in logic | | 1 Comment
Suppose we have sets $A$ and $B$ and a function $f : A \to B$ (that is, $f$’s domain is $A$ and its codomain is $B$). Suppose there is another function $g : B \to A$ such that $g(f(a)) = a$ for every $a \in A$. Is $f$ necessarily a bijection? That is, does $f$ necessarily match up each element of $A$ with a unique element of $B$ and vice versa? Or put yet another way, is $f$ necessarily invertible? | {
"domain": "mathlesstraveled.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145725743421,
"lm_q1q2_score": 0.8537969150909941,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 219.63282870187984,
"openwebmath_score": 0.8304086923599243,
"tags": null,
"url": "https://mathlesstraveled.com/?cpage=1"
} |
# Rotation Matrix 3d | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
The 3-dimensional versions of the rotation matrix A are the following matrices:. For counterclockwise rotation, the matrix has the following elements. Or you transpose the matrix first (mtmp=m; for x=0 to 2 for y=0 to 2 m[x,y] = mtmp[y,x]) and do your standard rotation calculation with the transposed matrix. a rotation around the z-axis wouldn't change the z-values of the vertices. However, I can only figure out how to do 1 and 4 using numpy. We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties. And thank you for taking the time to help us improve the quality of Unity Documentation. Primarily to support 3D rotations. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. We can project a point orthogonal down into one of the main planes by using a matrix that scale the axis normally onto the plane with 0. The 3D object is moved and rotated in the 3D space, and the new destination points become B1=, B2=, and B3=. • In 2D, a rotation just has an angle • In 3D, specifying a rotation is more complex -basic rotation about origin: unit vector (axis) and angle •convention: positive rotation is CCW when vector is pointing at you • Many ways to specify rotation -Indirectly through frame transformations -Directly through •Euler angles: 3 angles. When I look at the file, however, it appears that the inputs to the transformation are the trans x,y,z and the roll,pitch,yaw angles. Raises: ValueError: If the shape of angles is not supported. Rotation About an Arbitrary Axis and Avoiding Gimbal Lock - Cprogramming. Rotation Matrices Suppose that ↵ 2 R. translation, rotation, scale, shear etc. A 3 by 3 matrix sets the rotation and shear. The axis can be either x or y or z. 1 Matrix Representation A 2D rotation is a tranformation of the form 2 4 x 1 y 1 3 5 = 2 4 cos( ) sin( ) sin( ) cos( ) 3 | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
A 2D rotation is a tranformation of the form 2 4 x 1 y 1 3 5 = 2 4 cos( ) sin( ) sin( ) cos( ) 3 5 2 4 x 0 y 0 3 5 (1) where is the angle of rotation. We will try to enter into the details of how the matrices are constructed and why, so this article is not meant for absolute beginners. euler_rotation = mathutils. Raises: ValueError: If the shape of quaternion is not supported. The rotation matrix for this transformation is as follows. Convert a Rotation Matrix to Euler Angles in OpenCV. Each has its own uses and drawbacks. The only thing new in the C++ code is the usage of GetConsoleScreenBufferInfo and SetConsoleTextAttribute which gets the size of the console and sets the text color. We can think of rotations in another way. Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4. −Composition of geometric transformations in 2D and 3D. Equations ()-() effectively constitute the definition of a vector: i. A short derivation to basic rotation around the x-, y- or z-axis by Sunshine2k- September 2011 1. A matrix with M rows and N columns is defined as a MxN matrix. How to combine rotation in 2 axis into one matrix. It's so clever that it's worth sharing in full detail. In 2D it's much simpler. And Rotation is done with trigonometric functions in the matrix. If we add. Simple rotation – formulas were derived for rotation of a shape centered in origin by a certain angle. When a transformation takes place on a 2D plane, it is called 2D transformation. This matrix class is used for 3D object rotate along a specifid axis. The only difference is the signs for sinθ are reversed. matrix() Describes a homogeneous 2D transformation matrix. gives the column matrix corresponding to the point (a+ dx, b+ dy, c+ dz). Initially I hoped that. I have a point, in that mesh, that i must calculate mannualy. The rotation operation consists of multiplying the transformation matrix by a matrix whose elements are derived from the angle | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
of multiplying the transformation matrix by a matrix whose elements are derived from the angle parameter. Rotation matrix from axis and angle First rotate the given axis and the point such that the axis lies in one of the coordinate planes Then rotate the given axis and the point such that the axis is aligned with one Use one of the fundamental rotation matrices to rotate the point depending. Greetings, I'm trying to apply a Rotation or Affine matrix to a spiral. The preview on the right will be updated when you compute, x’ (dash) points to the new x direction of the body, y’, z’ do the same for y and z axes. We can project a point orthogonal down into one of the main planes by using a matrix that scale the axis normally onto the plane with 0. As others stated in the comments you have to make sure you don´t mix row- and column-major matrices and dont mix any coordinate spaces. I have a 3D translation and rotation problem I am trying to solve using Excel 2010. As I understand, the rotation matrix around an arbitrary point, can be expressed as moving the rotation point to the origin, rotating around the origin and moving back to the original position. We can think of rotations in another way. @Sascha Grusche and @Elie Maalouf. As a unit quaternion, the same 3D rotation matrix. Learn more about rotation matrix, point cloud, 3d. They are represented in the matrix form as below −. The formula is , using the dot and cross product of vectors. Describing rotation in 3d with a vector. A matrix is an array of values that defines a transformation of coordinates. This is why also the 3D version has two of the three axes change simultaneously - because it is just a derivative from its 2D version. RotationMatrix[{u, v}] gives the matrix that rotates the vector u to the direction of the vector v in any dimension. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles. They are both ways of completely describing a | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
such, and only occur with the usage of Euler angles. They are both ways of completely describing a rotation in 3D, so they are freely convertible. You can also rotate, resize and stretch a 3D graph by dragging the mouse. Download our 100% free 3D Rotation templates to help you create killer PowerPoint presentations that will blow your audience away. 3D scaling matrix. Hi, I'm trying to transform a PET scan onto a CT scan based on an existing rotation and translation matrix. G-CNNs achieve equivariance with respect to nite subgroups of the rotation group, which constitutes a bottleneck in 3D. I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties. 3D rotations made easy in Julia. Appearance Scope: 6. When the Frobenius norm is taken as the measure of closeness, the solution is usually computed using the singular value decomposition (SVD). 1 Matrix Representation A 2D rotation is a tranformation of the form 2 4 x 1 y 1 3 5 = 2 4 cos( ) sin( ) sin( ) cos( ) 3 5 2 4 x 0 y 0 3 5 (1) where is the angle of rotation. Position Cartesian coordinates (x,y,z) are an easy and natural means of representing a position in 3D space …But there are many other representations such as spherical converted to matrix form to perform rotation. As we know $\cos(0) = 1$ and $\sin(0) = 0$. Three shears. Appearance: 7. ACM SIGGRAPH is a thriving international organization. det(R) != 1 and R. step file, it stores a rotation matrix that needs to be applied to the appropriate geometry, correct?. Rotation in 3D - The Rotation Matrix In this note, I investigate the rotation matrix that relates the image of a point p ⃗ \vec{p} p when it is rotated by an angle θ \theta θ about an axis a ⃗ \vec{a} a that passes through the origin. winter wheat (WW))is available to successive crops in reduced or no-till systems, uncover the | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
winter wheat (WW))is available to successive crops in reduced or no-till systems, uncover the mechanisms involved in this process, and ultimately develop management practices to maximize Po utilization from these sources. Serializable. If you have a rotation matrix, then this will transform from one coordinate system to another. So essentially quaternions store a rotation axis and a rotation angle, in a way that makes combining rotations easy. Just like the graphics pipeline, transforming a vector is done step-by-step. A 2-columns matrix or data frame containing a set of X and Y coordinates. Abstract transformations, such as rotations (represented by angle and axis or by a quaternion), translations, scalings. A rotation matrix can be built by using the axis of the coordinate system you're rotating into. There will be some repetition of the earlier analyses. Convert your quaternion to a rotation matrix, and use it in the Model Matrix. 3D programming in python. Posted September 16, 2017 · CuraEngine mesh_rotation_matrix to rotate 180 degrees Just a WAG here, but since the print origin is at the corner of the print bed (unless you're using a Delta printer), and the rotation matrix is rotating about the origin, then haven't you just rotated it off the print bed?. I've been all. I want this rotation matrix to perform a rotation about the X axis (or YZ pla. Rotation matrices, on the other hand, are the representation of choice when it comes to implementing efficient rotations in software. Coordinates of point p in two systems The elementary 3D rotation matrices are constructed to perform In order to be able to write the rotation matrix directly, imagine that the the z-axis is playing the role of the x-axis,. As their trunks were rotated passively without fixing the legs, each volunteer was instructed to keep their knees straight and rotate their legs in a fashion comparable to the trunk rotation, in order to minimize the effects of leg rotation on trunk rotation. The | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
to the trunk rotation, in order to minimize the effects of leg rotation on trunk rotation. The easiest rotation axes to handle are those that are parallel to the co-ordinate axes. , is an orthogonal matrix) is 1). New coordinates by 3D rotation of points Calculator. Basic steps needed to display 3D objects: 4. If the first body is only capable of rotation via a revolute joint, then a simple convention is usually followed. Email this Article Givens rotation. Rotation Transforms for Computer Graphics covers a wide range of mathematical techniques used for rotating points and frames of reference in the plane and 3D space. Rotation. explanation of quaternion from matrix. Find the characteristic function, eigenvalues, and eigenvectors of the rotation matrix. Tag: math,vector,lua,rotation I have two Vec3s, Camera Forward and Turret Forward. Turn your head left and right; that’s a rotation around the Y axis. The rotation operation is a 3x3 matrix. Then, the tutorials will move on to give you the matrices for rotation over the x and y axes, tell you how to use them, and then give you a matrix which will allow rotations around an arbitrary axis. With a chain of rotations, roundoff errors accumulate. The rotation matrix for this transformation is as follows. 3D Transformations, Translation, Rotation, Scaling The Below program are for 3D Transformations. Unless specified, the rest of this page uses implies rotation to be a rotation of points about the origin. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n ⁄ 2 of them. represent data on 3D rotation groups. muting any, we clearly need a negative unit matrix, namely, Proper Rotation. Because Rotation can be done either along the x, y, or z axis, there is a different rotation matrix for each of the axises: Figure 6a - Rotation around the X axis. Its first 3 dimensional vectors(3*3 submatrix) contain the rotated X, | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
- Rotation around the X axis. Its first 3 dimensional vectors(3*3 submatrix) contain the rotated X, Y and Z axes. Examples in 3D computer graphics Rotation. In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The most general three-dimensional rotation matrix represents a counterclockwise rotation by an angle θ about a fixed axis that lies along the unit vector ˆn. Under rotations, vector lengths are preserved as well as the angles between vectors. The formula of this operations can be described in a simple multiplication of. You can probably use the axis angle representation to build a rotation matrix, with whatever math library you have available. G-CNNs achieve equivariance with respect to nite subgroups of the rotation group, which constitutes a bottleneck in 3D. This means that no rotation has taken place around any of the axes. In the above code, since the rotation transformation is prepended to the matrix, the rotation transformation would be performed first. Lecture 08. 2+) were then focused to capture these y-z′ images at >800 frames per second as the sheet was repeatedly scanned across the sample in the x direction. Each has its own uses and drawbacks. This object get transformed with the following matrix transformation: glRotated and glTranslate. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. A rotation matrix describes the relative orientation of two such frames. rotqrmean Find the average of several rotation quaternions rotqrvec Apply a quaternion rotation to an array of 3D vectors skew3d Convert between vectors and skew symmetric matrices: 3x3 matrix <-> 3x1 vector and 4x4 Plucker matrix <-> 6x1 vector. the product of a viewing matrix and a modeling matrix. If you are uncomfortable with the thought of 4D matrix rotations, then I | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
and a modeling matrix. If you are uncomfortable with the thought of 4D matrix rotations, then I recommend reading Wikipedia, or checking out my article about 3D graphing, which can be found here. where M is a constant 3x3 matrix, is the 3x3 identity matrix, and we are solving for the 3x3 matrix R. Using 3D Rotation Matrices in Practice By confuted So, now that you more or less know how to rotate a point in any arbitrary manner in three dimensions, generating matrices along the way, it's time to learn what you should do with each of these matrices. We found that this was the rotation transformation matrix about an x-axis rotation. First, suppose that all eigenvalues of the 3D rotation matrix A are real. We will try to enter into the details of how the matrices are constructed and why, so this article is not meant for absolute beginners. If you want to rotate around an arbitary axis you can use the equations under "Rotation matrix from axis and angle" To understand how a rotation matrix works you should know that each column represents a 3D Vector which in turn represents one of the axis of the rotated coordinate system. ' (as long as the translation is ignored). Again, we must translate an object so that its center lies on the origin before scaling it. We'll call the rotation matrix for the X axis matRotationX, the rotation matrix for the Y axis matRotationY, and the rotation matrix for the Z axis matRotationZ. The eigenvalues of A are. Therefore, an arbitrary 3D rotation can be decomposed into only two 3D beamshears. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. I'm looking for a SO(3) rotation representation that lends itself to energy minimization. Rotation matrices are used for computations in aerospace, image processing, and other technical computing applications. We conclude that every rotation matrix, when expressed in | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
other technical computing applications. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n / 2 of them. Click the 3D graph (the white space of the graph layer, not the 3D data plot), eight resizing handles appear around the 3D graph. Matrix representation. This axis, in this work, will be represented by the supporting line of the directed segment S ab (a 1D simplex), where ( , , (0)) 3 (0) 2 (0) a a1 a a and ( , (0)) 3 (0) 2 (0) b 1 b b are two non-coincident 3D points which we will refer as the. •In 3D, specifying a rotation is more complex –basic rotation about origin: unit vector (axis) and angle •convention: positive rotation is CCW when vector is pointing. The Rotation 3D page. The easiest rotation axes to handle are those that are parallel to the co-ordinate axes. Now, according to the equation, multiplying the transformation matrix with a coordinate would result in a coordinate but if is [9,1] for example, if i multiply with the rotation matrix. Yes, [R|t] implies the rotation and translation. Serializable, java. Rotation matrices are used in computer graphics and in statistical analyses. calibration cube. 3D Matrices. The Rotation 3D page. The eigenvalues of A are. rotation matrix specifies a 3 × 3 matrix. ppt), PDF File (. Shortest distance between two lines. 0 License, and code samples are licensed under the Apache 2. A short derivation to basic rotation around the x-, y- or z-axis by Sunshine2k- September 2011 1. In 2D euclidean space, rotation matrix is a matrix that tilts every single vector in the 2D space, without changing the scale. pdf), Text File (. Second, this method means we can create our own css animations, and do something a bit more advanced. Computing Euler angles from a rotation matrix Gregory G. I understand the sentence except for the "rotation matrix" part, again. RotationMatrix[{u, v}] gives the matrix that rotates the | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
for the "rotation matrix" part, again. RotationMatrix[{u, v}] gives the matrix that rotates the vector u to the direction of the vector v in any dimension. 3DPrimitivesTransformations - Free download as Powerpoint Presentation (. I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. If you are reading this page in order to write a 3D computer program I suggest you read enough of this page to convince yourself of the problems with Euler angles and to get an intuitive understanding of 3D rotations and then move on to quaternion or matrix algebra representations. skip_checks (bool, optional) – If True avoid sanity checks on rotation_matrix for performance. Math for simple 3D coordinate rotation (python) Ask Question Asked 3 years, 5 months ago. Defining the rotation axis as the z axis, we note first that the z coordinate will be unchanged by any rotation about the z axis. A general homoge- neous matrix formulation to 3D rotation geometric transformations is proposed which suits for the cases when the rotation axis is unnecessarily through the coordinate system origin given their rotation axes and. When acting on a matrix, each column of the matrix represents a different vector. The three simultaneous orthogonal rotations measured with a 3D gyroscope represent a single rotation around a certain axis for a certain angle. 3D rotations • A 3D rotation can be parameterized with three numbers • Common 3D rotation formalisms - Rotation matrix • 3x3 matrix (9 parameters), with 3 degrees of freedom - Euler angles • 3 parameters - Euler axis and angle • 4 parameters, axis vector (to scale) - Quaternions • 4 parameters (to scale). multiplying rotation matrices is a noisyrotation matrix [1]. M modelview = M viewing * M modeling. We can now write a transformation for the rotation of a point about this line. And so here's the rotation transformation matrix. RotationMatrix[{u, v}] gives the matrix that rotates the | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
here's the rotation transformation matrix. RotationMatrix[{u, v}] gives the matrix that rotates the vector u to the direction of the vector v in any dimension. R = rotx(ang) creates a 3-by-3 matrix for rotating a 3-by-1 vector or 3-by-N matrix of vectors around the x-axis by ang degrees. 2+) were then focused to capture these y-z′ images at >800 frames per second as the sheet was repeatedly scanned across the sample in the x direction. 1 1 3 Lecture Video 3 of 4 Rotation Matrix Example 1 - Duration: 10:20. Ask Question With r = RotationMatrix[a, {x, y, z}] I can compute a 3D rotation matrix from its axis/angle representation. all types of rotation calculation will eventually yield a 4x4 matrix, as this is always the required form for expressing the final transformation. Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. The 3-dimensional versions of the rotation matrix A are the following matrices:. ˇ, rotation by ˇ, as a matrix using Theorem 17: R ˇ= cos(ˇ) sin(ˇ) sin(ˇ) cos(ˇ) = 1 0 0 1 Counterclockwise rotation by ˇ 2 is the matrix R ˇ 2 = cos(ˇ 2) sin(ˇ) sin(ˇ 2) cos(ˇ 2) = 0 1 1 0 Because rotations are actually matrices, and because function composition for matrices is matrix multiplication, we’ll often multiply. 3D Rotation Matrix. From the Cartesian grid (left grid), we can see the blue point is located at (2, 1). Rotation Matrix. 3D rotation matrix around vector. is the rotation matrix already, when we assume, that these are the normalized orthogonal vectors of the local coordinate system. 0 License , and code samples are licensed under the Apache 2. I know that in 3D space the matrix product order is important - changing the order of the matrices can effect the rotate result. winter wheat (WW))is available to successive crops in reduced or no-till systems, uncover the mechanisms involved in this process, and ultimately develop management practices to maximize Po utilization from these sources. Euler angles can be defined with many | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
practices to maximize Po utilization from these sources. Euler angles can be defined with many different combinations (see definition of Cardan angles ). As I understand, the rotation matrix around an arbitrary point, can be expressed as moving the rotation point to the origin, rotating around the origin and moving back to the original position. matrix3d() Describes a 3D transformation as a 4×4 homogeneous matrix. 3D Rotations Rotation about z-axis. When acting on a matrix, each column of the matrix represents a different vector. This is the currently selected item. Learn more about rotation matrix, point cloud, 3d. A vector is a direction in a space (like for GPS), that is the result of the transformation (math operators) of a point by a rotation matrix. 3D rotation) that minimizes some objective function. Transformations is a Python library for calculating 4x4 matrices for translating, rotating, reflecting, scaling, shearing, projecting, orthogonalizing, and superimposing arrays of 3D homogeneous coordinates as well as for converting between rotation matrices, Euler angles, and quaternions. 3D Rotation • Counterclockwise •The matrix M transforms the UVW vectors to the XYZ vectors y z x u=(u x,u y,u z) v=(v x,v y,v z) Change of Coordinates. 8 (b): What constraints must the elements R_{ij} of the three-dimensional rotation matrix satisfy, in order to preserve the length of vector A (for all vectors A)? Homework Equations The. find angles , , which make the two matrices equal. Euler3D transform, how to set the rotation matrix?. This is done by multiplying the vertex with the matrix : Matrix x Vertex (in this order. Let T be a linear transformation from R^2 to R^2 given by the rotation matrix. Quaternions are just more compact and easier to interpolate. I want to rotate an object by 60 degrees around the y axis, counter-clockwise. Three shears. The solution is not unique in most cases. In matrix form, the infinitesimal rotation has the representation (4) where and (5) | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
in most cases. In matrix form, the infinitesimal rotation has the representation (4) where and (5) To first order in , it can be shown from that. Position Cartesian coordinates (x,y,z) are an easy and natural means of representing a position in 3D space …But there are many other representations such as spherical converted to matrix form to perform rotation. Thus, we have H O = [I O] ω ,. They are represented in the matrix form as below −. Scale and Rotate. Derivation of the 3D transformation matrix. Follow 34 views (last 30 days). A rotation matrix describes the relative orientation of two such frames. Angular velocity. Although the inverse process requires a choice of rotation axis between the two alternatives, it is a straightforward procedure to retrieve the rotation axis and angle (see Appendix A). Euler returns a 3D vector containing the XYZ Euler angles. Rotations in 3D applications are usually represented in one of two ways: Quaternions or Euler angles. Do you know any reference how to derive this rotation matrix? It seems clear to me the rotation matrix in planar truss, 3D truss and planar frame are pretty similar (only different a bit), the form has same appearance to a simple rotation matrix, or when using directional cosine. To get in right-handed coordinate system, replace the angle with negative. rotation from the scanner coordinates to the camera coordinates will use the following transformation matrix. −Matrix representation of affine transformations. Please try again in a few minutes. In order to export actual quaternion data to your ASCII log file, you will have to set the Orientation output to Quaternions in MT Manager under Tools > Preferences > Exporters. it is called 3D transformation. The table of direction cosines relating the femoral (F) and pelvic (P) reference frames is obtained most simply via matrix multiplication, which yields the rotation matrix F R P and its corresponding table of direction cosines, Direction cosines for virtually any | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
matrix F R P and its corresponding table of direction cosines, Direction cosines for virtually any compound rotation can be found easily by using this exact methodology. A matrix is an array of values that defines a transformation of coordinates. C 3d Rotation Codes and Scripts Downloads Free. Slabaugh Abstract This document discusses a simple technique to find all possible Euler angles from a rotation matrix. Using localOrientation for rotation values, however, returns a 3×3 matrix. To produce a sequence of transformations with these equations, rotation is followed by translation, we must calculate the transformed coordinates one step at a time, thereby eliminating the calculation of intermediate coordinate values. (x_x, x_y, x_z) is a 3D vector that represents only the direction of the X-axis with respect to the coordinate system 1. Consider a counter-clockwise rotation of 90 degrees about the z-axis. Object implements java. Rotation estimation is a fundamental step for object motion estimation, alignment and registration, in image processing, whereas, 3-D shape reconstruction, object recognition, autonomous navigation and ego-motion are typical applications of rotation estimation in computer vision and robotics. I have a similar class. Matrix M 1 is a 2x2 rotation matrix, M 2 is translation vector. Normalize those vectors. winter wheat (WW))is available to successive crops in reduced or no-till systems, uncover the mechanisms involved in this process, and ultimately develop management practices to maximize Po utilization from these sources. A rotation matrix describes the relative orientation of two such frames. 1 1 3 Lecture Video 3 of 4 Rotation Matrix Example 1 - Duration: 10:20. 3D Rotation Matrix. A rotation followed by a translation is very different from a translation followed by a rotation, as illustrated below: Using the Matrix Object. If you want to rotate around an arbitary axis you can use the equations under "Rotation matrix from axis and angle" To | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
around an arbitary axis you can use the equations under "Rotation matrix from axis and angle" To understand how a rotation matrix works you should know that each column represents a 3D Vector which in turn represents one of the axis of the rotated coordinate system. The matrix to the left is a parallel projection down into the xy-plane. They will allow us to transform our (x,y,z,w) vertices. This article discusses the different types of matrices including linear transformations, affine transformations, rotation, scale, and translation. The three simultaneous orthogonal rotations measured with a 3D gyroscope represent a single rotation around a certain axis for a certain angle. The rotation matrix has the following properties: A is a real, orthogonal matrix, hence each of its rows or columns represents a unit vector. CSS also supports 3D transformations. With c++ (win32). Alternatively, you can set the Orientation output to Rotation Matrix directly. Ask Question Asked 3 years, 10 months ago. But why would this 3D frame rotation seems much different from those?. In this, the first of two articles I will show you how to encode 3D transformations as a single 4×4 matrix which you can then pass into the appropriate. Details EulerMatrix is also known as Euler rotation matrix or Euler rotation, and the angles α , β , and γ are often referred to as Euler angles. Quaternions represent a single rotation. math on December 25, 2008. When acting on a matrix, each column of the matrix represents a different vector. You might use this when applying the same rotation to a number of different objects,. It allows you to examine different rotation sequences. @Sascha Grusche and @Elie Maalouf. In August 1987, in Vancouver, Canada, almost all of those who worked in the paleomagnetic group at the University College of Rhodesia and Nyasaland, Salisbury, Southern Rhodesia (now the University of Zimbabwe, Harare, Zimbabwe) were by chance attending the International Union of Geodesy and | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
of Zimbabwe, Harare, Zimbabwe) were by chance attending the International Union of Geodesy and Geophysics meeting. We'll call the rotation matrix for the X axis matRotationX, the rotation matrix for the Y axis matRotationY, and the rotation matrix for the Z axis matRotationZ. 3D Rotations Rotation about z-axis. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. det(R) != 1 and R. Composing a rotation matrix. A single precision floating point 4 by 4 matrix. Video transcript - What I want to do in this video is get some. I want this rotation matrix to perform a rotation about the X axis (or YZ pla. This example shows how to do rotations and transforms in 3D using Symbolic Math Toolbox™ and matrices. Online tools - vector rotation in 3D. 1 Representation Elements of the 3D rotation group, SO(3), are represented by 3D rotation matrices. New coordinates by 3D rotation of points Calculator - High accuracy calculation Welcome, Guest. The general rotation is much the same, with the up vector taken randomly, the desired rotation applied after the initial viewing transformation, and then the inverse of the viewing transformation is applied. Serializable. The Java 3D model for 4 X 4 transformations is: [ m00 m01 m02 m03 ] [ x ] [ x' ] [ m10 m11 m12 m13 ]. 3D Rotations are used everywhere in Computer Graphics, Computer Vision, Geometric Modeling and Processing, as well as in many other related areas. Now suppose we are given a matrix and are required to extract Euler angles corresponding to the above rotation sequence, i. C 3d Rotation Codes and Scripts Downloads Free. Just like the graphics pipeline, transforming a vector is done step-by-step. Examples in 3D computer graphics Rotation. This code checks that the input matrix is a pure rotation matrix and does not contain any scaling factor or reflection for example /** *This checks that the | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
matrix and does not contain any scaling factor or reflection for example /** *This checks that the input is a pure rotation matrix 'm'. What is the correct order of transformations scale, rotate and translate and why? 3. Moreover, there are similar transformation rules for rotation about and. is the orthogonal projection of onto. We will try to enter into the details of how the matrices are constructed and why, so this article is not meant for absolute beginners. Follow 34 views (last 30 days). In 2D it's much simpler. Primarily to support 3D rotations. The center of a Cartesian coordinate frame is typically used as that point of rotation. As David Joyce points out, this fact is true in odd dimensions (including 3) but not even dimensions. (b) Because all the columns of U are mutually orthogonal, we can conclude that U is an orthogonal matrix. Consider the original set of basis vectors, i, j, k, and rotate them all using the rotation matrix A. Rotation matrices are orthogonal as explained here. You might use this when applying the same rotation to a number of different objects,. Kelly! above x2: screenshots from here. Composing a rotation matrix. The matrix takes a coordinate in the inner coordinate system described by the 3 vectors and and finds its location in the outer coordinate system. Moreover, there are similar transformation rules for rotation about and. Hi, I am doing optimization on a vector of rotation angles tx,ty and tz using scipy. Parameters: matrix - double[][]. Auckland's prof. When acting on a matrix, each column of the matrix represents a different vector. Appearance Mixed: 8. Generally speaking any matrix in the group SO(3) represents a rotation in 3d. That matrix isn't exactly symmetric, but a rotation matrix that is symmetric is a 180 degree rotation. What does POSIT require to be able to do 3D pose estimation? First it requires image coordinates of some object's points (minimum 4 points). For the rotation matrix R and vector v, the rotated | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
of some object's points (minimum 4 points). For the rotation matrix R and vector v, the rotated vector is given by R*v. This example shows how to do rotations and transforms in 3D using Symbolic Math Toolbox™ and matrices. General Case Computer Graphics 3d Rotation Software Rendering Computer Graphics 3d Rotation 3d Coordinate Systems Powerpoint Presentation Axis Rotation 3d Rotation Matrix Derivation Machines Vertical Center Number Umc-750 Sample Trials Coordinate Geometry Basics. A general homoge- neous matrix formulation to 3D rotation geometric transformations is proposed which suits for the cases when the rotation axis is unnecessarily through the coordinate system origin given their rotation axes and. The View Matrix: This matrix will transform vertices from world-space to view-space. I have 2 known 3d points which are the origin of 2 axis plot in the space and I need to compute the 3D rotation matrix between them. Unless specified, the rest of this page uses implies rotation to be a rotation of points about the origin. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. The default polygon is a square that you can modify. all types of rotation calculation will eventually yield a 4x4 matrix, as this is always the required form for expressing the final transformation. You need to pass an angle of rotation and x, y, z axes as parameters to this method. However, manipulating 3D Rotations is always confusing, and debugging code that involves 3D rotation is usually quite time consuming. The rotation matrix is not parametric, created via eigendecomposition, I can't use angles to easily create an inverse matrix. 3D Transformations, Translation, Rotation, Scaling The Below program are for 3D Transformations. Find more Mathematics widgets in Wolfram|Alpha. Active 3 years, 7 months ago. However, now that I started implementing it | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
widgets in Wolfram|Alpha. Active 3 years, 7 months ago. However, now that I started implementing it myself, I came to the point where I'm really confused. for Java and C++ code to implement these rotations click here. In a two-dimensional cartesian coordinate plane system, the matrix R rotates the points in the XY-plane in the counterclockwise through an angle θ about the origin. math on December 25, 2008. Then transform a vector by that rotation matrix to get your result. 3D Matrices. CE503 Rotation Matrices Derivation of 2D Rotation Matrix Figure 1. Rotation Matrix. rotation of one frame into the other, Pj is the direction cosine for X2 with respect to Xk, and the rotation transformation matrix is an orthogonal matrix: P R12 = [flk] (5-5) fjk = Cos(X], Xi) JM In Sec. In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The rotation matrix is displayed for the current angle. In an analogous fashion, orientations are described relative to a standard orientation by a rotation, with the identity rotation describing the standard. In this, the first of two articles I will show you how to encode 3D transformations as a single 4×4 matrix which you can then pass into the appropriate. Accordingly, A v = v {\displaystyle Av=v} , and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. Matrix multiplications always have the origin as a fixed point. This way to do the inverse rotation works only with pure normalized rotation matrices, should be noticed. 3D Reflection in Computer Graphics- Reflection is a kind of rotation where the angle of rotation is 180 degree. Getting Started with the Java 3D API written in Java 3D: 9. 0° (rotation happens on the XY plane in 3D). Understanding transforamtion matrix requires some knoledge of math… R. 3D Rotation The easiest rotation axes are those that parallel to the coordinate axis. This | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
math… R. 3D Rotation The easiest rotation axes are those that parallel to the coordinate axis. This article shows how to implement three-dimensional rotation matrices and use them to rotate a 3-D point cloud. The homogeneous transformation matrix for 3D bodies As in the 2D case, a homogeneous transformation matrix can be defined. How does OCC read in and apply a rotation matrix in the STEP file? That is, if I define a rotation with BRepBuilderAPI_Transform, save the resulting shape to a. This method prepends or appends the transformation matrix of the Graphics by the rotation matrix according to the order parameter. I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. com Starting out. (x_x, x_y, x_z) is a 3D vector that represents only the direction of the X-axis with respect to the coordinate system 1. We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties. Primarily to support 3D rotations. Now, my understanding of your original question is that this unit vector, Ur, represents a rotation from U1, so you want to know how to find the rotation matrix that will transform U1 to Ur, i. Orthogonal matrices represent rotations (more precisely rotations, reflections, and compositions thereof) because, in a manner of speaking, the class of orthogonal matrices was defined in such a way so that they would represent rotations and refle. The rotation matrix has the following properties: A is a real, orthogonal matrix, hence each of its rows or columns represents a unit vector. Try a 90 degree rotation and then check. rotate3d() Rotates an element around a fixed axis in 3D space. 3D Transformations – Part 1 Matrices Transformations are fundamental to working with 3D scenes and something that can be frequently confusing to those that haven’t worked in 3D before. 1 Eigenvalues An n× nmatrix Ais orthogonal if its columns are unit vectors and orthogonal to. | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
1 Eigenvalues An n× nmatrix Ais orthogonal if its columns are unit vectors and orthogonal to. Lecture 08. The typical operations are translation, rotation. Here atan2 is the same arc tangent function, with quadrant checking, you typically find in C or Matlab. 1 The matrix for rotation about an arbitrary line. A Rotation instance can be initialized in any of the above formats and converted to any of the others. 3d curl intuition, part 1. Matrix for rotation is a clockwise direction. Geometrical Rotation Videos 9,222 royalty free stock videos and video clips of Geometrical Rotation. Now I would like to add a "tropism" command to simulate gravity pulling on the elements of an L-System, similar to how Laurens Laprés LParser does, and this is where I'm stuck. Homogeneous transforms contain BOTH rotation and translation information. R = rotx(ang) creates a 3-by-3 matrix for rotating a 3-by-1 vector or 3-by-N matrix of vectors around the x-axis by ang degrees. ANGLE DECOMPOSITION Recall that the rotation submatrix of the transformation is a multiplication matrix of the dot products of the unit vectors of the two body coordinate systems, and therefore includes trigonometric functions of the three angles of rotation, denoting flexion, abduction, and external rotation. Consider the 3-D rotation matrix U =-0. Ask Question Asked 3 years, 10 months ago. I'm struggling to understand the relation between the angles used to compose a rotation matrix and the angular velocity vector of the body expressed in the body frame. R = Rx*Ry*Rz. Both of these vectors are on different planes where Camera Forward is based on a free-look camera and Turret Forward is determined by the tank it sits on, the terrain the tank is on, etc. it only request for value and display the putout in text format on the screen. Moreover, the rotation axis in the 3D space coincides with the normal vector of the rotation plane. , is an orthogonal matrix) is 1). For x-axis rotation, we have the matrix:. This | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
the rotation plane. , is an orthogonal matrix) is 1). For x-axis rotation, we have the matrix:. This property allows you to rotate, scale, move, skew, etc. Home / Mathematics / Space geometry; Calculates the new coordinates by rotation of points around the three principle axes (x,y,z). 7 Transformation Matrix and Stiffness Matrix in Three-Dimensional Space. This article might seem exceedingly obvious to some but I'll build up to a point in a few articles. I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. Because rotation matrices. 3: geometry of the 2D coordinate transformation The 2 2 matrix is called the transformation or rotation matrix Q. As we know $\cos(0) = 1$ and $\sin(0) = 0$. two antiparallel axes and angles (one axis and angle is negation of the other). square matrix – For 2D, 3x3 matrix – For 3D, 4x4 matrix. Lecture 5: 3-D Rotation Matrices. In order to calculate the rotation about any arbitrary point we need to calculate its new rotation and translation. 3D rotations made easy in Julia. The rotation matrices SO(3) form a group: matrix multiplication of any two rotation matrices produces a third rotation matrix; there is a matrix 1 in SO(3) such that 1M= M; for each Min SO(3) there is an inverse matrix M 1such that M M= MM 1 = 1. This format is definitely less intuitive than Euler angles, but it's still readable: the xyz components match roughly the rotation axis, and w is the acos of the rotation angle (divided by 2). , the three quantities are the components of a vector provided that they transform under rotation of the coordinate axes about in accordance with Equations ()-(). Then apply the following rules. skip_checks (bool, optional) – If True avoid sanity checks on rotation_matrix for performance. This package implements various 3D rotation parameterizations and defines conversions between them. In this example, I will only show the 4D rotation matrices. pdf), Text File (. | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
between them. In this example, I will only show the 4D rotation matrices. pdf), Text File (. rotation of one frame into the other, Pj is the direction cosine for X2 with respect to Xk, and the rotation transformation matrix is an orthogonal matrix: P R12 = [flk] (5-5) fjk = Cos(X], Xi) JM In Sec. to_matrix() this is the equivalent of. Rotation matrices can be constructed from elementary rotations about the In this lecture, I extend the 2D rotation matrix of SO(2) from Lecture 2. Rotation matrix visualization [5] 2018/09/29 17:08 Male / 20 years old level / High-school/ University/ Grad student / Very /. All input is normalized to unit quaternions and may therefore mapped to different ranges. I want to rotate an object by 60 degrees around the y axis, counter-clockwise. Analogously, we can define the tensor of inertia about point O, by writing equation(4) in matrix form. The matrix takes a coordinate in the inner coordinate system described by the 3 vectors and and finds its location in the outer coordinate system. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. Given 3 Euler angles , the rotation matrix is calculated as follows: Note on angle ranges. The rotation is performed clockwise, if you are looking along the direction of the rotation axis vector. A great amount of work on this topic is available in literature (see , , , , ). Representation of orientation • Homogeneous coordinates (review) • 4X4 matrix used to represent translation, scaling, and rotation • a point in the space is represented as • Treat all transformations the same so that they can be easily combined p= x y z 1. void: preMultiply(Matrix mb) Premultiplies the object matrix by mb and stores the result in the object; As a result, the. This is the currently selected item. Matrix transposition - if we have a matrix M with n rows and m columns, the transpose of , | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
item. Matrix transposition - if we have a matrix M with n rows and m columns, the transpose of , denoted is a matrix with m rows and n columns, with the first column of equal to the first row of and so on. If you want to rotate around an arbitary axis you can use the equations under "Rotation matrix from axis and angle" To understand how a rotation matrix works you should know that each column represents a 3D Vector which in turn represents one of the axis of the rotated coordinate system. The Mathematics of the 3D Rotation Matrix fastgraph. For some reason your suggested change could not be submitted. Call R v(θ) the 2x2 matrix corresponding to rotation of all vectors by angle +θ. This package implements various 3D rotation parameterizations and defines conversions between them. Stretching 3D Graphs. However, I can only figure out how to do 1 and 4 using numpy. 2) The rotation angles. Figure 6c - Rotation around the Z axis. It can be useful to notice that this can be done with a matrix operation. Euler angles can be defined with many different combinations (see definition of Cardan angles ). WebGL - Cube Rotation. Cartesian coordinates are typically used to represent the world in 3D programming. Transformations, continued 3D Rotation 23 r r r x y z Full 3D Rotation 0 sin cos 0 cos sin 1 0 0 – Multiply the current matrix by the rotation matrix that. Do we need to subtract the translation vector (t) from matrix M. 1 Introduction. A tensor of shape [A1, , An, 3, 3], where the last two dimensions represent a 3d rotation matrix. I want this rotation matrix to perform a rotation about the X axis (or YZ pla. Thus, we have H O = [I O] ω ,. First, suppose that all eigenvalues of the 3D rotation matrix A are real. ) and perspective transformations using homogenous coordinates. )? Ask Question Asked 3 years, It is quit lengthy but you can search for decomposing a rotation matrix. 7 The 3D Rotation Toolbar. Given a 3×3 rotation matrix. So you know how a 3D rotation matrix can | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
7 The 3D Rotation Toolbar. Given a 3×3 rotation matrix. So you know how a 3D rotation matrix can be expressed in mathematical form. % example: % rotate around a random direction a random amount and then back % the result should be an Identity matrix. This is given by the product T P 1 − 1 T xz − 1 T z − 1 R z (θ) T z T xz T P 1. And so here's the rotation transformation matrix. This means they can rotate your 3D game geometry. Download our 100% free 3D Rotation templates to help you create killer PowerPoint presentations that will blow your audience away. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional. $\endgroup$ – Federico Poloni Nov 18 at 14:23 |. The arrows denote eigenvectors corresponding to eigenvalues of the same color. −Composition of geometric transformations in 2D and 3D. Find something interesting to watch in seconds. But the plane rotation is not realistic. 3 Rotation Matrix We have seen the use of a matrix to represent a rotation. When modelling three dimensions on a two-dimensional computer screen, you must project each point to 2D. Introduction As with strain, transformations of stress tensors follow the same rules of pre and post multiplying by a transformation or rotation matrix regardless of which stress or strain definition one is using. e you want matrix C such that [C] U1 = Ur. Defining the rotation axis as the z axis, we note first that the z coordinate will be unchanged by any rotation about the z axis. Matrix Rotations and Transformations. Try your hand at some online MATLAB problems. The underlying object is independent of the representation used for initialization. 3D Rotations Rotation about z-axis. Rotation (rotation_matrix, skip_checks=False) [source] ¶ Bases: DiscreteAffine, Similarity. This is defined in the Geometry module. Rotation Matrix Java. Both proposed algorithms aim at smoothing 3D rotation matrix | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
Geometry module. Rotation Matrix Java. Both proposed algorithms aim at smoothing 3D rotation matrix sequences in a causal way. Euler angles provide a way to represent the 3D orientation of an object using a combination of three rotations about different axes. 3D Rotation Matrix. CS4620/5620: Lecture 5 3D Transforms (Rotations) •A rotation in 3D is around an axis -so 3D rotation is w. I've been following a tutorial for creating a game engine and when I got to calculating the 3D Rotation matrix I ran into the problem that I believe the matrix isn't being calculated properly. Again, the righmost matrix is the operation that occurs first. Again, we must translate an object so that its center lies on the origin before scaling it. HelloJava3Db renders a single, rotated cube: 10. After the first rotation you can specify which rotation shall follow in the 2nd column and for the third rotation you the remaining axis will be automatically selected for you in the 3rd column. pptx), PDF File (. I have a point, in that mesh, that i must calculate mannualy. This article discusses the different types of matrices including linear transformations, affine transformations, rotation, scale, and translation. Thus, we have H O = [I O] ω ,. If you don't want any rotation you can use the built-in constant Matrix3dIdentity. Rotation around any given axis Rotation from normal vector to normal vector Apparently the 5th function is enough, because for example "Rotation around X axis" can be replace by rotation around (1,0,0), and "Rotation around all axes" is merely the product of 3 matrices. Practice: Rotate 2D shapes in 3D. Euler returns a 3D vector containing the XYZ Euler angles. Rotation in 3D That works in 2D, while in 3D we need to take in to account the third axis. What do the vectors mean in T? T is a 4*4 column-major matrix. 0625 rz = -0. Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4. ggb file along with images of a red arrow | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
page is licensed under the Creative Commons Attribution 4. ggb file along with images of a red arrow and ball (that represent the spiral) at different angles. represent data on 3D rotation groups. NEGATIVE_DETERMINANT - this matrix has a negative determinant. As I understand, the rotation matrix around an arbitrary point, can be expressed as moving the rotation point to the origin, rotating around the origin and moving back to the original position. Moreover, there are similar transformation rules for rotation about and. 3d curl intuition, part 2. Rotation direction is from the first towards the second axis. A rotation matrix is a matrix used to rotate an axis about a given point. Bobick Calibration and Projective Geometry 1 Projection equation • The projection matrix models the cumulative effect of all parameters • Useful to decompose into a series of operations **** **** **** 1 X sx Y sy Z s = =. Alternatively, you can set the Orientation output to Rotation Matrix directly. These are not the only possible rotations in 3-space, of course, but we will limit our. Coordinate axes rotations:-Three dimensional transformation matrix for each co-ordinate axes rotations with homogeneous co-ordinate are as given below. 3D rotation is not same as 2D rotation. My matrices are rows by columns, row 0 being the topmost and column 0 being the leftmost. Or is it possible to convert my 3x3 rotation matrix to a Quaternion which let me use4x4Matrix. A non-rotation is described by an identity matrix. This property allows you to rotate, scale, move, skew, etc. User Interfaces with Java. The Camera Transformation Matrix: The transformation that places the camera in the correct position and orientation in world space (this is the transformation that you would apply to a 3D model of the camera if you wanted to represent it in the scene). When modelling three dimensions on a two-dimensional computer screen, you must project each point to 2D. This is the currently selected item. | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
computer screen, you must project each point to 2D. This is the currently selected item. Transformation of Stresses and Strains David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 # Display transformation matrix for these angles: "evalf" evaluates the # matrix element, and "map" applies the evaluation to each element of # the matrix. Or is there a way to convert my 3x3 rotation matrix and translation to Unity 4x4Matrix since then i can use Matrix4x4. been stuck for about a month on this - i use "euler angles" (tait-bryan angles) to describe rotation coordinates. This list is useful for checking the accuracy of a rotation matrix if questions arise. Define the parametric surface x(u,v), y(u,v), z(u,v) as follows. R = rotx(ang) creates a 3-by-3 matrix for rotating a 3-by-1 vector or 3-by-N matrix of vectors around the x-axis by ang degrees. 0 License , and code samples are licensed under the Apache 2. This paper provides a basic introduction to the use of quaternions in 3D rotation applications. , robotics,. Define and Plot Parametric Surface. Properties of a rotation matrix In three dimensions, for any rotation matrix , where a is a rotation axis and a rotation angle, (i. If you want 3x3, just remove the last column and last row. Rotations in Three-Dimensions: Euler Angles and Rotation Matrices Part 2 - Summary and Sample Code. 33× rotation matrix equals a skew-symmetric matrix multiplied by the rotation matrix where the skew symmetric matrix is a linear (matrix-valued) function of the angular velocity and the rotation matrix represents the rotating motion of a frame with respect to a reference frame. Or is it possible to convert my 3x3 rotation matrix to a Quaternion which let me use4x4Matrix. Composition and inversion in the group correspond to matrix multiplication and inversion. public Matrix3DTransformation(double[][] matrix) Constructs a 3D transformation using the given matrix. Try a 90 degree | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
matrix) Constructs a 3D transformation using the given matrix. Try a 90 degree rotation and then check. The theory is given here. Now you have magically gotten out of a room with a closed door! Similarly, to rotate about a point that is not the origin, first you move all the points so the center is the origin, use the usual rotation matrix, and then move all the points back to where you found them. Auckland's prof. Model Rotation. It is moving of an object about an angle. 7 Transformation Matrix and Stiffness Matrix in Three-Dimensional Space. Although the inverse process requires a choice of rotation axis between the two alternatives, it is a straightforward procedure to retrieve the rotation axis and angle (see Appendix A). I've read on page 27 here that a 3x3 transform matrix can be just the nine dot products - thank you U. The rotation is performed clockwise, if you are looking along the direction of the rotation axis vector. Transformations is a Python library for calculating 4x4 matrices for translating, rotating, reflecting, scaling, shearing, projecting, orthogonalizing, and superimposing arrays of 3D homogeneous coordinates as well as for converting between rotation matrices, Euler angles, and quaternions. Under rotations, vector lengths are preserved as well as the angles between vectors. The function uses the Rodrigues formula for the computation. A rotation vector is a convenient and most compact representation of a rotation matrix (since any rotation matrix has just 3 degrees of freedom). skip_checks (bool, optional) – If True avoid sanity checks on rotation_matrix for performance. Transformations, continued 3D Rotation 23 r r r x y z r r r x y z r r r x y z z y x r r r r r r r r r, , , ,, , , ,, , , , 31 32 33 Full 3D Rotation 0 sin cos 0 cos sin 1 0 0 sin 0 cos 0 1 0 cos 0 sin 0 0 1 sin cos 0 - Multiply the current matrix by the rotation matrix that. Understanding transforamtion matrix requires some knoledge of math… R. A rotation matrix describes the | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
transforamtion matrix requires some knoledge of math… R. A rotation matrix describes the relative orientation of two such frames. This paper provides a basic introduction to the use of quaternions in 3D rotation applications. square matrix – For 2D, 3x3 matrix – For 3D, 4x4 matrix. This format is definitely less intuitive than Euler angles, but it's still readable: the xyz components match roughly the rotation axis, and w is the acos of the rotation angle (divided by 2). Gimbal lock When two rotational axis of an object pointing in the same direction, the rotation ends up losing one degree orientation matrix ( quaternion can be represented as matrix as well) quaternions or orientation matrix Euler angles, quaternion (harder) Summary. The center of a Cartesian coordinate frame is typically used as that point of rotation. Rotation rotate() Rotates an element around a fixed point on the 2D plane. AlignmentRotation (source, target, allow_mirror=False) [source] ¶ Bases: HomogFamilyAlignment, Rotation. jl package), and acts to rotate a 3-vector about the origin through matrix-vector multiplication. Eigen's Geometry module provides two different kinds of geometric transformations:. User Interfaces with Java. It turns out that the derivative R_ of a rotation matrix Rmust always be a skew symmetric matrix wb times R– any-thing else would be inconsistent with the contraints of orthogonality and determinant 1. A 3D rotation is a 2D rotation that is applied within a speci ed plane that contains the origin. When we multiply two rotation matrices, the result is a new matrix that is equivalent to performing the two rotations sequentially. The only thing new in the C++ code is the usage of GetConsoleScreenBufferInfo and SetConsoleTextAttribute which gets the size of the console and sets the text color. −OpenGL matrix operations and arbitrary geometric transformations. Rotation formalisms in three dimensions: | In |geometry|, various |formalisms| exist to express a |rotation| in | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
in three dimensions: | In |geometry|, various |formalisms| exist to express a |rotation| in three |dimensions| a World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Quat returns a quaternion representing the. Consider the original set of basis vectors, i, j, k, and rotate them all using the rotation matrix A. Orthonormalize a Rotation Matrix By Mehran Maghoumi in 3D Geometry , MATLAB If you use a 3×3 R matrix to store the result of the multiplication of a series of rotation transformations, it could be the case that sometimes you end up with a matrix that is not orthonormal (i. beams in 3D space, a transformation we call a3D beam shear. A matrix with M rows and N columns is defined as a MxN matrix. 0/Image/3D/Matrix/Rotation. 3D rotation matrix around vector. There are several modes available to specify rotation matrices. Position Cartesian coordinates (x,y,z) are an easy and natural means of representing a position in 3D space …But there are many other representations such as spherical converted to matrix form to perform rotation. Simply put, a matrix is an array of numbers with a predefined number of rows and colums. If you want to rotate around an arbitary axis you can use the equations under "Rotation matrix from axis and angle" To understand how a rotation matrix works you should know that each column represents a 3D Vector which in turn represents one of the axis of the rotated coordinate system. Transformations about a Plane. Prove that this linear transformation is an orthogonal transformation. muting any, we clearly need a negative unit matrix, namely, Proper Rotation. The default polygon is a square that you can modify. a unit quaternion 'q' can represent all 3d rotations by q=exp(p), where 'p' is a pure imaginary. Call R v(θ) the 2x2 matrix corresponding to rotation of all vectors by angle +θ. Raises: ValueError: If the shape of quaternion is not supported. This | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
of all vectors by angle +θ. Raises: ValueError: If the shape of quaternion is not supported. This form will allow you to rotate a vector along an arbitrary axis (in three dimensions), by an arbitrary angle. Rotation matrices are used in computer graphics and in statistical analyses. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles. | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
qu5c3h40by nfj8pk5c7n3tha 92r7rgmozqnm dp5u6lxh4v70xx 1inp6sxfegffy pyk42wowai98pl kdp7cn3147t0 7pqy6o37oydc qpb1j97983 h69xhpid3gzqra hu2ur9woanw7r2y aj6ssysohofbl3 khhhiwe2gm8f 1ii9mtnm0z8 dq19cqrrvl63 4fbhgegfefqsrhl q5e5lhey3pk 2f7sg9eq6ix m20teoy86fw 7zumpfes1aadq jjz6gmtdutdxh f10ctu3va8jjx vug3rydlwr 7le85daq5tr kxrgja3qqn8 bgaiv3kkatago | {
"domain": "valledelchieseonline.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986777179414275,
"lm_q1q2_score": 0.8537833831960149,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 608.277822995443,
"openwebmath_score": 0.5945829749107361,
"tags": null,
"url": "http://rfmb.valledelchieseonline.it/rotation-matrix-3d.html"
} |
# When is the moment of inertia of a smooth plane curve is maximum?
Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI = \int_0^L ((x(s)-x_{cm})^2 + (y(s)-y_{cm})^2) ds$$. What I predict from earlier discussions, and almost convinced is that if we fix length $L$, $MI$ is maximum when the curve is a straight line. I lack the faculty of mathematical machinery (I guess calculus of variations) to prove it, hence is my gentle request to help me out in proving it and thoroughly understanding the situation and all the corollaries and nuances. This not just the result I need, but I want to do more with it and hence would like understand all the things that are making this result and even more general ones (only to plane curves though).
• It certainly is essential that your curve be connected (which could be surmised from your explicit parametrization statement). – Ted Shifrin Dec 23 '13 at 5:03
• @Ted Shifrin : yes, agreed. thanks for pointing. – Rajesh Dachiraju Dec 23 '13 at 5:21
• Can you explain the relation between this question and a similar one ? – Tony Piccolo Dec 23 '13 at 16:54
• @TonyPiccolo : In Chris Culter 's answer of that question, If the curve is always an arc of a circle, he shows that the moment of inertia is an increasing function of $r$, the radius of curvature. But my question now here is that, the straight line has infinite radius of curvature, hence I wonder if the straight line is the global maximum? – Rajesh Dachiraju Dec 23 '13 at 17:13
• F.A.Valentine has written Curves of given length and minimum or maximum moments of inertia (1934) but I cannot read it. Can you ? – Tony Piccolo Jan 3 '14 at 8:33
There is no need for any calculus of variation. Ordinary calculus is enough. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
There is no need for any calculus of variation. Ordinary calculus is enough.
For simplicity of derivation, we will use complex numbers to represent points on the plane.
Let $z(s) = x(s) + i y(s)$ and WOLOG, we will assume $z(0) = 0$. We can express the position on our curve as an integral:
$$z(t) = \int_0^t z'(s)\;ds$$
Let $\theta(t) = \begin{cases} 1 & t > 0\\ 0 & t \le 0\end{cases}$ be the step function. The center of mass is given by
\begin{align}z_{cm} &= \frac{1}{L} \int_0^L z(t) dt = \frac{1}{L} \int_0^L \int_0^t z'(s)\;ds\; dt\\ &= \frac{1}{L} \iint_{[0,L]^2} \theta(t-s) z'(s)\;ds\; dt = \int_0^L \left(1-\frac{s}{L}\right) z'(s)\;ds \end{align}
The moment of inertia w.r.t. $z(0)$, the origin, is given by
\begin{align} \mathcal{M}_0 &= \int_0^L |z(s)|^2 ds = \int_0^L \left(\int_0^t z'(s_1)ds_1\right)\left(\int_0^t \bar{z}'(s_2) ds_2\right) dt\\ &=\iiint_{[0,L]^3} \theta(t-s_1)\theta(t-s_2) z'(s_1) \bar{z}'(s_2) ds_1 ds_2 dt\\ &=\iiint_{[0,L]^3} \theta(t-\max(s_1,s_2)) z'(s_1) \bar{z}'(s_2) ds_1 ds_2 dt\\ &= \iint_{[0,L]^2} \left(L - \max(s_1,s_2)\right) z'(s_1) \bar{z}'(s_2) ds_1 ds_2 \end{align} and hence the moment of inertia w.r.t. the center of mass is
$$\mathcal{M}_{cm} = \mathcal{M}_0 - L |z_{cm}|^2 = L \iint_{[0,L]^2} \Lambda(s_1,s_2) z'(s_1) \bar{z}'(s_2) ds_1 ds_2$$
where \begin{align}\Lambda(s_1,s_2) &= 1 - \max\left(\frac{s_1}{L},\frac{s_2}{L}\right) - \left(1-\frac{s_1}{L}\right)\left(1-\frac{s_2}{L}\right)\\ &= \left( 1 - \max\left(\frac{s_1}{L},\frac{s_2}{L}\right)\right)\min\left(\frac{s_1}{L},\frac{s_2}{L}\right) \end{align}
Since $|z'(s)| \equiv 1$ and $\Lambda(s_1,s_2) > 0$ for $(s_1,s_2) \in (0,L)^2$, we can bound $\mathcal{M}_{cm}$ as
$$\mathcal{M}_{cm} \le L \iint_{[0,L]^2} \Lambda(s_1,s_2) |z'(s_1)||z'(s_2)| ds_1 ds_2 = L \iint_{[0,L]^2} \Lambda(s_1,s_2) ds_1 ds_2$$ Notice the equality in above inequality is achieved when and only when $z'(s)$ is a constant.
We can conclude $\mathcal{M}_{cm}$ is largest for straight lines. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
• Impressive! The kernel $\Lambda$ would look much better if you took $L=1$ (obviously WOLOG). – Post No Bulls Dec 26 '13 at 6:43
• @achille : Its actually mind blowing how you arrived at this using algebra. I wonder whether there are any geometric interpretations of the expression for $\mathcal(M_{cm}$, or can we arrive at it using geometrical arguments. It would be very interesting to know about such a thing. – Rajesh Dachiraju Dec 26 '13 at 11:37
• @RajeshD I don't have any geometric interpretation for $\mathcal{M}_{cm}$. This approach is motivated by visualizing the curve as a chain of line segments with small fixed lengths. Since the only degree of freedoms are the directions of the line segments and we know in certain sense, the moment of inertia $\mathcal{M}_{cm}$ is a non-negative quadratic functions of these degree of freedoms. I try to express the dependence explicitly and see what can be done. – achille hui Dec 26 '13 at 11:56
• @Robjohn and achillehui : As per my perception Robjohn's answer seems to be elegant, but I somehow like the expression for moment of inertia $MI_cm$ derived in Achille's answer which explicitly shows the only degree of freedom angles of the tangents and says it is maximum when the angle is a constant function. – Rajesh Dachiraju Jan 2 '14 at 9:07
Assumptions:
1. the center of mass is $0$ $$0=\int_0^L\delta f\,\mathrm{d}s\tag{1}$$ 2. $f$ is parametrized by arc length: $f'\cdot f'=1$ \begin{align} 0&=\int_0^Lf'\cdot\delta f'\,\mathrm{d}s\\ &=\int_0^Lf'\cdot\,\mathrm{d}\delta f\\ &=\Big[\,f'\cdot\delta f\,\Big]_0^L-\int_0^Lf''\cdot\delta f\,\mathrm{d}s\\ &=-\int_0^Lf''\cdot\delta f\,\mathrm{d}s\tag{2} \end{align} Note that $\delta f$ can be adjusted in the direction of $f'$ near $0$ and $L$ without affecting the integral of $f''\cdot\delta f$ since $f'\cdot f''=0$. Thus, we can assume $\Big[\,f'\cdot\delta f\,\Big]_0^L=0$.
3. the moment of inertia is stationary $$0=\int_0^Lf\cdot\delta f\,\mathrm{d}s\tag{3}$$ Conclusions: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
3. the moment of inertia is stationary $$0=\int_0^Lf\cdot\delta f\,\mathrm{d}s\tag{3}$$ Conclusions:
For an extreme $f$, any $\delta f$ that satisfies $(1)$ and $(2)$ also satisfies $(3)$; thus, linearity says that there are constants $a$ and $b$ so that $$f=a+bf''\tag{4}$$ Since $f$ is parameterized by arclength $f'\cdot f''=0$, integrating the dot product of $(4)$ with $f'$ yields $$\frac12f\cdot f=a\cdot f+c\tag{5}$$ Equation $(5)$ represents an arc of the circle $$|f-a|=\left(2c+|a|^2\right)^{1/2}\tag{6}$$ or in the extreme case, a line segment.
Checking Possible Arcs:
The equation of an arc of radius $r$ is $$f=r(\cos(s/r),\sin(s/r))\tag{7}$$ The center of mass is $$\frac1L\int_{-L/2}^{L/2}r(\cos(s/r),\sin(s/r))\,\mathrm{d}s=\left(\frac{2r^2}{L}\sin\left(\frac{L}{2r}\right),0\right)\tag{8}$$ The moment of inertia is \begin{align} &\frac{r^2}{L}\int_{-L/2}^{L/2}\left[\left(\cos(s/r)-\frac{2r}{L}\sin\left(\frac{L}{2r}\right)\right)^2+\sin^2(s/r)\right]\,\mathrm{d}s\\ &=\frac{r^2}{L}\int_{-L/2}^{L/2}\left[1-\frac{4r}{L}\sin\left(\frac{L}{2r}\right)\cos(s/r)+\frac{4r^2}{L^2}\sin^2\left(\frac{L}{2r}\right)\right]\,\mathrm{d}s\\ &=r^2-\frac{4r^4}{L^2}\sin^2\left(\frac{L}{2r}\right)\\ &=r^2\left(1-\frac{\sin^2\left(\frac{L}{2r}\right)}{\left(\frac{L}{2r}\right)^2}\right)\tag{9} \end{align} $(9)$ increases to $\frac{L^2}{12}$ as $r\to\infty$. Thus, the maximal moment of inertia would be at the extreme case of a line segment. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
• Thanks for the answer. There are some things I don't understand. In the begining what you mean by $\delta f$ and other things, I need a bit explanation on your considerations. How you claim that $f$ need to be an arc? etc,. Also at the end I am not able to understand how we get $L^2/12$ as $r$ goes to $\infty$. – Rajesh Dachiraju Dec 27 '13 at 3:03
• @RajeshD: $\delta f$ is a small instantaneous perturbation of $f$. More precisely, $$\delta\int F(f(x))\,\mathrm{d}x=\int F'(f(x))\delta f(x)\,\mathrm{d}x$$ – robjohn Dec 27 '13 at 9:21
• \begin{align}\lim_{r\to\infty}r^2\left(1-\frac{\sin^2\left(\frac{L}{2r}\right)} {\left(\frac{L}{2r}\right)^2}\right) &=\lim_{r\to\infty}\frac{\frac{L^2}{4}} {\left(\frac{L}{2r}\right)^2} \left(1-\frac{\sin^2\left(\frac{L}{2r}\right)} {\left(\frac{L}{2r}\right)^2}\right)\\ &=\lim_{x\to0}\frac{L^2/4}{x^2}\left(1-\frac{\sin^2(x)} {x^2}\right)\\ &=\lim_{x\to0}\frac{L^2/4}{x^2}\left(1-\frac{\sin(x)} {x}\right)\left(1+\frac{\sin(x)} {x}\right)\\ &=\lim_{x\to0}\frac{L^2}{2}\left(\frac{x-\sin(x)} {x^3}\right)\\ &=\lim_{x\to0}\frac{L^2}{2}\left(\frac{\cos(x)} {6}\right)\\&=\frac{L^2}{12}\end{align} – robjohn Dec 27 '13 at 9:52
There is no maximum ( monotonous increase) , but only a minimum.
Using polar coordinates, when object and constraint functions are together, variational problem is
$\int r^2 ds - \lambda^2 \int ds,$ where $ds= \sqrt{(r^2 + r^{'2})} d \theta$
Lagrangian $(r^2 - \lambda^2)$ is independent of $r^{'}$ when considered with respect to arc $s$. So it is not a variational problem.
Minimum M of I when $r$ is independent of $\theta$, or when $r$ is a constant. If arc length L is given, minimizing solution is for constant radius loop $L/ (2 \pi).$( Cowboy lasso)
I am not forgetting about he previous post, but the attached image proves that circles are also extrema of the momentum of inertia. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
Although a few minutes testing will be sufficient to convince that probably circles are unstable extrema (minimum), while straight lines are stable extrema (maximum).
This is a classical problem of the calculus of variations, and its is surprising it is not in the collections of classical applications.
Intuitively, it is the shape taken by a lasso, a rope in rotation around a fixed point (which will also be the center of mass, as will be able to check on the result). In facts, playing with a lasso shows that we can expect circles as unstable solutions and straight lines as stable solutions.
The computation is hard to understand and involves sophisticated use of the calculus of variation. Lets, after choosing appropriate axes, maximize $$M = \int_a^b (x^2 + y^2) ds, \text{ subject to } L= \int_a^b ds \text{, in which }ds=\sqrt{1+y'^2}.$$
The first thing to do is to define the derivative of $M$ and $L$ when looked as function of the curve $y=y(x)$. The so called functional derivative of $$F = \int_a^b f(x,y,y') dx,$$ is shown to be $$\delta F = \frac {\partial f} {\partial y} - \frac d {dx} \frac{\partial f} {\partial y'}.$$
A complete demonstration can be found here. Basically, you replace $y$ by $y+\epsilon \: \eta$ where $\epsilon \rightarrow 0$ is a number and $\eta$ a fixed function. Then you compute the derivative is the usual way: $(f(x,y+\epsilon \: \eta, (y+\epsilon \: \eta)') - f(x,y, y')) / \epsilon$. Integrating by part replaces the $(y+\epsilon \: \eta)'$ term by $-\frac d {dx} \frac{\partial f} {\partial y'}$ and a term outside the integral sign, which vanishes because the end points $a$ and $b$ are fixed. Because the formulas are linear, the $\epsilon$ cancels, giving a result $\int {\delta F} \eta dx$, valid for any $\eta$ thus the result. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
The functional derivatives are second order differentials that can be computed formally. For $L$ we get: $$\delta L = \frac \partial {\partial y} \sqrt{1+y'^2} - \frac d {dx} \left[ \frac \partial {\partial y'} \sqrt{1+y'^2}\right]= 0 - \frac d {dx} \left[ \frac{y'}{\sqrt{1+y'^2}} \right],$$ the first term is null because $ds$ depend only on $y'$ and not on $y$, the second is a derivative when $ds$ is looked as a function of on $y'$. We can the pursue with the usual derivative as a function of $x$: $$\delta L=- \frac{y'' \sqrt{1+y'^2} - y' (\sqrt{1+y'^2})'}{1+y'^2} = \cdots = \frac{y''}{(1+y'^2)^{3/2}}.$$
For $M$ we have:
$$\delta M = \frac \partial {\partial y} \left[ (x^2+y^2) \sqrt{1+y'^2}\right]- \frac d {dx} \frac \partial {\partial {y'}} \left[ (x^2+y^2) \sqrt{1+y'^2}\right],$$ thus, $$\delta M = 2y \sqrt{1+y'^2} - \frac d {dx} \left[ (x^2+y^2) \frac \partial {\partial {y'}} \sqrt{1+y'^2}\right] \\= 2y \sqrt{1+y'^2} - \frac d {dx} \left[ (x^2+y^2) \frac {y'} {\sqrt{1+y'^2}}\right] \\= \frac{2y (1+y'^2)}{\sqrt{1+y'^2}} - (2x+2yy') \frac {y'} {\sqrt{1+y'^2}} - (x^2+y^2) \frac {y''} {(1+y'^2)^{3/2}} \\= \frac{2y-2x y'}{\sqrt{1+y'^2}} - \frac {(x^2+y^2)y''}{(1+y'^2)^{3/2}}$$
To solve the question of the function of a given length with the highest moment of inertia, we have to introduce a Lagrange multiplier. It express at the extremum of a function $M$ subject to a condition $L = C^{te}$, the tangents planes of $M$ and $L$ are parallels. Here, the condition means that is exist a constant $\mu$, called the Lagrange-multiplier, such that $\delta M = \mu \delta L$.
This equation, known as the Euler-Lagrange equation, says there exist $\mu$ such that $$\frac{2y-2x y'}{\sqrt{1+y'^2}} - \frac {(x^2+y^2)y''}{(1+y'^2)^{3/2}} = \mu \frac{y''}{(1+y'^2)^{3/2}},$$ which is the same as $$(x^2+y^2+\mu)y'' = 2(y-x y')(1+y'^2)$$
Which needs some checks to be continued | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
Which needs some checks to be continued
• How did you assume that the moment of inertia of the curve about its center of mass to be $$M = \int_a^b y^2 dx$$, ofcourse you might have chosen origin as center of mass, but it is still wrong as per the definition given in the question. – Rajesh Dachiraju Dec 31 '13 at 20:56
• I really don't get what you are trying to say, as per my understanding both the answers given by Robjohn and Achille are correct. They explicitly use arc length parameterization and also assume curve to be of fixed length $L$. – Rajesh Dachiraju Dec 31 '13 at 20:59
• Of course, I am choosing the origin at the center of mass, computations are difficult enough. – AlainD Jan 1 '14 at 11:47
• b) You are write, I am on the wrong question. I was looking for the curve minimizing the moment on inertia about an axis ("chosen" as Ox). I'll edit in the post. – AlainD Jan 1 '14 at 11:54
• c) No Robjohn and Achille did not took into account that the curve length is constant. They use ∫ds = L, not d(∫ds)=0. In facts, if you follow their reasoning to search the curve maximizing the include area (or the lowest center of gravity), you also find straight lines, not circles (or catenaries). – AlainD Jan 1 '14 at 12:05 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771759145342,
"lm_q1q2_score": 0.853783381882623,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 362.9206858080145,
"openwebmath_score": 0.9473633170127869,
"tags": null,
"url": "https://math.stackexchange.com/questions/616106/when-is-the-moment-of-inertia-of-a-smooth-plane-curve-is-maximum"
} |
# Finding Prime Numbers¶
## Summary¶
Here I explore coding a simple script for finding prime numbers using Python. I try my first attempt without any optimization by creating the script off the top of my head. As you will see, that script works but runs very slowly. I then follow some online lessons on how to optimize the script and get much faster results.
## Prime Finder version 1¶
Now, here is the code for figuring out if any one number is prime.
But let's allow the user to get all the prime numbers from 1 to the value they enter:
Cool! Now turn that into a function:
And now run this:
This seems to work, but let's time how long this takes for larger numbers since it has to search all those numbers and might take a while!
So, finding all primes between 1 and 10,000 required 8+ seconds! What about 100,000?
start = time.time() find_primes(100000) end = time.time() elapsed = end - start print(f"Time elapsed: ",{elapsed})
Ok, so after a few minutes I gave up and terminated the kernel. Is there a more efficient way to do this??
## Prime Finder 2.0¶
Need to optimize the prime number search space with each iteration so that the code doesn't just use brute force and check every single number. For example, we know that even numbers are not prime, so why not remove those first? The following methodology does that, but also removes multiples of every new prime number found.
Essentially we can work in reverse. Rather than finding each number that doesn't have any multiples, work through the list of multiples and remove all numbers from the final list that those multiples can create.
For example, the first prime number is 2, so we can remove all other values from the set that are multiples of 2. The next prime number is 3, so we can remove all other values from the set that are multiples of 3. The next is 5, since the 4 was already removed when the multiples of 2 were removed, and so on. Thus, the search list becomes smaller and smaller! | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771805808551,
"lm_q1q2_score": 0.8537833790613634,
"lm_q2_score": 0.8652240808393984,
"openwebmath_perplexity": 758.3815330236117,
"openwebmath_score": 0.5803171396255493,
"tags": null,
"url": "https://lukaneg.github.io/projects/prime-num-optim.html"
} |
Now each time you repeat that set of code, you add values to the actual_primes set and remove values from the potential_primes set, making the search space smaller and smaller.
So let's repeat that process with a while loop:
Great! Now time to package it up into a function again:
Now let's time it again!
## All primes through 10,000:¶
Wow!! Only 0.006 seconds compared to 8+ seconds with version 1!
## All primes through 100,000:¶
Only 0.035 seconds for 100,000 compared to several minutes + with version 1
## All primes through 1,000,000:¶
Now for the mega test: search for all prime numbers from one to one million:
Less. Than. A. Second. WOW
## Conclusion¶
So, as you can see, code optimization can make all the difference between having time for happy hour or not :P
This was a great experience in learning how to optimize code, and that it is not just about the types of data structures (though I'm sure using sets here helped a ton), it's also about optimizing the algorithm itself so that it does as little reduntant work as possible. I credit Andrew Jones (at Data Science Infinity) for sharing this very elegant but powerful algorithm for cutting through the redundancy and efficiently extracting a list of prime numbers. | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771805808551,
"lm_q1q2_score": 0.8537833790613634,
"lm_q2_score": 0.8652240808393984,
"openwebmath_perplexity": 758.3815330236117,
"openwebmath_score": 0.5803171396255493,
"tags": null,
"url": "https://lukaneg.github.io/projects/prime-num-optim.html"
} |
# Empty set, subsets, and vacuous truths
So I was trying to prove that a null set is a subset of any set.
First, to define when $A$ is a subset of $B$:
$$A \subseteq B \iff \forall x(x \in A \implies x \in B)$$
(At least I think that's right?)
So then consider the empty set $\emptyset$ for which $\forall x(x \not\in \emptyset)$ is true.
I tried to prove that the empty set is a subset of any other set, or:
$$\emptyset \subseteq B \iff \forall x(x \in \emptyset \implies x \in B) \vdash \text{T}$$
To me this seemed true because it was "vacuously true"... somehow. Like it makes sense to call it vacuously true that "all $0$ items in $\emptyset$ can be found in $B$, yep!" but that isn't satisfying to me, how do I "prove" this is the case?
Is $x \in \emptyset$ a false... statement? A false predicate? Something else? Something that results in false so that the implication itself is true. Is this a $(\text{F}\implies \text{F}) \vdash \text{T}$ thing?
Or is it the $\forall$ that makes it false somehow?
What if I had said $\exists x \in \emptyset$, this feels like it would certainly be false but again I can't prove why, it's just an intuition.
Can anyone clarify the correct definitions / implications and why they're true or false or what have you? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771798031351,
"lm_q1q2_score": 0.8537833698151196,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 382.2226008121524,
"openwebmath_score": 0.7642687559127808,
"tags": null,
"url": "https://math.stackexchange.com/questions/2721368/empty-set-subsets-and-vacuous-truths"
} |
• You don't seem to be using the symbol $\vdash$ in the usual way (see en.wikipedia.org/wiki/Turnstile_(symbol) ). Normally $P\vdash Q$ means you have proven $Q$ using assumptions $P$. In this case you want to show $\vdash\emptyset\subseteq B$, and the vacuous implication you are looking for would be $F\vdash P$ (or equivalently $\vdash F\Rightarrow P$) for any $P$. – stewbasic Apr 4 '18 at 4:46
• $x\in\emptyset$ is a predicate. It's neither true nor false until you substitute for $x$, but of course, whatever $x$ you substitute makes it false. So, for every $x$, the statement $x \in \emptyset \implies x \in B$ is true. I don't know if this will help you, but I hope so. – saulspatz Apr 4 '18 at 5:03
• "x∈∅ is a predicate. It's neither true nor false until you substitute for x". True. But saying $x \in \emptyset$ is false for all $x$ is perfectly legitimate. – fleablood Apr 4 '18 at 5:22
• @stewbasic I was trying to say that "(a iif for all x(p implies q)) is true" is there a better symbol to use? Equal sign? – user525966 Apr 4 '18 at 13:18
• @user525966 You could write $\vdash\emptyset\subseteq B\Leftrightarrow\forall x(x\in\emptyset\Rightarrow x\in B)$ or just $\emptyset\subseteq B\Leftrightarrow\forall x(x\in\emptyset\Rightarrow x\in B)$. – stewbasic Apr 4 '18 at 21:09
The proof relies on Ex falso :
$\vdash \lnot P \to (P \to Q)$.
We have to apply it in the form :
$\lnot (x \in \emptyset) \to (x \in \emptyset \to x \in B)$.
We have (axiom or theorem) : $\mathsf {ZF} \vdash \forall x \ \lnot (x \in \emptyset)$.
By Universal instantiation we get : $\lnot (x \in \emptyset)$ and thus from Ex falso, by Modus Ponens : $(x \in \emptyset \to x \in B)$.
Finally, by Universal generalization we conclude with :
$\mathsf {ZF} \vdash \forall x \ (x \in \emptyset \to x \in B)$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771798031351,
"lm_q1q2_score": 0.8537833698151196,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 382.2226008121524,
"openwebmath_score": 0.7642687559127808,
"tags": null,
"url": "https://math.stackexchange.com/questions/2721368/empty-set-subsets-and-vacuous-truths"
} |
$\mathsf {ZF} \vdash \forall x \ (x \in \emptyset \to x \in B)$.
• What does it mean to have the turnstile on the far left with nothing before it? – user525966 Apr 4 '18 at 15:33
• @user525966 - that is a "law of logic" (in this case: of propositional logic, i.e. a tautology) and thus we can use in every theory. – Mauro ALLEGRANZA Apr 4 '18 at 15:35
• In my original post was I using turnstile incorrectly? What's the correct symbol to use there instead? Equal sign? – user525966 Apr 4 '18 at 15:53
• The "turnstile" $\vdash$ means derivable (in a system). Thus $\mathsf {ZF} \vdash \varphi$ means that $\varphi$ is a theorem of Zermelo-Frenkel set theory. – Mauro ALLEGRANZA Apr 4 '18 at 15:56
• Thus, you want to prove: $\mathsf {ZF} \vdash \forall B \ (\emptyset \subseteq B)$. It is equivalent to say that $\forall B \ (\emptyset \subseteq B)$ is TRUE in every model of the theory. – Mauro ALLEGRANZA Apr 4 '18 at 15:58
1)
$A\subset B$ if all elements of $A$ are elements of $B$. As $\emptyset$ has no elements, then all of them are in $B$.
That's vacuously true.
So $\emptyset \subset B$.
2)
$A\subset B$ if any elements not in $B$ are not in $A$ either. As any element that is not in $B$ is not in $\emptyset$ either, $\emptyset \subset B$.
That's true-true; nothing vacuous about it.
3)
$A \subset B$ if $x \in A \implies x \in B$ is true for all $x$. As $x \in \emptyset$ is always false and $FALSE \implies P$ is always true, $x \in \emptyset \implies x \in B$ is always true.
So $\emptyset \subset B$.
===
So for the most part, yes, they are vacuously true statements, or they are a false premise implies anything true statements.
But it's not all smoke and mirrors. A subset is "embedded" in the superset and everything you can pull out of the subset most come directly from the superset, and there is nothing in the subset that isn't in the superset. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9867771798031351,
"lm_q1q2_score": 0.8537833698151196,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 382.2226008121524,
"openwebmath_score": 0.7642687559127808,
"tags": null,
"url": "https://math.stackexchange.com/questions/2721368/empty-set-subsets-and-vacuous-truths"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.