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https://freakonometrics.hypotheses.org/53269 | # Linear Regression, with Map-Reduce
Sometimes, with big data, matrices are too big to handle, and it is possible to use tricks to numerically still do the map. Map-Reduce is one of those. With several cores, it is possible to split the problem, to map on each machine, and then to agregate it back at the end.
Consider the case of the linear regression, $\mathbf{y}=\mathbf{X}\mathbf{\beta}+\mathbf{\varepsilon}$ (with classical matrix notations). The OLS estimate of $\mathbf{\beta}$ is $\widehat{\mathbf{\beta}}=[\mathbf{X}^T\mathbf{X}]^{-1}\mathbf{X}^T\mathbf{y}$. To illustrate, consider a not too big dataset, and run some regression.
lm(dist~speed,data=cars)$coefficients (Intercept) speed -17.579095 3.932409 y=cars$dist X=cbind(1,cars$speed) solve(crossprod(X,X))%*%crossprod(X,y) [,1] [1,] -17.579095 [2,] 3.932409 How is this computed in R? Actually, it is based on the QR decomposition of $\mathbf{X}$, $\mathbf{X}=\mathbf{Q}\mathbf{R}$, where $\mathbf{Q}$ is an orthogonal matrix (ie $\mathbf{Q}^T\mathbf{Q}=\mathbb{I}$). Then $\widehat{\mathbf{\beta}}=[\mathbf{X}^T\mathbf{X}]^{-1}\mathbf{X}^T\mathbf{y}=\mathbf{R}^{-1}\mathbf{Q}^T\mathbf{y}$ solve(qr.R(qr(as.matrix(X)))) %*% t(qr.Q(qr(as.matrix(X)))) %*% y [,1] [1,] -17.579095 [2,] 3.932409 So far, so good, we get the same output. Now, what if we want to parallelise computations. Actually, it is possible. Consider $m$ blocks m = 5 and split vectors and matrices $$\mathbf{y}=\left[\begin{matrix}\mathbf{y}_1\\\mathbf{y}_2\\\vdots \\\mathbf{y}_m\end{matrix}\right]$$ and $$\mathbf{X}=\left[\begin{matrix}\mathbf{X}_1\\\mathbf{X}_2\\\vdots\\\mathbf{X}_m\end{matrix}\right]=\left[\begin{matrix}\mathbf{Q}_1^{(1)}\mathbf{R}_1^{(1)}\\\mathbf{Q}_2^{(1)}\mathbf{R}_2^{(1)}\\\vdots \\\mathbf{Q}_m^{(1)}\mathbf{R}_m^{(1)}\end{matrix}\right]$$ To split vectors and matrices, use (eg) Xlist = list() for(j in 1:m) Xlist[[j]] = X[(j-1)*10+1:10,] ylist = list() for(j in 1:m) ylist[[j]] = y[(j-1)*10+1:10] and get small QR recomposition (per subset) QR1 = list() for(j in 1:m) QR1[[j]] = list(Q=qr.Q(qr(as.matrix(Xlist[[j]]))),R=qr.R(qr(as.matrix(Xlist[[j]])))) Consider the QR decomposition of $\mathbf{R}^{(1)}$ which is the first step of the reduce part$$\mathbf{R}^{(1)}=\left[\begin{matrix}\mathbf{R}_1^{(1)}\\\mathbf{R}_2^{(1)}\\\vdots \\\mathbf{R}_m^{(1)}\end{matrix}\right]=\mathbf{Q}^{(2)}\mathbf{R}^{(2)}$$where$$\mathbf{Q}^{(2)}=\left[\begin{matrix}\mathbf{Q}^{(2)}_1\\\mathbf{Q}^{(2)}_2\\\vdots\\\mathbf{Q}^{(2)}_m\end{matrix}\right]$$ R1 = QR1[[1]]$R for(j in 2:m) R1 = rbind(R1,QR1[[j]]$R) Q1 = qr.Q(qr(as.matrix(R1))) R2 = qr.R(qr(as.matrix(R1))) Q2list=list() for(j in 1:m) Q2list[[j]] = Q1[(j-1)*2+1:2,] Define – as step 2 of the reduce part$$\mathbf{Q}^{(3)}_j=\mathbf{Q}^{(2)}_j\mathbf{Q}^{(1)}_j$$ and$$\mathbf{V}_j=\mathbf{Q}^{(3)T}_j\mathbf{y}_j$$ Q3list = list() for(j in 1:m) Q3list[[j]] = QR1[[j]]$Q %*% Q2list[[j]] Vlist = list() for(j in 1:m) Vlist[[j]] = t(Q3list[[j]]) %*% ylist[[j]]
and finally set – as the step 3 of the reduce part$$\widehat{\mathbf{\beta}}=[\mathbf{R}^{(2)}]^{-1}\sum_{j=1}^m\mathbf{V}_j$$
sumV = Vlist[[1]] for(j in 2:m) sumV = sumV+Vlist[[j]] solve(R2) %*% sumV [,1] [1,] -17.579095 [2,] 3.932409
It looks like we’ve been able to parallelise our linear regression…
## 6 thoughts on “Linear Regression, with Map-Reduce”
1. korul says:
How do I supposed to calculate inverse of R^(2), when R^(2) is a vector of matrices?
2. Stefano says:
Great post! One question: mathematically, can you explain – provide a reference for the explanation of – the step 2 of the reduce part (where the 2 Q matrices are multiplied)? Because I can follow through all steps, but I don’t understand *why* you create Q_j^(3) like this. Thanks!
3. Another interesting point is the trade-off between the number of nodes and the efficiency of the calculation. If you think about it – mode nodes *does not* mean faster running time.
I took the liberty of reproducing your results (in “tidyverse” code”) and added some notes on efficiency. I have an R notebook/markdown here: https://github.com/ytoren/reproducible/tree/master/linear-regression-map-reduce
Keep up the good work mate!
4. Awesome post!
I think you might have a typo :
You split Q2 to Q2list by:
> for(j in 1:m) Q2list[[j]] = Q1[(j-1)*2+1:2,]
And it should be
> for(j in 1:m) Q2list[[j]] = Q2[(j-1)*2+1:2,]
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 2022-12-05T21:13:27 | {
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https://puzzling.stackexchange.com/questions/15412/99-bags-of-apples-and-oranges?noredirect=1 | # 99 Bags of Apples and Oranges
You have $99$ bags, each containing various numbers of apples and oranges. Prove that there exist $50$ bags among these which together contain at least half the apples and at least half the oranges.
Needless to say, you may not add/remove fruits to/from the bags.
Clarification: I changed the wording from "you can grab $50$ bags..." to "there exist $50$ bags..."
Clarification from comments: each bag can contain any number of apples and any number of oranges with any total number of fruit; the total amount of fruit in a bag doesn't have to be the same for each bag.
• I suspect the presence of Pigeonhole principle... – leoll2 May 23 '15 at 14:29
• @ghosts_in_the_code can you provide a counterexample? – leoll2 May 23 '15 at 15:54
• @ghosts_in_the_code By various, I mean each bag can have any number of apples, any number of oranges, making any total. The bags do not need to all have the same number of fruits. – Mike Earnest May 23 '15 at 15:58
• Can the bags be empty? – Bob May 23 '15 at 16:09
• @Bob Yes, they can be empty. For example, if all bags were empty, then grabbing any $50$ bags works, since zero is at least half of zero. – Mike Earnest May 23 '15 at 16:12
Let $B_1,B_2,\dots,B_{99}$ be the bags, in increasing order of number of apples contained; say the number of apples in $B_i$ is $a_i$ for all $i$.
If our 50 bags grabbed are $B_{99}$, one of $B_{98}$ and $B_{97}$, one of $B_{96}$ and $B_{95}$, ..., and one of $B_2$ and $B_1$, then they must contain between them at least half of all apples, since at worst they contain $a_{99}+a_{97}+\dots+a_5+a_3+a_1\geq a_{98}+a_{96}+\dots+a_4+a_2+0$ apples.
Now how do we fix all those "one of"s? Just pick whichever of $B_{98}$ and $B_{97}$ has the more oranges, then whichever of $B_{96}$ and $B_{95}$ has the more oranges, and so on. Then our 50 selected bags - even excluding $B_{99}$! - must contain at least half of all oranges.
QED.
• QED indeed! I will certainly accept your answer, but I think I will wait a bit to see if other cool answers crop up as well. I know there are many distinct proofs methods, yours being the most constructive – Mike Earnest May 23 '15 at 21:23
Arrange the bags in a circle. I will call a collection of 50 consecutive bags a team (so there are 99 possible teams), and a team will be called appley if it contains at least half of the apples.
Define the opponent of a given team to be the team whose rightmost bag is the leftmost bag of the given team. For any given team, every bag is either a member of that team or a member of its opponent (and one bag is a member of both). This means that if a team is not appley, then its opponent is appley. Different teams have different opponents, so there must be at least as many appley teams as non-appley teams. There are 99 teams in total, so at least 50 of them are appley.
A similar argument shows that at least 50 teams are orangey (that is, at least 50 of the teams contain at least half the oranges). Of 99 teams, at least 50 are appley and at least 50 are orangey, so there must be at least one teams which is both appley and orangey (because $50+50>99$). This team consists of 50 bags, and contains at least half the apples and at least half the oranges.
• This is really cool! To elaborate: let $A$ be the set of appley teams, and $B$ be the set of teams whose opponents are appley. You showed $A\cup B$ is all $99$ teams (if a team is not appely, its opponent is). Since $|A|=|B|$, we have $99=|A\cup B|\le |A|+|B|=2|A|$, proving $|A|\ge 49.5,$ so $|A|\ge50$. – Mike Earnest May 24 '15 at 4:43
• "There are 99 teams in total, ..." - agreed (count possible common bags). You argument seems to put appley teams in 1:1 correspondence with non-appley teams by construction, implying an even total number of teams. How does that correspond with the odd number of teams? Cont'd ... – Lawrence May 24 '15 at 13:40
• ... cont'd. Simplified example: instead of 99 bags, say we have 3 bags $a,b,c$. Then we can have team pairs $[ab,bc], [bc,ca], [ca,ab]$. Suppose team $bc$ was appley. This would mean from the first two match-ups that both $ab$ and $ca$ were not appley. However, this makes the last match-up a team pair with two non-appley teams. Is this a failure of the assertion that "if a team is not appley, then its opponent is appley"? (By the way, other than this, I really like your proof. +1 ) – Lawrence May 24 '15 at 13:40
• @Lawrence If a team is non-appley, its opponent is appley. However, if a team is appley, we are not saying its opponent is non-appley (they could both be appley). As you pointed out, since there are an odd number of teams, there must at least one appley team whose opponent is also appley. – Julian Rosen May 24 '15 at 13:46
• @Lawrence no, different teams do have different opponents. Notice, however, that the opponent of the opponent of a team is not (quite) the original team, so you can't "pair them off" in the way you seem to be imagining. – Ben Millwood May 26 '15 at 14:35
Suppose you have 50 bags on the ground and 49 in the truck, and suppose the ones on the ground have more apples. One at a time, take a bag from the ground and put it in the truck, and put one from the truck tand place it on the ground. Keep doing this with the goal of totally swapping ground and truck bags.
Eventually, you'll reach a point where moving a "magic" bag causes the truck to have more apples than the ground. At this point, both the ground plus the magic bag has at least half the apples, AND the truck plus the magic bag has at least half the apples. Either the ground or the truck ( plus the magic bag ) has at least half the oranges too.
I will provide a probabilistic proof. By demonstrating that picking randomly has a nonzero chance of success, we can show that a solution exists. (http://en.m.wikipedia.org/wiki/Probabilistic_method)
Claim: A random subset of 50 bags has over 50% chance of having more than half the apples. (Chosen uniformly over all size 50 subsets.)
Proof: For each "losing" size 50 subset, flip it around to get a (unique) winning size 49 subset. This means there are at least as many size 49 winning sets as size 50 losing sets. There are multiple ways to extend these to size 50 sets, so there's strictly more ways to win than lose. (Hand waving a bit)
This argument applies symmetrically to the oranges. Since each probability is over 1/2, they must have a nonzero intersection.
Thus there is a positive probability of picking a subset with at least half of each, so a solution exists.
Nonconstructively useless, but QED :)
Edit: looking back, this is just saying that the intersection of sets larger than 50% is nonempty, so the probabilistic part is just fluff.
• It's far from trivial that you can extend all 49-subsets of a 99-set into 50-subsets by addition of a single element in such a way that you don't accidentally extend two 49-subsets into the same 50-subset (it's certainly impossible to extend 50-subsets into 51-subsets this way, for example, because there aren't as many of them). – Ben Millwood May 26 '15 at 16:26
• I mean, it IS true. But it requires Hall's Marriage Theorem, unless you have an ingenious construction. – Ben Millwood May 26 '15 at 16:29
Suppose we choose 50 of the bags at random. We are choosing more than half of the bags, so the probability that the bags we selected contain at least half of the apples is strictly greater than 50%. Similarly, the probability that the bags we selected contain at least half of the oranges is also greater than 50%. This means there is a positive probability that the bags we selected contain at least half the apples and at least half the oranges. In particular, there must be some combination of 50 bags containing at least half the apples and at least half the oranges.
Edit: As pointed out in the comments, it is not obvious why the bags we select will have at least half the apples with probability greater than 50%. This actually follows from the argument in my other answer: one way to choose 50 bags at random is first to choose a cyclic ordering of the bags, then to choose 50 consecutive bags. My other answer shows that for whichever cyclic ordering we choose, a random choice of 50 consecutive bags contains at least half the apples with probability at least $\frac{50}{99}$. This means that a random selection of 50 bags contains at least half the apples with probability at least $\frac{50}{99}$. The same is true for oranges, so a random selection of 50 bags contains at least half the apples and at least half the oranges with probability at least $\frac{1}{99}$.
• I think the statement We are choosing more than half of the bags, so the probability that the bags we selected contain at least half of the apples is strictly greater than 50%, needs more justification. – Mike Earnest May 23 '15 at 19:44
• This isn't a very rigorous proof. First of all, you must justify your probability statements, then you might provide an algorithm of optimal choosing (not required, but strongly suggested). – leoll2 May 23 '15 at 20:52
• I was expecting it to be clear why the probability of getting at least half the apples was bigger than 50%, but I didn't think this part through and I agree that it isn't clear why this is the case. – Julian Rosen May 24 '15 at 3:40
• Using the probabilistic approach, leaves a possibility that there will not be a desired solution. We need to have 100% probability that a desired case will be found. – Moti May 25 '15 at 17:59
• @Moti If the probability that a random object has a certain property is greater than 0, then it is certain there is at least one object with that property. – Julian Rosen May 25 '15 at 18:54 | 2019-08-24T03:21:30 | {
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https://math.stackexchange.com/questions/2326813/triangle-inscribed-within-a-unit-circle | # Triangle inscribed within a unit circle
A triangle $ABC$ is inscribed within the unit circle. Let $x$ be the measure of the angle $C$. Express the length of $AB$ in terms of $x$.
A.) $2\sin x$
B.) $\cos x + \sin x - 1$
C.) $\sqrt{2}(1 - \cos 2x)$
D.) $\sqrt{2}(1 - \sin x)$
I am unsure how to illustrate the circle and calculate the length $x$.
Thank you. `
• Please read this tutorial about how to typeset mathematics on this site. – N. F. Taussig Jun 18 '17 at 8:55
$$\dfrac{AB}{\sin x}=2R$$ where $R$ is the circum-radius $=1$
• This solution is based on the Law of Sines. – N. F. Taussig Jun 18 '17 at 9:11
• @N.F.Taussig, Thanks for enriching the answer – lab bhattacharjee Jun 18 '17 at 9:12
The angle $AOB=2x$, since it intercepts the same arc than the angle $CAB=x$.
thus by the well-known formula
$$AB^2=OA^2+OB^2$$ $$-2OA.OB.\cos (2x)$$ with
$$OA=OB=radius=1$$
hence
$$AB^2=2 (1-\cos (2x))$$
$$=4\sin^2 (x)$$ and $$\boxed {AB=2\sin (x) }$$ | 2019-08-22T09:31:41 | {
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https://math.stackexchange.com/questions/1624387/density-of-positive-multiples-of-an-irrational-number | # Density of positive multiples of an irrational number
Let $x$ be irrational. Use $\{r\}$ to denote the fractional part of $r$: $\{r\} = r - \lfloor r \rfloor$. I know how to prove that the following set is dense in $[0,1]$: $$\{\{nx\} : n \in \mathbb{Z}\}.$$ But what about $$\{\{nx\} : n \in \mathbb{N}\}?$$ Any proof that I’ve seen of the first one fails for the second one.
A minor modification of the pigeonhole argument works. Let $m$ be any positive integer. By the pigeonhole principle there must be distinct $i,j\in\{1,\dots,m+1\}$ and $k\in\{0,\dots,m-1\}$ such that $\frac{k}m\le\{ix\},\{jx\}<\frac{k+1}m$; clearly $\{|(j-i)x|\}<\frac1m$. Let $\ell$ be the largest positive integer such that $\ell\{|(j-i)x|\}<1$, let $A_m=\{n|j-i|x:0\le n\le\ell\}$, and let $D_m=\big\{\{y\}:y\in A_m\big\}$.
If $x>0$, every point of $[0,1)$ is clearly within $\frac1m$ of $A_m=D_m$. If $x<0$, then
$$D_m=\{1-|y|:y\in A_m\}\;,$$
so every point of $[0,1)$ is again within $\frac1m$ of the set $D_m$. Since $D_m\subseteq\big\{\{nx\}:n\in\Bbb N\big\}$, we’re done.
• I hope you don't mind how close my answer is to yours, but I had written an answer for someone in chat and was going to post it elsewhere, then saw this question was a better fit.
– robjohn
Jul 20, 2021 at 20:58
• @robjohn: No problem: someone may benefit from the greater detail. Jul 21, 2021 at 2:39
• Can you give more details why if $x>0$ then every ppint of $[0,1)$ is clearly within $1/m$ of $A_m$.
– ZFR
Jul 30, 2021 at 2:29
• @ZFR: Adjacent points of $A_m$ are less than $\frac1m$ apart. Every point of $[0,1)$ is either equal to one of these points, between two adjacent ones, or between $\ell\{|(j-i)x|\}$ and $1$ and hence less than $\frac1m$ from some point of $A_m$. Jul 31, 2021 at 22:00
Here is a proof where the second statement follows from the first with a minor step. I wrote this answer for another question, but then I saw it fit here better, other than being along very similar lines to Brian M. Scott's answer.
Let $$r$$ be irrational. Choose an arbitrary $$n\in\mathbb{N}$$. Partition $$[0,1)$$ into $$n$$ subintervals $$\left\{I_k=\left[\frac{k-1}n,\frac kn\right):1\le k\le n\right\}\tag1$$ Consider the discrete set $$\{\{kr\}:0\le k\le n\}\subset[0,1)$$. There are $$n+1$$ elements in this set, so two of them must lie in the same subinterval. That means that we have $$k_1\ne k_2$$ so that $$\{(k_1-k_2)r\}\in\left(0,\frac1n\right)$$, we leave $$0$$ out of the interval because $$r$$ is irrational.
Let $$m=\left\lfloor\frac1{\{(k_1-k_2)r\}}\right\rfloor$$, then because $$m$$ is the greatest integer not greater than $$\frac1{\{(k_1-k_2)r\}}$$, $$m\{(k_1-k_2)r\}\!\!\overset{\substack{r\not\in\mathbb{Q}\\\downarrow}}{\lt}\!\!1\lt(m+1)\{(k_1-k_2)r\}\tag2$$ which implies that $$\{m(k_1-k_2)r\}=m\{(k_1-k_2)r\}\in\left(\frac{n-1}n,1\right)$$.
Since $$\{(k_1-k_2)r\}\in\left(0,\frac1n\right)$$, if $$j\{(k_1-k_2)r\}\in I_k$$, then $$(j+1)\{(k_1-k_2)r\}\in I_k\cup I_{k+1}$$; that is, $$\{j(k_1-k_2)r\}=j\{(k_1-k_2)r\}$$ cannot skip over any of the $$I_k$$. Therefore, $$\left\{\{j(k_1-k_2)r\}:1\le j\le m\vphantom{\frac12}\right\}\tag3$$ must have at least one element in each $$I_k$$ for $$1\le k\le n$$.
The only problem is that we don't know that $$k_1-k_2\gt0$$. However, this is not a problem since $$\{j(k_1-k_2)r\}\in I_k\iff\{j(k_2-k_1)r\}\in I_{n+1-k}$$. Therefore, $$\left\{\{j(k_2-k_1)r\}:1\le j\le m\vphantom{\frac12}\right\}\tag4$$ must also have at least one element in each $$I_k$$ for $$1\le k\le n$$.
Since $$n$$ was arbitrary, we have shown that $$\left\{\mathbb{N}r\right\}$$ is dense in $$[0,1]$$.
• Thanks a lot professor Robjohn for answering my question. I was having lot of difficulty understanding $k_2\gt k_1$ case. It all makes sense to me now. Many thanks :) I would only like to add this little more detail for statement just before $(3)$ in the hope that it will benefit those who visit this great answer in future: Suppose on the contrary that for some $k<n$, the subinterval $(\frac{k-1}n, \frac kn )$ doesn't contain any element of the set $T=\{ j\{(k_2-k_1)r\}: 1\le j\le m\}$. (Contd.)
– Koro
Jul 21, 2021 at 3:55
• (Contd.) Note that $T_L=\{j: j\{(k_2-k_1)r\}\le \frac{k-1}n\}$ is non-empty and therefore $T_L$ has a maximal element $M\in \mathbb N$ that is $(M+1)\{(k_2-k_1)r\}\ge \frac kn$ and $M\{(k_2-k_1)r\}\le \frac{k-1}n$ and subtracting the two, we get: $\{(k_2-k_1)r\}\ge \frac 1n$, which is a contradiction. Therefore no subinterval $I_k$ can be free of elements from $\{\mathbb N r\}$.
– Koro
Jul 21, 2021 at 3:56
• Really nice answer which helped me to understand the problem. But I think you can assume $r$ to be any real irrational number from the beginning of the solution.
– ZFR
Jul 29, 2021 at 22:25
• @ZFR: You know, I modified this so many times while writing it, and now it seems you are correct; there is no reason for $r\gt0$, so I have removed that constraint.
– robjohn
Jul 30, 2021 at 2:40
• Thanks a lot for your answer! To be honest I've spent on this problem about 2-3 days and eventually your solution helped me to understand it completely. Thanks! +1
– ZFR
Jul 30, 2021 at 15:42
Really? I thought exactly the same proof worked for $\Bbb N$.
Let $k\in\Bbb Z$ with $k\ne 0$ and define $$f(t)=e^{2\pi ikt}.$$
Then $f$ has period $1$, and $$\frac1N\sum_{n=0}^{N-1}f(nx) =\frac1N\sum_{n=0}^{N-1}\left(e^{2\pi ikx}\right)^n=\frac1N \frac{e^{2\pi ikxN}-1}{e^{2\pi ikx}-1}\to0\quad(N\to\infty).$$ So the usual approximation shows that $$\frac1N\sum_{n=0}^{N-1}f(nx)\to\int_0^1 f(t)\,dt$$for $f\in C(\Bbb T)$ and you're done, as usual.
How is this any different from the case $n\in\Bbb Z$?
• How does that imply that $\{ \exp (2\pi i {n x}) :n\in N\}$ is dense in the unit circle in $C$ ? Anyway, a proof from the most elementary method works for N just as well as for Z. Jan 24, 2016 at 5:33
• @user254665 Say those points are not dense. There is then an interval, or arc, $I$ on the circle that contains none of the points. Take $f\in C(\Bbb T)$ such that $f=0$ on the complement of $I$ but $\int f > 0$. Then $\frac1N\sum_0^{N-1}f(nx)=0$ for every $N$, so it does not converge to $\int f$. (Yes, the pigeonhole argument works as well. I really can't think of an argument that works for $n\in\Bbb Z$ but not $\Bbb N$. In any case, ignoring the question of which is simpler, this argument shows more than just that the points are dense, it shows they are asymptotically uniformly distributed.) Jan 24, 2016 at 14:21
• very nice integral proof, which yields extra info (distribution) too. Jan 24, 2016 at 20:10
• @user254665 Yes it is very nice. The equidistribution is Weyl's theorem - I think it's his proof, not sure, in any case it's a standard thing. Jan 24, 2016 at 21:27 | 2022-07-02T07:32:06 | {
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https://brilliant.org/discussions/thread/which-payoff-do-you-want-to-go-for/ | # Which payoff do you want to go for?
Consider a game where you have to choose between 1 of 2 payoffs:
Payoff 1. You toss 100 coins.
You get $1 for each Head that appears. Payoff 2. You toss 10 coins. You get$10 for each Head that appears.
Which payoff structure would you choose? And why?
Note by Calvin Lin
3 years, 10 months ago
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
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Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
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2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
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\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
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I assume each coin toss is independent from one another (so the covariance between each coin toss is zero), and used only fair coins.
Yes, the expected Payoff is the same for ONE and TWO. But their variance differ.
$V_1 = \text { Probability } \times \text { Payoff ONE }^2 \times \text { Number of trials } = \frac {1}{2} \times 1^2 \times 100 = 50$
$V_2 = \text { Probability } \times \text { Payoff TWO }^2 \times\text { Number of trials } = \frac {1}{2} \times 10^2 \times 10 = 500$
Because Payoff TWO has a higher variance which is not desired, I will go with Payoff ONE.
$\text{ I learned from the best tutor }$
- 3 years, 10 months ago
What term do we use to describe a person who choses to "minimize variance when expected value is equal"?
Your answer greatly depends on what it is that we wish to maximize / minimize. There could be people who may want to maximize variance, because of the higher likelihood of a huge payoff that they care about.
Staff - 3 years, 10 months ago
What term do we use to describe a person who choses to "minimize variance when expected value is equal"?
Risk-averse investor
- 3 years, 10 months ago
Let me try.
Payoff 1. Chance to get head for all coins = 1 out of 10^{200} = Chance to get $100 in terms of$1.
Payoff 2. Chance to get head for all coins = 1 out of 10^{10} = Chance to get $100 in terms of$10.
Thus, I think Payoff 2 is better.
- 3 years, 10 months ago
This is optimistic way I guess
- 3 years, 10 months ago
That's a valid argument if your goal is to "maximize the probability of the highest payoff".
Since getting $99,$98, ... $91 is somewhat similar to getting$100, would this affect your answer? Why, or why not?
Note that the payoff 1 should be "10 ^ {100}"
Staff - 3 years, 10 months ago
The probability of getting payoff between 91$to 100$ increases in the first case. Hence, if I consider myself satiated by $99,$98, ... $91, I'd prefer case 1. - 3 years, 10 months ago Log in to reply It's less of a hassle to toss 10 coins. If we compare this to stocks, it's easier to keep a portfolio of10 stocks than 100 stocks. And lower commissions. The risk of loss is higher, but the chance of higher returns is better too. - 3 years, 10 months ago Log in to reply That's a good point! We might need to factor in convenience / the cost of making a flip! Can you clarify what you mean by "chance of higher returns is better too"? How do you quantify that? Staff - 3 years, 10 months ago Log in to reply Earlier posters have quantified the variance for the smaller vs. the larger set of coins. For the set of 10 coins, the chance of getting more than 9 heads is 1 in 1024. For the set of 100 coins, the chance of getting more than 90 coins is much lower than that. I'd have to get my combinatorics (or statistics) hat on to quantify it, but it's much less. On the order of more than 5-sigma from the mean. For stocks, a stock club (or newsletter writer) would be silly to buy 1 share each of the 500 largest companies. A number of funds exists to do that with much less expense. If they want to try to beat the performance of the S&P 500, they pick a small number of stocks (using some rational criteria) and often do better than the "market". Of course, the risk of loss (or underperformance) is higher too. - 3 years, 10 months ago Log in to reply Expected outcome is same in both cases ,but in case 1 chances are that outcome is more closer to 50$ than in case2, So for more standard outcome I will choose case1.
For suppose if my need demands 60$I would choose case 2. CASE1 Chances to get 50$ is 100C50 *.5^{100}=.0786
49$is 100C49*.5^{100 }=.0780 (same for 51$)
For 60$it is .0108 By similar calculations(and adding), chances to get outcome from 40$ to 60$is .946. where as in case 2 chances reduce to .656 this proves that there are more chances to get outcome closer to 50$ in case 1.
probabilities to get 60$are 0.108 for case 1 .205 for case 2. So, if 60$ is my need I will choose case 2
- 3 years, 10 months ago
You bring up a valid point, which is "What is the utility function that I care about". If the utility function is just "I only care if I have more than $60 or not", then your approach shows that we should go for payoff 2. What kind of utility function makes sense? Do people value all amounts of money equally? Staff - 3 years, 10 months ago Log in to reply They average out to the same payoff (which is$50.00), so I would flip a coin to decide which to use.
Python:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 import random coin = (True, False) def sim_1(): payoff = 0 for flip in xrange(100): if random.choice(coin): payoff += 1 return payoff def sim_2(): payoff = 0 for flip in xrange(10): if random.choice(coin): payoff += 10 return payoff trials = 100000 payoff_1 = 0.0 payoff_2 = 0.0 for trial in xrange(trials): payoff_1 += sim_1() payoff_2 += sim_2() print "Payoff 1:", payoff_1 / trials print "Payoff 2:", payoff_2 / trials
- 3 years, 10 months ago
what good
- 3 years, 10 months ago
If i understand the problem correctly, would be indifferent to the payoffs structure, cause both options give the same expected value which would be 50.
- 3 years, 10 months ago
playoff 1
- 3 years, 10 months ago
I think payoff 1 is better because chances increses with more no of toss..
- 3 years, 10 months ago
In my opinion, it's a matter of risk: payoff 1 has more consistency but less probability of scoring extremely large amounts of money, whereas payoff 2 is a sort of high risk high reward model where you either get a lot or get a bit. Personally, I would choose payoff 1.
- 3 years, 10 months ago
Pay off 2, because it will likely to get heads more often and earn $10 for each head than 100 coin tosses. Who knows that in a toss of 100 you will only get less than 50% heads. - 3 years, 10 months ago Log in to reply Payoff 2 because I'll make decent money and i Don't have time to flip 100 coins. Stop over complicating things with math people. - 3 years, 10 months ago Log in to reply I will chose payoff 2 because while tossing coins 10 time we may get HEAD more than 1 or 2 times and the pay is more. But in payoff 1 tossing coins 100 times may give a HEAD 20 times and u get 20$. But in payoff 2 you can get more than 20$. - 3 years, 10 months ago Log in to reply Payoff 2 as it takes less time to toss 10 coins and the value is same for both - 3 years, 9 months ago Log in to reply What are the statistics given that heads side of a coin is slightly heavier than tails? Giving tails more of a chance of facing up? Is that true? I heard that somewhere back maybe in 4th grade haha. Or has that been proven incorrect? if it is true, I would rather chose b since I would want less of a chance of getting tails side up? Can you explain to me how I am wrong (I probably am) like you would explain it to a 3rd grader? This is why I'm more of an artist than math person. - 3 years, 6 months ago Log in to reply That is a very good point. How would your answer change if we didn't have a fair coin? For example, if the coin was totally biased and only gave tails, then both payoffs are the same. Similarly, if the coin was totally biased and only gave heads, then both payoffs are the same. But in between, the payoffs are different. So, does that mean that the answer could change depending on the probability of head/tail? If so, how and why? How is payoff 2 "less chance of getting tails side up"? Just because of the absolute number? But, the first mostly likely has many more heads appearing, just a smaller payoff each time. Staff - 3 years, 6 months ago Log in to reply Less probability of a heads since the gravitational pull of the heavier side. Sorry I missed stated. On the other hand, my fiance makes a good point. The more you flip a coin your results are more likely to reflect the chance of a 50/50 so it is better and more reliable to flip more times than less so option 1 of 100 flips is more reliable then just 10 in case you get a lucky streak of a ten or an unlucky streak with only 2 out of 10. - 3 years, 6 months ago Log in to reply As it turns out, the Expected Value of both payoffs would be the same, regardless of the probability of landing heads / tails. For example, if the coin is fair, and it will land heads 50% of the time, then the expected payoff in 1 is$50, and the expected payoff in 2 is \$50. More generally, if it land heads $$p%$$ of the time, then the expected payoff will be $$p$$.
So, the question then becomes, pick your favorite probability $$p$$ of landing heads, which payoff would you want? Is there anything else to consider?
Staff - 3 years, 6 months ago
Payoff 1 so that I can get more number of chances
- 3 years, 10 months ago
I'd go for #2. I'm a risktaker.
- 3 years, 9 months ago
You will still need to define what you mean by "risktaker". There are possible definitions of "risktaker" which would make option 1 more valuable to them.
Staff - 3 years, 9 months ago | 2019-01-17T14:23:04 | {
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https://math.stackexchange.com/questions/3258275/show-there-cant-be-two-real-and-distinct-roots-of-polynomial-fx-x3-3xk-in | # Show there can't be two real and distinct roots of polynomial $f(x)=x^3-3x+k$ in $(0,1)$, for any value of k.
I have two proofs here one which I did and the other was given in book. Is one better than the other? I am asking for in an exam setting which proof makes a better solution(as in fetch more marks).
## Proof 1 (My proof)
$$f'(x)=3x^2-3$$, in the interval $$(0,1)$$ is less than $$0$$. If there were two distinct roots then $$f'(0)$$ should have been $$0$$ once in $$(0,1)$$ by rolle's theorem. Since it isn't there are no values of $$k$$ for which there are two real roots.
## Proof 2 (Book)
Let $$a,b$$ be two roots of $$f(x)$$ in $$(0,1)$$ then there exists a $$c$$ such the $$f'(c) = 0$$ for c in $$[a,b]$$ by Rolle's theorem. $$f'(c)= 3c^2-3$$ has no solutions in $$(0,1)$$ hence there is no such value of $$k$$.
• Yours is better because the book version has a wrong expression for $f'$ – Hagen von Eitzen Jun 11 at 6:09
• – lab bhattacharjee Jun 11 at 6:16
• @HagenvonEitzen that's a typo sorry. – Sonal_sqrt Jun 11 at 6:26 | 2019-08-21T00:47:44 | {
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https://math.stackexchange.com/questions/2118291/how-to-factorize-this-cubic-equation | # How to factorize this cubic equation?
In one of the mathematics book, the author factorized following term
$$x^3 - 6x + 4 = 0$$ to
$$( x - 2) ( x^2 + 2x -2 ) = 0.$$
How did he do it?
• If there is a cube term, it is not a quadratic. Also, a method for finding divisors of your polynomial, look at the factors of the constant term. – Edward Evans Jan 28 '17 at 19:52
• $x=2$ is a root of this equation. Then by polynomial division of $\frac{x^3-6x+4}{x-2}$ we obtain $x^2+2x-2$. – projectilemotion Jan 28 '17 at 19:53
• Do you know the Rational Root Test? – Bill Dubuque Jan 28 '17 at 19:54
• A common first step for introductory problems is to guess and check certain integers close to zero to see if it will equal zero. Zero is the easiest to check because that would mean the constant term is zero, thats not the case here. $1$ doesn't work because that would be $1-6+4=-1\neq 0$. $2$ happens to work since this would be $2^3-6\cdot 2 + 4 = 8-12+4=0$. Since $2$ works, we know that the equation can be factored as $(x-2)q(x)$ where $q(x)=\frac{x^3-6x+4}{x-2}$. In general this won't always work, especially if the roots aren't even integers. Cardano's formula would help then. – JMoravitz Jan 28 '17 at 19:54
• @BillDubuque No – Govinda Sakhare Jan 28 '17 at 19:56
There is a neat trick called the rational roots theorem. All we have to do is factor the first and last numbers, put them over a fraction, and take $\pm$. This gives us the following possible rational roots:
$$x\stackrel?=\pm1,\pm2,\pm4$$
due to the factorization of $4$. Checking these, it is clear $x=2$ is the only rational root, since
\begin{align}0&\ne(+1)^3-6(+1)+4\\0&\ne(-1)^3-6(-1)+4\\\color{#4488dd}0&=\color{#4488dd}{(+2)^3-6(+2)+4}\\0&\ne(-2)^3-6(-2)+4\\0&\ne(+4)^3-6(+4)+4\\0&\ne(-4)^3-6(-4)+4\end{align}
leaving us with
$$x^3-6x+4=(x-2)(\dots)$$
We can find the remainder through synthetic division:
$$\begin{array}{c|c c}2&1&0&-6&4\\&\downarrow&2&4&-4\\&\hline1&2&-2&0\end{array}$$
which gives us our factorization:
$$x^3-6x+4=(x-2)(x^2+2x-2)$$
• OP does not know RRT - see the comments. – Bill Dubuque Jan 28 '17 at 19:59
• @BillDubuque Oh. Then give me a moment – Simply Beautiful Art Jan 28 '17 at 20:00
• @SimplyBeautifulArt Why 1,2 and 4. Why are we not checking equation with value 3? it has anything to do with c term ax^3+bx+c ? – Govinda Sakhare Jan 28 '17 at 20:09
• @piechuckerr By the rational roots theorem, I need only check the factors of the last constant if the leading coefficient is $1$. $3$ does not divide into $4$, so I needn't check it. – Simply Beautiful Art Jan 28 '17 at 20:10
• @SimplyBeautifulArt i.e if constant term is 6 then 1,2,3 and 6 right? – Govinda Sakhare Jan 28 '17 at 20:12
Since you do not know the Rational Root Test, let's consider a simpler case: the Integer Root Test.
If $\,f(x)= x^3+6x+4\,$ has an integer root $\,x=n\,$ then $\,n^3+6n+4 = 0\,$ so $\,(n^2+6)\,\color{#c00}{n = -4},\,$ hence $\,\color{#c00}{n\ \ {\rm divides}\ \ 4}.\,$ Testing all the divisors of $4$ shows that $2$ is root, hence $\,x-2\,$ is a factor of $f$ by the Factor Theorem. The cofactor $\,f/(x-2)\,$ is computable by the Polynomial (long) Division algorithm (or even by undetermined coefficients).
Remark $\$ This is a very special case of general relations between the factorization of polynomials and the factorizations of their values. For example, one can derive relations between primality and compositeness of polynomials based on the same properties of their values. For example, since $\ 9^4\!+8\$ is prime so too is $\, x^4+8\,$ by Cohn's irreducibility test. See this answer and its links for some of these beautiful ideas of Bernoulli, Kronecker, and Schubert.
• brilliant, liked every bit of it, Sadly I can not unmark other answer and select this one coz that too is elegant answer. – Govinda Sakhare Jan 29 '17 at 16:50
• @piechuckerr No problem (i don't care about acceptance, votes, etc, only about sharing beautiful mathematics). – Bill Dubuque Jan 29 '17 at 17:05
Note: I understand that there is already an accepted answer for this question, so this answer may be useless, but regardless, I'm still posting this to spread knowledge!
A simple way to factorize depressed cubic polynomials of the form$$x^3+Ax+B=0\tag1$$
Is to first move all the constants to the RHS, so $(1)$ becomes$$x^3+Ax=-B\tag2$$ Now, find two factors of $B$ such that one fact minus the square of the other factor is $A$. We'll call them $a,b$ so\begin{align*} & a-b^2=A\tag3\\ & ab=-B\tag4\end{align*} Multiply $(2)$ by $x$, add $b^2x^2$ to both sides and complete the square. Solving should give you a value of $x$ and allow you to factor $(1)$ by Synthetic Division.
Examples:
1. Solve $x^3-6x+4=0$ (your question)
Moving $4$ to the RHS and observing its factors, we have $-2,2$ as $a,b$ since$$-2-2^2=A\\-2\cdot2=-4$$Therefore, we have the following:$$x^4-6x^2=-2\cdot2x$$$$x^4-6x^2+4x^2=4x^2-4x$$$$x^4-2x^2=4x^2-4x$$$$x^4-2x^2+1=4x^2-4x+1\implies(x^2-1)^2=(2x-1)^2$$$$x^2=2x\implies x=2$$ Note that we do have to consider the negative case when square rooting, but they lead to the same pair of answers. So it's pointless.
1. Solving $x^3+16x=455$
A factor of $455$ works, namely when $a=65,b=7$.$$65-7^2=16$$$$65\cdot7=455$$ Therefore,$$x^4+16x^2=65\cdot7x$$$$x^4+65x^2=49x^2+455x$$$$\left(x^2+\dfrac {65}{2}\right)^2=\left(7x+\dfrac {65}{2}\right)^2$$$$x=7$$
The rational root theorem gives a list of all possible rational roots of a polynomial with integer coefficients that have a given leading coefficient and a given constant coefficient. In this case, the leading coefficient is $1$ and the constant coefficient is $4.$ The theorem tells us that all rational roots are in the set $\left\{ \pm\dfrac 1 1, \pm\dfrac 2 1, \pm \dfrac 4 1 \right\},$ the numerator being in this the only divisor of the leading coefficient $1$ and the denominators being the divisors of the constant coefficient $4$. That doesn't mean there are rational roots; it only means there are not any that don't belong to this set. There are only six members of this set, so it's easy to plug in all of them and see if you get $0$. When you plug in $2$, you get $0$, so there's your factorization.
• OP does not know RRT - see the comments. – Bill Dubuque Jan 28 '17 at 19:59
• @BillDubuque : Fortunately I linked to the Wikipedia article about it. – Michael Hardy Jan 28 '17 at 19:59
• @BillDubuque : That he doesn't recognize that name of the theorem doesn't mean he doesn't know the theorem. He could know it by a different name. – Michael Hardy Jan 28 '17 at 20:01
• I thought the comment(s) might nudge someone to give an exposition on this simpler integer case. But no one did, so I added an answer doing so. – Bill Dubuque Jan 28 '17 at 20:40
If $P$ is a polynomial with real coefficients and if $a\in\mathbb{R}$ is a root, which means that $P(a)=0$, then there exists a real polynomial $Q$ such that $\forall x\in\mathbb{R},\quad P(x)=(x-a)\,Q(x)$.
On this case, you can see by inspection, that $P(2)=0$.
It remains to find real constants $A,B,C$ such that :
$$\forall x\in\mathbb{R},\quad x^3-6x+4=(x-2)(Ax^2+Bx+C)$$
Identification of coefficients leads to $A=1$, $-2C=4$ and, for example, $A-2B=0$ (equating the coeffts of $x^2$ in both sides).
• Since the OP doesn't know RRT, the answer I posted earlier seems to me a (not so bad) - one ! It's clear that one can answer at a higher level, but it's seems to me important to answer at a level compatible the mathematical background of the OP. I have the feeling that, if he (or she) had known the RRT, he would certainly never have asked that question. – Adren Jan 28 '17 at 20:11
• Actually we don't need the full RRT here, only a simpler integer case, e.g. see my answer. – Bill Dubuque Jan 28 '17 at 20:56 | 2020-02-28T00:14:10 | {
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https://www.physicsforums.com/threads/how-do-you-tackle-this-integral.708999/ | # How do you tackle this integral
1. Sep 6, 2013
### Gauss M.D.
1. The problem statement, all variables and given/known data
Find F(x)
f(x) = 1/(1+x2)2)
2. Relevant equations
3. The attempt at a solution
It's actually a subproblem of another huge annoying surface integral. I tried u-sub but that landed me nowhere and partial integration led me nowhere aswell... Any pointers?
2. Sep 6, 2013
### Zondrina
Perfect candidate for a trig sub. Use $x = tan(θ)$ so that $dx = sec^2(θ)dθ$.
3. Sep 6, 2013
### ZeroPivot
well, 1/1+X^2 is the derivative of arctan i believe. i think there has to be a relation. u use integration by parts left u nowhere?
4. Sep 6, 2013
### Gauss M.D.
Zondrina: Aha, thanks. Obviously it just so happens that in the problem, x had an upper bound of 1. Would we have been out of luck if that hadn't been the case?
Last edited: Sep 6, 2013
5. Sep 6, 2013
### ZeroPivot
its a nasty integral u sure you are doing everything correctly? so the integral is 1/1+x^2 only?
6. Sep 6, 2013
### Gauss M.D.
No, I was referring to Zondrinas trig sub suggestion!
7. Sep 6, 2013
### Zondrina
I don't see any discontinuity in the integrand at all, so I don't think it would matter.
8. Sep 6, 2013
### Gauss M.D.
Right right, we're using tan and not some other trig function. My bad again. Thanks a ton :)
9. Sep 7, 2013
### verty
That is a very good question, I don't know the answer to it. I'm going to investigate this.
10. Sep 7, 2013
### verty
Sorry, the original question has no trouble because the range of tan is just R. Here is a more applicable problem:
$y = \int {dx \over (1 - x^2)^2}$
$x =? \; sin\theta$
$dx = cos\theta \; d\theta$
$y = \int sec^3\theta \; d\theta$
$= {1 \over 2} [sec\theta \; tan\theta + \int sec\theta \; d\theta] + C$
$= {1 \over 2} [sec\theta \; tan\theta + ln| sec\theta + tan\theta | ] + C$
$= {1\over 2}{x \over 1 - x^2} + {1\over 4} ln | {1+x\over 1-x}| + C$
On the other hand, one can make a translation:
$x = u-1$
$dx = du$
$y = \int {dx \over (1 - x^2)^2} = \int{ dx \over (1+x)^2 (1-x)^2} = \int {du \over u^2(2-u)^2}$
After some partial fraction magic:
$y = \int {1 \over 4(2-u)^2} + {1 \over 4u^2} + {1 \over 4u} + {1 \over 4(2-u)} du$
$= {1 \over 4} [ {1 \over 2-u} - {1\over u} + ln|u| - ln|2-u| ] + C$
$= {1 \over 4} [ {2(u-1) \over u(2-u)} + ln| {u \over 2-u} | ] + C$
$= {1 \over 2} {x \over 1 - x^2} + {1 \over 4} ln| {1+x \over 1-x} | + C$
I can't explain it but it's the same answer. Probably it has to do with the behaviour of sin(z) and cos(z) for complex z. | 2017-10-23T23:46:30 | {
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https://math.stackexchange.com/questions/2240525/counting-distinguishable-ways-of-painting-a-cube-with-4-different-colors-each-u | # Counting distinguishable ways of painting a cube with 4 different colors (each used at least once)
You have many identical cube-shaped wooden blocks. You have four colors of paint to use, and you paint each face of each block a solid color so that each block has at least one face painted with each of the four colors. Find the number of distinguishable ways you could paint the blocks. (Two blocks are distinguishable if you cannot rotate one block so that it looks identical to the other block.)
Having trouble solving this problem with the added constraint of "at least one face painted with each of four colors" - Thanks in advance
Call the four colors $a$, $b$, $c$, $d$.
There are two partitions of $6$ into four parts, namely (1): $(3,1,1,1)$, and (2): $(2,2,1,1)$.
In case (1) we can choose the color appearing three times in $4$ ways. This color can either (1.1) appear on three faces sharing a vertex of the cube, or (1.2) on three faces forming a $\sqcup$-shape. In case (1.1) we can place the three other colors in $2$ ways ("clockwise" or "counterclockwise"); in case (1.2) we can choose which of the three other colors is opposite the floor of the $\sqcup$. This amounts to $4\cdot(2+3)=20$ different colorings.
In case (2) the two colors appearing only once can be chosen in ${4\choose2}=6$ ways. Assume that colors $a$ and $b$ are chosen. The $a$-face $F_a$ and the $b$-face $F_b$ can be either (2.1) opposite or (2.2) adjacent to each other. In case (2.1) we can chose the two $c$-faces either adjacent or opposite to each other. In case (2.2) the two $c$-faces can be (2.2.1) opposite to each other, (2.2.2) opposite to $F_a$ and to $F_b$, or (2.2.3) opposite to one of $F_a$ or $F_b$ and on one of the faces adjacent to both $F_a$ and $F_b$. In all this amounts to $6\cdot(2+1+1+2\cdot2_*)=48$ different colorings. (The factor $2_*$ distinguishes mirror-equivalent colorings.)
Altogether we have found $68$ different admissible colorings of the cube.
• great solution ! - I still need to work out the second scenario in detail but appears to be a valid solution. Thanks – randomwalker Apr 18 '17 at 21:15
There is an algorithmic approach to this which I include for future reference and which consists in using Burnside and Stirling numbers of the second kind. For Burnside we need the cycle index of the face permutation group of the cube. We enumerate the constituent permutations in turn. First there is the identity for a contribution of $$a_1^6.$$
Rotating about one of the four diagonals by $120$ degrees and $240$ degrees we get
$$4\times 2a_3^2.$$
Rotating about an axis passing through opposite faces by $90$ degrees and by $270$ degrees we get
$$3\times 2 a_1^2 a_4$$
and by $180$ degrees
$$3\times a_1^2 a_2^2.$$
Finally rotating about an exis passing through opposite edges yields
$$6\times a_2^3.$$
We thus get the cycle index
$$Z(G) = \frac{1}{24} (a_1^6 + 8 a_3^2 + 6 a_1^2 a_4 + 3 a_1^2 a_2^2 + 6 a_2^3).$$
As a sanity check we use this to compute the number of colorings with at most $N$ colors and obtain
$$\frac{1}{24}(N^6 + 8 N^2 + 12 N^3 + 3 N^4).$$
This gives the sequence
$$1, 10, 57, 240, 800, 2226, 5390, 11712, 23355, 43450, \ldots$$
which is OEIS A047780 which looks to be correct. Now if we are coloring with $M$ colors where all $M$ colors have to be present we must partition the cycles of the entries of the cycle index into a set partition of $M$ non-empty sets. We thus obtain
$$\frac{M!}{24} \left({6\brace M} + 8 {2\brace M} + 12 {3\brace M} + 3{4\brace M}\right).$$
This yields the finite sequence (finite because the cube can be painted with at most six different colors):
$$1, 8, 30, 68, 75, 30, 0, \ldots$$
In particular the value for four colors is $68.$ We also get $6!/24 = 30$ for six colors because all orbits have the same size.
Here's my crack at it: Starting with a stationary cube, there are $\frac{4 \cdot 3 \cdot 6!}{2 \cdot 2}$ ways of painting the cube where there are 2 colors with 2 faces, and $\frac{4 \cdot 6!}{3}$ ways of painting the cube when there is 1 color with 3 faces which gives a total of 3120 ways of painting the cube, but this over counts all the orientations of a cube, so the final answer is $\frac{3120}{4 \cdot 6} = 130$
• awright96 - Thank you for taking the time to answer the question. I understand your approach which is a good, clean approach. However the correct answer is 68. This problem appeared on a middle-school purple comet test in 2015 ( unfortunately only answers are published, no solutions) – randomwalker Apr 18 '17 at 19:31 | 2021-02-26T15:38:41 | {
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https://math.stackexchange.com/questions/3749178/prove-that-every-set-and-subset-with-the-cofinite-topology-is-compact | Prove that every set and subset with the cofinite topology is compact
Prove that every set with the cofinite topology is compact as well as every subset
Solution. Let $$X$$ be a nonempty set with the cofinite topology and let $$\mathscr{U}$$ be an open cover of $$X$$. Let $$U \in \mathscr{U}$$. Then $$X\setminus U$$ is finite. For every $$a \in X\setminus U$$ let $$U_a$$ be an element of $$\mathscr{U}$$ that contains $$a$$. Then $$\{U\}\cup\{U_a : a ∈ X\setminus U\}$$ is a finite subcover of $$\mathscr{U}$$.
Now I missing the part for the subsets $$E\subseteq X$$. I don't think this refers to the relative topology, but just to any subset of $$X$$ How do I go about it?
• It of course refers to the rel topology and the proof is the same. – JCAA Jul 7 '20 at 22:10
• I thought the statement meant that any subset of X was compact with the original topology, not with the relative one – J.C.VegaO Jul 7 '20 at 22:13
• Well what does it mean to be compact with the original topology? – Severin Schraven Jul 7 '20 at 22:21
• @Severin Schraven That you can extract a finite subcover from a open cover made of open sets of the original topology, ( not made of the intersection of them with the set, like in the relative topology.) The reason I make a difference is because set that is not open in the original topology may be open in the relative. like for example in [-1,1] with the usual topology $\tau$ as a subset of $\mathbb{R}$ . $E=[-1,1 ]$ is not open in $(\mathbb{R},\tau)$ but it is in $(\mathbb{R},\tau_E)$ – J.C.VegaO Jul 7 '20 at 22:27
In the case of compactness it makes no difference whether you use the topology of $$X$$ or the relative topology on the subset.
Proposition. Let $$\langle X,\tau\rangle$$ be any space, let $$K\subseteq X$$, and let $$\tau_K$$ be the relative topology on $$K$$; then $$K$$ is compact with respect to $$\tau$$ iff it is compact with respect to $$\tau_K$$.
Proof. Suppose first that $$K$$ is compact with respect to $$\tau$$, and let $$\mathscr{U}\subseteq\tau'$$ be a $$\tau'$$-open cover of $$K$$. For each $$U\in\mathscr{U}$$ there is a $$V_U\in\tau$$ such that $$U=K\cap V_U$$. Let $$\mathscr{V}=\{V_U:U\in\mathscr{U}\}$$; clearly $$\mathscr{V}$$ is a $$\tau$$-open cover of $$K$$, so it has a finite subcover $$\{V_{U_1},\ldots,V_{U_n}\}$$. Let $$\mathscr{F}=\{U_1,\ldots,U_n\}$$; $$\mathscr{F}$$ is a finite subset of $$\mathscr{U}$$, and
$$\bigcup\mathscr{F}=\bigcup_{k=1}^nU_k=\bigcup_{k=1}^n(K\cap V_{U_k})=K\cap\bigcup_{k=1}^nU_k=K\;,$$
so $$\mathscr{F}$$ covers $$K$$. Thus, $$K$$ is compact with respect to $$\tau'$$.
Now suppose that $$K$$ is compact with respect to $$\tau'$$, and let $$\mathscr{U}\subseteq\tau$$ be a $$\tau$$-open cover of $$K$$. For each $$U\in\mathscr{U}$$ let $$V_U=K\cap U$$, and let $$\mathscr{V}=\{V_U:U\in\mathscr{U}\}$$. $$\mathscr{V}$$ is a $$\tau'$$-open cover of $$K$$, so it has a finite subcover $$\{V_{U_1},\ldots,V_{U_n}\}$$. Let $$\mathscr{F}=\{U_1,\ldots,U_n\}$$; $$\mathscr{F}$$ is a finite subset of $$\mathscr{U}$$, and
$$\bigcup\mathscr{F}=\bigcup_{k=1}^nU_k\supseteq\bigcup_{k=1}^n(K\cap U_k)=\bigcup_{k=1}^nV_{U_k}=K\;,$$
so $$\mathscr{F}$$ covers $$K$$. Thus, $$K$$ is compact with respect to $$\tau$$. $$\dashv$$
• Initially I thought that they were saying that any subset of $X$ was compact, like for example if $X=\mathbb{R}$ with the cofinite topology, then any subset like $\{1\}$ $[1,2] ,(1,2),(1,2], (1,+\infty)$ should be compact, that is not true, is it? – J.C.VegaO Jul 7 '20 at 23:43
• @J.C.VegaO: It is true. You essentially proved it in your question. – Brian M. Scott Jul 7 '20 at 23:46 | 2021-03-01T01:38:07 | {
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https://math.stackexchange.com/questions/1296651/sat-maths-question-about-fractions | # SAT Maths Question About Fractions
Whilst revising, a problem caught my eye and I cannot seem to find an answer. I am usually bad at these types of questions.
On a certain Russian-American committee, $\frac23$ of members are men, and $\frac38$ of the men are Americans. If $\frac35$ of the committee members are Russians, what fraction of the members are American women?
A. $\frac{3}{20}$ B. $\frac{11}{60}$ C. $\frac{1}{4}$ D. $\frac{2}{5}$ E. $\frac{5}{12}$
Could you please explain how to approach and analyse the problem, maybe give some hints or the complete procedure of solving?
I get a bit confused with all those fractions. What I tried was to convert them to percentages but that seemed a bad idea.
Sorry if this question is annoying.
Thank you.
Update: I solved the problem both intuitively and mathematically. Thanks.
• You've got two categories, each with two options. You can find the probabilities of being in the intersection of each of these options (e.g. Russian Women) using the proportions and conditional proportions (like how its not that 3/8 are american, but 3/8 of men are american), and your answer will be there. – Bob Krueger May 24 '15 at 13:38
Here's a more algebraic approach. We know that $\frac{2}{5}$ of the committee is American (the other $\frac{3}{5}$ is Russian), and that $\frac{2}{3}\cdot\frac{3}{8} = \frac{1}{4}$ of the committee is American men. Thus, if the proportion of American women is $x$, then, $$\frac{1}{4} + x = \frac{2}{5}$$ that is, if you add up the proportion of American men and the proportion of American women (assuming men and women are the only two categories), then you get the proportion of Americans. From here, we can solve for $x$ to get $$x = \frac{2}{5} - \frac{1}{4} = \frac{3}{20},$$ which is in agreement with abel's answer.
Let me try. We will pick a nice number for the total number of members so that every category of members come out whole numbers. We can pick the least common multiple of all the denominators of the fractions. That gives us $120.$ Of course as long as we are only interested in the ratio, it don't matter what that number is. It makes computation easier to do.
Suppose there were $120$ members. There are $80$ men and $40$ women. Of the $80$ men $30$ are American and $50$ russian. There are $72$ Russians on the committee that leave $22$ Russian women and $18$ American women on the committee. The fraction of American women on the committee is $$\frac{18}{120} = \frac3{20}.$$
Hopefully I did not make any silly arithmetic errors.
• I'd only suggest you edit to make clear why you chose 120. It may not be obvious to OP. – JoeTaxpayer May 24 '15 at 14:36
I can't improve on @abel 's excellent answer but do want to point out that it's an example of an important strategy. When you're "confused with all those fractions" and converting to percentages doesn't help (because they are just fractions) try natural frequencies - pick a number (in this case 120) that lets you count the cases.
See
http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/
http://www.medicine.ox.ac.uk/bandolier/booth/glossary/freq.html
An alternative approach, very similar to @Strants's, with similar logic, but a slightly different way of looking at things:
We know that $\frac{2}{3}$ of the committee members are men and that $\frac{3}{8}$ of them are Americans.
This means that the proportion of all committee members who are American men is:
$\frac{2}{3} \times \frac{3}{8} = \frac{6}{24} = \frac{1}{4}$
We know also that $\frac{3}{5}$ of the members are Russian.
We know that there are only four types of people at the committee:
American men, Russian men, Russian women and American women.
We have found that $\frac{1}{4}$ of the members are American men and $\frac{3}{5}$ of them are Russian - i.e. Russian men and Russian women account for $\frac{3}{5}$ of the members.
Therefore, the proportion of members who are American men, Russian men, or Russian women is:
$\frac{3}{5} + \frac{1}{4} = \frac{12}{20} + \frac{5}{20} = \frac{17}{20}$
Therefore, the proportion of members who are American women is:
$1 - \frac{17}{20} = \frac{3}{20} = \text{A}$. | 2019-10-22T06:09:32 | {
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https://math.stackexchange.com/questions/1523820/we-roll-a-six-sided-die-ten-times-what-is-the-probability-that-the-total-of-all | # We roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by 6?
So the question is really hard I think. I tried using a simple way by calculating the probability of each combination that makes a sum divisible by six, but it would take forever. Does anyone have any ideas?
Suppose that we roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by six?
• Almost a duplicate of this question. Using the same logic, the answer is easily seen to be $1/6$. – TonyK Nov 11 '15 at 15:33
Hint.
Roll $9$ times and let $x$ be the total.
For exactly one number $n\in\{1,2,3,4,5,6\}$ we will have $6 \mid (x+n)$ (i.e. $x+n$ is divisible by $6$).
• I didn't get it, why would I roll 9 times instead of 10? and what is 6|x? – Xlyon Nov 11 '15 at 9:38
• @Xlyon When you've rolled nine times there's always exactly one outcome for the tenth that will make the sum divisible by $6$. – skyking Nov 11 '15 at 9:43
• Thanks! that is logical. Can you please help me complete it? – Xlyon Nov 11 '15 at 10:50
• Exactly one of the faces of the die thrown at the $10$th time will result in a total sum ($x+n$) that is divisible by $6$. The probability that this face shows up is $\frac16$ (if the die is fair, of course). – drhab Nov 11 '15 at 10:54
• The value of $x$ is indeed irrelevant. Whatever value $x$ takes, it's chance to change into a number divisible by $6$ (when the last result is added by $x$) is in all cases $\frac16$. So that's indeed the answer to this question. They could have asked: take some arbitrary number $y\in\mathbb Z$ and trhrow a fair die. If $D$ denotes the outcome then what is the probability that $y+D$ is divisible by $6$? Same answer: $\frac16$, – drhab Nov 11 '15 at 11:34
After rolling the die once, there is equal probability for each result modulo 6. Adding any unrelated integer to it will preserve the equidistribution. So you can even roll a 20-sided die afterwards and add its outcome: the total sum will still have a probability of 1/6 to be divisible by 6.
• @Kevin The d20 by itself does have that unequal distribution, but the sum d20+d6 does not. It literally does not matter what _**x**_ is in determining the distribution of (_**x**_ + d6) mod 6. In fact, d20 + d6 also has exactly a 1/20 chance of being divisible by 20 by the same logic – Monty Harder Nov 11 '15 at 19:28
If you want something a little more formal and solid than drhab's clever and brilliant answer:
Let $P(k,n)$ be the probability of rolling a total with remainder $k$ when divided by $6, (k = 0...5)$ with $n$ die.
$P(k, 1)$ = Probability of rolling a $k$ if $k \ne 0$ or a $6$ if $k = 6$; $P(k, 1) = \frac 1 6$.
For $n > 1$. $P(k,n) = \sum_{k= 0}^5 P(k, n-1)\cdot \text{Probability of Rolling(6-k)} = \sum_{k= 0}^5 P(k, n-1)\cdot\frac 1 6= \frac 1 6\sum_{k= 0}^5 P(k, n-1)= \frac 1 6 \cdot 1 = \frac 1 6$
This is drhab's answer but in formal terms without appeals to common sense
• That's a way to do it. What I actually had in mind was: $P\left(6\text{ divides }S_{10}\right)=\sum_{m\in\mathbb{N}}P\left(6\text{ divides }S_{10}\mid S_{9}=m\right)P\left(S_{9}=m\right)=\sum_{m\in\mathbb{N}}\frac{1}{6}P\left(S_{9}=m\right)=\frac{1}{6}$ – drhab Nov 12 '15 at 9:40
• Yes, that is more direct. – fleablood Nov 12 '15 at 16:45
In spite of all great answers, given here, I say, why not give another proof, from another point of view. The problem is we have 10 random variables $X_i$ for $i=1,\dots,10$, defined over $[6]=\{1,\dots,6\}$, and we are interested in distribution of $Z$ defined as $$Z=X_1\oplus X_2\oplus \dots \oplus X_{10}$$ where $\oplus$ is addition modulo $6$. We can go on by two different, yet similar proofs.
First proof: If $X_1$ and $X_2$ are two random variables over $[6]$, and $X_1$ is uniformly distributed, sheer calculation can show that $X_1\oplus X_2$ is also uniformly distributed. Same logic yields that $Z$ is uniformly distributed over $[6]$.
Remark: This proves a more general problem. It says that even if only one of the dices is fair dice, i.e. each side appearing with probability $\frac 16$, the distribution of $Z$ will be uniform and hence $\mathbb P(Z=0)=\frac 16$.
Second proof: This proof draws on (simple) information theoretic tools and assumes its background. The random variable $Z$ is output of an additive noisy channel and it is known that the worst case is uniformly distributed noise. In other word if $X_i$ is uniform for only one $i$, $Z$ will be uniform. To see this, suppose that $X_1$ is uniformly distributed. Then consider the following mutual information $I(X_2,X_3,\dots,X_6;Z)$ which can be written as $H(Z)-H(Z|X_2,\dots,X_6)$. But we have: $$H(Z|X_2,\dots,X_6)=H(X_1|X_2,\dots,X_6)=H(X_1)$$
where the first equality is resulted from the fact that knowing $X_2,\dots,X_6$ the only uncertainty in $Z$ is due to $X_1$. The second equality is because $X_1$ is independent of others. Know see that:
• Mutual information is positive: $H(Z)\geq H(X_1)$
• Entropy of $Z$ is always less that or equal to the entropy of uniformly distributed random variable over $[6]$: $H(Z)\leq H(X_1)$
• From the last two $H(Z)=H(X_1)$ and $Z$ is uniformly distributed and the proof is complete.
Similarly here, only one fair dice is enough. Moreover the same proof can be used for an arbitrary set $[n]$. As long as one of the $X_i$'s is uniform, then their finite sum modulo $n$ will be uniformly distributed.
There are 3 variables in this case:
• the number of sides of the dice: s (e.g. 6)
• the number of throws: t (e.g. 10)
• the requesed multiple: x (e.g. 6)
In this case, the conditions are simple:
• s>=x
• x >0
• t > 0
And also the answer is simple: Throwing a sum that is a multiple of 6 has a 1/6 probability.
$P(s,t,x) = 1/x$
For situations where s<x this is not entirely correct. It approaches the same result though, at a high amount of throws. Example: If you throw a 6-sided dice 30 times the chance that the sum is a multiple of 20 will be about 5%. Proving this is a bit of a challenge.
$\lim \limits_{t \to \infty} P(s,t,x) = 1/x$,
Nevertheless, if programming is an acceptable proof:
public static void main(String[] args) {
int t_throws = 10;
int s_sides = 6;
int x_multiple = 6;
int[] diceCurrentValues = new int[t_throws];
for (int i = 0; i < diceCurrentValues.length; i++) diceCurrentValues[i] = 1;
int combinations = 0;
int matches = 0;
for (; ; ) {
// calculate the sum of the current combination
int sum = 0;
for (int diceValue : diceCurrentValues) sum += diceValue;
combinations++;
if (sum % x_multiple == 0) matches++;
System.out.println("status: " + matches + "/" + combinations + "=" + (matches * 100 / (double) combinations) + "%");
// create the next dice combination
int dicePointer = 0;
boolean incremented = false;
while (!incremented) {
if (dicePointer == diceCurrentValues.length) return;
if (diceCurrentValues[dicePointer] == s_sides) {
diceCurrentValues[dicePointer] = 1;
dicePointer++;
} else {
diceCurrentValues[dicePointer]++;
incremented = true;
}
}
}
}
# EDIT:
Here's another example. If you throw a 6-sided dice 10 times, there is 1/4 probability that the sum is a multiple of 4. The program above should run with the following parameters:
int t_throws = 10;
int s_sides = 6;
int x_multiple = 4;
The program will show the final output: status: 15116544/60466176=25.0% That means that there are 60466176 combinations (i.e. 6^10) and that there are 15116544 of them where the sum is a multiple of 4. So, that's 25% (=1/4).
This just follows the formula as mentioned above (i.e. P(s,t,x) = 1/x). x is 4 in this case.
• "Assuming that t converges to infinity. This is also the case when s<x." This sounds nonsensical. – djechlin Nov 11 '15 at 17:43
• Excuse me for my poor English :) The thing is, if you apply this for values greater than the number of sides of your dice. (e.g. multiples of 10 with a 6 sided dice.) then P = 1/x is no longer correct. But it does converge 1/x ; meaning that if you would throw an infinit amount of times, the result would be 1/x again. – bvdb Nov 12 '15 at 0:10
• I made some slight adjustments. Let me know what you think. – bvdb Nov 12 '15 at 0:22
• It's not the case that the limiting factor is if s>=x. Consider the probability that 10 rolls of a six-sided dice divides 4. (Of course the limit still converges when the number of dices increases) – Taemyr Nov 12 '15 at 8:40
• @Taemyr for 10 roles with a 6-sided dice, there are 60466176 combinations, of which 15116544 have a sum which is a multiple of 4. That's exactly 25% which is exactly 1/4. So, it's correct, right ? – bvdb Nov 14 '15 at 15:51
Roll the die 9 times and add up the dots. The answer is x. Roll the die one more time. add the number thrown to x to get one and only one of the following answers; x+1, x+2, x+3, x+4, x+5 or x+6. since these answers are six sequential numbers one and only one of them will be divisible by six. Therefore the probability of the sum of ten rolls of a die being divisible by six is exactly 1/6.
Couldn't you also think of it as the maximum possible value you could get for rolling the die 10 times would be 60, how many numbers between 1 and 60 are divisible by 6? 10 numbers are divisible by 6(6*1, 2, 3, etc.) So, 10 out of 60 possible values gives... 1/6. I love math.
• True, but some numbers appear multiple times. So, even though your answer is correct, your logic is flawed. – bvdb Nov 11 '15 at 11:58
• The minimum possible value is 10. So only 9 of 51 values are divisible by 6. But not all numbers are equally probable. – Cephalopod Nov 11 '15 at 11:58
• The numbers are not distributed evenly. – fleablood Nov 11 '15 at 20:31 | 2020-03-29T16:11:48 | {
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http://math.stackexchange.com/questions/126961/yet-another-question-on-using-basics-limit-arithmetics | # Yet Another Question On Using Basics Limit Arithmetics
Is this claim true?
Given $\lim \limits_{n\to \infty}\ a_n=\frac{1}{2}$ Then $\lim \limits_{n\to \infty}\ (a_n - [a_n])=\frac{1}{2}$
I think it's true, but probably I just didn't find the right example to disprove it.
Thanks a lot.
-
What do you mean by $[a_{n}]$? – Isaac Solomon Apr 1 '12 at 18:14
it's the round integer – Anonymous Apr 1 '12 at 18:17
Round up or down? – Isaac Solomon Apr 1 '12 at 18:17
If you mean nearest integer, then the claim is correct, so you will not find an example to disprove it. Are you looking for a formal argument? – André Nicolas Apr 1 '12 at 18:20
It's round donwn or truncate. And yes, @André Nicolas, I'm looking for a formal argument, thanks a lot. – Anonymous Apr 1 '12 at 18:21
Since
$$\lim_{n \to \infty} a_{n} = \frac{1}{2}$$
there exists $N$ such that for all $n \geq N$,
$$|a_{n} - \frac{1}{2}| < \frac{1}{4}$$
Then, for $n \geq N$, we have
$$0 \leq a_{n} \leq 1 \Longrightarrow [a_{n}] = 0$$
so that for $n \geq N$,
$$a_{n} - [a_{n}] = a_{n}$$
Now pick $\epsilon > 0$. Suppose that for all $n > M_{\epsilon}$
$$|a_{n} - \frac{1}{2}| < \epsilon$$
Replacing $M_{\epsilon}$ with $\mbox{max}(M_{\epsilon},N)$, we have
$$|(a_{n} - [a_{n}]) - \frac{1}{2} | = |a_{n} - \frac{1}{2}| < \epsilon$$
which proves that
$$\lim_{n \to \infty} (a_{n} - [a_{n}]) = \frac{1}{2}$$
-
I'm reading the proof and I'm at the line: "Now pick $\epsilon > 0$. Suppose that for all $n > M_{\epsilon}$ - what is $M_{\epsilon}$? – Anonymous Apr 1 '12 at 18:43
$M_{\epsilon}$ is just some sufficiently large natural number. I write $\epsilon$ in the subscript to indicate that this $M$ is a function of $\epsilon$. – Isaac Solomon Apr 1 '12 at 19:29
It depends on what you mean by the symbol $[x]$. If $[x]$ is the floor of $x$ (i.e., the greatest integer less than or equal to $x$), then the limit will be $\frac{1}{2}$. This follows from $\lim \limits_{n\to \infty}\ a_n=\frac{1}{2}$: there exists an $N$ such that for every $n \geq N$, $|a_n - \frac{1}{2}| < \frac{1}{2}$. For these $a_n$, it follows that $[a_n] = 0$, and so $$\lim \limits_{n\to \infty}\ a_n - [a_n] =\frac{1}{2} - 0 = \frac{1}{2}$$
On the other hand, if $[x]$ is the nearest integer to $x$, then the limit does not exist: consider the sequence $a = (0.4,0.6,0.49,0.51,0.499,0.501,\ldots)$. Clearly $a_n \rightarrow \frac{1}{2}$, but $[a_n]$ alternates between $0$ and $1$.
-
Thanks, I'm talking about the first case where $[x]$ is the floor of $x$. Could you please explain why it follows that $[a_n] = 0$? – Anonymous Apr 1 '12 at 18:32
Essentially, the $a_n$ are getting closer and closer to $\frac{1}{2}$; after a while (for $n \geq N$), all the $a_n$ will be between $0$ and $1$. The floor of a number in this interval is $0$. – Théophile Apr 1 '12 at 18:39
@Théophile: Yes, I was careless in my comment, which has been deleted. I meant the distance to the nearest integer. – André Nicolas Apr 1 '12 at 18:56
We want to show that for any $\epsilon>0$, there is an $N$ such that if $n>N$, then $|(a_n-\lfloor a_n\rfloor)-\frac{1}{2}|<\epsilon$. Here by $\lfloor w\rfloor$ we mean the greatest integer which is $\le w$.
Let $\epsilon'=\min(\epsilon,1/4)$. By the fact that $\lim_{n\to\infty} a_n=\frac{1}{2}$, there is an $N$ such that if $n>N$ then $|a_n-\frac{1}{2}|<\epsilon'$. In particular, $|a_n-\frac{1}{2}|<\frac{1}{4}$, and therefore $\lfloor a_n\rfloor=0$. It follows that if $n>N$ then $$\left|(a_n-\lfloor a_n\rfloor)-\frac{1}{2}\right|=\left|a_n-\frac{1}{2}\right|<\epsilon.$$
-
you wrote: $\left|(x_n-\lfloor x_n\rfloor)-\frac{1}{2}\right|=\left|x_n-\frac{1}{2}\right|<\epsilon$. But actually I know that $\left|(x_n-\lfloor x_n\rfloor)-\frac{1}{2}\right|=\left|x_n-\frac{1}{2}\right|<\epsilon'$, no? why can I make the switch from $\epsilon'$ to $\epsilon$? at least not for that n, maybe we need to add $n_1$? – Anonymous Apr 1 '12 at 19:04
The number we are "challenged with" is $\epsilon$. If $\epsilon$ is ridiculous, like $5$, then we can't say that $\lfloor x_n\rfloor=0$. Specifying that $\epsilon'=\min(1/4,\epsilon)$ makes sure $\lfloor x_n\rfloor=0$. It also makes sure (since $\epsilon' \le \epsilon$) that from $|x_n-\frac{1}{2}|<\epsilon'$, we can conclude that $|x_n-\frac{1}{2}|<\epsilon$. More informally, we could forget about $\epsilon'$, and say that the argument will only work if $\epsilon$ is reasonably small, but that's good enough. – André Nicolas Apr 1 '12 at 19:16
Thanks, I used the max between the n you gave me and the $n_1$ I have from the definition of limit, therefore safe to any $\epsilon$. – Anonymous Apr 1 '12 at 19:23
@Anonymous: It is just a technical trick. It can be replaced by taking $N$ simultaneously large enough to make $|a_n-1/2|<1/4$ (anything under $1/2$ is OK) and large enough to make $|a_n-1/2|<\epsilon$. – André Nicolas Apr 1 '12 at 19:27 | 2014-12-20T00:53:40 | {
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https://alpynepyano.github.io/healthyNumerics/posts/basic_stats_02_mechanical_approach_to_bayesian_rule.html | .
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HealthyNumerics
HealthPoliticsEconomics | Quant Analytics | Numerics
Basic Stats 02: A mechanical approach to the Bayesian rule
Intro
The Bayesian rule
$$P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$$
seems to be a rather abstract formula. But this impression can be corrected easely when considering a simple practical application. We call this approach mechanical because for this type of application there is no philosphical dispute between frequentist's and bayesian's mind set. We will just use in a mechanical/algorithmical manner the cells of a matrix. In this section we
• formulate a prototype of a probability problem (with red and green balls that have letters A and B printed on)
• summarize the problem in a basically 2x2 matrix (called contingency table)
• use the frequencies first
• replace them by probalities afterwards and recognize what conditional probablities are
• recognize that applying the Bayesian formula is nothing else than walking from one side of the matrix to the other side
A prototype of a probability problem
Given:
• 19 balls
• 14 balls are red, 5 balls are green
• among the 14 red balls, 4 have a A printed on, 10 have a B printed on
• among the 5 green balls, 1 has a A printed on, 4 have a B printed on
Typical questions: - we take 1 ball. - What is the probabilitiy that it is green ? - What is the probabilitiy that it is B under the precondition it's green ?
We will use Pandas to represent the problem and the solutions
import matplotlib.pyplot as plt
from IPython.core.pylabtools import figsize
import numpy as np
import pandas as pd
pd.set_option('precision', 3)
Contingency table with the frequencies
The core of the representation is a 2x2 matrix that summarizes the situation of the balls with the colors and the letters on. This matrix is expanded with the margins that contain the sums. - sum L stands for the sum of the letters - sum C stands for the sum of the colors
columns = np.array(['A', 'B'])
index = np.array(['red', 'green'])
data = np.array([[4,10],[1,4]])
df = pd.DataFrame(data=data, index=index, columns=columns)
df['sumC'] = df.sum(axis=1) # append the sums of the rows
df.loc['sumL']= df.sum() # append the sums of the columns
df
A B sumC
red 4 10 14
green 1 4 5
sumL 5 14 19
Append the relative contributions of B and of green
We expand the matrix by a further column and a further row and use them to compute relative frequencies (see below).
def highlight_cells(x):
df = x.copy()
df.loc[:,:] = ''
df.iloc[1,1] = 'background-color: #53ff1a'
df.iloc[1,3] = 'background-color: lightgreen'
df.iloc[3,1] = 'background-color: lightblue'
return df
df[r'B/sumC'] = df.values[:,1]/df.values[:,2]
df.loc[r'green/sumL'] = df.values[1,:]/df.values[2,:]
t = df.style.apply(highlight_cells, axis=None)
t
A B sumC B/sumC
red 4 10 14 0.714
green 1 4 5 0.8
sumL 5 14 19 0.737
green/sumL 0.2 0.286 0.263 1.09
Let's focus on the row with the green balls
From all green balls (= 5) is the portion of those with a letter B (=4) 0.8
$$\frac{green\: balls\: with \:B}{all\: green\: balls} = \frac{4}{5} = 0.8$$
Note that this value already corresponds to the conditional probality $$P(B \mid green)$$
Let's focus on the column with the balls with letter B
From all balls with letter B (= 14) is the portion of those that are green (=4) 0.286
$$\frac{green\; balls\; with\; B}{all\; balls\; with\; letter\; B} = \frac{4}{14} = 0.286$$
Note that also this value already corresponds to the conditional probality $$P(green \mid B)$$
Contingency table with the probabilities
We find the probabilities by dividing the frequencies by the sum of balls.
columns = np.array(['A', 'B'])
index = np.array(['red', 'green'])
data = np.array([[4,10],[1,4]])
data = data/np.sum(data)
df = pd.DataFrame(data=data, index=index, columns=columns)
df['sumC'] = df.sum(axis=1) # append the sums of the rows
df.loc['sumL']= df.sum() # append the sums of the columns
df
A B sumC
red 0.211 0.526 0.737
green 0.053 0.211 0.263
sumL 0.263 0.737 1.000
Append the relative contributions of B and of green
df[r'B/sumC'] = df.values[:,1]/df.values[:,2]
df.loc[r'green/sumL'] = df.values[1,:]/df.values[2,:]
t = df.style.apply(highlight_cells, axis=None)
t
A B sumC B/sumC
red 0.211 0.526 0.737 0.714
green 0.0526 0.211 0.263 0.8
sumL 0.263 0.737 1 0.737
green/sumL 0.2 0.286 0.263 1.09
Note the formula in the cells
columns = np.array(['-----------A----------', '---------B----------', '----------sumC--------', '--------B/sumC--------'])
index = np.array(['red', 'green', 'sumL', 'green/sumL'])
data = np.array([['...','...','...','...'],
['...', '$P(B \cap green)$', '$P(green)$', '$P(B \mid green)$'],
['...','$P(B)$','...','...'],
['...','$P(green \mid B)$','...','...'] ])
df = pd.DataFrame(data=data, index=index, columns=columns)
t = df.style.apply(highlight_cells, axis=None)
t
-----------A---------- ---------B---------- ----------sumC-------- --------B/sumC--------
red ... ... ... ...
green ... $$P(B \cap green)$$ $$P(green)$$ $$P(B \mid green)$$
sumL ... $$P(B)$$ ... ...
green/sumL ... $$P(green \mid B)$$ ... ...
Conditional probability: Let's focus on the row with the green balls
The probability to get a ball with B out of all green balls is 0.8
$$P(B \mid green) = \frac{P(green\: balls\: with \:B)}{P(all\: green\: balls)} = \frac{P(green \cap B)}{P(green)} = \frac{0.211}{0.263} = 0.8$$
Conditional probability: Let's focus on the column with the balls with letter B
The probability to get a green ball out of all balls with a B is 0.286
$$P(green \mid B) = \frac{P(green\; balls\; with\; B)}{P(all\; balls\; with\; letter\; B)} = \frac{P(green \cap B)}{P(B)} = \frac{0.211}{0.737} = 0.286$$
Bayes rule
Given $$P(green \mid B$$) find $$P(B \mid green)$$ :
$$P(B \mid green) = \frac{P(green \mid B) \cdot P(B)}{P(green)} = \frac{0.286 \cdot 0.737}{0.263} = \frac{0.211}{0.263}= 0.8$$
and given $$P(B \mid green)$$ find $$P(green \mid B$$):
$$P(green \mid B) = \frac{P(B \mid green) \cdot P(green)}{P(B)} = \frac{0.8 \cdot 0.263}{0.737} = \frac{0.211}{0.737} = 0.286$$
Applying the Bayes rule means that we walk from the element most right in the matrix to the element at the bottom of the matrix and vice versa:
show_frequencies()
def show_frequencies():
px = 4; py = 4
figsize(10, 10)
fontSize1 = 15
fontSize2 = 12
fontSize3 = 20
A1 = 4; A2 = 10; A3 = A1+A2
B1 = 1; B2 = 4; B3 = B1+B2
C1 = A1+B1; C2 = A2+B2; C3 = A3+B3
data = np.array([[A1, A2, A3],
[B1, B2, B3],
[C1, C2, C3] ])
clr = np.array([ ['#ffc2b3', '#ff704d', '#ff0000'],
['#b3ff99', '#53ff1a', '#208000'],
['#00bfff', '#0000ff', '#bf00ff'] ])
title = np.array([['A and red', 'B and red', 'sum of reds',
'$P(B|red) = \\frac {P(A\\cap red)}{P(red)}$' ],
['A and green', 'B and green', 'sum greens', '% (B of greens)' ],
['sum of A', 'sum of B', 'sum of all balls',' ' ],
[' ', '% (greens of B)', '', ' ' ],
] )
xlabel = np.array([ ['A', 'B', 'A+B', 'B/(A+B)'],
['A', 'B', 'A+B', 'B/(A+B)'],
['A red + A green', 'B red + B green', 'A+B', 'B/(A+B)'],
['A', 'B green/(B red + B green)', 'A+B', 'B/(A+B)'] ] )
ylabel = np.array([ ['red', 'red', 'red', 'B of reds'],
['green', 'green', 'green','B of greens'],
['sum A', 'sum B', 'Total', '' ],
['green', 'greens of B', 'green', ' '], ])
f, ax = plt.subplots(px, py, sharex=True, sharey=True, edgecolor='none') #, facecolor='lightgrey'
for i in range(px-1):
for j in range(py-1):
patches, texts =ax[i,j].pie([data[i,j]], labels=[str(data[i,j])], #autopct='%1.1f%%',
colors = [clr[i,j]])
texts[0].set_fontsize(fontSize3)
ax[i,j].set_title(title[i,j], position=(0.5,1.2), bbox=dict(facecolor='#f2f2f2', edgecolor='none'), fontsize= fontSize1)
if i*j==1 :
ax[i,j].set_title(title[i,j], position=(0.5,1.2), bbox=dict(facecolor='#bfbfbf', edgecolor='none'), fontsize=fontSize1)
ax[i,j].set_xlabel(xlabel[i,j], fontsize=fontSize2, color='c')
ax[i,j].xaxis.set_label_position('top')
ax[i,j].set_ylabel(ylabel[i,j], fontsize=fontSize2, color='c')
ax[i,j].axis('equal')
j += 1
if (i == 1):
p = data[i,-2]/data[i,-1]; o = p/(1-p)
patches, texts =ax[i,j].pie([o,1], #autopct='%1.1f%%',
colors = [clr[i,1], clr[i,2] ] )
texts[0].set_fontsize(fontSize3)
ax[i,j].set_title(title[i,j], position=(0.5,1.2), bbox=dict(facecolor='#bfbfbf', edgecolor='none'), fontsize=fontSize1)
ax[i,j].set_xlabel(xlabel[i,j], color='c', fontsize=fontSize2)
ax[i,j].xaxis.set_label_position('top')
ax[i,j].set_ylabel(ylabel[i,j], fontsize=fontSize2, color='c')
ax[i,j].axis('equal')
else:
ax[i,j].plot(0,0)
ax[i,j].set_frame_on(False)
i = px-1;
for j in range(py):
if j==1:
p = data[1,1]/data[2,1]; o = p/(1-p)
patches, texts =ax[i,j].pie([o,1], #autopct='%1.1f%%',
colors = [clr[1,1], clr[2,1] ])
texts[0].set_fontsize(fontSize3)
ax[i,j].set_title(title[i,j], position=(0.5,1.2), bbox=dict(facecolor='#bfbfbf', edgecolor='none'), fontsize=fontSize1)
ax[i,j].set_xlabel(xlabel[i,j], color='c', fontsize=fontSize2)
ax[i,j].xaxis.set_label_position('top')
ax[i,j].set_ylabel(ylabel[i,j], fontsize=fontSize2, color='c')
ax[i,j].axis('equal')
ax[i,j].set_facecolor('y')
else:
ax[i,j].plot(0,0)
ax[i,j].set_frame_on(False)
plt.tight_layout()
plt.show()
from IPython.display import HTML
# HTML('''<script> $('div .input').hide()''') #lässt die input-zellen verschwinden HTML('''<script> code_show=true; function code_toggle() { if (code_show){$('div.input').hide();
} else {
$('div.input').show(); } code_show = !code_show }$( document ).ready(code_toggle);
</script> | 2020-07-03T10:04:01 | {
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"url": "https://alpynepyano.github.io/healthyNumerics/posts/basic_stats_02_mechanical_approach_to_bayesian_rule.html",
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https://www.physicsforums.com/threads/permutations-hard-question.669846/ | # Permutations : hard question !
1. Feb 6, 2013
### hms.tech
1. The problem statement, all variables and given/known data
In how many ways can the three letters from the word " GREEN " be arranged in a row if atleast one of the letters is "E"
2. Relevant equations
Permutations Formula
3. The attempt at a solution
The total arrangements without restriction: 5P3/2! = $\frac{5!}{2! * 2!}$
The number of arrangements in which there is no "E" = 3!
Ans : 5P3/2! - 3! = 24 (wrong)
Here is another approach :
The arrangements with just one "E" = $\frac{2!*4!}{2!}$
The arrangements with two "E" = $\frac{3*2}{1}$
I think I making a mistake due to the repetition of "E" ... Can any one of you tell me a better way which avoids the problem I am being having .
2. Feb 6, 2013
### HallsofIvy
If "at least one letter must be E", that means that the other two letters must be chosen from "GREN". That is, the first letter must be one of those 4 letters, the second one of the three remaining letters. How many is that?
But then we can put the "E" that we took out into any of three places: before the two, between them, or after the two letters so we need 3 times that previous number.
3. Feb 6, 2013
### hms.tech
4P2 = 4*3
According to your method the answer should be 12 * 3 = 36 (The correct answer in the solutions is "27")
Clearly your method (as did mine) repeats some of the permutations :
You didn't take onto account that the two "E" are not distinct. Thus in those permutations where we chose two "E" were repeated. see :
GEE , EEG, EGE
ENE, EEN, NEE
REE, ERE , EER
4. Feb 6, 2013
### CAF123
Alternatively, split the problem into two: Count those combinations with only 1 E and then count separately those with two E's.
For one E combination: (1C1)*(3C2)*3! = 18
For two E combination: (2C2)*(3C1)*(3!/2!) = 9. Then add. | 2018-02-21T06:13:23 | {
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https://forum.math.toronto.edu/index.php?PHPSESSID=kqv56eqotgiu56ejlj64jkai53&action=printpage;topic=1204.0 | # Toronto Math Forum
## MAT334-2018F => MAT334--Lectures & Home Assignments => Topic started by: Ende Jin on September 08, 2018, 03:39:53 PM
Title: About the definition of Argument (in book)
Post by: Ende Jin on September 08, 2018, 03:39:53 PM
I found that the definition of "arg" and "Arg" in the book is different from that introduced in the lecture (exactly opposite) (on page 7).
I remember in the lecture, the "arg" is the one always lies in $(-\pi, \pi]$
Which one should I use?
Title: Re: About the definition of Argument (in book)
Post by: Victor Ivrii on September 08, 2018, 04:58:09 PM
Quote
Which one should I use?
This is a good and tricky question because the answer is nuanced:
Solving problems, use definition as in the Textbook, unless the problem under consideration requires modification: for example, if we are restricted to the right half-plane $\{z\colon \Re z >0\}$ then it is reasonable to consider $\arg z\in (-\pi/2,\pi/2)$, but if we are restricted to the upper half-plane $\{z\colon \Im z >0\}$ then it is reasonable to consider $\arg z\in (0,\pi)$ and so on.
Title: Re: About the definition of Argument (in book)
Post by: Ende Jin on September 09, 2018, 12:48:34 PM
I am still confused. Let me rephrase the question again.
In the textbook, the definition of "arg" and "Arg" are:
$arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta$
which means $arg(z) \in \mathbb{R}$
while
$Arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta \land \theta \in [-\pi, \pi)$
which means $Arg(z) \in [-\pi, \pi)$
While in the lecture, as you have introduced, it is the opposite and the range changes to $(-\pi, \pi]$ instead of $[-\pi, \pi)$ (unless I remember incorrectly):
Arg is defined to be
$Arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = (cos\theta + isin\theta)$
which means $arg(z) \in \mathbb{R}$
while arg is
$arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta \land \theta \in (-\pi, \pi]$
I am confused because if I am using the definition by the book,
when $z \in \{z : Re (z) > 0\}$
then $arg(z) \in (-\frac{\pi}{2} + 2\pi n,\frac{\pi}{2} + 2\pi n), n \in \mathbb{Z}$
Title: Re: About the definition of Argument (in book)
Post by: Victor Ivrii on September 09, 2018, 04:40:38 PM
BTW, you need to write \sin t and \cos t and so on to have them displayed properly (upright and with a space after): $\sin t$, $\cos t$ and so on
Title: Re: About the definition of Argument (in book)
Post by: Ende Jin on September 10, 2018, 10:03:33 AM
Thus in a test/quiz/exam, I should follow the convention of the textbook, right?
Title: Re: About the definition of Argument (in book)
Post by: Victor Ivrii on September 10, 2018, 01:34:22 PM
Thus in a test/quiz/exam, I should follow the convention of the textbook, right?
Indeed
Title: Re: About the definition of Argument (in book)
Post by: oighea on September 12, 2018, 04:24:46 PM
The $\arg$ of a complex number $z$ is an angle $\theta$. All angles $\theta$ have an infinite number of "equivalent" angles, namely $\theta =2k\pi$ for any integer $k$.
Equivalent angles can be characterized by that they exactly overlap when graphed on a graph paper, relative to the $0^\circ$ mark (usually the positive $x$-axis). Or more mathematically, they have the same sine and cosine. It also makes sine and cosine a non-reversible function, as given a sine or cosine, there are an infinite number of angles that satisfy this property.
$\Arg$, on the other hand, reduces the range of the possible angles such that it always lie between $0$ (inclusive) to $2\pi$ (exclusive). That is because one revolution is $2\pi$, or $360$ degrees. That is called the principal argument of a complex number.
We will later discover that complex logarithm also have a similar phenomenon.
Title: Re: About the definition of Argument (in book)
Post by: Victor Ivrii on September 12, 2018, 04:34:21 PM
oighea
Please fix your screen name and use LaTeX command (rendered by MathJax) to display math, not paltry html commands.
I fixed it in this post | 2022-12-02T23:28:35 | {
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http://mathhelpforum.com/discrete-math/123546-equation-problem-others.html | # Math Help - Equation problem and others...
1. ## Equation problem and others...
Could any1 help me a bit with this equation ?
1) X1+X2+X3=11 , So how many are the solutions?
2) We 've got these letters -> A,B,C,D,E,F,G,H , how many permutations
of these letters can we make that include ABC ?
3) How many ways are possible in order 3 girls and 8 boys sit in a row so that Mary and John NOT to sit next to each other ?
4) I can understand at all what this question wants, but anyway ill ask here maybe you can understand ...
How many bytes exist with 1) Exactly 2 aces
2) Exactly 4 aces
3) Exactly 6 aces
4) At least 6 aces
i dont get it , a byte consists of 8 bits, whats that ace? any ideas?
Thanks in advance for your help =)
2. Hi primeimplicant,
where are you getting stuck with these?
For 1)
$X_1,\ X_2, X_3$ probably need to be positive integers >0.
Are any of the numbers allowed to be zero?
If they are allowed to be negative or non-integers, that leaves infinitely
many solutions.
-14+1+2, -15+1+3 etc
Hence, start with 1, add 2 and find out what $X_3$ is.
Then start again at 1, add 3 and find out what $X_3$ is.
When finished with 1, go on to 2 and make sure you leave out all numbers
less than 2.
Then go on to 3, leaving out all numbers less than 3.
Continue until done, there are only a few.
3. Originally Posted by primeimplicant
[snip]
4) I can understand at all what this question wants, but anyway ill ask here maybe you can understand ...
How many bytes exist with 1) Exactly 2 aces
2) Exactly 4 aces
3) Exactly 6 aces
4) At least 6 aces
i dont get it , a byte consists of 8 bits, whats that ace? any ideas?
Thanks in advance for your help =)
My guess is that an "ace" is a 1 bit, although this is unusual terminology I have not seen before. If my guess is correct then 1) is asking you how many 8-bit sequences there are which consist of two 1's and six 0's.
4. Hello, primeimplicant!
$1)\;x_1+x_2+x_3\:=\:11$
So how many are the solutions?
The problem is NOT clearly stated.
I must assume that the $x$'s are positive integers.
Consider an 11-inch board marked in 1-inch intervals.
. . $\square\square\square\square\square\square
\square\square\square\square\square$
We will cut the board at two of the ten marks.
Then: . $\square\square\square\square\square |
\square\square | \square\square\square\square$
.represents: . $5+2+4$
And:- - $\square \square \square | \square \square \square \square \square \square \square | \square$ .represents: . $3 + 7 + 1$
Therefore, there are: . $_{10}C_2 \:=\:45$ solutions.
2) We 've got these letters: . $A,B,C,D,E,F,G,H$
How many permutations of these letters can we make that include $ABC$ ?
I assume you mean in that exact order: $ABC$
Duct-tape $ABC$ together.
Then we have 6 "letters" to arrange: . $\boxed{ABC}\;D\;E\;F\;G\;H$
There are: . $6! \,=\,720$ permutations.
3) How many ways are possible in order 3 girls and 8 boys sit in a row
so that Mary and John do NOT sit next to each other?
With no restrictions, the children can be seated in: $11!$ ways.
Suppose Mary and John DO sit together.
Duct-tape them together.
Then we have 10 "people" to arrange: . $\boxed{MJ}\;A\;B\;C\;D\;E\;F\;G\;H\;I$
. . They can be seated in: $10!$ ways.
But Mary and John could be taped like this: . $\boxed{JM}$
. . This makes for another $10!$ ways.
So there are: $2(10!)$ ways that Mary and John DO sit together.
Therefore, there are: . $11! - 2(10!) \:=\:39,916,800 - 2(3,628,800) \:=\:32,659,200$ ways
. . that Mary and John do NOT sit together.
5. Thanks a lot Soroban , awesome, i totally understood!!
6. Hi Primeimplicant,
For Q1...
that was fabulous work by Soroban!!
He's an artist at breaking long calculations into neat compact solutions.
Here's just a bit i wanted to add.
For your Q1, i interpreted it as....
how many different additions of 3 unequal positive non-zero integers sum to 11.
Hence, i recommended you try
1+2+(11-3)=11, which is 1+2+8=11
then 1+3+(11-4)=1+3+7,
1+4+6=11,
1+5+5...but this contains 2 fives.
Starting with 2,
2+3+6,
2+4+5,
2+5+4 is the same as 2+4+5 so move on to 3,
3+4+4 has two fours, so that's out.
3+5+3 out
3+6+2 is ok
3+7+1 counted
start with 4
4+5+2 has been found already as has 4+6+1
The list then is
1+2+8
1+3+7
1+4+6
2+3+6
2+4+5
3+6+2
If you meant that
$X_1,\ X_2, X_3$ could be variables that can have any value
from 1 to 9, since 1+1+9=11, so 9 is the largest positive integer useable,
then we have, starting with 1
1+1+9
1+2+8
1+3+7
1+4+6
1+5+5
Since any variable can be any of these values, these can be arranged in 3! ways, therefore there are 5(3!) = 30 combinations with the digit 1,
with 119 and 191 and 911 all double-counted, along with 155, 515, 551 double-counted, so we subtract 6.
30-6 =24.
starting with 2, not containing 1
2+2+7
2+3+6
2+4+5
these can also be arranged in (2)3!+3!/2! ways to give 12+3=15 additional combinations.
starting with 3, not containing 1 or 2
3+3+5
3+4+4
3+5+3 has been counted
3+6+2 has 2 in it
so there are 2 new ones which can be arranged in (2)3!/2! ways to account for
each of the 3 variables having those values.
this is another 3! combinations =6.
starting with 4, not containing 1, 2 or 3.
4+4+3 is out so we have found them all as there are no further combinations due to the fact that we will need smaller digits for the sum.
Hence the total is 24+15+6=45
This is the total for this interpretation of the question.
If the variables must have different values, then we must subtract the three with the repeated digit 1, the three with the repeated digit 2, the three with the repeated digit 3, the three with the repeated digit 4, and the three with the repeated digit 5,
getting 45-15=30 with all 3 digits different,
in other words.... combinations of the digits, where the digits are selections of 3 from the 8 available in that case, in such a way that they sum to 11.
Looking at Soroban's analysis in another way
*****|*|*****
The red lines can move 4 places left and 4 places right, giving an additional 8 solutions to the 1 shown....8+1=9 solutions containing the digit 1.
****|**|*****
The red lines can move 3 places left and 4 places right,
giving 7+1=8 solutions containing the digit 2.
****|***|****
The red lines can move 3 places left and 3 places right giving 6+1=7 solutions with the digit 3.
Continuing that we get 9+8+7+6+5+4+3+2+1=45 solutions containing repeated digits. | 2015-11-27T15:52:59 | {
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https://byjus.com/rd-sharma-solutions/class-7-maths-chapter-25-data-handling-iv-probability-ex-25-1/ | # RD Sharma Solutions Class 7 Data Handling Iv Probability Exercise 25.1
## RD Sharma Solutions Class 7 Chapter 25 Exercise 25.1
### RD Sharma Class 7 Solutions Chapter 25 Ex 25.1 PDF Free Download
#### Exercise 25.1
Q 1.A coin is tossed 1000 times with the following frequencies:
Head : 445, Tail : 555
When a coin is tossed at random, what is the probability of getting
(ii).a tail?
SOLUTION:
Total number of times a coin is tossed = 1000
Number of times a head comes up = 445
Number of times a tail comes up = 555
(i) Probability of getting a head = $\frac{No.\;of\;heads}{Total\;No.\;of\;trails}$ = $\frac{445}{1000}$=0.445
(ii) Probability of getting a tail = $\frac{No.\;of\;tails}{Total\;No.\;of\;trails}$ = $\frac{555}{1000}$ = 0.555
Q 2.A die is thrown 100 times and outcomes are noted as given below:
If a die is thrown at random, find the probability of getting a/an:
(i) 3 (ii) 5 (iii) 4 (iv) Even number (v) Odd number (vi) Number less than 3.
SOLUTION:
Total number of trials = 100
Number of times “1” comes up = 21
Number of times “2” comes up = 9
Number of times “3” comes up = 14
Number of times “4” comes up = 23
Number of times “5” comes up = 18
Number of times “6” comes up = 15
(i) Probability of getting 3 = $\frac{Frequency\;of\;3}{Total\;No.\;of\;trails}$ = $\frac{14}{100}$ = 0.14
(ii) Probability of getting 5 = $\frac{Frequency\;of\;5}{Total\;No.\;of\;trails}$ = $\frac{18}{100}$ = 0.18
(iii) Probability of getting 4 = $\frac{Frequency\;of\;4}{Total\;No.\;of\;trails}$ = $\frac{23}{100}$ = 0.23
(iv) Frequency of getting an even no. = Frequency of 2 + Frequency of 4 + Frequency of 6 = 9 + 23 + 15 = 47
Probability of getting an even no. = $\frac{Frequency\;of\;even\;number}{Total\;No.\;of\;trails}$ = $\frac{47}{100}$ = 0.47
(v) Frequency of getting an odd no. = Frequency of 1 + Frequency of 3 + Frequency of 5 = 21 + 14 + 18 = 53
Probability of getting an odd no. = $\frac{Frequency\;of\;odd\;number}{Total\;No.\;of\;trails}$ = $\frac{53}{100}$ = 0.53
(vi) Frequency of getting a no. less than 3 = Frequency of 1 + Frequency of 2 = 21 + 9 = 30
Probability of getting a no. less than 3 = $\frac{Frequency\;of\;number\;less\;than\;3}{Total\;No.\;of\;trails}$ = $\frac{30}{100}$ = 0.30
Q 3.A box contains two pair of socks of two colours (black and white). I have picked out a white sock. I pick out one more with my eyes closed. What is the probability that I will make a pair?
SOLUTION:
No. of socks in the box = 4
Let B and W denote black and white socks respectively. Then we have:
S = {B,B,W,W}
If a white sock is picked out, then the total no. of socks left in the box = 3
No. of white socks left = 2 – 1 = 1
Probability of getting a white sock = $\frac{no.\;of\;white\;socks\;left\;in\;the\;box}{total\;no.\;of\;socks\;left\;in\;the\;box}$ = $\frac{1}{3}$
Q 4.Two coins are tossed simultaneously 500 times and the outcomes are noted as given below:
If same pair of coins is tossed at random, find the probability of getting:
SOLUTION:
Number of trials = 500
Number of outcomes of two heads (HH) = 105
Number of outcomes of one head (HT or TH) = 275
Number of outcomes of no head (TT) = 120
(i) Probability of getting two heads = $\frac{Frequency\;of\;getting\;2\;heads}{Total\;No.\;of\;trails}$ = $\frac{105}{500}$ = $\frac{21}{100}$
(ii) Probability of getting one head = $\frac{Frequency\;of\;getting\;1\;heads}{Total\;No.\;of\;trails}$ = $\frac{275}{500}$ = $\frac{11}{20}$
(iii) Probability of getting no head = $\frac{Frequency\;of\;getting\;no\;heads}{Total\;No.\;of\;trails}$ = $\frac{120}{500}$ = $\frac{6}{25}$
#### Practise This Question
The lines shown below never meet at any point. So, these lines are not parallel. Say true or false. | 2019-02-21T14:33:15 | {
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http://qjsk.agenzialenarduzzi.it/fourier-transform-of-cos-wt-in-matlab.html | Sound Waves. The expression you have is an person-friendly remodel, so which you will detect the inverse in a table of Laplace transforms and their inverses. It exploits the special structure of DFT when the signal length is a power of 2, when this happens, the computation complexity is significantly reduced. This is a good point to illustrate a property of transform pairs. Let f ( x ) be a function defined and integrable on. Discrete Fourier transform (DFT) is the basis for many signal processing procedures. Fn = 1 shows the transform of damped exponent f(t) = e-at. Think about this intuitively. These advantages are particularly important in climate science. Recently, methods for solving this drawback by using a wavelet transform (WT) [25,26] have been reported. Likewise, the amplitude of sine waves of wavenumber in the superposition is the sine Fourier transform of the pulse shape, evaluated at wavenumber. Discrete Fourier Transform (DFT) The frequency content of a periodic discrete time signal with period N samples can be determined using the discrete Fourier transform (DFT). The DFT: An Owners' Manual for the Discrete Fourier Transform William L. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. One side of this was discussed in the last chapter: time domain signals can be convolved by multiplying their frequency spectra. We shall show that this is the case. Learn more about fourier transform. • Fourier invents Fourier series in 1807 • People start computing Fourier series, and develop tricks Good comes up with an algorithm in 1958 • Cooley and Tukey (re)-discover the fast Fourier transform algorithm in 1965 for N a power of a prime • Winograd combined all methods to give the most efficient FFTs. (14) and replacing X n by. yes my signal is load-time signal with T0=0. As described in details in Stewart et al. The Segment of Signal is Assumed Stationary A 3D transform ()(t f ) [x(t) (t t )]e j ftdt t ω ′ = •ω* −′•−2π STFTX , ω(t): the window function A function of time. It exploits the special structure of DFT when the signal length is a power of 2, when this happens, the computation complexity is significantly reduced. The amplitude, A, is the distance measured from the y-value of a horizontal line drawn through the middle of the graph (or the average value) to the y-value of the highest point of the sine curve, and B is the number of times the sine curve repeats itself within 2π, or 360 degrees. Fourier Transform and Inverse Fourier Transform of Lists I am trying to compute the Fourier transform of a list, say an array of the form {{t1, y[t1]},{tn, y[tn]}}; apply some filters in the spectral components, and then back transform in time domain. The Dual-Tree Complex Wavelet Transform [A coherent framework for multiscale signal and image processing] T he dual-tree complex wavelet transform (CWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher. The following MATLAB. Digital signal processing (DSP) vs. Let Y(s)=L[y(t)](s). Joseph Fourier appears in the Microwaves101 Hall of Fame! Fourier transforms of regular waveforms can be found in textbooks. Cu (Lecture 7) ELE 301: Signals and Systems Fall 2011-12 22 / 22. The N-D transform is equivalent to computing the 1-D transform along each dimension of X. The DFT of the sequence x(n) is expressed as 2 1 ( ) ( ) 0 N jk i X k x n e N i − − Ω = ∑ = (1) where = 2Π/N and k is the frequency index. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. Gómez Gil, INAOE 2017 22. 1) is called the inverse Fourier integral for f. The sine and cosine transforms are useful when the given function x(t) is known to be either even or odd. Most of the following equations should not be memorized by the reader; yet, the reader should be able to instantly derive them from an understanding of the function's characteristics. Inverted frequency spectrum, also called cepstrum, is the result of taking the inverse Fourier transform of the logarithm of a signal estimated spectrum. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Fourier Transform. Online Fast Fourier Transform Calculator. Cu (Lecture 7) ELE 301: Signals and Systems Fall 2011-12 22 / 22. To learn more, see our tips on writing great. 1 Properties of the Fourier transform Recall that. Si X es un array multidimensional, fft(X) trata los valores a lo largo de la primera dimensión del array cuyo tamaño no sea igual a 1 como vectores y devuelve la transformada de Fourier de cada vector. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas. Make it an integer number of cycles long (e. Fourier transform t=[1:4096]*10e-3/4096; % Time axis T=1024*10e-3/4096; % Period of a periodic function f0=1/T omega0=2*pi*f0; % Sampling Frequency dt=t(2)-t(1) fsample=1/dt f0 = 400 dt = 2. • cos(2 )πfcτ term is constant (τ is independent, the integral is over t). We denote Y(s) = L(y)(t) the Laplace transform Y(s) of y(t). MATLAB has a built-in sinc function. Has the form [ry,fy,ffilter,ffy] = FouFilter(y, samplingtime, centerfrequency, frequencywidth, shape, mode), where y is the time. To illustrate determining the Fourier Coefficients, let's look at a simple example. Daileda Fourier transforms. The Fourier Transform is a method to single out smaller waves in a complex wave. A Phasor Diagram can be used to represent two or more stationary sinusoidal quantities at any instant in time. $\endgroup$ – Robert Israel Jan 19 '17 at 21:33. Basic theory and application of the FFT are introduced. 1 Frequency Analysis Remember that we saw before that when a sinusoid goes into a system, it comes out as a sinusoid of the same frequency,. The component of x ( t ) at frequency w , X ( w ) , can be considered a density: if the units of x ( t ) are volts, then the units of X ( w ) are volt-sec (or volt/Hz if we had been using Hz. The Fourier transform is sometimes denoted by the operator Fand its inverse by F1, so that: f^= F[f]; f= F1[f^] (2) It should be noted that the de. at the MATLAB command prompt. In practice, when doing spectral analysis, we cannot usually wait that long. The Fast Fourier Transform (FFT) is an efficient way to do the DFT, and there are many different algorithms to accomplish the FFT. The Discrete Fourier Transformation (DFT): Definition and numerical examples — A Matlab tutorial; The Fourier Transform Tutorial Site (thefouriertransform. The Laplace transform is used to quickly find solutions for differential equations and integrals. prior to entering the outer for loop. Always keep in mind that an FFT algorithm is not a different mathematical transform: it is simply an efficient means to compute the DFT. Amyloid Hydrogen Bonding Polymorphism Evaluated by (15)N{(17)O}REAPDOR Solid-State NMR and Ultra-High Resolution Fourier Transform Ion Cyclotron Resonance Mass Spectrometry. The coe cients in the Fourier series of the analogous functions decay as 1 n, n2, respectively, as jnj!1. So for the Fourier Series for an even function, the coefficient b n has zero value: b_n= 0 So we only need to calculate a 0 and a n when finding the Fourier Series expansion for an even function f(t): a_0=1/Lint_(-L)^Lf(t)dt a_n=1/Lint_(-L)^Lf(t)cos{:(n pi t)/L:}dt An even function has only cosine terms in its Fourier expansion:. If any argument is an array, then fourier acts element-wise on all elements of the array. First the frequency grid has to be fine enough so the peaks are all resolved. If we \squeeze" a function in t, its Fourier transform \stretches out" in !. The wavelet transform and other linear time-frequency analysis methods decompose these signals into their components by correlating the signal with a dictionary of time-frequency atoms. Fast Fourier Transform in MATLAB ®. Cal Poly Pomona ECE 307 Fourier Transform The Fourier transform (FT) is the extension of the Fourier series to nonperiodic signals. In modo analogo possiamo ricavare la DFT inversa come N −1 1 X 2π x[n] = X[k]ejkn N (58) N k=0. Skip to content. The input, x, is a real- or complex-valued vector, or a single-variable regularly sampled timetable, and must have at least four samples. txt) or read book online for free. These systems are based on neural networks [1] with wavelet transform based feature extraction. Note that in the summation over n = 0, 1, … N-1, the value of the basis function is computed ("sampled") at the same times 'n' as your recorded signal x[n] was sampled. I then wish to calculate the imaginary and real parts of the fourier transform. These coefficients can be calculated by applying the following equations: f(t)dt T a tT t v o o = 1!+ f(t)ktdt T a tT t n o o o =!+cos" 2 f(t)ktdt T b tT t n o o o =!+sin" 2 Answer Questions 1 – 2. The Trigonometric Fourier Series is an example of Generalized Fourier Series with sines and cosines substituted in as the orthogonal basis set. Posts about Fast Fourier Transform of 16-point sequence written by kishorechurchil. In MATLAB: sinc(x)= sin(πx) πx Thus, in MATLAB we write the transform, X, using sinc(4f), since the π factor is built in to the function. Topics include: The Fourier transform as a tool for solving physical problems. How can we use Laplace transforms to solve ode? The procedure is best illustrated with an example. Since it is time shifted by 1 to the. 下载 数字信号处理科学家与工程师手册 （非常实用，含有大量C实用代码）. FFT Software. 4Hz in the spectrum corresponding to. Moreover, as cosine and sine transform are real operations (while Fourier transform is complex), they can be more efficiently implemented and are widely used in various applications. Sine and cosine waves can make other functions! Here two different sine waves add together to make a new wave: Try "sin(x)+sin(2x)" at the function grapher. Orthogonal Properties of sinusoids and cosinusoids. Fourier Series is very useful for circuit analysis, electronics, signal processing etc. Let W f ( u , s ) denote the wavelet transform of a signal, f(t) , at translation u and scale s. MATLAB - using Signal Processing Toolbox features to analyze PicoScope data Introduction. What you have given isn't a Fourier remodel; it particularly is a Laplace remodel with jw=s. Thus, the DFT formula basically states that the k'th frequency component is the sum of the element-by-element products of 'x' and ' ', which is the so-called inner product of the two vectors and , i. To establish these results, let us begin to look at the details first of Fourier series, and then of Fourier transforms. Fourier transform infrared (FT-IR) spectroscopy, principal component analysis (PCA), two-dimensional correlation spectroscopy (2D-COS), and X-ray diffraction, while the sorption properties were evaluated by water vapor isotherms using the gravimetric method coupled with infrared spectroscopy. Before delving into the mechanics of the Fourier transform as implemented on a computer, it is important to see the origin of the technique and how it is constructed. La transformada de Fourier es básicamente el espectro de frecuencias de una función. Taylor Series A Taylor Series is an expansion of some function into an infinite sum of terms , where each term has a larger exponent like x, x 2 , x 3 , etc. The Fast Fourier Transform (FFT) is a fascinating algorithm that is used for predicting the future values of data. (i) We must calculate the Fourier coefficients. The Fourier transform is a mathematical formula that relates a signal sampled in time or space to the same signal sampled in frequency. Monitoring Light Induced Structural Changes of Channelrhodopsin-2 by UV/Vis and Fourier Transform Infrared Spectroscopy* Eglof Ritter‡1, Katja Stehfest§1, Andre Berndt §, Peter Hegemann§¶, Franz J. Bartl‡¶ From the ‡Institut für medizinische Physik und Biophysik, Charité-Universitätsmedizin Berlin,. 2; % gravitational acceleraton (ft/s^2) beta = 180; % relative wave direction (deg) z = 0; % depth below the surface (ft) rho = 1. When we do this, we would end up with the Fourier transform of y(t). The finite, or discrete, Fourier transform of a complex vector y with n ele-ments y j+1;j = 0;:::n •1 is another complex. The Fourier transform is defined for a vector x with n uniformly sampled points by. The resulting transform pairs are shown below to a common horizontal scale: Cu (Lecture 7) ELE 301: Signals and Systems Fall 2011-12 8 / 37. Visit http://ilectureonline. 4 Ms Sampling Period. This process is equal to apply 1D WT on Radon slices (Chen, 2007). Electronics and Circuit Analysis Using MATLAB - John O. How to complete the fourier Analysis using Learn more about fourier, fft, fourier transform, plotting, digital signal processing, signal processing, transform MATLAB. Since real phase noise data cannot be collected across a continuous spectrum, a summation must be performed in place of the integral. The period is taken to be 2 Pi, symmetric around the origin, so the. The expression you have is an person-friendly remodel, so which you will detect the inverse in a table of Laplace transforms and their inverses. Taylor Series A Taylor Series is an expansion of some function into an infinite sum of terms , where each term has a larger exponent like x, x 2 , x 3 , etc. Topics include: The Fourier transform as a tool for solving physical problems. This property, together with the fast Fourier transform, forms the basis for a fast convolution algorithm. Home / ADSP / MATLAB PROGRAMS / MATLAB Videos / Discrete Fourier Transform in MATLAB Discrete Fourier Transform in MATLAB 18:48 ADSP , MATLAB PROGRAMS , MATLAB Videos. pattern of signs, the summation being taken over all odd positive integers n that are not multiples of 3. Goldberg, Kenneth A. The discrete Fourier transform or DFT is the transform that deals with a nite discrete-time signal and a nite or discrete number of frequencies. That is, all the energy of a sinusoidal function of frequency A is entirely localized at the frequencies given by |f|=A. 1 an RC circuit 2. 分数阶傅里叶变换（fractional fourier transform） matlab代码. 下载 数字信号处理科学家与工程师手册 （非常实用，含有大量C实用代码）. We denote Y(s) = L(y)(t) the Laplace transform Y(s) of y(t). It is unusual in treating Laplace transforms at a relatively simple level with many examples. We'll save FFT for another day. The CWT is obtained using the analytic Morse wavelet with the symmetry parameter (gamma) equal to 3 and the time-bandwidth product equal to 60. Integration in the time domain is transformed to division by s in the s-domain. The convergence criteria of the Fourier transform (namely, that the function be absolutely integrable on the real line) are quite severe due to the lack of the exponential decay term as seen in the Laplace transform, and it means that functions like polynomials, exponentials, and trigonometric functions all do not have Fourier transforms in the. Title: 315_DSP_DWT Author: Christos Faloutsos school2 Created Date: 11/4/2019 9:06:39 AM. Fourier Transform Example #2 MATLAB Code % ***** MATLAB Code Starts Here ***** % %FOURIER_TRANSFORM_02_MAT % fig_size = [232 84 774 624]; m2ft = 3. The Fourier Transform As we have seen, any (sufficiently smooth) function f(t) that is periodic can be built out of sin's and cos's. Especially important are the solutions to the Fourier transform of the wave equation, which define Fourier series, spherical harmonics, and their generalizations. Since it is u(t-1), the cos(wt) function will be zero till 1. Fourier analysis is used in electronics, acoustics, and communications. Hilbert transform, short-time Fourier transform (more about this later), Wigner distributions, the Radon Transform, and of course our featured transformation, the wavelet transform, constitute only a small portion of a huge list of transforms that are available at engineer's and mathematician's disposal. Using MATLAB, determine the Fourier transform numerically and plot the result. edu is a platform for academics to share research papers. This is the simple code for FFT transform of Cos wave using Matlab. Fit Fourier Models Interactively. a finite sequence of data). com for more math and science lectures! In this video I will find the Fourier transform F(w)=? given the input function f(t)=cos(w0t. Turn in your code and plot. Inverted frequency spectrum, also called cepstrum, is the result of taking the inverse Fourier transform of the logarithm of a signal estimated spectrum. 1, 2017 ROTOR FAULT DETECTION OF WIND TURBINE SQUIRREL CAGE INDUCTION GENERATOR USING TEAGER–KAISER ENERGY OPERATOR Lahc`ene Noured. Sound Waves. En 1822, Fourier expose les séries et la transformation de Fourier dans son traité Théorie analytique de la chaleur. All I can do is give you a hint. yes my signal is load-time signal with T0=0. This is a good point to illustrate a property of transform pairs. We will now derive the complex Fourier series equa-tions, as shown above, from the sin/cos Fourier series using the expressions for sin() and cos() in terms of complex exponentials. no hint Solution. A truncated Fourier series, where the amplitude and frequency do not vary with time, is a special case of these signals. The custom Matlab/Octave function FouFilter. - [Voiceover] Many videos ago, we first looked at the idea of representing a periodic function as a set of weighted cosines and sines, as a sum, as the infinite sum of weighted cosines and sines, and then we did some work in order to get some basics in terms of some of these integrals which we then started to use to derive formulas for the various coefficients, and we are almost there. % Input: % X - 1xM - complex vector - data points (signal discretisation). ylim: the y limits of the plot. One hardly ever uses Fourier sine and cosine transforms. Edward Donley Mathematics Department Indiana University of Pennsylvania Basics of Sound. DFT needs N2 multiplications. Expression (1. Visit Stack Exchange. We need to compute the Fourier transform of the product of two functions: $f(t)=f_1(t)f_2(t)$ The Fourier transform is the convolution of the Fourier transform of the two functions: $F(\omega)=F_1(\omega)*F_2(\omega)=$ [math. Fourier Transform Solution 0. We have fb(w)= 1 √ 2π Z1 −1 xe−ixw dx = 1 √ 2π Z1 −1 x coswx−isinwx dx = −i √ 2π Z1 −1 x sinwxdx = −2i √ 2π Z1 0 x sinwxdx = −2i √ 2π 1 w2 sinwx− x w coswx 1 0 = −i r 2 π sinw − wcosw w2. That is G k=g j exp− 2πikj N ⎛ ⎝⎜ ⎞ ⎠⎟ j=0 N−1 ∑ (7-6) Scaling by the sample interval normalizes spectra to the continuous Fourier transform. Karris - Free ebook download as PDF File (. Symmetry in Exponential Fourier Series. orthogonal functions fourier series. you can ask your doubt in the comment box. If we de ne s 1(t) = s(t t 0); then S 1(f) = Z 1 1 s(t t 0)e j2ˇftdt; Z 1 1 s(u)e j2ˇf(u+t 0)du; = e j2ˇft 0 Z 1 1 s(u)e j2ˇfudu; = e j2ˇft 0S(f): There is a similar dual relationshp if a signal is scaled by an exponential in the time domain. 下载 常用傅里叶变换对. The Fourier transform. The detection process is mainly based on the wavelet transform. Digital signal processing (DSP) vs. Fourier transform that f max is f 0 plus the bandwidth of rect(t - ½). If any argument is an array, then fourier acts element-wise on all elements of the array. We describe this FFT in the current section. The trigonometric Fourier series of the even signal cos(t) + cos(2. Thus, the DFT formula basically states that the k'th frequency component is the sum of the element-by-element products of 'x' and ' ', which is the so-called inner product of the two vectors and , i. Fourier analysis is a method of defining periodic waveform s in terms of trigonometric function s. Inverted frequency spectrum converts the periodic signal and sidebands in FFT results to spectral lines, thus making it easier to detect the complex periodic component of the spectrum. Now multiply it by a complex exponential at the same frequency. The 3-D FrFT can independently compress and image radar data in each dimension for a broad set of parameters. Therefore, we will start with the continuous Fourier transform,. Fourier transform (FT), time–frequency analysis such as the STFT and Cohen-class quadratic distributions, and time-scale analysis based on the wavelet trans-form (WT) are often applied to investigate the hidden properties of real signals. In practice, when doing spectral analysis, we cannot usually wait that long. 5t) = αn cos(2π t) n=1 T0 ∞ n = αn cos( t) 2 n=1 By equating the coefficients of cos( n t) 2 of both sides we observe that an = 0 for all n unless n = 2, 5 in which case a2 = a5 = 1. Unfortunately, the FT can represent the meaningful spectrum property of only lin-. 3 Fourier Series Using Euler’s rule, can be written as X n 10 1 0 11 00 00 11 ( )cos( ) ( )sin( ) tT tT n tt X xt n tdt j xt n tdt TT ωω ++ =−∫∫ If x(t) is a real-valued periodic signal, we have. 999; % water density (lbm/ft^3). My function is intended for just plain Fourier series expansion (a_k cos(k*x)). Matlab经典教程——从入门到精通 - 第一章 基础准备及入门 本章有两个目的：一是讲述 MATLAB 正常运行所必须具备的基础条件；二是简明系统 地介绍高度集成的 Desktop 操作桌面的. Fast Transforms in Audio DSP; Related Transforms. Using the Fourier transform formula directly to compute each of the n elements of y requires on the order of n 2 floating-point operations. Table 4: Basic Continuous-Time Fourier Transform Pairs Fourier series coefficients Signal Fourier transform (if periodic) +∞ k=−∞ ake jkω0t 2π k=−∞ akδ(ω −kω0) ak ejω0t 2πδ(ω −ω0). Using MATLAB, determine the Fourier transform numerically and plot the result. The usual computation of the discrete Fourier transform is done using the Fast Fouier Transform (FFT). I tried using the following definition Xn = 1/T integral ( f(t) e^-jwnt ) I converted the cos(wt) to its exponential form, then multiplied and combined and. So you either need to install Matlab to your own laptop or connect to cloud version of MatLab or use a Lamar lab which has MatLab, most engineering and CS labs do, so does GB 113. $\endgroup$ – Robert Israel Jan 19 '17 at 21:33. The coefficient in the Fourier cosine series expansion of is by default given by. The Cosine Function. where A k (t) is the slowly varying amplitude and ϕ k (t) is the instantaneous phase. Fast Fourier Transform(FFT) • The Fast Fourier Transform does not refer to a new or different type of Fourier transform. For this case, when performing the Fourier transform in equation 31, the coefficients c lm and s lm with m > L are simply discarded. If there is any solution at all, it would have to involve a transformation of variables. This computational efficiency is a big advantage when processing data that has millions of data points. 2808; % conversion from meters to feet g = 32. cosh() sinh() 22 tttt tt +---== eeee 3. It is often easier to calculate than the sin/cos Fourier series because integrals with exponentials in are usu-ally easy to evaluate. The Fourier transform is defined for a vector x with n uniformly sampled points by. 下载 数字信号处理科学家与工程师手册 （非常实用，含有大量C实用代码）. you can ask your doubt in the comment box. A Phasor Diagram can be used to represent two or more stationary sinusoidal quantities at any instant in time. The can be accomplished in the Fourier domain just as a multiplication. The 3-D Fractional Fourier Transformation (FrFT) has unique applicability to multi-pass and multiple receiverSynthetic Aperture Radar (SAR) scenarios which can collect radar returns to create volumetric reflectivity data. To establish these results, let us begin to look at the details first of Fourier series, and then of Fourier transforms. Sketch by hand the magnitude of the Fourier transform of c(t) for a general value of f c. I have a time series where I want to do a real Fourier transform Since the data has missing values, I cannot use a FFT which requires equidistant data. It exploits the special structure of DFT when the signal length is a power of 2, when this happens, the computation complexity is significantly reduced. The Segment of Signal is Assumed Stationary A 3D transform ()(t f ) [x(t) (t t )]e j ftdt t ω ′ = •ω* −′•−2π STFTX , ω(t): the window function A function of time. For a general real function, the Fourier transform will have both real and imaginary parts. • Fourier invents Fourier series in 1807 • People start computing Fourier series, and develop tricks Good comes up with an algorithm in 1958 • Cooley and Tukey (re)-discover the fast Fourier transform algorithm in 1965 for N a power of a prime • Winograd combined all methods to give the most efficient FFTs. Tech ECE 5th semester can be seen by clicking here. MATLAB uses notation derived from matrix theory where the subscripts run from 1 to n, so we will use y j+1 for mathemat-ical quantities that will also occur in MATLAB code. Control and Intelligent Systems, Vol. Skip to content. Il énonce qu'une fonction peut être décomposée sous forme de série trigonométrique, et qu'il est facile de prouver la convergence de celle-ci. The Fourier transform is essential in mathematics, engineering, and the physical sciences. information in the Matlab manual for more specific usage of commands. Vectors, Phasors and Phasor Diagrams ONLY apply to sinusoidal AC alternating quantities. Pure tone — sine or cosine function frequency determines pitch (440 Hz is an A note) amplitude determines volume. The usual computation of the discrete Fourier transform is done using the Fast Fouier Transform (FFT). I don't know what is not working in my code, but I got an output image with the same number. wavelet transform) offer a huge variety of applications. The SST approach in [8, 7] is based on the continuous wavelet transform (CWT). Explain the effect of zero padding a signal with zero before taking the discrete Fourier Transform. Fn = 1 shows the transform of damped exponent f(t) = e-at. These coefficients can be calculated by applying the following equations: f(t)dt T a tT t v o o = 1!+ f(t)ktdt T a tT t n o o o =!+cos" 2 f(t)ktdt T b tT t n o o o =!+sin" 2 Answer Questions 1 – 2. Table of Discrete-Time Fourier Transform Pairs: Discrete-Time Fourier Transform : X() = X1 n=1 x[n]e j n Inverse Discrete-Time Fourier Transform : x[n] =. Digital signal processing (DSP) vs. We denote Y(s) = L(y)(t) the Laplace transform Y(s) of y(t). This is because the limits of the integral are from -∞ to +∞. The Fourier Transform: Examples, Properties, Common Pairs Example: Fourier Transform of a Cosine Spatial Domain Frequency Domain cos (2 st ) 1 2 (u s)+ 1 2 (u + s) 0. Ø Many software packages are capable of computing the Fourier transform of input signal (e. DFT needs N2 multiplications. coz they are just using fourier. It is represented in either the trigonometric form or the exponential form. I have a time series where I want to do a real Fourier transform Since the data has missing values, I cannot use a FFT which requires equidistant data. Fourier Transform of aperiodic and periodic signals - C. Example: cos(pi/4*(0:159))+randn(1,160) specifies a sinusoid embedded in white Gaussian noise. It may be possible, however, to consider the function to be periodic with an infinite period. The filter portion will look something like this b = fir1(n,w,'type'); freqz(b,1,512); in = filter(b,1,in);. I suggest that you generate a cosine with a frequency of about 1/16th sample per cycle. In signal processing, the Fourier transform can reveal important characteristics of a signal, namely, its frequency components. The phase noise plot of a PRS10 shows a 42 dB reduction in phase noise at 10 Hz offset from the carrier at 10 MHz when compared to a conventional rubidium standard. For this application, we find that. Outline Introduction to the Fourier Transform and Frequency Domain Magnitude of frequencies Phase of frequencies Fourier transform and DFT Filtering in the frequency domain Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation of Fourier transform Background 1807, French math. I didnot get any feedback from fft2. The Fourier transform of the product of two signals is the convolution of the two signals, which is noted by an asterix (*), and defined as: This is a bit complicated, so let's try this out. Note that this is similar to the definition of the FFT given in Matlab. The approach to characterize the image texture are investigated based on WT other than DOST. Laplace transform allows us to convert a differential equation to an algebraic equation. Re There are two important implications of Im(j) the Fourier Transform as defined in this equation here is applicable only to aperiodic signals. Briggs , Van Emden Henson Just as a prism separates white light into its component bands of colored light, so the discrete Fourier transform (DFT) is used to separate a signal into its constituent frequencies. Integration in the time domain is transformed to division by s in the s-domain. Analog signal processing (ASP) The theory of Fourier transforms is applicable irrespective of whether the signal is continuous or discrete, as long as it is "nice" and absolutely integrable. For more information about the Fourier series, refer to Fourier Analysis and Filtering (MATLAB). Windowed Fourier Transform: Represents non periodic signals. For this case, when performing the Fourier transform in equation 31, the coefficients c lm and s lm with m > L are simply discarded. So, generally, we use this property of linearity of Laplace transform to find the Inverse Laplace transform. Home / ADSP / MATLAB PROGRAMS / MATLAB Videos / Discrete Fourier Transform in MATLAB Discrete Fourier Transform in MATLAB 18:48 ADSP , MATLAB PROGRAMS , MATLAB Videos. The N-D transform is equivalent to computing the 1-D transform along each dimension of X. If f(t) is non-zero with a compact support, then its Fourier transform cannot be zero on a whole interval. Matlab时频分析工具箱及函数应用说明. The Laplace Transform. All I can do is give you a hint. The forward transform converts a signal from the time domain into the frequency domain, thereby analyzing the frequency components, while an inverse discrete Fourier transform, IDFT, converts the frequency components back into the time domain. Using MATLAB to Plot the Fourier Transform of a Time Function The aperiodic pulse shown below: has a Fourier transform: X(jf)=4sinc(4πf) This can be found using the Table of Fourier Transforms. The following MATLAB. Re There are two important implications of Im(j) the Fourier Transform as defined in this equation here is applicable only to aperiodic signals. Conclusion As we have seen, the Fourier transform and its 'relatives', the discrete sine and cosine transform provide handy tools to decompose a signal into a bunch of partial waves. The can be accomplished in the Fourier domain just as a multiplication. f and f^ are in general com-plex functions (see Sect. Matlab uses the FFT to find the frequency components of a. Equation [2] states that the fourier transform of the cosine function of frequency A is an impulse at f=A and f=-A. d/dt[-cos(mt)/m] =? sin(mt) Since -1/m is constant with respect to t, bring it out front. This is soo confusing u know. Note: The FFT-based convolution method is most often used for large inputs. The sum of signals (disrupted signal) As we created our signal from the sum of two sine waves, then according to the Fourier theorem we should receive its frequency image concentrated around two frequencies f 1 and f 2 and also its opposites -f 1 and -f 2. The Segment of Signal is Assumed Stationary A 3D transform ()(t f ) [x(t) (t t )]e j ftdt t ω ′ = •ω* −′•−2π STFTX , ω(t): the window function A function of time. The - dimensional Fourier cosine coefficient is given by. Tech ECE 5th semester can be seen by clicking here. Matlab时频分析工具箱及函数应用说明. 4414e-06 fsample = 409600. However, computationally efficient algorithms can require as little as n log2(n) operations. For the input sequence x and its transformed version X (the discrete-time Fourier transform at equally spaced frequencies around the unit circle), the two functions implement the relationships. And, of course, everybody sees that e to the inx, by Euler's great formula, is a combination of cosine nx and sine nx. For example, the Fourier transform allows us to convert a signal represented as a function of time to a function of frequency. e jwt Cos wt jSen wt Time Fourier transform, DTFT) Matlab y uso de función òfft código aquí) (c) P. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 1 shows how increasing the period does indeed lead to a continuum of coefficients, and. an = 1 1 R1 1 (1 x 2) cos nπx dx 1 1cos nπx dx 1 1x 2cos nπx dx = 0 1 nπx 2sin (nπx) j 1 1 + 2 nπ R 1 1 xsin nπx dx = 0 0 2 n2π2xcos (nπx) j 1 1 + 2 n2π2 R 1 1 cos nπx dx = 4 n2π2 cos (nπ. I just had a look at what the Curve Fitting app is doing at its "Fourier" option includes the fundamental frequency as one of the fit parameters. Thus, the DFT formula basically states that the k'th frequency component is the sum of the element-by-element products of 'x' and ' ', which is the so-called inner product of the two vectors and , i. You can write a book review and share your experiences. Dilation and rotation are real-valued scalars and position is a 2-D vector with real-valued elements. (b) We can directly observe from the plot that given function is a multiplication of sin(2πt) and the sign (signum) function sgn(t) which has the Fourier transform F(sgn)(s)= 1/(πis). Fast Fourier Transform in MATLAB ®. Fn = 5 and 6 shows the function reconstructed from its spectrum. Full text of "The Fourier Transform And Its Applications Bracewell" See other formats. (i) We must calculate the Fourier coefficients. Lab 5 Fourier Series I. f and f^ are in general com-plex functions (see Sect. Then, use the duality function to show that 1 t 2 j 2sgn j sgn j sgn. The expression you have is an person-friendly remodel, so which you will detect the inverse in a table of Laplace transforms and their inverses. Discrete Time Fourier Transform (DTFT) Fourier Transform (FT) and Inverse. 7) Synthèse de Fourier L’opération inverse de la décomposition de Fourier peut également être faite, et s’appelle la synthèse de Fourier. However, the definition of the MATLAB sinc. 19 Two-dimensional (2D) Fourier. Matlab时频分析工具箱及函数应用说明. Vectors, Phasors and Phasor Diagrams ONLY apply to sinusoidal AC alternating quantities. For particular functions we use tables of the Laplace. We expressed ChR2 in COS-cells, purified it, and subsequently investigated this unusual photoreceptor by flash photolysis and UV-visible and Fourier transform infrared difference spectroscopy. As an example, if we are given the phasor V = 5/36o, the expression for v(t) in the time domain is v(t) = 5 cos(ωt + 36o), since we are. Abstract In this paper, a MATLAB model of a Digital Fourier Transform (DFT)-based digital distance relay was developed and then its behavior was analyzed when it is applied on distance protection of a real series compensated transmission system, belonging to the Chilean generation and transmission utility Colbún S. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. 5 BACKGROUND In the following sections, we discuss the effect of amplitude modulation and sampling on the signal spectrum using common properties of the Fourier transform. To learn how to use the fft function type >> help fft at the Matlab command line. I tried a number of different representations of tanh() but none of them had an analytical solution for the fourier cosine transform. Visit Stack Exchange. txt) or view presentation slides online. The Laplace Equation; The Wave Equation; The Heat Equation; Bibliography. There are various implementations of it, but a standard form is the Radix-2 FFT. The Fourier transform is essential in mathematics, engineering, and the physical sciences. Ive tried to write matlab code that takes in a grayscale image matrix, performs fft2() on the matrix and then calculates the magnitude and phase from the transform. The following options can be given:. The Cosine Function. FFT(X) is the discrete Fourier transform of vector X. Fourier Transform of Array Inputs. Fourier Transform Z. I tried using the following definition Xn = 1/T integral ( f(t) e^-jwnt ) I converted the cos(wt) to its exponential form, then multiplied and combined and. THE DISCRETE COSINE TRANSFORM (DCT) 3. the fourier transform and its applications. Basic theory and application of the FFT are introduced. MATLAB provides command for working with transforms, such as the Laplace and Fourier transforms. A sinusoidal function can be expressed in either Fourier transform (Fourier series) or phasor representation: (31) where (32) We see that the phasor and the Fourier coefficients and are essentially the same, in the sense that they are both coefficients representing the amplitude and phase of the complex exponential function. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Хотя формула, задающая преобразование Фурье, имеет ясный смысл только для функций класса (), преобразование Фурье может быть определено и для более широкого класса функций и даже обобщённых функций. where a 0 models a constant (intercept) term in the data and is associated with the i = 0 cosine term, w is the fundamental frequency of the signal, n is the number of terms (harmonics) in the series, and 1 ≤ n ≤ 8. Discrete Fourier Transform (DFT) The frequency content of a periodic discrete time signal with period N samples can be determined using the discrete Fourier transform (DFT). The main idea is to extend these functions to the interval and then use the Fourier series definition. Let's overample: f s = 100 Hz. Fourier analysis is used in electronics, acoustics, and communications. Use integration by parts to evaluate the. To address this issue, short-time Fourier transform (STFT) is proposed to exhibit the time-varying information of the analyzed signal. Let Y(s)=L[y(t)](s). transform a signal in the time or space domain into a signal in the frequency domain. The Fourier transform of a Gaussian is a Gaussian and the inverse Fourier transform of a Gaussian is a Gaussian f(x) = e −βx2 ⇔ F(ω) = 1 √ 4πβ e ω 2 4β (30) 4. This is the simple code for FFT transform of Cos wave using Matlab. The Laplace Transform Example: Using Frequency Shift Find the L[e-atcos(wt)] In this case, f(t) = cos(wt) so, The Laplace Transform Time Integration: The property is: The Laplace Transform Time Integration: Making these substitutions and carrying out The integration shows that. The custom Matlab/Octave function FouFilter. Here is a simple implementation of the Discrete Fourier Transform: myFourierTransform. The four Fourier transforms that comprise this analysis are the Fourier Series, Continuous-Time Fourier Transform, DiscreteTime Fourier Transform and Discrete Fourier Transform. Cal Poly Pomona ECE 307 Fourier Transform The Fourier transform (FT) is the extension of the Fourier series to nonperiodic signals. Given the F. In the form FourierCosCoefficient [expr, t, n], n can be symbolic or a non - negative integer. 18 Applying the Fourier transform to the image of a group of people. To illustrate determining the Fourier Coefficients, let's look at a simple example. at the MATLAB command prompt. The finite, or discrete, Fourier transform of a complex vector y with n ele-ments y j+1;j = 0;:::n •1 is another complex. Signals having finite energy are energy signals. Fourier analysis transforms a signal from the. We can use MATLAB to plot this transform. orthogonal functions fourier series. The Fourier transform is sometimes denoted by the operator Fand its inverse by F1, so that: f^= F[f]; f= F1[f^] (2) It should be noted that the de. When the energy is finite, the total power will be zero. What you have given isn't a Fourier remodel; it particularly is a Laplace remodel with jw=s. 2001-01-01. I have a time series where I want to do a real Fourier transform Since the data has missing values, I cannot use a FFT which requires equidistant data. Expression (1. you can ask your doubt in the comment box. The period is taken to be 2 Pi, symmetric around the origin, so the. com for more math and science lectures! In this video I will find the Fourier transform F(w)=? given the input function f(t)=cos(w0t. This function is a cosine function that is windowed - that is, it is multiplied by the box or rect function. 1 The upper plot shows the magnitude of the Fourier series spectrum for the case of T=1 with the Fourier transform of p(t) shown as a dashed line. The approach to characterize the image texture are investigated based on WT other than DOST. Can you please send proper solution of this question. With more than 5,000 lines of MATLAB code and more than 700 figures embedded in the text, the material teaches readers how to program in MATLAB and study signals and systems concepts at the same time, giving them the tools to harness the power of computers to quickly assess problems and then visualize their solutions. Labview, MATLAB and EXCEL, many others) 21 Discrete Sampling Sampling Period To accurately reproduce spectrum for periodic waveforms, the measurement period must be an integer multiple of the fundamental period, T 1 : mT 1 =n δ t DFT Specta Waveform. ω=2π, we expand f (t) as a Fourier series by ( ) ( ) + + + = + + + b t b t f t a a t a t ω ω ω ω sin sin 2 ( ) cos cos 2 1 1 0 1 2 (Eqn 1) 2. SFORT TIME FOURIER TRANSFORM (STFT) Dennis Gabor (1946) Used STFT ♥To analyze only a small section of the signal at a time -- a technique called Windowing the Signal. The Discrete Cosine Transform (DCT) Number Theoretic Transform. In this case F(ω) ≡ C[f(x)] is called the Fourier cosine transform of f(x) and f(x) ≡ C−1[F(ω)] is called the inverse Fourier cosine transform of F(ω). Sampled sound (digital audio) — discrete sequence of intensities CD Audio is 44100 samples per second. Fourier Transform of aperiodic and periodic signals - C. How can we use Laplace transforms to solve ode? The procedure is best illustrated with an example. Full text of "The Fourier Transform And Its Applications Bracewell" See other formats. 2-D Continuous Wavelet Transform. Fast Fourier Transform. First sketch the function. Here is a link to a video in YouTube that provides a nice illustration: Slinky. Find the Fourier cosine series and the Fourier sine series for the function f(x) = ˆ 1 if 0 iFFTUnderstanding FFTs and Windowing Overview Learn about the time and frequency domain, fast Fourier transforms (FFTs), and windowing as well. (b) We can directly observe from the plot that given function is a multiplication of sin(2πt) and the sign (signum) function sgn(t) which has the Fourier transform F(sgn)(s)= 1/(πis). Langton Page 3 And the coefficients C n are given by 0 /2 /2 1 T jn t n T C x t e dt T (1. The Fourier transform of a signal exist if satisfies the following condition. DOEpatents. cos(wt)의 힐베르트 변환은 sin(wt)이다. com) Fourier Series Applet (Tip: drag magnitude or phase dots up or down to change the wave form). The Laplace Transform Theorem: Initial Value If the function f(t) and its first derivative are Laplace transformable and f(t) Has the Laplace transform F(s), and the exists, then lim sF(s) 0 lim ( ) lim ( ) (0) o f o s t sF s f t f The utility of this theorem lies in not having to take the inverse of F(s). The integrals from the last lines in equation [2] are easily evaluated using the results of the previous page. a0 = 1 2 R 1 1 (1 x 2) dx 1 2 x 1 3x 3 j 1 1 2 3. [2] Inverse DFT is defined as: N -1 1 x ( n) = N å X ( k )e k =0 j 2pnk / N for 0 £ n £ N - 1. Normal images such as straw, wood, sand and grass are used in the analysis. 分数阶傅里叶变换（fractional fourier transform） matlab代码. Dct vs dft Dct vs dft. Discrete Fourier Transform See section 14. Explain the effect of zero padding a signal with zero before taking the discrete Fourier Transform. Fourier transform infrared (FT-IR) spectroscopy, principal component analysis (PCA), two-dimensional correlation spectroscopy (2D-COS), and X-ray diffraction, while the sorption properties were evaluated by water vapor isotherms using the gravimetric method coupled with infrared spectroscopy. Fourier Transform of Periodic Signals 0 0 /2 /2 0 1 o o jt n n T jn t n T x t c e c x t e dt T 2 ( ) jn t jn t oo nn nn no n. It is one commonly encountered form for the Fourier series of real periodic signals in continuous time. From Wikibooks, the open-content textbooks collection < Engineering Tables Jump to: navigation, search. Introduction to Fourier Transforms Fourier transform as a limit of the Fourier series The Fourier transform of a sine or cosine at a frequency f 0 only has energy exactly at f 0, which is what we would expect. Matlab经典教程——从入门到精通 - 第一章 基础准备及入门 本章有两个目的：一是讲述 MATLAB 正常运行所必须具备的基础条件；二是简明系统 地介绍高度集成的 Desktop 操作桌面的. for fourier transform we need to define w. The Fourier transform is defined for a vector x with n uniformly sampled points by. Find the transfer function of the following RC circuit. Fourier Transform of the Gaussian Konstantinos G. It is often easier to calculate than the sin/cos Fourier series because integrals with exponentials in are usu-ally easy to evaluate. [2] Inverse DFT is defined as: N -1 1 x ( n) = N å X ( k )e k =0 j 2pnk / N for 0 £ n £ N - 1. Ø Many software packages are capable of computing the Fourier transform of input signal (e. information in the Matlab manual for more specific usage of commands. I tried a number of different representations of tanh() but none of them had an analytical solution for the fourier cosine transform. Finally, I am supposed to create a filter using the basic MATLAB commands and filter the noise out of the plot of the signal and then do the Fourier Transform of the signal again and plot the results. Basic theory and application of the FFT are introduced. Van Loan and K. The knowledge of Fourier Series is essential to understand some very useful concepts in Electrical Engineering. function [ft] = myFourierTransform (X, n) % Objective: % Apply the Discrete Fourier Transform on X. The Laplace Equation; The Wave Equation; The Heat Equation; Bibliography. Recall the definition of hyperbolic functions. 1946: Gabor 开发了短时傅立叶变换(Short time Fourier transform, STFT) STFT的时间-频率关系图 小波分析发展历程 (1900 - 1979) where，s(t) is a signal, and g(t) is the windowing function. Grundlagen und Begriffsabgrenzungen, Rechtecksignal, Dreieckfunktion und Fourier-Transformation - Holger Schmid - Ausarbeitung - Ingenieurwissenschaften - Wirtschaftsingenieurwesen - Arbeiten publizieren: Bachelorarbeit, Masterarbeit, Hausarbeit oder Dissertation. Existence of the Fourier Transform; The Continuous-Time Impulse. Pure tone — sine or cosine function frequency determines pitch (440 Hz is an A note) amplitude determines volume. Using the Fourier transform formula directly to compute each of the n elements of y requires on the order of n 2 floating-point operations. Un buen ejemplo de eso es lo que hace el oído humano, ya que recibe una onda auditiva y la transforma en una descomposición en distintas frecuencias (que es lo que finalmente se escucha). That is, all the energy of a sinusoidal function of frequency A is entirely localized at the frequencies given by |f|=A. The DFT of the sequence x(n) is expressed as 2 1 ( ) ( ) 0 N jk i X k x n e N i − − Ω = ∑ = (1) where = 2Π/N and k is the frequency index. 5 Signals & Linear Systems Lecture 10 Slide 11 Fourier Transform of any periodic signal ∑Fourier series of a periodic signal x(t) with period T 0 is given by: Take Fourier transform of both sides, we get: This is rather obvious!. (i) We must calculate the Fourier coefficients. The trigonometric Fourier series of the even signal cos(t) + cos(2. If the analyzing wavelet is analytic, you obtain W f ( u , s ) = 1 2 W f a ( u , s ) , where f a (t) is the analytic signal corresponding to f(t). 2 p693 PYKC 8-Feb-11 E2. Hey everyone, i know that matlab have the method for fourier transform implemented but i was wondering if there is anything that could give me coefficients of fourier transfrom. The Fourier series is a sum of sine and cosine functions that describes a periodic signal. Note that this is similar to the definition of the FFT given in Matlab. 3 251 with a + - - + + - - + +. In the paragraphs that follow we illustrate this approach using Maple, Mathematica, and MATLAB. It uses real DFT, that is, the version of Discrete Fourier Transform which uses real numbers to represent the input and output signals. Today I want to start getting "discrete" by introducing the discrete-time Fourier transform (DTFT). To learn more, see our tips on writing great. Engineering Tables/Fourier Transform Table 2 From Wikibooks, the open-content textbooks collection < Engineering Tables Jump to: navigation, search Signal Fourier transform unitary, angular frequency Fourier transform unitary, ordinary frequency Remarks 10 The rectangular pulse and the normalized sinc function 11 Dual of rule 10. The Laplace Equation; The Wave Equation; The Heat Equation; Bibliography. Example: cos(pi/4*(0:159))+randn(1,160) specifies a sinusoid embedded in white Gaussian noise. Esta función de MATLAB calcula la transformada discreta de Fourier (DFT) de X usando un algoritmo de transformada rápida de Fourier (FFT). Provide Plots For The Time Domain Signal And The Magnitude Of Its FT. Bartl‡¶ From the ‡Institut für medizinische Physik und Biophysik, Charité-Universitätsmedizin Berlin,. Derivation in the time domain is transformed to multiplication by s in the s-domain. This is my attempt in hoping for a way to find it without using the definition: x(t) = c. 5 Signals & Linear Systems Lecture 10 Slide 11 Fourier Transform of any periodic signal ∑Fourier series of a periodic signal x(t) with period T 0 is given by: Take Fourier transform of both sides, we get: This is rather obvious!. selection, high computation speed, and extensive application. The DFT: An Owners' Manual for the Discrete Fourier Transform William L. The top equation de nes the Fourier transform (FT) of the function f, the bottom equation de nes the inverse Fourier transform of f^. FFT Software. This is the simple code for FFT transform of Cos wave using Matlab. Fourier-transform and global contrast interferometer alignment methods. com) Fourier Series Applet (Tip: drag magnitude or phase dots up or down to change the wave form). Repeat the example in Section II. 小波变换（dwt）源代码. If f(t) is non-zero with a compact support, then its Fourier transform cannot be zero on a whole interval. We need to compute the Fourier transform of the product of two functions: $f(t)=f_1(t)f_2(t)$ The Fourier transform is the convolution of the Fourier transform of the two functions: $F(\omega)=F_1(\omega)*F_2(\omega)=$ [math. To illustrate determining the Fourier Coefficients, let's look at a simple example. Vectors, Phasors and Phasor Diagrams ONLY apply to sinusoidal AC alternating quantities. Fourier Series. The Laplace Transform. Fourier Transform Example #2 MATLAB Code % ***** MATLAB Code Starts Here ***** % %FOURIER_TRANSFORM_02_MAT % fig_size = [232 84 774 624]; m2ft = 3. Explain the effect of zero padding a signal with zero before taking the discrete Fourier Transform. On the second plot, a blue spike is a real (cosine) weight and a green spike is an imaginary (sine) weight. The 3-D Fractional Fourier Transformation (FrFT) has unique applicability to multi-pass and multiple receiverSynthetic Aperture Radar (SAR) scenarios which can collect radar returns to create volumetric reflectivity data. Note that this is similar to the definition of the FFT given in Matlab. The can be accomplished in the Fourier domain just as a multiplication. There are various implementations of it, but a standard form is the Radix-2 FFT. Fourier Transform of Periodic Signals 0 0 /2 /2 0 1 o o jt n n T jn t n T x t c e c x t e dt T 2 ( ) jn t jn t oo nn nn no n. Generally speaking, the more concentrated g(t) is, the more spread out its Fourier transform G^(!) must be. Using MATLAB to Plot the Fourier Transform of a Time Function The aperiodic pulse shown below: has a Fourier transform: X(jf)=4sinc(4πf) This can be found using the Table of Fourier Transforms. The Fourier transform is sometimes denoted by the operator Fand its inverse by F1, so that: f^= F[f]; f= F1[f^] (2) It should be noted that the de. FFT onlyneeds Nlog 2 (N). This signal will have a Fourier. For the input sequence x and its transformed version X (the discrete-time Fourier transform at equally spaced frequencies around the unit circle), the two functions implement the relationships. The Fourier series is a sum of sine and cosine functions that describes a periodic signal. csdn已为您找到关于小波变换与图像、图形处理技术相关内容，包含小波变换与图像、图形处理技术相关文档代码介绍、相关教学视频课程，以及相关小波变换与图像、图形处理技术问答内容。. Fast Fourier Transform. Other readers will always be interested in your opinion of the books you've read. Write a second version that first sets up a transform matrix (with rows corresponding to the various values of k) and then multiplies this matrix by the input to perform the transform. y = Sin (300t) fits the form y = sin(ωt + ϴ) ω = frequency in radians/second. Fourier Transform. Computational Efficiency. The forward transform converts a signal from the time domain into the frequency domain, thereby analyzing the frequency components, while an inverse discrete Fourier transform, IDFT, converts the frequency components back into the time domain. efine the Fourier transform of a step function or a constant signal unit step what is the Fourier transform of f (t)= 0 t< 0 1 t ≥ 0? the Laplace transform is 1 /s, but the imaginary axis is not in the ROC, and therefore the Fourier transform is not 1 /jω in fact, the integral ∞ −∞ f (t) e − jωt dt = ∞ 0 e − jωt dt = ∞ 0 cos. , make it 8 cycles long). The DTFT is often used to analyze samples of a continuous function. Example: cos(pi/4*(0:159))+randn(1,160) specifies a sinusoid embedded in white Gaussian noise. Over a time range of 0 400< 0. 5 BACKGROUND In the following sections, we discuss the effect of amplitude modulation and sampling on the signal spectrum using common properties of the Fourier transform. Here's the 100th column of X_rows: plot(abs(X_rows(:, 100))) ylim([0 2]) As I said above, the Fourier transform of a constant sequence is an impulse. In signal processing, the Fourier transform can reveal important characteristics of a signal, namely, its frequency components. Amyloid Hydrogen Bonding Polymorphism Evaluated by (15)N{(17)O}REAPDOR Solid-State NMR and Ultra-High Resolution Fourier Transform Ion Cyclotron Resonance Mass Spectrometry. The Fast Fourier Transform (FFT) is an efficient way to do the DFT, and there are many different algorithms to accomplish the FFT. Most of the following equations should not be memorized by the reader; yet, the reader should be able to instantly derive them from an understanding of the function's characteristics. These systems are based on neural networks [1] with wavelet transform based feature extraction. The Fourier series is a sum of sine and cosine functions that describes a periodic signal. • Hence, even if the Heisenberg constraints are verified, it is. Question: MATLAB Problem: Fourier Transform (FT) Of A Cosine Signal This Problem Analyzes A Transmitted Sinusoidal Signal In The Frequency Domain. Combines traditional methods such as discrete Fourier transforms and discrete cosine transforms with more recent techniques such as filter banks and wavelet Strikes an even balance in emphasis between the mathematics and the applications with the emphasis on linear algebra as a unifying theme. For functions of two variables that are periodic in both variables, the. 数字信号处理科学家与工程师. Always keep in mind that an FFT algorithm is not a different mathematical transform: it is simply an efficient means to compute the DFT. filtering the spectrum and regenerating the signal using the filtered spectrum is done at the end Rayleigh theorem is proved by showing that the energy content of both time domain and frequency domain signals are equal. The Fourier Transform Introduction In the Communication Labs you will be given the opportunity to apply the theory learned in Communication Systems. where a 0 models a constant (intercept) term in the data and is associated with the i = 0 cosine term, w is the fundamental frequency of the signal, n is the number of terms (harmonics) in the series, and 1 ≤ n ≤ 8. The idea is that one can. The Fast Fourier Transform (FFT) is an efficient way to do the DFT, and there are many different algorithms to accomplish the FFT. Karris - Free ebook download as PDF File (. We describe this FFT in the current section. Dilation and rotation are real-valued scalars and position is a 2-D vector with real-valued elements. Como nos comenta el mismo Sergey en su sitio Web, la tarea básica en el procesamiento de electro-cardiogramas (ECG) es la detección de los picos R. A truncated Fourier series, where the amplitude and frequency do not vary with time, is a special case of these signals. ϴ = angle in degrees. Then, use the duality function to show that 1 t 2 j 2sgn j sgn j sgn. is equal to the frequency integral of the square of its Fourier Transform. For particular functions we use tables of the Laplace. That is G k=g j exp− 2πikj N ⎛ ⎝⎜ ⎞ ⎠⎟ j=0 N−1 ∑ (7-6) Scaling by the sample interval normalizes spectra to the continuous Fourier transform. , Bracewell) use our −H as their definition of the forward transform. If the input to the above RC circuit is x(t) cos(2Sf 0, find the output t) y(t). Next, on defining τ = t− s and writing c τ = t (y t − y¯)(y t−τ − y¯)/T, we can reduce the latter expression to (16) I(ω j)=2 T−1 τ=1−T cos(ω jτ)c τ, which is a Fourier transform of the sequence of empirical autocovariances. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. 19 Two-dimensional (2D) Fourier. THE DISCRETE COSINE TRANSFORM (DCT) 3. 34 matlab programs here! Please click here to see all the matlab programs present in this blog. Time Complexity • Definition • DFT • Cooley-Tukey’s FFT 6. FFT Discrete Fourier transform. Compare with the previous result. orthogonal functions fourier series. Grundlagen und Begriffsabgrenzungen, Rechtecksignal, Dreieckfunktion und Fourier-Transformation - Holger Schmid - Ausarbeitung - Ingenieurwissenschaften - Wirtschaftsingenieurwesen - Arbeiten publizieren: Bachelorarbeit, Masterarbeit, Hausarbeit oder Dissertation. It may be possible, however, to consider the function to be periodic with an infinite period. 2) Here 0 is the fundamental frequency of the signal and n the index of the harmonic such. Note that this is similar to the definition of the FFT given in Matlab. I tried using the following definition Xn = 1/T integral ( f(t) e^-jwnt ) I converted the cos(wt) to its exponential form, then multiplied and combined and. The resulting transform pairs are shown below to a common horizontal scale: Cu (Lecture 7) ELE 301: Signals and Systems Fall 2011-12 8 / 37. | 2020-10-23T03:28:55 | {
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http://mathhelpforum.com/advanced-algebra/183398-point-best-fit.html | # Thread: Point of best fit?
1. ## Point of best fit?
So we all know that you can apply the "Line of best fit" to a set of points such that the square differences between the estimated values and the real values is minimised but how do you go about finding the "point of best fit" and what does that point describe?
For example I have 3 lines:
$3x + 4y = 12$
$3x + 6y = 9$
$x + y = 2$
I want to find the point that intersects all 3 lines. This point clearly does not exist. So instead I want to find the point that is 'closest' to being a triple intersection. I imagine this could be called a 'point of best fit'. How would I go about finding this point?
2. ## Re: Point of best fit?
That's a very interesting problem. I have a couple of questions for you:
1. Does the "point of best fit" have to be on at least one of the lines? On at least two of the lines? Or could it be anywhere?
2. How are you defining "closest"? Since you only have two variables, x and y, in your equations, are you in just the xy plane? If so, are you measuring distance using Euclidean distance?
3. ## Re: Point of best fit?
I see a triangle fromed by the 3 lines and the closet point to the intersection being the middle of the triangle.
4. ## Re: Point of best fit?
Originally Posted by Ackbeet
That's a very interesting problem. I have a couple of questions for you:
1. Does the "point of best fit" have to be on at least one of the lines? On at least two of the lines? Or could it be anywhere?
2. How are you defining "closest"? Since you only have two variables, x and y, in your equations, are you in just the xy plane? If so, are you measuring distance using Euclidean distance?
1. The point can be anywhere at all.
2. Yes, this is only in 2-dimensions with variables x and y. Defining "closest" is the main problem I have with this. What would the point have to satisfy for it to be considered the "closest" to a triple intersection?
I have found a few set of points that satisfy different things:
1. $(1, 17/12)$ is the point such that it has the minimum squared distances from the point to the y coordinate of the lines (ie, its the point on the line of best fit that has the smallest error).
2. $(1, 1)$ is the point such that the summed distance (perpendicular distance) from each line to the point is minimised.
3. $(53/38, 43/38)$ is the point such that the summed squared distance (perpendicular distance) from each line to the point is minimised.
4. $(1, 11/6)$ is the centre of the triangle formed by the 3 lines.
5. ## Re: Point of best fit?
Originally Posted by pickslides
I see a triangle fromed by the 3 lines and the closet point to the intersection being the middle of the triangle.
Which of the myriad definitions of "middle" or "center" of a triangle do you mean? Incenter? Circumcenter? Orthocenter? Centroid? See below: I'm not sure you can pick one center over another until we know the context of the problem.
Originally Posted by Corpsecreate
1. The point can be anywhere at all.
2. Yes, this is only in 2-dimensions with variables x and y. Defining "closest" is the main problem I have with this. What would the point have to satisfy for it to be considered the "closest" to a triple intersection?
I have found a few set of points that satisfy different things:
1. $(1, 17/12)$ is the point such that it has the minimum squared distances from the point to the y coordinate of the lines (ie, its the point on the line of best fit that has the smallest error).
2. $(1, 1)$ is the point such that the summed distance (perpendicular distance) from each line to the point is minimised.
3. $(53/38, 43/38)$ is the point such that the summed squared distance (perpendicular distance) from each line to the point is minimised.
4. $(1, 11/6)$ is the centre of the triangle formed by the 3 lines.
It looks like you've found a number of solutions to your problem. I haven't checked your numbers, but I think that before you choose one solution over any of the others, you need to examine the context of your problem (which wouldn't be a bad idea to post here, actually). Is this problem the same problem, exactly, that you were given? Or have you performed a number of operations on the given problem to reduce it down to what you posted?
6. ## Re: Point of best fit?
I created the problem from a real world situation. The answer to the problem is purely out of interest (it was something that could have been useful to me earlier but not anymore) so it is of no 'real importance'. The problem is as follows:
Suppose you have 2 paints, paint x and paint y. All paints are composed of white paint with added tints of 3 different colours. Call these 3 colours A, B and C. Adding tints does not affect the volume of paint.
if paint x has the composition - A:3 B:3 C:1
and paint y has the composition - A:4 B:6 C:1
Then how much of paint x and y should you add together to yield as close as possible to a composition of A:12 B:9 C:2?
This situation required you to solve the above linear equations however as we know, there is no solution so we are left with finding the 'closest' solution. I am quite capable of finding the 'best point' given that I know what I'm actually looking for. Now that you know the context of the problem, hopefully I could hear your opinion on what the properties of the 'best point' may be.
I'm leaning towards the shortest combined distances from the point to the lines simply because when there IS a solution to the linear equations, this distance is 0. In fact, thinking about it now, the point (1, 17/12) is certainly not what we're looking for since it must be on the line of best fit and that line wont necessarily pass through the triple intersection if there is one. All the other points are still possibilities though. | 2017-11-21T12:47:54 | {
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https://sieuthitranh.net/4ylq3/archive.php?id=49f9bd-equivalence-class-in-relation | X ] Theorem 3.6: Let F be any partition of the set S. Define a relation on S by x R y iff there is a set in F which contains both x and y. Non-equivalence may be written "a ≁ b" or " In general, if ∼ is an equivalence relation on a set X and x∈ X, the equivalence class of xconsists of all the elements of X which are equivalent to x. a Prove that the relation $$\sim$$ in Example 6.3.4 is indeed an equivalence relation. is the intersection of the equivalence relations on a { (c) $$[\{1,5\}] = \big\{ \{1\}, \{1,2\}, \{1,4\}, \{1,5\}, \{1,2,4\}, \{1,2,5\}, \{1,4,5\}, \{1,2,4,5\} \big\}$$. } ∣ The equivalence cl… In both cases, the cells of the partition of X are the equivalence classes of X by ~. From this we see that $$\{[0], [1], [2], [3]\}$$ is a partition of $$\mathbb{Z}$$. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. Thus, is an equivalence relation. Exercise $$\PageIndex{2}\label{ex:equivrel-02}$$. hands-on exercise $$\PageIndex{2}\label{he:samedec2}$$. b) find the equivalence classes for $$\sim$$. Therefore, \begin{aligned} R &=& \{ (1,1), (3,3), (2,2), (2,4), (2,5), (2,6), (4,2), (4,4), (4,5), (4,6), \\ & & \quad (5,2), (5,4), (5,5), (5,6), (6,2), (6,4), (6,5), (6,6) \}. . {\displaystyle x\sim y\iff f(x)=f(y)} For any $$i, j$$, either $$A_i=A_j$$ or $$A_i \cap A_j = \emptyset$$ by Lemma 6.3.2. , ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈. A relation that is all three of reflexive, symmetric, and transitive, is called an equivalence relation. If $$R$$ is an equivalence relation on $$A$$, then $$a R b \rightarrow [a]=[b]$$. ) . This occurs, e.g. So, $$A \subseteq A_1 \cup A_2 \cup A_3 \cup ...$$ by definition of subset. { Define the relation $$\sim$$ on $$\mathscr{P}(S)$$ by \[X\sim Y \,\Leftrightarrow\, X\cap T = Y\cap T, Show that $$\sim$$ is an equivalence relation. In the example above, [a]=[b]=[e]=[f]={a,b,e,f}, while [c]=[d]={c,d} and [g]=[h]={g,h}. X {\displaystyle \pi (x)=[x]} Then if ~ was an equivalence relation for ‘of the same age’, one equivalence class would be the set of all 2-year-olds, and another the set of all 5-year-olds. Example $$\PageIndex{3}\label{eg:sameLN}$$. X → c We have demonstrated both conditions for a collection of sets to be a partition and we can conclude , $$\exists x (x \in [a] \wedge x \in [b])$$ by definition of empty set & intersection. $$[S_4] = \{S_4,S_5,S_6\}$$ ] The equivalence relation is usually denoted by the symbol ~. Equivalence Class Testing, which is also known as Equivalence Class Partitioning (ECP) and Equivalence Partitioning, is an important software testing technique used by the team of testers for grouping and partitioning of the test input data, which is then used for the purpose of testing the software product into a number of different classes. If R (also denoted by ∼) is an equivalence relation on set A, then Every element a ∈ A is a member of the equivalence class [a]. {\displaystyle {a\mathop {R} b}} Let X be a set. that contain , the equivalence relation generated by [ ) a) $$m\sim n \,\Leftrightarrow\, |m-3|=|n-3|$$, b) $$m\sim n \,\Leftrightarrow\, m+n\mbox{ is even }$$. ) Transitive Two sets will be related by $$\sim$$ if they have the same number of elements. We saw this happen in the preview activities. , (Since { And so, $$A_1 \cup A_2 \cup A_3 \cup ...=A,$$ by the definition of equality of sets. It is, however, a, The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. This relation turns out to be an equivalence relation, with each component forming an equivalence class. , aRa ∀ a∈A. Any relation ⊆ × which exhibits the properties of reflexivity, symmetry and transitivity is called an equivalence relation on . All elements of X equivalent to each other are also elements of the same equivalence class. c We can refer to this set as "the equivalence class of $1$" - or if you prefer, "the equivalence class of $4$". a b Now we have that the equivalence relation is the one that comes from exercise 16. the equivalence classes of R form a partition of the set S. More interesting is the fact that the converse of this statement is true. . Having every equivalence class covered by at least one test case is essential for an adequate test suite. $$\exists i (x \in A_i \wedge y \in A_i)$$ and $$\exists j (y \in A_j \wedge z \in A_j)$$ by the definition of a relation induced by a partition. x See also invariant. The equivalence classes are $\{0,4\},\{1,3\},\{2\}$. a [ [x]R={y∈A∣xRy}. Let $$x \in A.$$ Since the union of the sets in the partition $$P=A,$$ $$x$$ must belong to at least one set in $$P.$$ ∣ That is, for all a, b and c in X: X together with the relation ~ is called a setoid. = Then: No equivalence class is empty. a } ) The advantages of regarding an equivalence relation as a special case of a groupoid include: The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. An equivalence class is defined as a subset of the form {x in X:xRa}, where a is an element of X and the notation "xRy" is used to mean that there is an equivalence relation between x and y. This equivalence relation is referred to as the equivalence relation induced by $$\cal P$$. Every number is equal to itself: for all … Each part below gives a partition of $$A=\{a,b,c,d,e,f,g\}$$. {\displaystyle [a]=\{x\in X\mid x\sim a\}} For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number. Every equivalence relation induces a partitioning of the set P into what are called equivalence classes. , X Equivalence class definition is - a set for which an equivalence relation holds between every pair of elements. {\displaystyle a} {\displaystyle A} Since $$a R b$$, we also have $$b R a,$$ by symmetry. In this case $$[a] \cap [b]= \emptyset$$ or $$[a]=[b]$$ is true. An equivalence class is a subset whose elements are related to each other by an equivalence relation.The equivalence classes of a set under some relation form a partition of that set (i.e. Let us consider that R is a relation on the set of ordered pairs that are positive integers such that … {\displaystyle \{\{a\},\{b,c\}\}} [ Let X be a finite set with n elements. Since $$xRb, x \in[b],$$ by definition of equivalence classes. Let The power of the concept of equivalence class is that operations can be defined on the equivalence classes using representatives from each equivalence class. Given an equivalence relation $$R$$ on set $$A$$, if $$a,b \in A$$ then either $$[a] \cap [b]= \emptyset$$ or $$[a]=[b]$$, Let $$R$$ be an equivalence relation on set $$A$$ with $$a,b \in A.$$ , { Transcript. Equivalence relations are a ready source of examples or counterexamples. := An equivalence class is a complete set of equivalent elements. were given an equivalence relation and were asked to find the equivalence class of the or compare one to with respect to this equivalents relation. . is an equivalence relation, the intersection is nontrivial.). "Is equal to" on the set of numbers. , The equivalence classes of ~—also called the orbits of the action of H on G—are the right cosets of H in G. Interchanging a and b yields the left cosets. A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive. Hence, $\mathbb{Z} = [0] \cup [1] \cup [2] \cup [3].$ These four sets are pairwise disjoint. {\displaystyle X\times X} Each equivalence class consists of all the individuals with the same last name in the community. It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of X. In this case $$[a] \cap [b]= \emptyset$$ or $$[a]=[b]$$ is true. Symmetric "Has the same absolute value" on the set of real numbers. ) Thus, $$\big \{[S_0], [S_2], [S_4] , [S_7] \big \}$$ is a partition of set $$S$$. ,[1] is defined as $$xRa$$ and $$xRb$$ by definition of equivalence classes. ~ is finer than ≈ if every equivalence class of ~ is a subset of an equivalence class of ≈, and thus every equivalence class of ≈ is a union of equivalence classes of ~. ( In the previous example, the suits are the equivalence classes. $$\therefore R$$ is transitive. For any x ∈ ℤ, x has the same parity as itself, so (x,x) ∈ R. 2. ⟺ to see this you should first check your relation is indeed an equivalence relation. Case 1: $$[a] \cap [b]= \emptyset$$ Define the relation $$\sim$$ on $$\mathbb{Q}$$ by $x\sim y \,\Leftrightarrow\, 2(x-y)\in\mathbb{Z}.$ $$\sim$$ is an equivalence relation. Find the equivalence relation (as a set of ordered pairs) on $$A$$ induced by each partition. ) \end{array}\], $\mathbb{Z} = [0] \cup [1] \cup [2] \cup [3].$, $a\sim b \,\Leftrightarrow\, \mbox{a and b have the same last name}.$, $x\sim y \,\Leftrightarrow\, x-y\in\mathbb{Z}.$, $\mathbb{R}^+ = \bigcup_{x\in(0,1]} [x],$, $R_3 = \{ (m,n) \mid m,n\in\mathbb{Z}^* \mbox{ and } mn > 0\}.$, $\displaylines{ S = \{ (1,1), (1,4), (2,2), (2,5), (2,6), (3,3), \hskip1in \cr (4,1), (4,4), (5,2), (5,5), (5,6), (6,2), (6,5), (6,6) \}. Exercise $$\PageIndex{4}\label{ex:equivrel-04}$$. , : The equivalence kernel of an injection is the identity relation. x A strict partial order is irreflexive, transitive, and asymmetric. { in the character theory of finite groups. π , Define a relation $$\sim$$ on $$\mathbb{Z}$$ by \[a\sim b \,\Leftrightarrow\, a \mbox{ mod } 3 = b \mbox{ mod } 3.$ Find the equivalence classes of $$\sim$$. Now we have $$x R a\mbox{ and } aRb,$$ Read this as “the equivalence class of a consists of the set of all x in X such that a and x are related by ~ to each other”.. Consider the following relation on $$\{a,b,c,d,e\}$$: $\displaylines{ R = \{(a,a),(a,c),(a,e),(b,b),(b,d),(c,a),(c,c),(c,e), \cr (d,b),(d,d),(e,a),(e,c),(e,e)\}. , Define $$\sim$$ on $$\mathbb{R}^+$$ according to \[x\sim y \,\Leftrightarrow\, x-y\in\mathbb{Z}.$ Hence, two positive real numbers are related if and only if they have the same decimal parts. Notice an equivalence class is a set, so a collection of equivalence classes is a collection of sets. {\displaystyle [a]} a) True or false: $$\{1,2,4\}\sim\{1,4,5\}$$? Over $$\mathbb{Z}^*$$, define $R_3 = \{ (m,n) \mid m,n\in\mathbb{Z}^* \mbox{ and } mn > 0\}.$ It is not difficult to verify that $$R_3$$ is an equivalence relation. $$[S_2] = \{S_1,S_2,S_3\}$$ Much of mathematics is grounded in the study of equivalences, and order relations. We define a rational number to be an equivalence classes of elements of S, under the equivalence relation (a,b) ’ (c,d) ⇐⇒ ad = bc. which maps elements of X into their respective equivalence classes by ~. / ∀a ∈ A,a ∈ [a] Two elements a,b ∈ A are equivalent if and only if they belong to the same equivalence class. Practice: Congruence relation. $$[S_7] = \{S_7\}$$. ∼ The relation "≥" between real numbers is reflexive and transitive, but not symmetric. Example $$\PageIndex{6}\label{eg:equivrelat-06}$$. The equivalence classes are the sets \begin{array}{lclcr} {[0]} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 0 \} &=& 4\mathbb{Z}, \\ {[1]} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 1 \} &=& 1+4\mathbb{Z}, \\ {[2]} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 2 \} &=& 2+4\mathbb{Z}, \\ {[3]} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 3 \} &=& 3+4\mathbb{Z}. , \end{aligned}, $X\sim Y \,\Leftrightarrow\, X\cap T = Y\cap T,$, $x\sim y \,\Leftrightarrow\, 2(x-y)\in\mathbb{Z}.$, $x\sim y \,\Leftrightarrow\, \frac{x-y}{2}\in\mathbb{Z}.$, $\displaylines{ R = \{(a,a),(a,c),(a,e),(b,b),(b,d),(c,a),(c,c),(c,e), \cr (d,b),(d,d),(e,a),(e,c),(e,e)\}. We have shown if $$x \in[a] \mbox{ then } x \in [b]$$, thus $$[a] \subseteq [b],$$ by definition of subset. × In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are the equivalence classes of A under ~. For other uses, see, Well-definedness under an equivalence relation, Equivalence class, quotient set, partition, Fundamental theorem of equivalence relations, Equivalence relations and mathematical logic, Rosen (2008), pp. x Have questions or comments? A binary relation ~ on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. ". Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows. Thus, if we know one element in the group, we essentially know all its “relatives.”. The following relations are all equivalence relations: If ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~. Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). A relation on a set $$A$$ is an equivalence relation if it is reflexive, symmetric, and transitive. b Exercise $$\PageIndex{8}\label{ex:equivrel-08}$$. If (x,y) ∈ R, x and y have the same parity, so (y,x) ∈ R. 3. Related thinking can be found in Rosen (2008: chpt. In a sense, if you know one member within an equivalence class, you also know all the other elements in the equivalence class because they are all related according to $$R$$. [9], Given any binary relation Here's a typical equivalence class for : A little thought shows that all the equivalence classes look like like one: All real numbers with the same "decimal part". Then the equivalence class of a denoted by [a] or {} is defined as the set of all those points of A which are related to a under the relation … If $$A$$ is a set with partition $$P=\{A_1,A_2,A_3,...\}$$ and $$R$$ is a relation induced by partition $$P,$$ then $$R$$ is an equivalence relation. Define $$\sim$$ on a set of individuals in a community according to \[a\sim b \,\Leftrightarrow\, \mbox{a and b have the same last name}.$ We can easily show that $$\sim$$ is an equivalence relation. Both $$x$$ and $$z$$ belong to the same set, so $$xRz$$ by the definition of a relation induced by a partition. X Find the ordered pairs for the relation $$R$$, induced by the partition. , After this find all the elements related to $0$. So, if $$a,b \in A$$ then either $$[a] \cap [b]= \emptyset$$ or $$[a]=[b].$$. For example 1. if A is the set of people, and R is the "is a relative of" relation, then A/Ris the set of families 2. if A is the set of hash tables, and R is the "has the same entries as" relation, then A/Ris the set of functions with a finite d… if $$R$$ is an equivalence relation on any non-empty set $$A$$, then the distinct set of equivalence classes of $$R$$ forms a partition of $$A$$. Now WMST $$\{A_1, A_2,A_3, ...\}$$ is pairwise disjoint. Two elements related by an equivalence relation are called equivalent under the equivalence relation. ] 10). Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples: Properties definable in first-order logic that an equivalence relation may or may not possess include: Euclid's The Elements includes the following "Common Notion 1": Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). Examples. / "Has the same birthday as" on the set of all people. Let $$x \in [b], \mbox{ then }xRb$$ by definition of equivalence class. ∈ Next we will show $$[b] \subseteq [a].$$ Thus $$x \in [x]$$. {\displaystyle A\subset X\times X} The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids. (a) $$\mathcal{P}_1 = \big\{\{a,b\},\{c,d\},\{e,f\},\{g\}\big\}$$, (b) $$\mathcal{P}_2 = \big\{\{a,c,e,g\},\{b,d,f\}\big\}$$, (c) $$\mathcal{P}_3 = \big\{\{a,b,d,e,f\},\{c,g\}\big\}$$, (d) $$\mathcal{P}_4 = \big\{\{a,b,c,d,e,f,g\}\big\}$$, Exercise $$\PageIndex{11}\label{ex:equivrel-11}$$, Write out the relation, $$R$$ induced by the partition below on the set $$A=\{1,2,3,4,5,6\}.$$, $$R=\{(1,2), (2,1), (1,4), (4,1), (2,4),(4,2),(1,1),(2,2),(4,4),(5,5),(3,6),(6,3),(3,3),(6,6)\}$$, Exercise $$\PageIndex{12}\label{ex:equivrel-12}$$. An equivalence relation is a relation that is reflexive, symmetric, and transitive. The converse is also true: given a partition on set $$A$$, the relation "induced by the partition" is an equivalence relation (Theorem 6.3.4). An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. × For example, 7 ≥ 5 does not imply that 5 ≥ 7. Each individual equivalence class consists of elements which are all equivalent to each other. That is why one equivalence class is $\{1,4\}$ - because $1$ is equivalent to $4$. So, $$\{A_1, A_2,A_3, ...\}$$ is mutually disjoint by definition of mutually disjoint. x If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient space for the details. Two integers will be related by $$\sim$$ if they have the same remainder after dividing by 4. Define three equivalence relations on the set of students in your discrete mathematics class different from the relations discussed in the text. $$[x]=A_i,$$ for some $$i$$ since $$[x]$$ is an equivalence class of $$R$$. } x ∈ The equivalence class of an element $$a$$ is denoted by $$\left[ a \right].$$ Thus, by definition, The canonical example of an equivalence relation individuals within a class the given set are to. May be written a ≢ b { \displaystyle a\not \equiv b } '' are three familiar properties reflexivity! Structure of order relations class \ ( \cal P\ ) class ) Advanced relation … equivalence differs... Muturally exclusive equivalence classes is a set of ordered pairs ( S\.... An adequate test suite equality too obvious to warrant explicit mention an important property of equivalence relations (... 1 $is equivalent to another given object prove Theorem 6.3.3, look at example 6.3.2 for example, elements! 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Of numbers ≠ ϕ ) all equivalence relations other element in set \ ( [ ( a A_1. Relation, we also acknowledge previous National Science Foundation support under grant 1246120! In mathematics, an equivalence relation R is symmetric, i.e., aRb ⟹ bRa Transcript are of. Φ ) classes of an equivalence relation relation is equal to '' is the identity relation...,! The cells of the lattice theory captures the mathematical structure of order relations and join are elements of X to... { Z } \ ] this is an equivalence class of X the! Content is licensed by CC BY-NC-SA 3.0 already seen that and are relations. ( since X × X { \displaystyle a, b\in X } sameLN } \ ) for any \ T=\! ] \ ) thus \ ( \sim\ ) in example 6.3.4 is indeed an equivalence.! '' is meant a binary relation, this article was adapted from an original article by V.N 4 are by... Relation by studying its ordered pairs ( A\ ) is pairwise disjoint are equivalent ( under that ). 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A ready source of examples or counterexamples with each component forming an equivalence relation by studying its ordered for! Contact us at info @ libretexts.org or check out our status page at https: //status.libretexts.org, y_1-x_1^2=y_2-x_2^2\.. Let a, b ∈ X { \displaystyle a\not \equiv b } '' and are equivalence.... Represented by any element in set \ ( \PageIndex { 10 } \label { ex: }., y_1-x_1^2=y_2-x_2^2\ ) on equivalence relation is referred to as the equivalence is the of! They belong to the same remainder after dividing by 4 are related to each other also \... That every integer belongs to exactly one of these equivalence relations over, equivalence is! ≥ 5 does not imply that 5 ≥ 7 the universe or underlying:... Other, if and only if they have the same birthday as '' on the set P what... That \ ( X ) ∈ R. 2 b\ ) to denote a relation is. Elements are related to $4$ natural bijection between the set of equivalent.. Reflexive, symmetric, and Keyi Smith all belong to the same as! Individual equivalence class of X such that: 1 created by equivalence class in relation each... Illustration of Theorem 6.3.3, look at example 6.3.2 out our status page at https:.. 6 } \label { ex: equivrelat-01 } \ ) all a, called representative... Substitute for one another, but not individuals within a class the lattice theory operations and. To $4$ aRb\ ) by the definition of equality of sets, i.e., aRb ⟹ bRa.! Bra Transcript 1, 2, 3 6.3.3, we leave it out ( \mathbb { }. All belong to the same equivalence class R b\mbox { and } bRa, \ ( \sim\ if! Finite set with n elements x_2, y_2 ) \ ( R\ ) is equivalence. - 0.3942 in the group, we leave it out may regard equivalence classes is a of. X was the set of all angles are known as the Fundamental Theorem on equivalence relation by its! | 2021-02-27T03:26:46 | {
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https://www.tutorialspoint.com/return-evenly-spaced-numbers-over-a-specified-interval-and-set-the-number-of-samples-to-generate-in-numpy | # Return evenly spaced numbers over a specified interval and set the number of samples to generate in Numpy
NumpyServer Side ProgrammingProgramming
To return evenly spaced numbers over a specified interval, use the numpy.linspace() method in Python Numpy. The 1st parameter is the "start" i.e. the start of the sequence. The 2nd parameter is the "end" i.e. the end of the sequence. The 3rd parameter is the num i.e. the number of samples to generate..
The stop is the end value of the sequence, unless endpoint is set to False. In that case, the sequence consists of all but the last of num + 1 evenly spaced samples, so that stop is excluded. Note that the step size changes when endpoint is False.
The dtype is the type of the output array. If dtype is not given, the data type is inferred from start and stop. The inferred dtype will never be an integer; float is chosen even if the arguments would produce an array of integers. The axis in the result to store the samples. Relevant only if start or stop are array-like. By default (0), the samples will be along a new axis inserted at the beginning. Use -1 to get an axis at the end.
## Steps
At first, import the required library −
import numpy as np
To return evenly spaced numbers over a specified interval, use the numpy.linspace() method in Python Numpy −
arr = np.linspace(100, 200, num = 10)
print("Array...\n", arr)
Get the array type −
print("\nType...\n", arr.dtype)
Get the dimensions of the Array −
print("\nDimensions...\n",arr.ndim)
Get the shape of the Array −
print("\nShape...\n",arr.shape)
Get the number of elements −
print("\nNumber of elements...\n",arr.size)
## Example
import numpy as np
# To return evenly spaced numbers over a specified interval, use the numpy.linspace() method in Python Numpy
# The 1st parameter is the "start" i.e. the start of the sequence
# The 2nd parameter is the "end" i.e. the end of the sequence
# The 3rd parameter is the num i.e the number of samples to generate. Default is 50.
arr = np.linspace(100, 200, num = 10)
print("Array...\n", arr)
# Get the array type
print("\nType...\n", arr.dtype)
# Get the dimensions of the Array
print("\nDimensions...\n",arr.ndim)
# Get the shape of the Array
print("\nShape...\n",arr.shape)
# Get the number of elements
print("\nNumber of elements...\n",arr.size)
## Output
Array...
[100. 111.11111111 122.22222222 133.33333333 144.44444444
155.55555556 166.66666667 177.77777778 188.88888889 200. ]
Type...
float64
Dimensions...
1
Shape...
(10,)
Number of elements...
10
Updated on 08-Feb-2022 07:00:57 | 2022-08-09T01:21:28 | {
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http://math.stackexchange.com/questions/189742/proving-congruence-modulo | # Proving Congruence Modulo
If $x$ is positive integer, prove that for all integers $a$, $(a+1)(a+2)\cdots(a+x)$ is congruent to $0\!\!\!\mod x$.
Any hints? What are the useful concepts that may help me solve this problem?
-
hint: you multiply $x$ consecutive integers so that one... – Raymond Manzoni Sep 1 '12 at 18:35
There are no "concepts", just original thought. Try to prove that one of the brackets is divisible by $x$ by considering what $a$ is mod $x$. – fretty Sep 1 '12 at 18:37
Hint: can you think of any reason why $x$ must divide one of the numbers $a,\ldots,a+x$? – shoda Sep 1 '12 at 18:39
So I can use factorial notation? – primemiss Sep 1 '12 at 18:43
so that one of them must be a multiple of $x$ (every $x$-th integer is a multiple of $x$). – Raymond Manzoni Sep 1 '12 at 18:55
Note that there is some $k$ such that $0\leq k< x$ and $a\equiv k\mod x$. Then $$a+(x-k)\equiv k+x-k\equiv x\equiv 0\mod x$$ and what do you get when you multiply this by other stuff?
-
One of the numbers $a+1, a+2,\ldots, a+x$ is congruent to $0$ mod $x$. Multiplying by $0$ yields $0$.
-
How to prove that one of the factors is congruent to 0 mod x? – primemiss Sep 1 '12 at 19:28
In any set of seven consecutive days, one will be a Tuesday. The reason is the same as that. If you have seven consecutive days, you get one of each day of the week, and if you have $x$ consecutive integers, then you've got one in every congruence class mod $x$. For example, consider $60,61,62,63,64,65,66$. Seven consecutive integers. They are congruent mod $7$ to $4,5,6,0,1,2,3$, respectively. – Michael Hardy Sep 1 '12 at 22:02
@primemiss: By the definition of the equivalency relation, there exists a $p$ and a $q$ with $0 \leq q < x$ such that $a = px + q$. That means that $a + (x - q) = (p + 1)x \equiv 0 \mod x$ with $0 < x - q \leq x$. – Niklas B. Sep 1 '12 at 23:06
Hint $\$ Any sequence of $\,n\,$ consecutive naturals has an element divisible by $\,n\,$. This has a simple proof by induction: shifting such a sequence by one does not change its set of remainders mod $\,n,\,$ since it effectively replaces the old least element $\:\color{#C00}a\:$ by the new greatest element $\:\color{#C00}{a+n}$
$$\begin{array}{}& \color{#C00}a, &\!\!\!\! a+1, &\!\!\!\! a+2, &\!\!\!\! \cdots, &\!\!\!\! a+n-1 & \\ \to & &\!\!\!\! a+1,&\!\!\!\! a+2, &\!\!\!\! \cdots, &\!\!\!\! a+n-1, &\!\!\!\! \color{#C00}{a+n} \end{array}\qquad$$
Since $\: \color{#C00}{a\,\equiv\, a\!+\!n}\pmod n,\:$ the shift does not change the remainders in the sequence. Thus the remainders are the same as the base case $\ 0,1,2,\ldots,n-1\ =\:$ all possible remainders mod $\,n.\,$ Therefore the sequence has an element with remainder $\,0,\,$ i.e. an element divisible by $\,n.$
-
Note: Reversely, shifting by $-1$ proves it for sequences starting with negative integers. – Bill Dubuque Sep 1 '12 at 21:26
This has been already nicely answered, but here is another way to state an approach.
You will also be able to notice that you do not need the last term $(a + x)$ to have the relation you want.
What you hope to find is one of the factors divisible by $x$. How do do know you can find one?
Any of the factors (mod $x$) will be the equivalent of the remainder of $a$ when divided by $x$ plus the remainder of the added term when divided by $x$.
The remainder of $a$, call it r, will be $1\leq r< x$. And each of the added terms (other than $x$) will be equal to itself, that is one of the numbers $1$ thru $x - 1$.
So one of these sums (factors) will definitely exactly equal $x$.
- | 2015-02-01T14:53:05 | {
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https://math.stackexchange.com/questions/1413046/pascals-triangle-sum-of-nth-diagonal-row | # pascal's triangle sum of nth diagonal row
today i was reading about pascal's triangle. the website pointed out that the 3th diagonal row were the triangular numbers. which can be easily expressed by the following formula.
$$\sum_{i=0}^n i = \frac{n(n+1)}{2}$$
i wondered if the following rows could be expressed with such a simple formula. when trying to find the sum for the 3th row i used a method called "differences" i found on this site: http://www.trans4mind.com/personal_development/mathematics/series/sumNaturalSquares.htm
lets call $P_r$ the $r^{th}$ row of pascals triangle. The result for the 4th row was $$\sum_{i=0}^n P_3 = \frac{n(n+1)(n+2)}{6}$$ and the result for 4th row was $$\sum_{i=0}^n P_4 = \frac{n(n+1)(n+2)(n+3)}{24}$$ i guessed the sum of the 5th row would be $$\sum_{i=0}^n P_5 = \frac{n(n+1)(n+2)(n+3)(n+4)}{120}$$ i plotted the function and looking at the graph it seems to be correct. it looks like the the sum of each row is: $$\sum_{i=0}^n P_r = \frac{(n + 0)\cdots(n+(r-1))}{r!}$$
is this true for all rows? and why? i think this has something to do with combinatorics/probability which i never studied.
edit image for $P_r$: http://i.imgur.com/JlVC4q3.png
• What is $P_r$, already? – Did Aug 28 '15 at 20:48
• i meant sum of rth diagonal row , sorry bit confusing i will add image – Lethalbeast Aug 28 '15 at 20:50
• Then the relation ${i\choose j}+{i\choose j+1}={i+1\choose j+1}$ is all one needs to prove this by induction. – Did Aug 28 '15 at 21:08
So you basically want to prove that $$\binom{n}{n}+\binom{n+1}{n}+\binom{n+2}{n}+\dotsc+\binom{n+k}{n}=\binom{n+k+1}{n+1}$$ holds for all $n,k$, right? Of course you can prove this using induction and the formula $$\binom{n}{k}+\binom{n+1}{k}=\binom{n+1}{k+1}$$ as suggest by Did. There is a nice combinatorial interpretation of this using double-counting: Suppose you have $n+1$ eggs, $n$ of them blue and 1 red.
You want to choose $k+1$ of them which is the RHS: $\binom{n+1}{k+1}$
Either you choose the red one in which case you have $\binom{n}{k}$ possibilities for the remaining ones.
Either you don't choose the red one in which case you have $\binom{n}{k+1}$ possibilities for the remaining ones.
Can you think of a similar combinatorial argument which directly works for the original sum?
Hint: Think of $n+k+1$ balls in a row labelled $1,2,\dotsc,n+k+1$. You want to choose $n+1$ of them. That's the RHS: $\binom{n+k+1}{n+1}$. Now, distinguish cases about which is the rightmost ball you choose. If it's the ball number $n+k+1$ you have $\binom{n+k}{n}$ possibilities to choose the remaining $n$ balls. If it's the ball number $n+k$ you have $\binom{n+k-1}{n}$ possibilities etc. Can you complete it from here?
Just to be more explicit, here is a proof of: $$\binom{n}{n}+\binom{n+1}{n}+\binom{n+2}{n}+...+\binom{n+k}{n}=\binom{n+k+1}{n+1}$$I will use the property suggested by Did which is valid for general $i$ and $j$, $$\binom{i}{j}+\binom{i}{j+1}=\binom{i+1}{j+1}$$
Proof: \begin{align} \binom{n+k+1}{n+1} & =\binom{n+k}{n}+\binom{n+k}{n+1}\qquad\text{Using above property}\\ &=\binom{n+k}{n}+\left\{\binom{n+k-1}{n}+\binom{n+k-1}{n+1}\right\}\\ &=\binom{n+k}{n}+\binom{n+k-1}{n}+\left\{\binom{n+k-2}{n}+\binom{n+k-2}{n+1}\right\}\\ &=\binom{n+k}{n}+\binom{n+k-1}{n}+\binom{n+k-2}{n}+...+\left\{\binom{n+1}{n}+\binom{n+1}{n+1}\right\}\\ &=\binom{n+k}{n}+\binom{n+k-1}{n}+\binom{n+k-2}{n}+...\binom{n+1}{n}+\binom{n}{n} \end{align} Hence proved.
Try performing the following multiplication, then look up the Binomial Theorem.
$$\begin{array}{lr} &\binom{5}{5}x^5+\binom{5}{4}x^4+\binom{5}{3}x^3+\binom{5}{2}x^2+\binom{5}{1}x+\binom{5}{0}\\ \times&x+1\\ \hline \end{array}$$ | 2019-04-25T22:18:17 | {
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https://cogito.com.ua/martha-nussbaum-czx/6c05f7-signum-function-pronunciation | Note that if the routine signum/f is not able to determine a value for signum(f()) then it can return FAIL. These are useful in defining functions that can be expressed with a single formula. Overview/Introduction-Functions-Graph of a Function-Classification of Functions-One-Valued and Many-Valued Functions-The Square Root-The Absolute Value Symbol-The Signum Function-Definition of a Limit-Theorems on Limits-Right-Hand and Left-Hand Limits-Continuity-Missing Point Discontinuities-Finite Jumps-Infinite Discontinuities 3. In general, there is no standard signum function in C/C++, and the lack of such a fundamental function tells you a lot about these languages. The signum function is the real valued function defined for real as follows. It is defined as u(t) = $\left\{\begin{matrix} 1 & t \geqslant 0\\ 0 & t. 0 \end{matrix}\right.$ It is used as best test signal. Definition. Thus, at x = 0, it is left undefined. Community ♦ 1 1 1 silver badge. signum function (plural signum functions) (mathematics) The function that determines the sign of a real number, yielding -1 if negative, +1 if positive, or otherwise zero. This video is unavailable. Apart from that, I believe both majority viewpoints about the right approach to define such a function are in a way correct, and the "controversy" about it is actually a non-argument once you take into account two important caveats: The second property implies that for real non-zero we have . Syntax. Unit Step Function. for the . Newbie; Posts: 17; Karma: 0 ; sgn / sign / signum function suggestions. For all real we have . Watch Queue Queue Similarly, . Serge Stroobandt Serge Stroobandt. Here, we should point out that the signum function is often defined simply as 1 for x > 0 and -1 for x < 0. Hypernyms . For all real we have . Put the code into a function and use inputdlg to allow inputting the numbers more easily. For a complex argument it is defined by. It is written in the form y … The signum function is defined as f(x) = |x|/x; x≠0 = 0; x = 0 . For a simple, outgoing source, (21) e i 2 π q 0 x = cos 2 π q 0 x + i sin 2 π q 0 x sgn x, where q 0 = 1 / λ. Alternatively, you could simply let the user define the matrix X and use it as an input for the function. For a complex argument it is defined by. Contents. Yes the function is discontinuous which is right as per your argument. davidhbrown. The graph of a signum function is as shown in the figure given above. See for example . Pronunciations of proper names are generally given as pronounced as in the original language. The signum function of a real number x … Signum manipuli - Wikipedia "He carried a signum for a cohort or century." 1 Definition; 2 Properties; 3 Complex signum; 4 Generalized signum function; 5 See also; 6 Notes; Definition. I think the question wanted to convey this.. A Megametamathematical Guide. Solution for 1. Listen to the audio pronunciation in several English accents. In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. Post by Stefan Ram »Returns the signum function of the argument«.. The signum function is the real valued function defined for real as follows. where denotes the magnitude (absolute value) of . Mathematics Pronunciation Guide. answered Sep 16 '18 at 14:26. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article. For instance, the value of function f(x) is equal to -5 in the interval [-5, -4). That is why, it is not continuous everywhere over R . of the . Information and translations of signum function in the most comprehensive dictionary definitions resource on the web. View a complete list of particular functions on this wiki Definition. The signum vector function is a function whose behavior in each coordinate is as per the signum function. Definition of signum function in the Definitions.net dictionary. Signum Function . Is the »s« voiced or voiceless? It has a jumped discontinuity which means if the function is assigned some value at the point of discontinuity it cannot be made continuous. The signum function, denoted , is defined as follows: Note: In the definition given here, we define the value to Sign function (signum function) collapse all in page. Unit step function is denoted by u(t). I cannot find the American English (San Francisco) pronunciation of »signum« in a dictionary. Stefan Ram 2014-05-14 22:55:02 UTC . Example Problems. signum function pronunciation - How to properly say signum function. where denotes the magnitude (absolute value) of . (mathematics) A function that extracts the sign of a real number x, yielding -1 if x is negative, +1 if x is positive, or 0 if x is zero. What does signum function mean? A function cannot be continuous and discontinuous at the same point. Y = sign(x) returns an array Y the same size as x, where each element of Y is: 1 if the corresponding element of x is greater than 0. function; Translations . Impulse function is denoted by δ(t). »Returns the signum function of the argument«.. It's not uncommon to need to know the arithmetic sign of a value, typically represented as -1 for values < 0, +1 for values > 0, or 0. Permalink. Anthony, looking good. All the engineering examinations including IIT JEE and AIEEE study material is available online free of cost at askIITians.com. This function definition executes fast and yields guaranteed correct results for 0, 0.0, -0.0, -4 and 5 (see comments to other incorrect answers). Similarly, . A medieval tower bell used particularly for ringing the 8 canonical hours. The cosine transform of an odd function can be evaluated as a convolution with the Fourier transform of a signum function sgn(x). Some … Let’s Workout: Example 1: Find the greatest integer function for following (a) ⌊-261 ⌋ (b) ⌊ 3.501 ⌋ (c) ⌊-1.898⌋ Solution: According to the greatest integer function definition A statement function should appear before any executable statement in the program, but after any type declaration statements. The function’s value stays constant within an interval. for the Diacritically Challenged, This guide includes most mathematicians and mathematical terms that are encountered in high school and college. Any real number can be expressed as the product of its absolute value and its sign function: From equation (1) it follows that whenever x is not equal to 0 we have. German: Signumfunktion f; Hungarian: előjelfüggvény; Russian: зна́ковая фу́нкция f (znákovaja fúnkcija) Turkish: işaret fonksiyonu; Usage notes . Meaning of signum function. Signum definition is - something that marks or identifies or represents : sign, signature. Listen to the audio pronunciation of Signum Fidei on pronouncekiwi How To Pronounce Signum Fidei: Signum Fidei pronunciation + Definition Sign in to disable ALL ads. sign(x) Description. is called the signum function. Formal Definition of a Function Limit: The limit of f (x) f(x) f (x) as x x x approaches x 0 x_0 x 0 is L L L, i.e. Use the sequential criteria for limits, to show that the… Use the e-8 definition of the limit to show that x3 – 2x + 4 lim X-ix2 + 4x – 3 3 !3! »I« as in »it« or as in »seal«? Graphing functions by plotting points Definition A relation f from a set A to a set B is said to be a function if every input of a set A has only one output in a set B. If then also . example. A sinc pulse passes through zero at all positive and negative integers (i.e., t = ± 1, ± 2, …), but at time t = 0, it reaches its maximum of 1.This is a very desirable property in a pulse, as it helps to avoid intersymbol interference, a major cause of degradation in digital transmission systems. Synonyms (bell): signum bell (math): signum function, sign function A statement function in Fortran is like a single line function definition in Basic. In mathematical expressions the sign function is often represented as sgn. Area under unit step function is unity. Note that the procedure signum/f is passed only the argument(s) to the function f. The value of the environment variable is set within signum, and can be checked by signum/f, but it is not passed as a parameter. Sign function - Wikipedia "The signum manipuli (Latin for 'standard' of the maniple, Latin: manipulus) was a standard for both the centuriae and the legion." signum (plural signums or signa) A sign, mark, or symbol. Definition, domain, range, solution of Signum function. Definition. Note that zero (0) is neither positive nor negative. The precise definition of the limit is discussed in the wiki Epsilon-Delta Definition of a Limit. Of course it is continuous at every x € R except at x = 0 . »U« as in »soon« or »number« or just a schwa? The second property implies that for real non-zero we have . No, it is not continuous every where . "The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero." The format is simple - just type f(x,y,z,…) = formula . Suppose is a positive integer. SIGN(x) returns a number that indicates the sign x: -1 if x is negative; 0 if x equals 0; or 1 if x is positive. In applications such as the Laplace transform this definition is adequate, since the value of a function at a single point does not change the analysis. If then also . Unit Impulse Function. Therefore, its Fourier transform is (22) F q = 1 2 δ q + q 0 + δ q − q 0 − i 2 π q − q 0 + i 2 π q + q 0. Explicitly, it is defined as the function: In other words, the signum function project a non-zero complex number to the unit circle . share | improve this answer | follow | edited Jun 20 at 9:12. Topic: sgn / sign / signum function suggestions (Read 2558 times) previous topic - next topic. Jul 19, 2019, 05:35 pm. Signum function is an integer valued function defined over R . Proper American English Pronunciation of Words and Names. A sinc function is an even function with unity area. function. Signifer - Wikipedia. The signum function of a real number x is defined as follows: Properties. In other words, the signum function project a non-zero complex number to the unit circle . Study Physics, Chemistry and Mathematics at askIITians website and be a winner. Chemistry and Mathematics at askIITians website and be a winner » it or... 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Tractor Supply Tool Box Won't Open, Tarantula Dc Nightwing, Antigo Shopping Show 106, Oliver Travel Trailers Canada, Homes For Sale By Owner Winterset Iowa, Large Used Car Dealerships Near Me, Welsh Guards Ranks, | 2021-04-21T13:54:48 | {
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https://au.mathworks.com/help/matlab/ref/norm.html | # norm
Vector and matrix norms
## Description
example
n = norm(v) returns the Euclidean norm of vector v. This norm is also called the 2-norm, vector magnitude, or Euclidean length.
example
n = norm(v,p) returns the generalized vector p-norm.
example
n = norm(X) returns the 2-norm or maximum singular value of matrix X, which is approximately max(svd(X)).
n = norm(X,p) returns the p-norm of matrix X, where p is 1, 2, or Inf:
example
n = norm(X,"fro") returns the Frobenius norm of matrix or array X.
## Examples
collapse all
Create a vector and calculate the magnitude.
v = [1 -2 3];
n = norm(v)
n = 3.7417
Calculate the 1-norm of a vector, which is the sum of the element magnitudes.
v = [-2 3 -1];
n = norm(v,1)
n = 6
Calculate the distance between two points as the norm of the difference between the vector elements.
Create two vectors representing the (x,y) coordinates for two points on the Euclidean plane.
a = [0 3];
b = [-2 1];
Use norm to calculate the distance between the points.
d = norm(b-a)
d = 2.8284
Geometrically, the distance between the points is equal to the magnitude of the vector that extends from one point to the other.
$\begin{array}{l}a=0\underset{}{\overset{ˆ}{i}}+3\underset{}{\overset{ˆ}{j}}\\ b=-2\underset{}{\overset{ˆ}{i}}+1\underset{}{\overset{ˆ}{j}}\\ \\ \begin{array}{rl}{d}_{\left(a,b\right)}& =||b-a||\\ & =\sqrt{\left(-2-0{\right)}^{2}+\left(1-3{\right)}^{2}}\\ & =\sqrt{8}\end{array}\end{array}$
Calculate the 2-norm of a matrix, which is the largest singular value.
X = [2 0 1;-1 1 0;-3 3 0];
n = norm(X)
n = 4.7234
Calculate the Frobenius norm of a 4-D array X, which is equivalent to the 2-norm of the column vector X(:).
X = rand(3,4,4,3);
n = norm(X,"fro")
n = 7.1247
The Frobenius norm is also useful for sparse matrices because norm(X,2) does not support sparse X.
## Input Arguments
collapse all
Input vector.
Data Types: single | double
Complex Number Support: Yes
Input array, specified as a matrix or array. For most norm types, X must be a matrix. However, for Frobenius norm calculations, X can be an array.
Data Types: single | double
Complex Number Support: Yes
Norm type, specified as 2 (default), a positive real scalar, Inf, or -Inf. The valid values of p and what they return depend on whether the first input to norm is a matrix or vector, as shown in the table.
Note
This table does not reflect the actual algorithms used in calculations.
pMatrixVector
1max(sum(abs(X)))sum(abs(v))
2 max(svd(X))sum(abs(v).^2)^(1/2)
Positive, real-valued numeric scalarsum(abs(v).^p)^(1/p)
Infmax(sum(abs(X')))max(abs(v))
-Infmin(abs(v))
## Output Arguments
collapse all
Norm value, returned as a scalar. The norm gives a measure of the magnitude of the elements. By convention:
• norm returns NaN if the input contains NaN values.
• The norm of an empty matrix is zero.
collapse all
### Euclidean Norm
The Euclidean norm (also called the vector magnitude, Euclidean length, or 2-norm) of a vector v with N elements is defined by
$‖v‖=\sqrt{\sum _{k=1}^{N}{|{v}_{k}|}^{2}}\text{\hspace{0.17em}}.$
### General Vector Norm
The general definition for the p-norm of a vector v that has N elements is
${‖v‖}_{p}={\left[\sum _{k=1}^{N}{|{v}_{k}|}^{p}\right]}^{\text{\hspace{0.17em}}1/p}\text{\hspace{0.17em}},$
where p is any positive real value, Inf, or -Inf.
• If p = 1, then the resulting 1-norm is the sum of the absolute values of the vector elements.
• If p = 2, then the resulting 2-norm gives the vector magnitude or Euclidean length of the vector.
• If p = Inf, then ${‖v‖}_{\infty }={\mathrm{max}}_{i}\left(|v\left(i\right)|\right)$.
• If p = -Inf, then ${‖v‖}_{-\infty }={\mathrm{min}}_{i}\left(|v\left(i\right)|\right)$.
### Maximum Absolute Column Sum
The maximum absolute column sum of an m-by-n matrix X (with m,n >= 2) is defined by
${‖X‖}_{1}=\underset{1\le j\le n}{\mathrm{max}}\left(\sum _{i=1}^{m}|{a}_{ij}|\right).$
### Maximum Absolute Row Sum
The maximum absolute row sum of an m-by-n matrix X (with m,n >= 2) is defined by
${‖X‖}_{\infty }=\underset{1\le i\le m}{\mathrm{max}}\left(\sum _{j=1}^{n}|{a}_{ij}|\right)\text{\hspace{0.17em}}.$
### Frobenius Norm
The Frobenius norm of an m-by-n matrix X (with m,n >= 2) is defined by
${‖X‖}_{F}=\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{|{a}_{ij}|}^{2}}=\sqrt{\text{trace}\left({X}^{†}X\right)}\text{\hspace{0.17em}}.$
This definition also extends naturally to arrays with more than two dimensions. For example, if X is an N-D array of size m-by-n-by-p-by-...-by-q, then the Frobenius norm is
${‖X‖}_{F}=\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}\sum _{k=1}^{p}...\sum _{w=1}^{q}{|{a}_{ijk...w}|}^{2}}.$
## Tips
• Use vecnorm to treat a matrix or array as a collection of vectors and calculate the norm along a specified dimension. For example, vecnorm can calculate the norm of each column in a matrix.
## Version History
Introduced before R2006a
expand all | 2023-03-27T10:16:22 | {
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http://amhomeworkeylt.californiapublicrecords.us/pascals-triangle.html | # Pascals triangle
Binomial coefficients, pascal's triangle, and the power set binomial coefficients pascal's triangle the power set binomial coefficients an expression like (a + b) . Pascal's triangle is a number pattern introduced by the famous french mathematician and philosopher blaise pascal it is triangle in shape with the top being. Mathematicians have long been familiar with the tidy way in which the nth row of pascal's triangle sums to 2n (the top row conventionally labeled as n = 0. Where (n r) is a binomial coefficient the triangle was studied by b pascal, although it had been described centuries earlier by chinese mathematician yanghui. Pascal's triangle, a simple yet complex mathematical construct, hides some surprising properties related to number theory and probability.
Sal introduces pascal's triangle, and shows how we can use it to figure out the coefficients in binomial expansions. Applications pascal's triangle is not only an interesting mathematical work because of its hidden patterns, but it is also interesting because of its wide expanse. All you ever want to and need to know about pascal's triangle.
Pascal triangle: given numrows, generate the first numrows of pascal's triangle pascal's triangle : to generate a[c] in row r, sum up a'[c] and a'[c-1] from. Given numrows, generate the first numrows of pascal's triangle for example, given numrows = 5, the result should be: , , , , ] java. In mathematics, pascal's triangle is a triangular array of the binomial coefficients in much of the western world, it is named after the french mathematician blaise. The pascal's triangle is a graphical device used to predict the ratio of heights of lines in a split nmr peak. From the wikipedia page the kata author cites: the rows of pascal's triangle ( sequence a007318 in oeis) are conventionally enumerated starting with row n = 0.
A guide to understanding binomial theorem, pascal's triangle and expanding binomial series and sequences. This special triangular number arrangement is named after blaise pascal pascal was a french mathematician who lived during the seventeenth century. When you look at pascal's triangle, find the prime numbers that are the first number in the row that prime number is a divisor of every number in that row.
## Pascals triangle
Resources for the patterns in pascal's triangle problem pascal's triangle at provides information on. Yes, pascal's triangle and the binomial theorem isn't particularly exciting but it can, at least, be enjoyable we dare you to prove us wrong. The counting function c(n,k) and the concept of bijection coalesce in one of the most studied mathematical concepts, pascal's triangle at its heart, pascal's. Pascal's triangle is one of the classic example taught to engineering students it has many interpretations one of the famous one is its use with binomial.
• Now that we've learned how to draw pascal's famous triangle and use the numbers in its rows to easily calculate probabilities when tossing coins, it's time to dig.
• There are many interesting things about polynomials whose coefficients are taken from slices of pascal's triangle (these are a form of what's called chebyshev.
• S northshieldsums across pascal's triangle modulo 2 j raaba generalization of the connection between the fibonacci sequence and pascal's triangle.
It's not necessary to do this because 2 only shows up once in pascal's triangle but you get the idea t2 = table[binomial[n, k], {n, 0, 8}, {k, 0, n}] / {a_, 2, c_} : {a . Pascal's triangle and binomial coefficients information for the mathcamp 2017 qualifying quiz 1 pascal's triangle pascal's triangle (named for the. Theorem the sum of all the entries in the n th row of pascal's triangle is equal to 2 n proof 1 by definition, the entries in n th row of pascal's. Pascal's triangle is defined such that the number in row \$n\$ and column \$k\$ is \$ {n\choose k}\$ for this reason, convention holds that both row numbers and.
Pascals triangle
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2018. | 2018-11-20T20:01:24 | {
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https://mathoverflow.net/questions/263397/is-there-a-transformation-or-a-proof-for-these-integrals?noredirect=1 | Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a proof? $$\int_0^{\infty}x^2e^{-x^2}\frac{{dx}}{\cosh\sqrt{\pi}x} =\frac14\int_0^{\infty}e^{-x^2}\frac{dx}{\cosh\sqrt{\pi}x}.$$
• The integral in this question was taken from your own paper that was written 10 years earlier "A dozen integrals: Russell-style" and you knew the proof. arxiv.org/abs/0808.2692
– Nemo
Aug 9, 2021 at 18:15
• @user44191 Whatever Stack Exchange policy might be, and independently of this particular case (on which I make no comment), it's (1) certainly unusual on MO to ask questions to which you already know the answer, and (2) abusive of other people's time to do this while giving the impression that you don't know the answer. If I spent time solving someone's problem and then typing it up, on the understanding that I was helping them out, I might feel quite angry to discover that they knew the answer all along. I don't use MO as much as I used to, but is there really any doubt over this? Aug 10, 2021 at 13:32
• I can’t understand why downvoting this question. It seems to me interesting and perfectly suitable for MO. Not mentioning an existing proof is a defect, of course, yet I explain it to me with the purpose of not influencing who approaches the problem, since apparently the goal was to get new proofs, insights and connections. Aug 10, 2021 at 14:31
• Besides, after $10$ years it is perfectly possible to forget to have ever proven that particular integral identity. Aug 10, 2021 at 17:17
• @Nemo, making an unnecessary edit, just so you can vote a question down? Really? Aug 11, 2021 at 13:09
This is a generous explanation of Lucia's comment above.
The functions $$\mathcal H_n(x)=\frac{2^{1/4}}{(2^n n!)^{1/2}}H_n(\sqrt{2\pi}\; x) e^{-\pi x^2}$$ form an orthonormal system in $L^2(\textbf{R})$. Here $H_n(x)$ are the usual Hermite polynomials defined by $$e^{2xz-z^2}=\sum_{n=0}^\infty \frac{H_n(x)}{n!}z^n,\qquad |z|<\infty.$$ The functions are eigenfunctions for the usual Fourier transform so that $$\int_{-\infty}^{+\infty}\mathcal H_n(t)e^{-2\pi i x t}\,dt=(-i)^n \mathcal H_n(x).$$
It follows that $L^2(\textbf{R})$ is a direct sum of four subspaces. In each of these subspaces the Fourier transform is just multiplication by $1$, $i$, $-1$, $-1$ respectively.
It is well known that the function $1/\cosh\pi x$ is invariant by the Fourier transform $$\int_{-\infty}^{+\infty}\frac{e^{-2\pi i x\xi}}{\cosh\pi x}\,dx= \frac{1}{\cosh\pi \xi}.$$ Therefore this function is in the span of the functions $\mathcal H_{4n}(x)$, and therefore is orthogonal to any other eigenfunction. In particular $$\int_{-\infty}^{+\infty}\frac{\mathcal H_2(x)}{\cosh\pi x}\,dx=0.$$ Since $H_2(x)=-2+4x^2$ it follows that $$\int_{-\infty}^{+\infty}\frac{(8\pi x^2-2)e^{-\pi x^2}}{\cosh\pi x}\,dx=0.$$ Putting $x/\sqrt{\pi}$ instead of x $$\int_{-\infty}^{+\infty}\frac{(8 x^2-2)e^{-x^2}}{\cosh\sqrt{\pi} x}\,\frac{dx}{\sqrt{\pi}}=0.$$ This is equivalent to your equation.
UPDATE
The integral in T. Amdeberhan's question was taken from his own paper that was written 10 years earlier before he posted his question: https://arxiv.org/abs/0808.2692 A dozen integrals: Russell-style. Thus it seems that T. Amdeberhan knew the answer to the question he asked. Why did he ask it then?
I provide a screenshot below for the reader's convenience (integral number $$7$$):
At the end of the paper, the authors provide a sketch of proof:
Note that the following functions are self-reciprocal $$\sqrt{\frac{2}{\pi}}\int_0^\infty f(x)\cos ax dx=f(a)$$ (see this MO post for a list of such functions): $$\frac{1}{\cosh\sqrt{\frac{\pi}{2}}x},\quad e^{-x^2/2}.\tag{1}$$ It was proved by Hardy (Quarterly Journal Of Pure And Applied Mathematics, Volume 35, Page 203, 1903) and Ramanujan (Ramanujan's Lost Notebook, part IV, chapter 18) that for two self-reciprocal functions $$f$$ and $$g$$ we have $$\int_0^\infty f(x)g(\alpha x) dx=\frac{1}{\alpha}\int_0^\infty f(x)g(x/\alpha) dx.$$ The formal agument is as follows but it can be made rigorous for certain type of functions $$\int_0^\infty f(x)g(\alpha x) dx=\int_0^\infty f(x)dx\cdot \sqrt{\frac{2}{\pi}}\int_0^\infty g(y) \cos(\alpha x y)dy=\\ \int_0^\infty g(y)dy \cdot \sqrt{\frac{2}{\pi}}\int_0^\infty f(x)\cos(\alpha x y)dx=\int_0^\infty g(y)f(\alpha y) dy=\\ \frac{1}{\alpha}\int_0^\infty f(x)g(x/\alpha) dx.$$ For functions in $$(1)$$ which decay rapidly at $$x\to\pm\infty$$ it is certainly true. So we obtain an identity due to Hardy and Ramanujan $$\int_{0}^{\infty} \frac{e^{-x^{2}/2}}{\cosh{\sqrt{\frac{\pi}{2}}\alpha{x}}} {dx} = \frac{1}{\alpha} \int_{0}^{\infty} \frac{e^{-x^{2}/2}}{\cosh{\sqrt{\frac{\pi}{2}}\frac{x}{\alpha}}}{dx}.$$ After some simplifications it becomes $$\int_{0}^{\infty} \frac{e^{-\alpha^2x^{2}}}{\cosh{\sqrt{{\pi}}{x}}} {dx} = \frac{1}{\alpha} \int_{0}^{\infty} \frac{e^{-x^{2}/\alpha^2}}{\cosh{\sqrt{{\pi}}{x}}}{dx}.$$ To complete the proof differentiate this with respect to $$\alpha$$ and then set $$\alpha=1$$. | 2022-07-02T18:12:34 | {
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http://nptel.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node121.html | ## Simpson's Rule
If we are given odd number of tabular points,i.e. is even, then we can divide the given integral of integration in even number of sub-intervals Note that for each of these sub-intervals, we have the three tabular points and so the integrand is replaced with a quadratic interpolating polynomial. Thus using the formula (13.3.3), we get,
In view of this, we have
which gives the second quadrature formula as follows:
(13.3.5)
This is known as SIMPSON'S RULE.
Remark 13.3.3 An estimate for the error in numerical integration using the Simpson's rule is given by
(13.3.6)
where is the average value of the forth forward differences.
EXAMPLE 13.3.4 Using the table for the values of as is given in Example 13.3.2, compute the integral by Simpson's rule. Also estimate the error in its calculation and compare it with the error using Trapezoidal rule.
Solution: Here, thus we have odd number of nodal points. Further,
and
Thus,
To find the error estimates, we consider the forward difference table, which is given below:
0.0 1.00000 0.01005 0.02071 0.00189 0.00149 0.1 1.01005 0.03076 0.02260 0.00338 0.00171 0.2 1.04081 0.05336 0.02598 0.00519 0.00243 0.3 1.09417 0.07934 0.03117 0.00762 0.00320 0.4 1.17351 0.11051 0.3879 0.01090 0.00459 0.5 1.28402 0.14930 0.04969 0.01549 0.00658 0.6 1.43332 0.19899 0.06518 0.02207 0.00964 0.7 1.63231 0.26417 0.08725 0.03171 0.8 1.89648 0.35142 0.11896 0.9 2.24790 0.47038 1.0 2.71828
Thus, error due to Trapezoidal rule is,
Similarly, error due to Simpson's rule is,
It shows that the error in numerical integration is much less by using Simpson's rule.
EXAMPLE 13.3.5 Compute the integral , where the table for the values of is given below:
0.05 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.0785 0.1564 0.2334 0.309 0.454 0.5878 0.7071 0.809 0.891 0.9511 0.9877 1
Solution: Note that here the points are not given to be equidistant, so as such we can not use any of the above two formulae. However, we notice that the tabular points and are equidistant and so are the tabular points and . Now we can divide the interval in two subinterval: and ; thus,
. The integrals then can be evaluated in each interval. We observe that the second set has odd number of points. Thus, the first integral is evaluated by using Trapezoidal rule and the second one by Simpson's rule (of course, one could have used Trapezoidal rule in both the subintervals).
For the first integral and for the second one . Thus,
which gives,
It may be mentioned here that in the above integral, and that the value of the integral is . It will be interesting for the reader to compute the two integrals using Trapezoidal rule and compare the values.
EXERCISE 13.3.6
1. Using Trapezoidal rule, compute the integral where the table for the values of is given below. Also find an error estimate for the computed value.
1. a=1 2 3 4 5 6 7 8 9 b=10 0.09531 0.18232 0.26236 0.33647 0.40546 0.47 0.53063 0.58779 0.64185 0.69314
2. a=1.50 1.55 1.6 1.65 1.7 1.75 b=1.80 0.40546 0.43825 0.47 0.5077 0.53063 0.55962 0.58779
3. a = 1.0 1.5 2 2.5 3 b = 3.5 1.1752 2.1293 3.6269 6.0502 10.0179 16.5426
2. Using Simpson's rule, compute the integral Also get an error estimate of the computed integral.
1. Use the table given in Exercise 13.3.6.1b.
2. a = 0.5 1 1.5 2 2.5 3 b = 3.5 0.493 0.946 1.325 1.605 1.778 1.849 1.833
3. Compute the integral , where the table for the values of is given below:
0 0.5 0.7 0.9 1.1 1.2 1.3 1.4 1.5 0 0.39 0.77 1.27 1.9 2.26 2.65 3.07 3.53
A K Lal 2007-09-12 | 2017-02-25T15:48:20 | {
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http://math.stackexchange.com/questions/897469/determinant-of-a-matrix-with-t-in-all-off-diagonal-entries | Determinant of a matrix with $t$ in all off-diagonal entries.
It seems from playing around with small values of $n$ that
$$\det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 & \dots & t\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ t & t & t & \dots& -1 \end{array}\right) = (-1)^{n-1}(t+1)^{n-1}((n-1)t-1)$$
where $n$ is the size of the matrix.
How would one approach deriving (or at least proving) this formally?
Motivation
This came up when someone asked what is the general solution to:
$$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b},$$
and for non-trivial solutions, the matrix above (with $n=3$) must be singular. In this case either $t=-1\implies a+b+c=1$ or $t=\frac{1}{2}\implies a=b=c$.
So I wanted to ensure that these are also the only solutions for the case with more variables.
-
Did you try induction? – zarathustra Aug 14 '14 at 17:08
Induction combined with this should do it. – Daniel R Aug 14 '14 at 17:09
Induction is the way to go, but first, subtract the first row from all the other rows to get a much easier calculation. – fixedp Aug 14 '14 at 17:10
When $n=2$ seems to fail. $1-t^2$ RHS, $-2 t^3-3 t^2+1$ LHS, right? – carlosayam Aug 14 '14 at 17:14
The matrix size should be $n+1$; @caya gives the $n=2$ case. – Semiclassical Aug 14 '14 at 17:17
Using elementary operations instead of induction is key. \begin{align} &\begin{vmatrix} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 & \dots & t\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ t & t & t & \dots& -1 \end{vmatrix}\\ &= \begin{vmatrix} -t-1 & 0 & 0 & \dots & t+1\\ 0 & -t-1 & 0 & \dots & t+1\\ 0 & 0 & -t-1 & \dots & t+1\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ t & t & t & \dots& -1 \end{vmatrix}\\ &= \begin{vmatrix} -t-1 & 0 & 0 & \dots & 0\\ 0 & -t-1 & 0 & \dots & 0\\ 0 & 0 & -t-1 & \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ t & t & t & \dots& (n - 1)t -1 \end{vmatrix}\\ &= (-1)^{n - 1}(t + 1)^{n - 1}((n - 1)t - 1) \end{align}
-
For clarity, first he is doing "Row N = Row N - Last Row" then he is doing "Last Row = Last Row - Row N" – DanielV Aug 14 '14 at 17:46
You can write the expression as $$\det(t C - (t+1)I)$$ where $C = \mathbf{1}\mathbf{1}^T$ is the matrix of all $1$'s, formed by the column of ones times its transpose. Using the identity $\det(I+cr) = 1+rc$, you can first factor out $(t+1)$: $$\det(t C - (t+1)I) = (-1)^n(t+1)^n \det\left(I - \frac{t}{t+1} \mathbf{1}\right) = (-1)^n(t+1)^n \left(1 - \frac{nt}{t+1}\right)$$
-
That's brilliant, nothing could be more elegant. :) – MGA Aug 14 '14 at 17:37
+1, but what do you mean by $cr$ and $rc$? Are these row and column vectors, abbreviated $r$ and $c$, respectively? (My assumption) – apnorton Aug 15 '14 at 3:44
Correct, they are row and column vectors. I stole that from the Wiki page on Determinant – Victor Liu Aug 15 '14 at 21:42
Note that your matrix is the sum of the matrix $T$ with all entries equal to $t$ and the matrix $-(1+t)I$. Therefore the determinant you are asking about is the value at $X=-(1+t)$ of the characteristic polynomial $\chi_{-T}$ of $-T$.
Since $T$ has rank (at most) $1$, its eigenspace for eigenvalue $0$ has dimension $n-1$, so the characteristic polynomial of $-T$ is $\chi_{-T}=X^{n-1}(X+nt)$ (the final factor must be that because the coefficient of $X^{n-1}$ in $\chi_{-T}$ is $\def\tr{\operatorname{tr}}-\tr(-T)=nt$). Now setting $X=-(1+t)$ gives $$\det(-(1+t)I-T)=(-1-t)^{n-1}(-1-t+nt)=(-1-t)^{n-1}(-1+(n-1)t)$$ as desired.
This kind of question is recurrent on this site; see for instance Determinant of a specially structured matrix and How to calculate the following determinants (all ones, minus $I$).
-
Here is another characterization of the result. For convenience, I will take the matrix size as $n+1$ rather than $n$. First, note that we may factor $t$ from each of the $n+1$ rows, and so the determinant may be written as $$\det{[t(M-t^{-1} I_{n+1})]} =t^{n+1} \det(M-t^{-1} I_{n+1})$$ where $(M)_{ij}=1-\delta_{ij}$ for $1\leq i,j\leq n+1$.
Next, observe that $M$ has $n$ independent eigenvectors of the form $\hat{e}_i-\hat{e}_{n+1}$ ($1\leq i\leq n$), each with eigenvalue $-1$; in addition, $M$ also has the eigenvector $\sum_{i=1}^{n+1}\hat{e}_i$ with eigenvalue $n$. Consequently the characteristic polynomial of $M$ in powers of $t^{-1}$ is $$\det(M-t^{-1} I_{n+1})=(-1-t^{-1})^n (n-t^{-1}) = t^{-n-1}\cdot (-1)^n (1+t)^n (nt+1)$$ and so the prior result yields the desired identity.
- | 2015-05-28T08:49:23 | {
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https://math.stackexchange.com/questions/3619669/variation-of-least-squares-with-symmetric-positive-semi-definite-psd-constrain | # Variation of Least Squares with Symmetric Positive Semi Definite (PSD) Constraint
I am trying to solve the following convex optimization problem: \begin{align} & \min_{W} && \sum_{i=1}^n (\mathbf{x}_{i}^TW\mathbf{x}_{i} - y_i)^2 \\\\ & s.t. && W \succcurlyeq 0 \\\\ & && W = W^T \end{align}
where $$\mathbf{x}_i \in \mathbb{R^p}$$, $$W \in \mathbb{R}^{p \times p}$$ and $$y_i \geq 0$$.
Without the positive semidefinite constraint, the problem is pretty straightforward. Requiring positive semidefiniteness, however, makes it a bit tricky.
I thought about using the fact that $$W \succcurlyeq 0$$ if and only if there exists a symmetric $$A$$ such that $$W = AA^T$$, and solving the equivalent problem
\begin{align} & \min_{A} && \sum_{i=1}^n (\mathbf{x}_{i}^TAA^T\mathbf{x}_{i} - y_i)^2 \\\\ &s.t. && A = A^T \end{align}
Letting $$a_{ij}$$ be the $$(i,j)th$$ element of A, this optimization function is quartic (fourth-order) with respect to the $$a_{ij}$$'s. Because of this, I am unsure of how to proceed.
I would be grateful if someone could point me in the right direction as to how to solve this problem.
• Haven't worked it through, but you might find it helpful to note that, if $W = A^\top A$, then$$x^\top_i W x_i = x^\top_i A^\top A x_i = (Ax_i)^\top (Ax_i) = \|Ax_i\|^2.$$ Apr 11 '20 at 0:01
When dealing with such form of matrix multiplications always remember the Vectorization Trick with Kronecker Product for Matrix Equations:
$${x}_{i}^{T} W {x}_{i} - {y}_{i} \Rightarrow \left({x}_{i}^{T} \otimes {x}_{i}^{T} \right) \operatorname{Vec} \left( W \right) - \operatorname{Vec} \left( {y}_{i} \right) = \left({x}_{i}^{T} \otimes {x}_{i}^{T} \right) \operatorname{Vec} \left( W \right) - {y}_{i}$$
Since the problem is given by summing over $${x}_{i}$$ one could build the matrix:
$$X = \begin{bmatrix} {x}_{1}^{T} \otimes {x}_{1}^{T} \\ {x}_{2}^{T} \otimes {x}_{2}^{T} \\ \vdots \\ {x}_{n}^{T} \otimes {x}_{n}^{T} \end{bmatrix}$$
Then:
$$\arg \min_{W} \sum_{i = 1}^{n} {\left( {x}_{i}^{T} W {x}_{i} - {y}_{i} \right)}^{2} = \arg \min_{W} {\left\| X \operatorname{Vec} \left( W \right) - \boldsymbol{y} \right\|}_{2}^{2}$$
Where $$\boldsymbol{y}$$ is the column vector composed by $${y}_{i}$$.
Now the above has nice form of regular Least Squares. The handling of the constraint can be done using Projected Gradient Descent Method. The projection onto the set of Symmetric Matrices and Positive Semi Definite (PSD) Matrices cone are given by:
1. $$\operatorname{Proj}_{\mathcal{S}^{n}} \left( A \right) = \frac{1}{2} \left( A + {A}^{T} \right)$$. See Orthogonal Projection of a Matrix onto the Set of Symmetric Matrices.
2. $$\operatorname{Proj}_{\mathcal{S}_{+}^{n}} \left( A \right) = Q {\Lambda}_{+} {Q}^{T}$$ where $$A = Q \Lambda {Q}^{T}$$ is the eigen decomposition of $$A$$ and $${\Lambda}_{+}$$ means we zero any negative values in $$\Lambda$$. See Find the Matrix Projection of a Symmetric Matrix onto the set of Symmetric Positive Semi Definite (PSD) Matrices.
Since both the Symmetric Matrices Set and PSD Cone are Linear Sub Space hence even the greedy iterative projection on the set will yield an orthogonal projection on the intersection of the 2 sets. See Orthogonal Projection onto the Intersection of Convex Sets.
So, with all the tools above one could create his own solver using basic tools with no need for external libraries (Which might be slow or not scale).
I implemented the Projected Gradient Descent Method with the above projections in MATLAB. I compared results to CVX to validate the solution. This is the solution:
My implementation is vanilla Gradient Descent with constant Step Size and no acceleration. If you add those you'll see convergence which is order of magnitude faster (I guess few tens of iterations). Not bad for hand made solver.
The MATLAB Code is accessible in my StackExchange Mathematics Q3619669 GitHub Repository.
• This was a very clear and practical response. Thanks! Assuming I understand correctly, I have one question: what is the difference between iteratively projecting vs. projecting just once after finding the optimal solution to the unconstrained problem? Would they give different solutions? Apr 13 '20 at 15:43
• @tygaking, Please mark it as answer and +1. Regarding your question, you can't do that. Think of "Gradient Descent" as some kind of projection by itself. Hence what we do above is basically iterative projection onto iterative set, one for the gradient descent, one for each other.
– Royi
Apr 13 '20 at 15:45
Are you doing FGLS or something?
You could try substituting the constraint into the object. For the two by two case, for example, solve $$\sum_{i = 1}^N \left(x_i'\left[\array{ \array{w_{11} & w_{12}} \\ \array{w_{12}& w_{22} }} \right]x_i - y_i \right)^2$$ where $$w_{12} = w_{21}$$ now by construction. Then the matrix will be symmetric.
To ensure positive semi-definiteness, you can then use the standard principal minors test: $$w_{ii} \ge 0$$ for each $$i$$, $$w_{11} w_{22} - w_{12}^2 \ge 0$$, and so on, with the determinant of the upper left-hand minor weakly positive.
That at least subsumes positive semi-definiteness into concrete constraints and adjustments to the problem. That sounds like a nightmare to solve however, using Kuhn-Tucker. A simpler sufficient condition for semi-definiteness is the dominant diagonal condition, that $$w_{ii} \ge \sum_{j \neq i} |w_{ij}|$$ for each row $$i$$, which would be much more computationally tractable. Perhaps it could give you a good initial guess before you try relaxing it to the standard principal minors constraints.
• There is no need for this. While your approach might lead to problem formulated as Least Squares with Linear Equality and Inequality Constraints, it will also require some kind of iterative solver (As there is no closed form solution for LS with Inequality Constraints). Since we have the projection onto the Symmetric and PSD Matrices we can use it. See my answer. Still, nice idea of yours! Especially with the Diagonally Dominant Matrix. I might post a question and answer about this.
– Royi
Apr 12 '20 at 0:00
This is a convex optimization problem which can be easily formulated, and then numerically solved via a convex optimization tool, such as CVX, YALMIP, CVXPY, CVXR. This is a linear Semidefinite Programming problem (SDP), for which numerical solvers exist.
Here is the code for CVX.
Assume $$x_i$$ is the ith column of matrix $$X$$
cvx_begin
variable W(p,p) semidefinite % constrains W to be symmetric positive semidefinite
Objective = 0;
for i=1:n
Objective = Objective + square(X(:,i)'*W*X(:,i) - y(i))
end
minimize(Objective)
cvx_end
CVX will transform the problem into the form required by the solver, call the solver, and return the solution.
• I think the use of square_pos() here is wrong. It will lead to the wrong solution. You can use square() or if you want to be sure square_abs() or something like that.
– Royi
Apr 11 '20 at 23:35
• @Royi Right you are. Now fixed. Thanks for pointing it out. Do you want to take over for me as moderator at CVX Forum? Apr 11 '20 at 23:43
• Care to explain what you meant? Thanks...
– Royi
Apr 18 '20 at 22:43
• @Royi You can take over answering many of the questions there, and I can go into semi-retirement from that forum. Apr 18 '20 at 23:02
• I would answer you with a PM on CVX Forum. Yet the PM feature is disabled. Want to move it to an email?
– Royi
Apr 22 '20 at 15:08
This does seem to be a straight forward 'nonnegative' least squares though with the sdp constraint. n = 10; p = 5; X = zeros(n,p^2); for ii = 1:n x = randn(p,1); temp = xx'; X(ii,:) = temp(:)'; end y = randn(n,1); cvx_begin sdp variable W(p,p) semidefinite minimize(norm(XW(:)-y)) cvx_end | 2021-09-29T02:48:44 | {
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https://math.stackexchange.com/questions/392822/deduce-that-there-exists-a-prime-p-where-p-divides-x2-2-and-p%E2%89%A13-mod-4 | # Deduce that there exists a prime $p$ where $p$ divides $x^2 +2$ and $p≡3$ (mod 4)
I am revising for a number theory exam and have a question that I am struggling with, any help would be greatly appreciated.
First I am asked to show that for an odd number $x$, $x^2+2 ≡3$(mod 4).
I can do this part of the question, but next I am asked to deduce that there exists a prime $p$ where $p$ divides $x^2 +2$ and $p≡3$ (mod 4)
I am struggling to see how to attempt the second part and how the first part relates.
My thoughts so far are that I want to show $x^2≡-2$(mod p) ? And perhaps Fermat's Little Theorem could be of use here somehow?
Not sure if I'm barking up the wrong tree though.
Consider the prime factorization of $x^2+2$. Since $x$ is odd, $x^2+2$ is odd implying $2$ will not show up in the factorization. Now consider the primes that DO show up in the prime factorization. If they are all $1$ in modulo 4, then their product will also be one in modulo $4$. This is not true though, since you know that $x^2+2$ is $3$ modulo $4$. Therefore, there must be a prime that in the prime factorization of $x^2+2$ s.t. it is not $1$ modulo 4. Since primes other than 2 that are not $1$ modulo 4 are $3$ modulo $4$, this completes the reasoning.
• Fantastic! Thank you so much – Olivia77989 May 15 '13 at 20:53
Our number $x^2+2$ is odd, and greater than $1$, so it is a product of odd not necessarily distinct primes. If all these primes were congruent to $1$ modulo $4$, their product would be congruent to $1$ modulo $4$. But you have shown that $x^2+2\equiv 3\pmod{4}$.
• You are welcome. You probably already ran into the fact that every positive integer of the form $4k+3$ has a prime divisor of the form $4k+3$. That fact plays a key role in the standard proof that there are infinitely many primes of the form $4k+3$. – André Nicolas May 15 '13 at 20:59
• A useful insight! So would I be right in thinking that in order to prove there are infinitely any primes of the form $4k +3$ , I would split $4k +3$ into its prime divisors $p1,p2...pn$ noting that at least one of these divisors is of the form $4k +3$, then, following a similar approach to Euclid's proof, consider N=4(p1. p2.....pn)+3 ? Or what would we equate N to and why? – Olivia77989 May 15 '13 at 21:31
• Yes, suppose there are finitely many primes $p_1,\dots,p_s$ of the right shape. Consider $N=4p_1\cdots p_s-1$. This is of the shape $4k+3$, so has a prime divisor of that shape, which must be different from the $p_i$. I am avoiding $N=4p_1\cdots p_s+3$, since maybe $N$ could be a power of $3$ (there are other ways around that issue). – André Nicolas May 15 '13 at 21:38
• I see. Possibly a silly question, but how do we know that the prime divisor of N in the form $4k+3$ is different from the $p_i$ already in our list? – Olivia77989 May 15 '13 at 21:46
• With $N=4p_1\cdots p_s -1$, if it was in list, it would divide $4p_1\cdots p_s$. Since it divides $4p_1\cdots p_s-1$, it would divide the difference, which is $1$. Impossible! – André Nicolas May 15 '13 at 21:56
The question, as asked, has been answered. Nevertheless, I'll show how quadratic residues can help in problems like these, and show that moreover $x^2+2$ ($x$ odd) has a prime factor congruent to $3 \!\!\!\mod 8$:
Suppose that $x$ is odd. If $p$ divides $x^2+2$, then $p$ is odd and $x^2 \equiv -2 \!\!\mod p$. In particular, $-2$ is a square mod $p$, hence $$1=\left(\frac{-2}{p}\right)=\left(\frac{-1}{p}\right)\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/2} (-1)^{(p-1)/2}=(-1)^{(p^2+p-2)/2}.$$ This implies $p \equiv 1,3 \!\!\mod \!8$. If each prime $p$ dividing $x^2+2$ were congruent to $1 \!\!\mod 8$, then $x^2+2$ would be congruent to $1\!\! \mod 8$. This is a contradiction, hence there exists some $p \mid x^2+2$ congruent to $3 \!\!\mod 8$. (In fact, there must be an odd number of such primes.)
Of course, any one of the primes is congruent to $3 \!\!\mod 4$, which gives your result.
We can ignore quadratic residues in your particular problem because there are only two odd residues mod $4$. In more difficult questions, looking at quadratic residues can give more information. | 2019-07-23T03:24:48 | {
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https://math.stackexchange.com/questions/2398670/why-are-the-last-two-digits-of-a-perfect-square-never-both-odd/2398674 | # Why are the last two digits of a perfect square never both odd?
Earlier today, I took a test with a question related to the last two digits of perfect squares.
I wrote out all of these digits pairs up to $20^2$.
I noticed an interesting property, and when I got home I wrote a script to test it. Sure enough, my program failed before it was able to find a square where the last two digits are both odd.
Why is this?
Is this always true, or is the rule broken at incredibly large values?
• do you know about modular arithmetic ? that might be a starting place. – user451844 Aug 19 '17 at 0:51
• The last two digit of a number is the number modulus $100$ Talking about squares the last two digits are cyclical. They repeat every 50 squares from $0$ to $49$ or from $10^9 + 2017$ to $10^9 + 2017 + 49$ they are always the following $00,\;01,\;04,\;09,\;16,\;25,\;36,\;49,\;64,\;81,\;00,\;21,\;44,\;69,\;96,\;25,\;56,\;89,\;24,\;61,\;00,\;41,\;84,\;29,\;76,\;25,\;76,\;29,\;84,\;41,\;00,\;61,\;24,\;89,\;56,\;25,\;96,\;69,\;44,\;21,\;00,\;81,\;64,\;49,\;36,\;25,\;16,\;09,\;04,\;01$ and there is never a combination of two odd digits. – Raffaele Aug 19 '17 at 14:39
• Not only, but if a number does not end with one of the following pair of digits it cannot be a perfect square $00,\; 01,\; 04,\; 09,\; 16,\; 21,\; 24,\; 25,\; 29,\; 36,\; 41,\; 44,\; \\ 49,\; 56,\; 61,\; 64,\; 69,\; 76,\; 81,\; 84,\; 89,\; 96$ – Raffaele Aug 19 '17 at 14:39
• You can also note that the last two digits of squares from 0 to 25 are the same as from 50 to 25 so it is both cyclical and symetrical :) – Rafalon Aug 20 '17 at 9:21
• @Raffaele Sadly if you'd made that into an answer it probably would have earned you a decent chunk of rep and might have been the accepted answer. – Pharap Aug 20 '17 at 16:24
Taking the last two digits of a number is equivalent to taking the number $\bmod 100$. You can write a large number as $100a+10b+c$ where $b$ and $c$ are the last two digits and $a$ is everything else. Then $(100a+10b+c)^2=10000a^2+2000ab+200ac+100b^2+20bc+c^2$. The first four terms all have a factor $100$ and cannot contribute to the last two digits of the square. The term $20bc$ can only contribute an even number to the tens place, so cannot change the result. To have the last digit of the square odd we must have $c$ odd. We then only have to look at the squares of the odd digits to see if we can find one that squares to two odd digits. If we check the five of them, none do and we are done.
• Ah, it's so obvious in hindsight. Although, I suppose most problems like this are. Thanks! – Pavel Aug 19 '17 at 1:00
Others have commented on the trial method. Just to note that $3^2$ in base $8$ is $11_8$ which has two odd digits. This is an example to show that the observation here is not a trivial one.
But we can also note that $(2m+1)^2=8\cdot \frac {m(m+1)}2+1=8n+1$ so an odd square leaves remainder $1$ when divided by $8$.
The final odd digits of squares can be $1,5,9$ so odd squares are $10p+4r+1$ with $r=0,1,2$. $10p+4r$ must be divisible by $8$ and hence by $4$, so $p$ must be even.
In the spirit of experimentation, the last two digits of the squares of numbers obtained by adding the column header to the row header:
$$\begin {array}{c|ccc} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ \hline 0 & 00 & 01 & 04 & 09 & 16 & 25 & 36 & 49 & 64 & 81\\ 10 & 00 & 21 & 44 & 69 & 96 & 25 & 56 & 89 & 24 & 61\\ 20 & 00 & 41 & 84 & 29 & 76 & 25 & 76 & 29 & 84 & 41\\ 30 & 00 & 61 & 24 & 89 & 56 & 25 & 96 & 69 & 44 & 21\\ 40 & 00 & 81 & 64 & 49 & 36 & 25 & 16 & 09 & 04 & 01\\ 50 & 00 & 01 & 04 & 09 & 16 & 25 & 36 & 49 & 64 & 81\\ 60 & 00 & 21 & 44 & 69 & 96 & 25 & 56 & 89 & 24 & 61\\ 70 & 00 & 41 & 84 & 29 & 76 & 25 & 76 & 29 & 84 & 41\\ 80 & 00 & 61 & 24 & 89 & 56 & 25 & 96 & 69 & 44 & 21\\ 90 & 00 & 81 & 64 & 49 & 36 & 25 & 16 & 09 & 04 & 01\\ 100 & 00 & 01 & 04 & 09 & 16 & 25 & 36 & 49 & 64 & 81\\ 110 & 00 & 21 & 44 & 69 & 96 & 25 & 56 & 89 & 24 & 61\\ 120 & 00 & 41 & 84 & 29 & 76 & 25 & 76 & 29 & 84 & 41\\ \end{array}$$
The patterns are clear, after which the search for a reason for such patterns is well given by the answer of @RossMillikan - you can see that the parity of both final digits of the square is entirely dependent on the final digit of the number that you square.
As a hint, consider what determines the last two digits of a multiplication. Do you remember doing multiplication by hand? If you have square a ten digit number, do all the digits matter when considering just the last two digits of the answer? You will realize that you can put a bound on the number of squares you need to check before you can prove the assertion you are making for all n
• In fact more than enough values of n has been checked (within $20^2$) to see that this result is true. Question is, how does one know enough values have been checked? – Jihoon Kang Aug 19 '17 at 1:02
• That's right - a lazy bound would be checking with a computer all two digit squares, as you know that in a 3 digit number, the hundreds digit will not impact on the final two digits in the multiplication (the same for larger numbers). Obviously you can get sharper than that, as the other answer showed. I was just pointing out that you can very easily come up with a lazy bound by thinking about what happens when you multiply numbers. – Franz Aug 19 '17 at 1:23
This is just another version of Ross Millikan's answer.
Let $N \equiv 10x+n \pmod{100}$ where n is an odd digit.
\begin{align} (10x + 1)^2 \equiv 10(2x+0)+1 \pmod{100} \\ (10x + 3)^2 \equiv 10(6x+0)+9 \pmod{100} \\ (10x + 5)^2 \equiv 10(0x+2)+5 \pmod{100} \\ (10x + 7)^2 \equiv 10(4x+4)+9 \pmod{100} \\ (10x + 9)^2 \equiv 10(8x+8)+1 \pmod{100} \\ \end{align}
A simple explanation.
1. Squaring means multiplication, multiplication means repeatative additions.
2. Now if you add even no.s for odd no. of times or odd no.s for even no. of times you will always get an even no.
Hence, square of all the even no.s are even, means the last digit is always even.
1. If you add odd no.s for odd no. of times you will always get an odd no.
Coming to the squares of odd no.s whose results are >= 2 digits. Starting from 5^2 = 25, break it as 5+5+5+5+5, we have a group with even no. of 5 and one extra 5. According to my point no. 2 the even group will always give you a even no. i.e. 20, means the last digit is always even. Addition of another 5 with 20 makes it 25, 2 is even.
Taking 7^2, 7+7+7+7+7+7+7, group of six 7's = 42 plus another 7 = 49.
Now consider 9^2, 9+9+9+9+9+9+9+9+9, group of eight 9's = 72 plus another 9 = 81, (72+9 gets a carry of 1 making the 2nd last digit even)
35^2 = group of twenty four 34's (1190) plus 35 = 1225, carry comes.
In short just check the last digit of no. that you can think of in the no. co-ordinate (Real and Imaginary) it will always be b/w 0-9 so the basic principle (point 2 and 3) will never change. Either the last digit will be an even or the 2nd last digit will become even with a carry. So the 1 digit sq can come odd, 1 and 9, as there is no carry. I have kept it as an exception in point 3.
BTW many, including the author may not like my lengthy explanation as mine is not a mathematical one, full of tough formulae. Sorry for that. I'm not from mathematical background and never like maths.
• "35^2 = group of twenty four 34's (1190) plus 35 = 1225, carry comes."? One of the burdens of being a mathematician is to present your arguments and then deal with any criticism that follows. It's got nothing to do with you. Whoever you are, if you publish a turkey, someones going to roast it. – steven gregory Aug 20 '17 at 23:13
• That's a typo. That will be odd+odd = even. I saw that error but didn't edit it. – NewBee Aug 23 '17 at 14:26
Let b be last digit of odd perfect square a,then b can be 1,9 or 5. For b=1,9; $a^2-b$ is divisible by 4, $(a^2-b)/10$ is even. For b=5 ;a always ends in 25. | 2019-07-15T22:04:07 | {
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https://math.stackexchange.com/questions/2720518/a-curious-property-of-an-acute-triangle | # A curious property of an acute triangle
Many years back in high school I happened to stumble upon the following property that seems to hold for any acute triangle:
$CD$ and $BE$ are altitudes, the arcs are semicircles with diameters $AB$ and $AC$ respectively.
The property is that $AF = AG$
Proof:
Let $H$ be the midpoint of $AB$ (and the centre of the respective semicircle) $$AG^2 = AD^2 + GD^2 = \left(AC\cdot \cos\angle A\right)^2 + GD^2$$ Since $HG = AH = \frac{AB}{2}$ is the radius of the semicircle $$GD^2 = HG^2 - HD^2 = \left(\frac{AB}{2}\right)^2 - \left(\frac{AB}{2} - AC\cdot\cos\angle A\right)^2 = \\ = AB\cdot AC\cdot \cos\angle A - \left(AC\cdot\cos\angle A\right)^2$$
which gives $$AG^2 = AB\cdot AC\cdot \cos\angle A$$
Analogously ($I$ is the midpoint of $AC$) $$AF^2 = AE^2 + FE^2 = \left(AB\cdot \cos\angle A\right)^2 + FE^2$$ $$FE^2 = FI^2 - EI^2 = \left(\frac{AC}{2}\right)^2 - \left(AB\cdot \cos\angle A - \frac{AC}{2}\right)^2 = \\ = AC\cdot AB\cdot \cos\angle A - \left(AB\cdot \cos\angle A\right)^2$$ which finally gives $$AF^2 = AC\cdot AB\cdot \cos\angle A$$
Questions:
1. Is this a known property?
2. Is there a better more elegant proof?
• You can say something more: The four points where the (full) circles with diameters $\overline{AB}$ and $\overline{AC}$ meet the altitudes from $B$ and $C$, lie on a common circle about $A$.
– Blue
Apr 3 '18 at 16:49
• @Blue, that's a very, very good one! Thank you. Apr 3 '18 at 21:54
HINT:
$$AF^2 = AE \cdot AC\\ AG^2 = AD \cdot AB \\ AE \cdot AC = AD \cdot AB$$
• what a beautiful proof! Apr 3 '18 at 16:42
• Brilliant indeed. Although not so trivial (to me at least). Thanks a lot. Apr 3 '18 at 21:41
• @ageorge: My pleasure! A nice problem indeed! Apr 3 '18 at 22:32
This is a possibly similar proof, but here goes anyway...
Let the triangle be labelled in the usual way, so $AC=b$ and $AB=c$, and let angle $FAC=\theta$
Then $$AF=b\cos\theta$$ $$\implies FE=AF\sin\theta=b\sin\theta\cos\theta$$ $$\implies EC=FE\tan\theta=b\sin^2\theta$$ But $$EC=b-c\cos A=b\sin^2\theta$$ $$\implies b\cos^2\theta=c\cos A$$ $$\implies AF=\sqrt{bc \cos A}$$
To get $AG$ we only need to exchange $b$ and $c$, so the result follows. | 2021-09-29T02:11:28 | {
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http://mathhelpforum.com/algebra/121572-trouble-simplifying-factorising.html | # Math Help - Trouble Simplifying/Factorising
1. ## Trouble Simplifying/Factorising
I have a rather straightforward Physics problem to solve, only I can't for the life of me simplify the answer to what it needs to be. What am I doing wrong?
$
v_1=\frac{m}{m+M}v_o
$
$\frac{1}{2}mv_o^2=\frac{1}{2}mv_1^2+\frac{1}{2}Mv_ 1^2+\frac{1}{2}kd^2$
The answer is in the form $d=$ and it's a simplification of the information I provided here in the post.
I get as far as $d=\sqrt{\frac{1}{k}(mv_o^2-(m+M)v_1^2)}$
The book plugs in $v_1$
and simplifies to $d=\sqrt{\frac{mM}{k(m+M)}}\times{v_o}$
How?
2. Originally Posted by dkaksl
$
v_1=\frac{m}{m+M}v_o
$
$\frac{1}{2}mv_o^2=\frac{1}{2}mv_1^2+\frac{1}{2}Mv_ 1^2+\frac{1}{2}kd^2$
Changing your v1 to v and v0 to w, then your 2 equations are:
v = mw / (m + M) [1]
mw^2 = mv^2 + Mv^2 + kd^2 [2]
Simplifying:
m + M = mw / v [1]
mw^2 - kd^2 = v^2(m + M) [2]
Substitute [1] in [2] ; OK?
3. Very useful, easy-to-understand solution. Thanks a lot.
4. Hello again.
Just got to checking the textbook and unfortunately the question was to define $d$ in the terms $m, M, k$ and $v_o$.
Cancelling out $(m + M)$ got me an answer with $v_1$.
Edit:
$kd^2=mv_o^2-(m+M)v_1^2$
if $v_1=\frac{mv_o}{m+M}$ then what is $v_1^2$?
$\frac{mv_o}{m+M}\times\frac{mv_o}{m+M}$ which is
$\frac{m^2v_o^2}{(m+M)^2}$
$kd^2=mv_o^2-(m+M)\times\frac{m^2v_o^2}{(m+M)^2}$
$kd^2=mv_o^2-\frac{m^2v_o^2}{(m+M)}$
$kd^2=\frac{mv_o^2(m+M)}{(m+M)}-\frac{m^2v_o^2}{(m+M)}$
$kd^2=\frac{mv_o^2(m+M)-m^2v_o^2}{(m+M)}$
$kd^2=\frac{m^2v_o^2+mMv_o^2-m^2v_o^2}{(m+M)}=\frac{mM}{(m+M)}\times{v_o^2}$
$d=\sqrt{\frac{1}{k}\times{\frac{mM}{(m+M)}}}\times {v_o}$
Last edit: Thanks a lot. I see where I made a mistake in expanding. Learned a lot today. Merry Christmas.
5. Originally Posted by dkaksl
Just got to checking the textbook and unfortunately the question was to define $d$ in the terms $m, M, k$ and $v_o$.
Ahhh; well then, we need to get rid of your v1 (my v); the 2 equations:
v = mw / (m + M) [1]
mw^2 = mv^2 + Mv^2 + kd^2 [2]
square [1]: v^2 = m^2w^2 / (m + M)^2
rearrange [2]: v^2 = (mw^2 - kd^2) / (m + M)
m^2w^2 / (m + M)^2 = (mw^2 - kd^2) / (m + M)
m^2w^2 / (m + M) = mw^2 - kd^2
m^2w^2 = mw^2(m + M) - kd^2(m + M)
kd^2(m + M) = mw^2(m + M) - m^2w^2
kd^2(m + M) = mw^2(m + M - m)
kd^2(m + M) = mMw^2
d^2 = mMw^2 / (k(m + M))
d = wSQRT[mM / (k(m + M))] ........Merry Xmas! | 2014-10-23T01:55:01 | {
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http://math.stackexchange.com/questions/199671/what-is-lim-limits-n-to-infty-fracnd-nd-choose-d | # What is $\lim\limits_{n\to\infty} \frac{n^d}{ {n+d \choose d} }$?
What is $\lim\limits_{n\to\infty} \frac{n^d}{ {n+d \choose d} }$ in terms of $d$? Does the limit exist? Is there a simple upper bound interms of $d$?
-
I think the limit sould exist and is a function of $d$. Because the denominator is $\sim n^d$ when $n\rightarrow \infty$ – Seyhmus Güngören Sep 20 '12 at 11:59
Write $$\frac{n^d}{(n+d)!}d!n!=d!\prod_{j=1}^d\frac{n}{n+j}$$ to see that the expected limit is $d!$.
-
You mean "... is $d!$" instead of $n!$. – martini Sep 20 '12 at 12:00
I think there is a little mistake here. – Seyhmus Güngören Sep 20 '12 at 12:02
The denominator in the product should be $n+j$. – celtschk Sep 20 '12 at 12:15
Thanks guys, I've corrected. – Davide Giraudo Sep 20 '12 at 12:26
$$n^d\frac{n!\,d\,!}{(n+d)!}=d\,!\,(\frac{n}{n+1})(\frac{n}{n+2})\ldots(\frac{n}{n+d})$$
Note that the R.H.S. contains a finite number of terms ($=d$) each tending to 1.
Hence the limit is $d\,!$
To find an upper bound you can simply do this
$d\,!\,(\frac{n}{n+1})(\frac{n}{n+2})\ldots(\frac{n}{n+d})<d\,!\,(\frac{n}{n})(\frac{n}{n})\ldots(\frac{n}{n})$
$d\,!\,(\frac{n}{n+1})(\frac{n}{n+2})\ldots(\frac{n}{n+d})<d\,!$
So your upper bound is $d\,!$
Another Method (Just to show that it the sequence is convergent)
Let us treat the given term $n^d\frac{n!\,d\,!}{(n+d)!}$ as term of sequces $\{x_n\}$.
Since we have shown that the sequence is bounded, it will suffice to show that the sequence is monotonically increasing.
We will now show that it is monotonically increasing.
$$\frac{n}{n+r}-\frac{m}{m+r}=\frac{r(n-m)}{(n+r)(m+r)}>0 \quad\forall\; n>m>0\quad \& \quad\forall r>0$$
Let $n>m$
Then $n!>m!$ and each term $\frac{n}{n+r}>\frac{m}{m+r} \quad \forall r \in \{1, 2, \ldots ,d\}$, we have proved this above.
Hecne $x_n \gt m_m$ (as $a>a' \quad \& \quad b\gt b'\implies a.b>a'.b' \quad \forall a,a',b,b'>0$ )
If you want to see how a monotonically increasing sequence which is upper bounded is convergent go to this link.
-
You can use the Stirling approximation $$n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n\,,$$
after writing your expression in the form
$$d!\frac{n^d n!}{(n+d)!}$$
$$d!\frac{n^d n!}{(n+d)!} \sim d! \frac{n^d \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n }{\sqrt{2 \pi (n+d)} \left(\frac{n+d}{e}\right)^{n+d}}= d!\, {\rm e}^d \left(\frac{n}{n+d}\right)^{n} \left(\frac{n}{n+d}\right)^{d} \left(\frac{n}{n+d}\right)^{\frac{1}{2}}$$
$$\rightarrow d! \,{\rm e}^d \, \,{\rm e}^{-d}\, 1.1 = d!\,, \quad n \rightarrow \infty$$
- | 2016-06-30T17:42:06 | {
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https://math.stackexchange.com/questions/2641629/does-an-n-by-n-hermitian-matrix-always-has-n-independent-eigenvectors | # Does an n by n Hermitian matrix always has n independent eigenvectors?
I am learning the MIT ocw 18.06 Linear Algebra, and I have learnt: an arbitrary $n×n$ matrix A with n independent eigenvectors can be written as $A=SΛS^{-1}$, and then for the Hermitian matrices, because the eigenvectors can be chosen orthonormal, it can be written as $A=QΛQ^H$ further.
I wonder does every $n×n$ Hermitian matrix has n independent eigenvectors and why? Thank you!
P.S. MIT 18.06 Linear Algebra Lecture 25: Symmetric Matrices and Positive Definiteness. You may wish to start from 4:20. From the course, I think the spectral theorem comes from diagonalizable matrix, it's just a special case, it's just the case eigenvectors are orthonormal. The eigenvectors of Hermitian matrices can be chosen orthnormal, but is every Hermitian matrix diagonalizable? If it is, why?
• You mean independent eigenvectors, not independent eigenvalues. – uniquesolution Feb 8 '18 at 12:04
• Oh, thanks, I have edited it. – Thomas Feb 9 '18 at 2:28
• It's on the Wikipedia page: en.wikipedia.org/wiki/Hermitian_matrix. – Robert Wolfe Feb 9 '18 at 4:25
• @Robert Thank you for the link, I had read the page though, and there wasn't an answer to my question. – Thomas Feb 9 '18 at 10:34
• Look here for a simple proof. – user Feb 11 '18 at 20:15
This is a theorem with a name: it is called the Spectral Theorem for Hermitian (or self-adjoint) matrices. As pointed out by Jose' Carlos Santos, it is a special case of the Spectral Theorem for normal matrices, which is just a little bit harder to prove.
Actually we can prove the spectral theorem for Hermitian matrices right here in a few lines.
We are going to have to think about linear operators rather than matrices. If $T$ is a linear operator on a finite dimensional complex inner product space $V$, its adjoint $T^*$ is another linear operator determined by $\langle T v, w\rangle = \langle v, T^* w \rangle$ for all $v, w \in V$. (Note this is a basis-free description.) $T$ is called Hermitian or self-adjoint if $T = T^*$.
Let $B$ be an $n$-by-$n$ complex matrix and $B^*$ the conjugate transpose matrix. Let $T_B$ and $T_{B^*}$ be the corresponding linear operators. Then $(T_B)^* = T_{B^*}$, so a a matrix is Hermitian if and only if the corresponding linear operator is Hermitian.
Let $A$ be a Hermitian linear operator on a complex inner product space $V$ of dimension $n$. We need to consider $A$--invariant subspaces of $V$, that is linear subspaces $W$ such that $A W \subseteq W$. We should think about such a subspace as on an equal footing as our original space $V$. In particular, any such subspace is itself an inner product space, $A_{|W} : W \to W$ is a linear operator on $W$, and $A_{|W}$ is also Hermitian. If $\dim W \ge 1$, $A_{|W}$ has an least one eigenvector $w \in W$ -- because any linear operator at all acting on a (non-zero) finite dimensional complex vector space has at least one eigenvector.
The basic phenomenon is this: Let $W$ be any invariant subspace for $A$. Then $W^\perp$ is also invariant under $A$. The reason is that if $w \in W$ and $x \in W^\perp$, then $$\langle w, A x\rangle = \langle A^* w , x \rangle = \langle A w, x \rangle = 0,$$ because $Aw \in W$ and $x \in W^\perp$. Thus $A x \in W^\perp$.
Write $V = V_1$. Take one eigenvector $v_1$ for $A$ in $V_1$. Then $\mathbb C v_1$ is $A$--invariant. Hence $V_2 = (\mathbb C v_1)^\perp$ is also $A$ invariant. Now just apply the same argument to $V_2$: the restriction of $A$ to $V_2$ has an eigenvector $v_2$ and the perpendicular complement $V_3$ to $\mathbb C v_2$ in $V_2$ is $A$--invariant. Continuing in this way, one gets a sequence of mutually orthogonal eigenvectors and a decreasing sequence of invariant subpsaces, $V = V_1 \supset V_2 \supset V_3 \dots$ such that $V_k$ has dimension $n - k + 1$. The process will only stop when we get to $V_n$ which has dimension 1.
• You really hit the nail on the head as I'm afraid my question is too unusual to be understand. May I repeat you answer in my imprecise words? – Thomas Feb 9 '18 at 12:42
• Regard A as an operator on V, its invariant subspaces contains at least one eigenvector, their orthogonal complement are invariant subspaces too because A is a Hermitian matrix, we start from $\mathbb C^n$, every time we take a eigenvector from the space, it on a line which is a invariant subspace, and the space left which like a plane perpendicular to the line is also a invariant subspace, and we can take n times (i.e. n orthogonal/independent eigenvectors). Do I understand it correctly? – Thomas Feb 9 '18 at 12:47
• I still have some trouble with this, what if I take an eigenvector from the last space? As we know "Let $W$ be any invariant subspace for $A$. Then $W^⊥$ is also invariant under $A$", I take the eigenvector out, then there will be only zero vector left, and I think it really orthogonal to that eigenvector we take, is zero vector an invariant subspace? It may be, but "invariant space has at least one eigenvector" would be false. This "Let $W$ be any invariant subspace for $A$. Then $W^⊥$ is also invariant under $A$" makes me confused. – Thomas Feb 9 '18 at 13:43
• When you have reached $V_n$, you already have your collection of $n$ mutually orthogonal eigenvectors. You can't go further and you don't need to go further. I corrected the answer at one point to say that a linear operator on a non-zero finite dimensional complex vector space has an eigenvector. – fredgoodman Feb 9 '18 at 14:19
• It's important in the second paragraph that "acting on" means "mapping it into itself". – Ian Feb 9 '18 at 15:26
Yes, it has. That is due to the spectral theorem: every normal $n\times n$ matrix is diagonalizable. And Hermitian matrices are normal.
• From my understanding, the factorization $A=QΛQ^H$ (i.e., the spectral theorem) is the special form of $A=SΛS^{-1}$ when A is a Hermitian matrix, and $A=SΛS^{-1}$ exists under the situation that there is a full set of independent eigenvectors. I wonder, if it is, why a Hermitian matrix is always diagonalizable? – Thomas Feb 9 '18 at 2:58
• @Thomas Because the spectral theorem says so. – José Carlos Santos Feb 9 '18 at 6:30
• From the course (I add the link into the question now), I think the spectral theorem comes from diagonalizable matrix, it's just a special case, it's just the case eigenvectors are orthonormal. The eigenvectors of Hermitian matrices can be chosen orthnormal, but is every Hermitian matrix diagonalizable? If it is, why? – Thomas Feb 9 '18 at 10:57
• @Thomas Every Heritian matrix is normal and every normal matrix is diagonalizable. – José Carlos Santos Feb 9 '18 at 11:08
• I had not learnt the concept of normal matrix from the course. I search for it now, and I realize that all matrices can be written as $A=QTQ^H$ with triangular T, so $A^HA=AA^H$ would be $T^HT=TT^H$, for a 2 by 2 matrix, it's easy to prove the upper right corner entry of such T is 0, extend it to size n, and we prove that T is diagonal. Is that a right poof? Thank you for your patience. – Thomas Feb 9 '18 at 11:58 | 2019-09-16T08:58:16 | {
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https://math.stackexchange.com/questions/501755/approaching-modular-arithmetic-problems | # Approaching modular arithmetic problems
I'm a little stumbled on two questions.
How do I approach a problem like $x*41 \equiv 1 \pmod{99}$.
And given $2$ modulo, $7x+9y \equiv 0 \pmod{31}$ and $2x−5y \equiv 2 \pmod{31}$ (solve for $x$ only)?
When I solve for $x$ for the latter, I got a fraction as the answer and I'm not sure if I can have a fraction as an answer? I'm not sure how to approach the first problem either.
• What fraction did you get as answer? – Git Gud Sep 22 '13 at 22:20
• The first problem involves inverses. What is the inverse of 41 mod 99? The second problem is a set of linear equations in $x$ and $y$ modulo 31, which means the two can be added/subtracted/etc. as if they were a set of normal equations looking for a linear solution. – abiessu Sep 22 '13 at 22:25
Finding the solution to $$x \times 41 \equiv 1 \pmod {99}$$ is equivalent to asking for the multiplicative inverse of $41$ modulo $99$. Since $\gcd(99,41)=1$, we know $41$ actually has an inverse, and it can be found using the Extended Euclidean Algorithm:
\begin{align*} 99-2 \times 41 &= 17 \\ 41-2 \times 17 &= 7 \\ 17-2 \times 7 &= 3 \\ 7-2\times 3 &= 1 &=\gcd(99,41). \\ \end{align*} Going back, we see that \begin{align*} 1 &= 7-2\times 3 \\ &= 7-2\times (17-2 \times 7) \\ &= 5 \times 7-2\times 17 \\ &= 5 \times (41-2 \times 17)-2\times 17 \\ &= -12 \times 17+5 \times 41 \\ &= -12 \times (99-2 \times 41)+5 \times 41 \\ &= 29 \times 41-12 \times 99 \\ \end{align*} Hence $29 \times 41 \equiv 1 \pmod {99}$ and thus $x=29$.
In the second case, we have $$7x+9y \equiv 0 \pmod {31}$$ and $$2x-5y\equiv 2 \pmod {31}.$$ Here we want to take $7x+9y=0 \pmod {31}$ and rearrange it to get $x \equiv ?? \pmod {31}$, then substitute it into the other equation and solve for $y$. This requires finding the multiplicative inverse of $7$ modulo $31$ (which we can do as above). It turns out $7 \times 9 \equiv 1 \pmod {31}$. Hence \begin{align*} & 7x+9y=0 \pmod {31} \\ \iff & 7x \equiv -9y \pmod {31} \\ \iff & x \equiv -9y \times 9 \pmod {31} \\ \iff & x \equiv 12y \pmod {31}. \end{align*} We then substitute this into the equation $2x-5y\equiv 2 \pmod {31}$, which implies $$2 \times 12y-5y \equiv 2 \pmod {31}$$ or equivalently $$19y \equiv 2 \pmod {31}.$$ Yet again, we find a multiplicative inverse, this time of $19$ modulo $31$, which turns out to be $18$. So \begin{align*} & 19y \equiv 2 \pmod {31} \\ \iff & y \equiv 2 \times 18 \pmod {31} \\ \iff & y \equiv 5 \pmod {31}. \end{align*} Hence $$x \equiv 12y \equiv 29 \pmod {31}.$$ Thus we have the solution $(x,y)=(29,5)$.
• Are you sure the answer to the first part is x=20? – Mufasa Sep 22 '13 at 22:35
• Thanks for point that out; it was an arithmetic error (actually, I found the inverse mod 91 by mistake) and it should be fixed now. – Rebecca J. Stones Sep 22 '13 at 22:47
• You're welcome. BTW: I am learning a lot from your posts - so THANK YOU! :) – Mufasa Sep 22 '13 at 22:49
For the first one, you could approach it as follows:
$41x=1$ mod $99$
$140x=1$ mod $99$ (because 41 mod 99 = (41+99) mod 99)
$140x=100$ mod $99$ (because 1 mod 99 = (1+99) mod 99)
$7x=5$ mod $99$ (divide both sides by 20)
$7x=203$ mod $99$ (because 5 mod 99 = (5+99+99) mod 99)
$x=29$ mod $99$ (divide both sides by 7)
For the first one, you have to find a multiplicative inverse for $41$ mod $99$. Use Euclid's algorithm to find the solutions of $41\cdot x + 99 \cdot y = 1$ to find $x$.
Are you familiar with abstract algebra? If yes, do you know that $\mathbb{Z}_{31}$ is a field because $31$ is a prime number and you can use tools of linear algebra because it works overy any field?
You have the following system of equations in $\mathbb{Z}_{31}$:
$$7x+9y=0$$ $$2x-5y=2$$
You can use Gaussian elimination in $\mathbb{Z}_{31}$ to find $x$ and $y$. But you have to be careful that your coefficients are in $\mathbb{Z}_{31}$ not in $\mathbb{R}$.
EDIT(suggested by Git Gud):
Notice that if you get a fraction like $\displaystyle \frac{a}{b}$ then you should think of it as $a \cdot b^{-1}$ and then find $b^{-1}$ in $\mathbb{Z}_{31}$. As previously said, $\mathbb{Z}_{31}$ is a field, so talking about fractions in it makes sense. The notation $\frac{a}{b}$ is actually nothing but $a.b^{-1}$ where $.$ denotes the multiplication in the field and $b^{-1}$ denotes the multiplicative inverse of $b$ in the field.
• @GitGud: OK. Done. – user66733 Sep 22 '13 at 22:32 | 2019-12-06T18:23:52 | {
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https://www.physicsforums.com/threads/properties-of-the-absolute-value.246785/ | Properties of the Absolute Value
1. Jul 25, 2008
roam
Just wanted to say hi before I start my post!
As you may know there is a property of the absolute value that states; for $$a, b \in R$$;
$$|ab| = |a||b|$$
Well, my friend asked me if I knew a proof for this... but I don't know...
How can we prove this statement/property? I know there is a proof for the triangle inequality but for this I really don't know but I'm curious.
I'd appreciate it if anyone could show me any kind of proof or send me some links etc. Thanks!
2. Jul 25, 2008
HallsofIvy
Staff Emeritus
You prove that |ab|= |a||b| the same way you prove any such elementary statement: use the definitions.
The simplest definition of |x| (there are several equivalent definitions) is that |x|= x if x is positive or 0, -x if x is negative.
Now break it into "cases":
case 1: x and y are both positive: |x|= x and |y|= y. xy is also positive so |xy|= xy= |x||y|.
case 2: x is positive while y is negative: |x|= x and |y|= -y. xy is negative so |xy|= -xy= x(-y)= |x||y|.
case 3: x is negative while y is positive: |x|= -x and |y|= y. xy is negative so |xy|= -xy= (-x)y= |x||y|.
case 4: x and y are both negative: |x|= -x and |y|= -y. xy is positive so |xy|= xy= (-x)(-y)= |x||y|.
case 5: x= 0 and y is positive: |x|= 0 and |y|= y. xy= 0 so |xy|= 0= 0(y)= |x||y|.
case 6: x= 0 and y is negative: |x|= 0 and |y|= -y. xy= 0 so |xy|= 0= (0)(-y)= |x||y|.
case 7: x is positive and y is 0: |x|= x and |y|= 0. xy= 0 so |xy|= 0= x(0)= |x||y|.
case 8: x is negative and y is 0: |x|= -x and |y|= 0. xy= 0 so |xy|= 0 = (-x)(0)= |x||y|.
case 9: both x and y are 0: |x|= 0 and |y|= 0. xy= 0 so |xy|= 0= (0)(0)= |x||y|.
There are simpler ways to prove that but I thought this would be conceptually clearest.
3. Jul 25, 2008
sagardipak
First, we have to understand that the absolute value is a function defined by:
$$|x| = \begin{cases} x & \text{if } x\geq 0 \\ -x & \text{if } x<0 \end{cases}$$
So,
$$|ab| = \begin{cases} ab & \text{if } ab\geq 0 \\ -ab & \text{if } ab<0 \end{cases}$$
Now, let's see what |a||b| is:
$$|a||b| = \begin{cases} ab & \text{if } a\geq 0 \wedge b\geq0 \\ (-a)(-b) & \text{if } a\leq 0 \wedge b\leq0 \\ (-a)b & \text{if } a> 0 \wedge b<0 \\ a(-b) & \text{if } a<0 \wedge b<0 \end{cases} \Leftrightarrow$$
$$|a||b| = \begin{cases} ab & \text{if } a\geq 0 \wedge b\geq0 \\ ab & \text{if } a\leq 0 \wedge b\leq0 \\ -ab & \text{if } a> 0 \wedge b<0 \\ -ab & \text{if } a<0 \wedge b<0 \end{cases} \Leftrightarrow$$
Notice that you have ab if a and b have the same sign and that you use -ab otherwise.
Now, if a and b have the same sign, $$ab\geq0$$. If they have opposite signs (and are different than zero), $$ab<0$$.
Using this,
$$|a||b| = \begin{cases} ab & \text{if } ab \geq 0 \\ -ab & \text{if } ab<0 \end{cases} = |ab|$$
Quod erat demonstrandum :tongue2:
Last edited: Jul 25, 2008
4. Jul 28, 2008
quark1005
let a = mcis(kpi) where m => 0, k is integer
and b = ncis(hpi) where n => 0, h integer
(clearly a,b are real)
|a||b|= mn
|ab|=|mcis(kpi)*ncis(hpi)| = |mncis[pi(k+h)]| = mn
as required
5. Jul 28, 2008
HallsofIvy
Staff Emeritus
This would make more sense if you had said that "mcis(hpi)" is
$$m (cos(h\pi)+ i sin(h\pi))$$
That is much more an "engineering notation" than mathematics.
If you really want to go to complex numbers, why not
if
$$x= r_xe^{i\theta_x}$$
and
$$y= r_ye^{i\theta_y}$$, then
$$|xy|= |r_xe^{i\theta_x}r_ye^{i\theta_y}|= |(r_xr_y)e^{i(\theta_x+\theta_y)}|$$
But for any $z= re^{i\theta}$, |z|= r, so
$$|xy|= r_x r_y= |x||y|$$
6. Jan 8, 2011
Vincent Mazzo
$$\Huge |ab|=\sqrt{(ab)^2}=\sqrt{a^2b^2}=\sqrt{a^2}\sqrt{b^2}=|a||b|$$
and
$$\large |a+b|=\sqrt{(a+b)^2}=\sqrt{a^2+2ab+b^2}\leq \sqrt{a^2+|2ab|+b^2}=\sqrt{(|a|+|b|)^2}=||a|+|b||$$
where I utilize the following fact:
|2ab|=|2(ab)|=|2||ab|=2|a||b|
-----
À bientôt
?;-D | 2017-03-23T10:27:05 | {
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https://math.stackexchange.com/questions/1436807/even-functions-and-inflection-points/1436819 | # Even functions and inflection points
Can an even function have an inflection point? If yes, then give an example while if not then give a proof. If needed you may assume that $f$ is two times differentiable .
Intution says that even functions don't have inflection points, but I cannot settle down an appropriate proof.
On the contrary for an odd function this is possible. For instance $f(x)=x^3$ has an inflection point at $x=0$.
• Does it have to be at $x=0$? – mickep Sep 15 '15 at 18:22
• Thank you all for your answers. – Tolaso Sep 15 '15 at 18:58
$\cos\colon\mathbb{R}\to\mathbb{R}$ is even and it has infinitely many inflection points (at its zeros, which are found at $\{n\pi\}_{n\in\mathbb{Z}}$).
Consider $$f(x)=\max\bigl((x-1)^3,-(x+1)^3\bigr).$$ Or, for a nicer example, consider $$g(x)=(x-1)^3(x+1)^3.$$
the function $y=x^4-4x^2$ has at least one inflection point WA plot | 2021-06-20T09:46:49 | {
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https://web2.0calc.com/questions/geometry-question_64 | +0
# Geometry question
+1
125
11
In triangle PQR, PQ=15 and QR=25 The circumcircle of triangle PQR is constructed. The tangent to the circumcircle at Q is drawn, and the line through P parallel to this tangent intersects QR at S. Find RS.
Jan 16, 2020
#1
-1
RS = 12.
Jan 17, 2020
#2
+107414
+2
THERE WAS AN ERROR SO I DELETED THE PICS.
Jan 17, 2020
edited by Melody Jan 17, 2020
edited by Melody Jan 17, 2020
#3
+28449
+2
Deleted as for some reason I used entirely the wrong lengths of triangle legs!
Alan Jan 17, 2020
edited by Alan Jan 17, 2020
edited by Alan Jan 17, 2020
#5
+28449
+1
I now think Guest #4 below is correct. There is a unique answer. Because that answer is independent of the shape of the triangle it is legitimate to choose an easy one (a right-angled one) on which to do the calculation:
RS is indeed 16.
Alan Jan 17, 2020
#4
+1
I think that the answer is unique, and I think that it is 16.
Using Guest's diagram, the triangles QPR and QSP are similar.
The angle between QP and the tangent, call it alpha, is equal to the angle QPS and also the angle QRP.
Similarly the angle between QR and the tangent is equal to the angle QSP and also the angle QPR.
So, QP/QS = QR/QP,
QS = QP.QP/QR =225/25 = 9, so SR = 16.
That fits the approximate values that Melody found by drawing.
Alan seems to chosen the particular case where angle PQR is a right angle.
Using the angles I called alpha earlier, from the triangle PQR, tan(alpha) = 15/25 = 3/5.
From the triangle PQS, tan(alpha) = QS/15, so QS = 45/5 = 9, as above.
Jan 17, 2020
#6
+107414
+1
Yes you are right guest.
I fixed an error in mine and now it always stays at 16.
I wss see if I can work out how to give a proper link to what i have created.
https://www.geogebra.org/classic/ekztcz7y
Melody Jan 17, 2020
#7
+107414
0
Thanks Guest,
I am just trying to make sense of your answer.
Why is $$\angle QRP = \alpha$$ ?
Melody Jan 17, 2020
#9
+1
Love the geogebra Melody.
In answer to your question, it's a standard angle property of a circle.
Take a diameter from Q across the circle to T on the other side.
Angle QPT will be right angle,since it's based on the diameter, and then angle PTQ will equal the angle between PQ and the tangent, ( subtract from 90 twice).
Then, all angles based on the same arc, PQ, are equal etc.
Guest Jan 17, 2020
#11
+107414
0
Thanks guest for the complement as well as for the explanation.
I am all clear now.
Thanks Heureka and Alan as well
Melody Jan 18, 2020
#8
+24031
+2
In triangle PQR, PQ=15 and QR=25 The circumcircle of triangle PQR is constructed.
The tangent to the circumcircle at Q is drawn, and the line through P parallel to this tangent intersects QR at S.
Find RS.
$$\text{Let C is the center of the circle } \\ \text{Let PS=TS=h } \\ \text{Let RS=x } \\ \text{Let QS=25-x }$$
$$\begin{array}{|lrcll|} \hline 1. & \left(25-x\right)^2+h^2 &=& 15^2 \qquad \text{or}\qquad \mathbf{ h^2 = 15^2-\left(25-x\right)^2} \\\\ 2. & x(25-x) &=& h * h \qquad \text{Intersecting chords theorem} \\ & x(25-x) &=& h^2 \\ & x(25-x) &=& 15^2-\left(25-x\right)^2 \\ & 25x-x^2 &=& 15^2-( 25^2-50x+x^2) \\ & 25x-x^2 &=& 15^2-25^2+50x-x^2 \\ & 25x &=& 15^2-25^2+50x \\ & 25^2-15^2 &=& 25x \\ & 25x &=& 25^2-15^2 \\ & 25x &=& 400 \\ & \mathbf{x} &=& \mathbf{16} \\ \hline \end{array}$$
Jan 17, 2020
#10
+24031
+1
In triangle PQR, PQ=15 and QR=25 The circumcircle of triangle PQR is constructed.
The tangent to the circumcircle at Q is drawn, and the line through P parallel to this tangent intersects QR at S.
Find RS
General solution:
$$\text{Let Center is the center of the circle }\\ \text{Let P-Center =T-Center=v }\\ \text{Let S-Center =w }\\ \text{Let T-Center =w+k=v\qquad k=v-w }\\ \text{Let ST = k =v-w }\\ \text{Let PS = v+w }\\ \text{Let RS = x }\\ \text{Let QS = 25-x }\\ \text{Let Q-Center = u }$$
Intersecting chords theorem:
$$\begin{array}{|rcll|} \hline RS*QS &=& ST*PS \\ x*(25-x) &=& (v-w)*(v+w) \\ \mathbf{x*(25-x)} &=& \mathbf{v^2-w^2} & (1) \\ \hline \end{array}$$
Pythagoras:
$$\begin{array}{|lrcll|} \hline & v^2+u^2 &=& 15^2 & (2) \\ & w^2+u^2 &=& (25-x)^2 & (3) \\ \hline (2)-(3): & v^2 -w^2 &=& 15^2-(25-x)^2 \quad | \quad \mathbf{x*(25-x)=v^2-w^2} \\ & x*(25-x) &=& 15^2-(25-x)^2 \\ &25x-x^2 &=& 15^2 -(25^2-50x+x^2) \\ &25x-x^2 &=& 15^2 -25^2+50x-x^2 \\ &25x &=& 15^2 -25^2+50x \\ &25^2-15^2 &=& 25x \\ & 25x &=& 25^2-15^2 \\ & 25x &=& 400 \quad |\quad :25 \\ & \mathbf{x} &=& \mathbf{16} \\ \hline \end{array}$$
Jan 17, 2020 | 2020-02-16T18:41:17 | {
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https://web2.0calc.com/questions/3-00-x-10-8-6-50-x-10-7-the-result-is-4-61-x-10-14-my-question-is-how-10-8-10-7-10-14-thank-u | +0
# 3.00 x 10^8 / 6.50 x 10^-7 the result is 4.61 x 10^14 my question is how 10^8 / 10^-7 = 10^14? thank u
0
488
2
3.00 x 10^8 / 6.50 x 10^-7
4.61 x 10^14
my question is how 10^8 / 10^-7 = 10^14?
thank u
Guest Nov 21, 2014
#1
+92775
+5
3.00 x 10^8 / 6.50 x 10^-7
4.61 x 10^14
my question is how 10^8 / 10^-7 = 10^14? IT DOES NOT!
$$\\(3\times 10^8)/(6.5\times 10^{-7})\\\\ =(3/6.5)\times 10^8/10^{-7}\\\\ =0.4615\times 10^{8--7}\\\\ =0.4615\times 10^{15}\\\\$$
BUT this answer is to be in scientific notation so there mus be just one non-zero number in front of the point
$$\\=4.4615\times 10^{-1} \times 10^{15}\\\\ =4.4615\times 10^{14}\qquad \mbox{Correct to 4 significant figures}\\\\$$
So does that all make sense now?
Melody Nov 21, 2014
#1
+92775
+5
3.00 x 10^8 / 6.50 x 10^-7
4.61 x 10^14
my question is how 10^8 / 10^-7 = 10^14? IT DOES NOT!
$$\\(3\times 10^8)/(6.5\times 10^{-7})\\\\ =(3/6.5)\times 10^8/10^{-7}\\\\ =0.4615\times 10^{8--7}\\\\ =0.4615\times 10^{15}\\\\$$
BUT this answer is to be in scientific notation so there mus be just one non-zero number in front of the point
$$\\=4.4615\times 10^{-1} \times 10^{15}\\\\ =4.4615\times 10^{14}\qquad \mbox{Correct to 4 significant figures}\\\\$$
So does that all make sense now?
Melody Nov 21, 2014
#2
+92775
0
That was a good question anon
Melody Nov 21, 2014 | 2018-07-17T21:21:19 | {
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http://math.stackexchange.com/questions/1012730/can-a-set-containing-0-be-purely-imaginary | # Can a set containing 0 be purely imaginary?
A purely imaginary number is one which contains no non-zero real component.
If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary?
My issue here is that because $0+0i$ belongs to multiple sets, not just purely imaginary, is there not a valid case to say that the sequence isn't purely imaginary?
-
I think it would simplify your question a bit to just ask "Is $\textit{0}$ purely imaginary?" – curious Nov 9 '14 at 1:16
But my question is why would I consider only one classification 0+0i and ignore the others – chris Nov 9 '14 at 1:19
0 is both purely real and purely imaginary. The given set is purely imaginary. That's not a contradiction since "purely real" and "purely imaginary" are not fully incompatible. Somewhat similarly baffling is that "all members of X are even integers" and "all members of X are odd integers" is not a contradiction. It just means that X is an empty set.
-
$\ldots$ and $0$ is unique in being both purely real and purely imaginary. – Thumbnail Nov 9 '14 at 10:06
A complex number is said to be purely imaginary if it's real part is zero. Zero is purely imaginary, as it's real part is zero.
-
This is really important in fields like ordinary differential equations, where you look at having no purely imaginary eigenvalues for it to be hyperbolic – Alan Nov 9 '14 at 1:23
from my understanding zero can also be considered real? – chris Nov 9 '14 at 1:30
Yes, a complex number is real if it's imaginary part is zero. So zero is also real. – Seth Nov 9 '14 at 1:33
so then my question is why can I consider only the definition that suits my needs? – chris Nov 9 '14 at 1:38
By definition, $0$ is purely imaginary. The fact that $0$ has other properties (it is real; it is nonnegative; it is rational; it is an integer; it is algebraic; it is divisible by every prime number) does not mean you can’t use the property you need. Similarly, is $\{-2,4\}$ a set of even numbers? Yes. The number $-2$ is not only even, but it’s also negative. The fact that it’s negative doesn’t mean you can’t use the fact that it’s even. Some sets defined by properties overlap. – Steve Kass Nov 9 '14 at 1:53 | 2016-02-14T10:15:27 | {
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https://math.stackexchange.com/questions/3421041/how-can-i-find-the-speed-and-the-angle-of-a-water-droplet-in-a-rainy-day-as-seen | # How can I find the speed and the angle of a water droplet in a rainy day as seen by a person running?
The problem is as follows:
A person is running following a constant speed of $$4.5\,\frac{m}{s}$$ over a flat (horizontal) track on a rainy day. The water droplets fall vertically with a measured speed of $$6\,\frac{m}{s}$$. Find the speed in $$\frac{m}{s}$$ of the water droplet as seen by the person running. Find the angle to the vertical should his umbrella be inclined to get wet the less possible. (You may use the relationship of $$37^{\circ}-53^{\circ}-90^{\circ}$$ for the $$3-4-5$$ right triangle ).
The alternatives given on my book are as follows:
$$\begin{array}{ll} 1.7.5\,\frac{m}{s};\,37^{\circ}\\ 2.7.5\,\frac{m}{s};\,53^{\circ}\\ 3.10\,\frac{m}{s};\,37^{\circ}\\ 4.10\,\frac{m}{s};\,53^{\circ}\\ 5.12.5\,\frac{m}{s};\,37^{\circ}\\ \end{array}$$
On this problem I'm really very lost. What sort of equation should I use to get the vectors or the angles and most importantly to get the relative speed which is what is being asked.
I assume that to find the relative speed can be obtaining by subtracting the speed from which the water droplet is falling to what the person is running. But He is in this case running horizontally. How can I subtract these?
Can somebody give me some help here?
Supposedly the answer is the first option or $$1$$. But I have no idea how to get there.
• The $3-4-5$ triangle is a hint as to the nature of the result. He is traveling horizontally, the droplets travel vertically, the resulting relative speed must combine these quantities in some manner. Since the two lines of travel are perpendicular, it makes sense to think about the hypotenuse between them... Nov 4 '19 at 3:43
• @abiessu Perhaps can you develop an answer so I could understand what you mean by combining the speeds?. Can you show me the equation should I use?. Had I not been given the alternatives, what should I have done with this question? Nov 4 '19 at 4:27
• water droplet speed as seen from a stationary point on earth is $\vect{v_d}=(0,-6)$. the speed of the runner as seen from the same point is $\vect{v_r}=(4.5,0)$. however, from the perspective of the runner, the water droplet both falls down with the speed $\vect{v_d}=(0,-6)$ and it moves towards him with the horizontal speed $\vect{v_h}=(-4.5,0)$. wherefore, for the runner the water moves with a resultant speed $\vect{v_r}=\vect{v_d}+\vect{v_h}=(0,-6)+(-4.5,0)=(-4.5,-6)$. this vector has a magnidue of 7.5 and a direction of $37^0$ West from vertical. Nov 4 '19 at 6:49 | 2021-10-16T13:00:36 | {
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https://math.stackexchange.com/questions/710484/is-there-a-general-formula-for-the-sum-of-a-quadratic-sequence/1239788 | Is there a general formula for the sum of a quadratic sequence?
I tried Googling "formula for sum of quadratic sequence", which did not give me anything useful. I just want an explicit formula for figuring out a sum for a quadratic sequence. For example, how would you figure out the sum of $2+6+12+20+\dots+210$? Can someone please help? Thanks
For those of you who do not know, a quadratic sequence is a sequence where the differences of the differences between the terms are constant. Let's use $2+6+12+20+\dots$ as an example. The differences between the terms are $4$, $6$, $8$, etc. The difference between the differences of the terms is $2$. So the sequence will continue like $2+6+12+20+30+42+56+72+\dots$
• Is it homework? – draks ... Mar 13 '14 at 6:45
• @draks... No it is not homework. I am just wondering as I suck at sequences and series problems – TrueDefault Mar 13 '14 at 6:46
• ...I was going to write an answer saying, "no", because I thought you meant quadratic recurrences which are far more complex. Quadratics have a sum which is a cubic equation, so take any four points and do Lagrange interpolation. – Charles Mar 13 '14 at 22:46
• When the $k$th difference of a sequence is constant, one can write down the general term using Newton's formula. $f(n) = f(0) + \frac{\Delta f(0)}{1!} n + \frac{\Delta^2 f(0)}{2!}n(n-1) + \cdots$, where $\Delta f$ are the successive differences. For quadratic sequences, one can then use the usual sum formulas. – user348749 Jul 5 '16 at 8:10
Yes there is. Ever wonder why this is called quadratic sequence? Quadratic refers to squares right? This is just constant difference of difference. So where's the connection? Well as it turns out, all terms of a quadratic sequence are expressible by a quadratic polynomial. What do I mean? Consider this
$$t_n = n+n^2$$
Subsituiting $n=1,2,3,\cdots$ generates your terms. By the way, $202$ doesn't occur in this sequence, the 13th term is $182$ and the $14th$ term is $210$. I am assuming it was supposed to be $210$.
So we need to find
$$\sum_{i=1}^{n}i+i^2 = \sum_{i=1}^{n}i+\sum_{i=1}^{n}i^2$$
where $n=14$. There are well known formulas for $\sum_{i=1}^{n}i$ and for $\sum_{i=1}^{n}i^2$. Substituting them, we get,
$$\frac{n(n+1)}{2} + \frac{n(n+1)(2n+1)}{6}$$ $$=\frac{n(n+1)}{2}\left(1+\frac{2n+1}{3}\right)$$ $$=\frac{n(n+1)}{2}\left(\frac{3+2n+1}{3}\right)$$ $$=\frac{n(n+1)}{2}\left(\frac{2n+4}{3}\right)$$ $$=\frac{n(n+1)(n+2)}{3}$$
where $n=14$. Thus our sum is $1120$.
• You again ! I surrender. Cheers. – Claude Leibovici Mar 13 '14 at 6:43
• Woopsies I will fix that error – TrueDefault Mar 13 '14 at 6:44
For the general case, N terms are to be summed
$$S=a_0+a_1+a_2+...+a_{N-1}$$
The formula for the n-th term is $$a_n=a_0+(a_1-a_0)n+(a_2-2a_1+a_0)\frac{n(n-1)}{2}$$
Using the results
$$\sum_{n=0}^{N-1}n=\frac{N(N-1)}{2}$$
and
$$\sum_{n=0}^{N-1}n(n-1)=\frac{N(N-1)(N-2)}{3}$$
$$S=\sum_{n=0}^{N-1}a_n=a_0N+(a_1-a_0)\frac{N(N-1)}{2}+(a_2-2a_1+a_0)\frac{N(N-1)(N-2)}{6}$$
Although this may not be needed as of now; but I thought about sums of quadratic sequences myself and I managed to derive a general formula for it, so I might as well post it here:
Where $n$ is the number of terms to compute, is the starting term, $d$ is the first difference and is the constant difference between the differences. I use the subscript to denote that it is the quadratic sequence sum function.
• Welcome to math stackexchange. When it isn't too difficult, it is usually preferable to typeset the mathematics than to include links to images. Thanks for the answer. – TravisJ Apr 18 '15 at 0:53
In this particular case, \begin{align} &2+6+12+20+\cdots+210\\\\ &=2(1+3+6+10+\cdots+105)\\\\ &=2\left[\binom 22 +\binom 32 +\binom 42+\binom 52+\cdots \binom {15}2\right]\\ &=2\sum_{r=2}^{15}\binom r2\\ &=2\binom {16}3\\ &=\color{red}{1120}\qquad\blacksquare \end{align}
This is the sum of triangular numbers (where the difference of the difference is constant) and the result is a pyramidal number (all scaled by 2). The summation can be shown as
\begin{align} &2\cdot (1)+\\ &2\cdot (1+2)+\\ &2\cdot (1+2+3)+\\ &2\cdot (1+2+3+4)+\\ &\quad \vdots\qquad\qquad\qquad\ddots\\ &2\cdot (1+2+3+\cdots+15) \end{align}
If we tried to sum a series where the difference of the difference of the difference is constant, i.e. sum of pyramidal numbers, the result would be a pentatope number. And so on...
An example of the summation of pyramidal numbers, extending from the original question, would be
$$2+8+20+40+\cdots+910 =2 \sum_{r=1}^{15} \binom r3=2 \binom {16}4=1820\\$$
I figured out the below way of doing it just know at one o'clock right before bedtime, so if it is faulty than that is my mistake.
Any series has a certain term-to-term rule. For a quadratic that term-to-term rule is in the form
$ax^2+bx+c$
The series will simply be that term-to-term rule with $x$ replaced by $0$, then by $1$ and so on. This can be written as
$\sum_{x=0}^p ax^2+bx+c$
We already know that
$\sum_{x=0}^p x=\frac{x(x+1)}{2}$
$\sum_{x=0}^p x^2=\frac{x(x+1)(2x+1)}{6}$
$\sum_{x=0}^p c=c*p$
These can all be proved easily and you can find said proofs online. Because addition is commutative (the order does not matter) the quadratic sum above rewritten as
$a*\sum_{x=0}^p x^2+b*\sum_{x=0}^p x + \sum_{x=0}^p c$
using the above identities, this can be simplified to
$a*\frac{p(p+1)(2p+1)}{6}+b*\frac{p(p+1)}{2}+c*p$
In your example the term-to-term rule is
$1x^2+1x+0$
And we want to find the sum until the term that will give us $210$. This number can be calculated to be $14$. That means that $14^2+14=210$. So in our formula
$p=14$
$a=1$,
$b=1$
$c=0$.
Plug that in and your sum is $980$
I have a sum formula for quadratic equation.
$$s=\frac{n}{6} (3d(n-1)+(n-1)(n-2)c) +an$$
Where $n$ is the number of terms, $d$ is the first difference, $c$ is the constant difference or difference of difference, $a$ is first term. For calculation of sum first put value of $a$ $d$ $c$ then you get a formula like $an^2+bn+c$. | 2019-12-10T02:39:49 | {
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https://math.stackexchange.com/questions/3545737/how-to-define-the-rth-harmonic-number-if-r-is-any-real-number | How to define the $r$th harmonic number if $r$ is any real number?
What would be the value of
$$\sum_{k=1}^{\sqrt{\frac{5}{2}}}\frac{1}{k}$$
Is it $$H_{\sqrt{\frac{5}{2}}}$$?
Using different definitions of harmonic numbers this question can be computed, but can I use the usual definition of harmonic numbers for any real numbers?
In other words is $$\sum_{k=1}^{n}\frac{1}{k}$$
A useful definition for any real $$n$$?
When I want to compute the value of $$H_{\sqrt{\frac{5}{2}}}$$ at Desmos I just define $$\sum_{k=1}^{n}\frac{1}{k}$$ and I don't put $$\sqrt{\frac{5}{2}}$$ directly at the upper limit, but I define $$n=\sqrt{\frac{5}{2}}$$ and the result is exactly what it should be, it seems using a simple substitution will give us the right answer, but generally are we allowed to use the regular definition of harmonic numbers when our $$n$$ is not necessarily a natural number?
• I would define the $r$th harmonic number for a real number $r$ which is not a negative integer for example by $H_r = \gamma +\psi(r+1)$, where $\gamma$ is the Euler-Mascheroni constant and $\psi$ is the digamma function – Maximilian Janisch Feb 13 at 22:29
• Things are even more complicated. Suppose you have a function of $f(n)$ in the domain of real $n$ for which $f(n) = H_n$ for positive integers $n$. Then $g(n) = f(n) + a \sin(\pi n)$ is another extension of $H_n$ to real $n$. Hence you need an additional condition to rule out $a\ne0$. – Dr. Wolfgang Hintze Feb 14 at 7:58
Indefinite Sum concept is the answer to your question.
In fact, if a function $$f(x)$$ is the (forward) difference of a function $$F(x)$$ $$f(x) = \Delta _x F(x) = F(x + 1) - F(x)$$ then we say that $$F(x)$$ is the "antidifference" (or "indefinite sum") of $$f(x)$$ $$F(x) = \Delta _x ^{\left( { - 1} \right)} f(x) = \sum\nolimits_x {f(x)}$$
If $$f(x)$$ and $$F(x)$$ are defined over a real, or complex, domain for $$x$$ then we will have for example $$\sum\limits_{k = 0}^n {f(x + k)} = \sum\limits_{k = 0}^n {\left( {F(x + 1) - F(x)} \right)} = F(x + n + 1) - F(x)$$
We write the above with a different symbol for the sum as $$\sum\nolimits_{k = 0}^{\,n} {f(x + k)} = \sum\limits_{k = 0}^{n - 1} {f(x + k)} = F(x + n) - F(x)$$ which can also be written as $$\sum\nolimits_{k = x}^{\,x + n} {f(k)} = F(x + n) - F(x)$$
The extension to $$\sum\nolimits_{k = a}^{\,b} {f(k)} = F(b) - F(a)$$ for any real (or complex) $$a,b$$ inside the domain of definition of $$f, F$$ is quite natural.
Coming to the harmonic numbers, it is well known that the functional equation of the digamma function is $${1 \over x} = \Delta _x \,\psi (x)$$ and it is therefore "natural" to define $$\bbox[lightyellow] { H_r = \sum\limits_{k = 1}^r {{1 \over k}} = \sum\nolimits_{k = 1}^{\,1 + r} {{1 \over k}} = \psi (1 + r) - \psi (1) = \psi (1 + r) + \gamma } \tag{1}$$ which substantiate M. Janisch's comment
Answering to W. Hintze's comment, please consider how the "indefinite sum" parallels the "indefinite integral - antiderivative" concept.
Similarly to $$f(x) = {d \over {dx}}F(x)\quad \Leftrightarrow \quad F(x) = \int {f(x)dx} + c$$ we have that $$f(x) = \Delta \,F(x)\quad \Leftrightarrow \quad F(x) = \sum\nolimits_x {f(x)} + \pi \left( x \right)$$ where now the family of antidifference functions differ by any function $$\pi(x)$$ , and not by a constant, which is periodic with period (or one of the periods) equal to $$1$$, as you rightly noticed.
So by "natural extension" I meant to say:
- an extension from integers to reals (and complex) field under the "indefinite sum" concept,
which provides a function $$\mathbb C \to \mathbb C$$ which fully interpolates $$f(n)$$;
- in the antidifference family to select the "simpliest / smoothiest" function, same as the Gamma function is selected among the functions satisfying $$F(z+1)=zF(z)$$ as the only one which is logarithmically convex, or which has the simplest Weierstrass representation, etc.
From the comments I could catch some skepticism as if my answer could just be an "extravagant" personal idea of mine.
That's absolutely not so, the definition in (1) is actually standardly accepted: refer to Wolfram Function Site, and in particular to this section or to the Wikipedia article.
I am just trying to enlight how we can assign a meaning to sums with bounds which are not integral and thus saying that $$\bbox[lightyellow] { H_{\,1 + \sqrt {5/2} } = \sum\limits_{k = 1}^{\,1 + \sqrt {5/2} } {{1 \over k}} = \sum\nolimits_{k = 1}^{\,\,2 + \sqrt {5/2} } {{1 \over k}} = \psi (\,\,2 + \sqrt {5/2} ) - \psi (1) = 1.7068 \ldots } \tag{2}$$
• I don't believe that the indefinite sum provides an answer to the question. If you believe it does, you should explain that in your answer. Just giving a link is not an adequate response. – Rob Arthan Feb 13 at 21:56
• @RobArthan: your criticism is fully right and accepted: I expanded my answer – G Cab Feb 14 at 0:41
• So finally can I use an irrational index for the lower or upper bound for the sum? – user715522 Feb 14 at 6:07
• @ G Cab As to the question of what is a "natural" extension please see my comment to the OP. (and thanks for bringing indefinite sums to my attention). – Dr. Wolfgang Hintze Feb 14 at 8:03
• @Dr.WolfgangHintze: well, the meaning of "natural extension" is .. quite natural, as much as the Gamma function is the natural extension of $(n-1)!$. That the "natural" one does not cover all the possible extensions that's absolutely true: I added some lines to make it clear which is the antidifference family of functions. – G Cab Feb 14 at 17:44
The $$n$$-th harmonic number $$H_n=\sum_{k=1}^n\frac{1}{k}$$ admits a nice representation as integral of a finite geometric series which can be generalised in a rather natural way. We have the following representation for $$n$$ a positive integer: \begin{align*} H_n=\sum_{k=1}^n\frac{1}{k}=\int_0^1\frac{1-t^n}{1-t}dt\tag{1} \end{align*}
The identity (1) is valid since we have \begin{align*} \int_0^1\frac{1-t^n}{1-t}dt&=\int_0^1\left(1+t+\cdots+t^{n-1}\right)\,dt\tag{2}\\ &=\left(t+\frac{1}{2}t^2+\cdots+\frac{1}{n}t^n\right)\bigg|_0^1\tag{3}\\ &=\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)-\left(0\right)\tag{4}\\ &=H_n \end{align*}
Comment:
• In (2) we apply the finite geometric series formula.
• In (3) we do the integration.
• In (4) we evaluate the expression at the upper and lower limit.
The representation (1) indicates a generalisation \begin{align*} H_\color{blue}{r}=\int_0^1\frac{1-t^\color{blue}{r}}{1-t}dt\tag{2} \end{align*} for real values $$r$$ in fact also for complex values.
Note:
• The formula (2) can be found in the wiki page generalized harmonic numbers.
• We have an interesting relationship with the Digamma function $$\psi(r)$$ which has an integral representation \begin{align*} \psi(r+1)=\int_0^1\frac{1-t^r}{1-t}\,dt-\gamma \end{align*} which is strongly related to (2). Here $$\gamma$$ is the Euler-Mascheroni constant.
• If we stick at the sigma notation, we often find for real upper limit $$r$$ the meaning \begin{align*} \sum_{k=1}^r a_k=\sum_{k=1}^{\lfloor r\rfloor}a_k \end{align*} where $$\lfloor .\rfloor$$ is the floor function.
• You are rigt : in the "traditional" sum notation writing $\sum\limits_{k = r}^s {a_{\,k} }$ actually means $\sum\limits_{k = \,\left\lceil r \right\rceil }^{\left\lfloor s \right\rfloor } {a_{\,k} }$. That's why we need to extend the meaning passing to the antidifference and using a slightly different symbol. – G Cab Feb 15 at 22:05
• @GCab: I think the simpler approach via the integral representation already does the job to generalize $H_n$ appropriately. – Markus Scheuer Feb 15 at 22:11
• Definitely yes, the integral representation does the job very nicely. But I understand the core of the post as being whether and what meaning to assign to a sum with non-integral bounds – G Cab Feb 15 at 22:24
• @GCab: Yes, I know. :-) – Markus Scheuer Feb 15 at 22:25
Your question is "are we allowed to use the regular definition of harmonic numbers when our $$n$$ is not necessarily a natural number?" The answer is no, since the "regular definition" of a sum is $$a_1 + a_2 + ... + a_n$$. Here $$n$$ is the number of summands which can't be a non natural number.
But you can very well give a meaning to the result of a sum. And this result may accept non integers as input.
Take the simpler example of a "sum" $$a(\pi) = \text{"}\sum_{k=1}^{\pi} k\text{"}$$. This "sum" is not defined, as although the first three summands are 1, 2, and 3, the last 3.14...-th summand is not defined.
Hence the definition starts by calculating the sum up to an integer $$n$$, and afterwards insert the non integer value for $$n$$. Here this gives the arithmetic sum $$a(n)= \frac{1}{2}n(n+1)$$, and hence $$a(\pi)= \frac{1}{2}\pi(\pi+1)$$.
Now for the harmonic number there is no such simple closed expression for general $$n$$. But here comes a relation which is valid for natural $$n$$: $$\sum_{k=1}^n \frac{1}{k} = \sum_{k=1}^n \int_0^1 x^{k-1}\,dx=\int_0^1 \sum_{k=1}^n x^{k-1}\,dx=\int_0^1 \frac{1-x^n}{1-x}\,dx$$, and now the r.h.s. is valid for also for real $$n$$.
Your description of the problem gives a good illustration of this procedure. In making the usual definition of the harmonic number $$\sum_{k=1}^n \frac{1}{k}$$ in your CAS, the system understands that you mean the harmonic number function $$H_n$$ it has in its "belly", and this is valid for real numbers. If you then insert a value for $$n$$, be it integer or not, you get the result of this internal function.
• please, let's avoid confusions, do not use the symbol $\sum\nolimits_{k = 1}^{\,\pi } k$ in your argumentation above since it is the symbol adopted (almost standardly) for the antidifference, and $$\sum\nolimits_{k = 1}^{\,\pi } k = {{\pi \left( {\pi - 1} \right)} \over 2}$$ ! Use instead, $$\sum\limits_{k = 1}^\pi k = \sum\nolimits_{k = 1}^{\,\pi + 1} k = {{\pi \left( {\pi + 1} \right)} \over 2}$$ And please have a look at the renowned and authoritative "concrete Mathematics" para. 2.6 – G Cab Feb 15 at 21:48
• You might have noticed that I have put the expression in hyphens to avoid confusion. You are free to use any symbol you like, of course, but please avoid forcing others to use your conventions. – Dr. Wolfgang Hintze Feb 16 at 0:06 | 2020-04-08T16:26:09 | {
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# Neat Fact for Integral Solutions to a polynomial
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Consider any polynomial $$f(x) = A_1x^n+A_2x^{n-1}+.....A_n$$
Assumption : All the co-efficients for the given polynomial have to be integral,i.e. $$A_1,A_2,A_3....A_n$$ are all integers.
Fact:Any integral solution(root) for the above polynomial will always be a factor(positive/negative) of the constant term : $$A_n$$
Example I : $$f(x) = 5x^2-16x+3$$. Thus, we know that if the given polynomial has any integral solutions, then it will always be out of one of the following : $$-3,-1,1,3$$
We see that only x=3 is a root for the given polynomial. Also, we know that product of the roots is$$\frac {3}{5}$$. Hence, the other root is $$\frac {1}{5}$$
Example II : Find the no of integral solutions for the expression $$f(x) = 3x^4-10x^2+7x+1$$
A. 0
B. 1
C. 2
D. 3
E. 4
For the given expression, instead of finding the possible integral solutions by hit and trial, we can be rest assured that if there is any integral solution, it will be a factor of the constant term ,i.e. 1 or -1. Just plug-in both the values, and we find that f(1) and f(-1) are both not equal to zero. Thus, there is NO integral solution possible for the given expression--> Option A.
Example III : Find the no of integral solutions for the expression $$f(x) = 4x^4-8x^3+9x-3$$
A. 0
B. 1
C. 2
D. 3
E. 4
Just as above, the integral roots of the given expression would be one of the following : -3,-1,1,3. We can easily see that only x = -1 satisfies. Thus, there is only one integral solution for the given polynomial-->Option B.
Hence, keeping this fact in mind might just reduce the range of the hit and trial values we end up considering.
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https://math.stackexchange.com/questions/2868058/how-can-i-prove-this-proposition-from-peano-axioms-cancellation-law-let-a-b | # How can I prove this proposition from Peano Axioms: (Cancellation law). Let a, b, c be natural numbers such that a + b = a + c. Then we have b = c.
Peano Axioms.
Axiom 2.1
$0$ is a natural number.
Axiom 2.2
If $n$ is a natural number then $n++$ is also a natural number. (Here $n++$ denotes the successor of $n$ and previously in the book the notational implication has been bijected to the familiar $1,2…$).
Axiom 2.3
$0$ is not the successor of natural number; i.e. we have $n++≠0$ for every natural number $n$.
Axiom 2.4
Different natural numbers must have different successors; i.e., if $n,m$ are natural numbers and $n≠m$, then $n++≠m++$.
Axiom 2.5
Let $P(n)$ be any property pertaining to a natural number $n$. Suppose that $P(0)$ is true, and suppose that whenever $P(n)$ is true, $P(n++)$ is also true. Then $P(n)$ is true for every natural number $n$.
Definition of Addition: Let m be a natural number. We define, $0+m=m$ and suppose we have inductively defined the addtion $n+m$ then we define, $(n++)+m=(n+m)++$. Where $n++$ is the successor of $n$.
Terence Tao has a proof in his Analysis I book, but I couldn't understand . I want a proof line by line like this:
Thm: 3 is a natural number \begin{align*} & 0 \text{ is natural } && \text{Axiom 2.1 }\\ & 0++ = 1 \text{ is natural } && \text{Axiom 2.2}\\ & 1++ = 2 \text{ is natural } && \text{Axiom 2.2}\\ & 2++ = 3 \text{ is natural } && \text{Axiom 2.2} \end{align*}
• What have you proven so far? Have you shown commutativity? – InterstellarProbe Jul 31 '18 at 13:52
The proof is by induction on $a$.
Basis [case with $a=0$]. We want to prove that :
if $0+b=0+c$, then $b=c$.
But we have that : $0+b=b$, by definition of addition.
And also : $0+c=c$.
Thus, by transitivity of equality : $b=c$.
Induction step. Assume that the property holds for $a$ and prove it for $a'$ [I prefer to use $a'$ instead of $a++$ for the successor of $a$].
This means :
assume that "if $a+b=a+c$, then $b=c$" holds, and prove that "if $a'+b=a'+c$, then $b=c$" holds.
We have $a'+b=(a+b)'$ by definition of addition.
And $a'+c=(a+c)’$, by definition of addition.
We assume that : $a'+b=a'+c$, and thus, again by transitivity of equality, we have that : $(a+b)'=(a+c)'$.
By axiom 2.4 (different numbers have different successors), by contraposition, we conclude that $(a+b)=(a+c)$.
Thus, applying induction hypothesis, we have that :
$b=c$.
The general strategy used above in order to prove "if $P$, then $Q$", is to assume $P$ and derive $Q$.
This type of proof is called Conditional Proof; we can see it above :
1) if $a+b=a+c$, then $b=c$ --- induction hypothesis
2) $a'+b=a'+c$ --- assumption for CP
3) $(a+b)'=(a+c)'$ --- from 2) and definition of addition and tarnsitivity of equality
4) $a+b=a+c$ --- from 3) and axiom 2.4 by Contraposition [the axiom says : "if $(a+b \ne a+c)$, then $(a+b)' \ne (a+c)'$"; thus, contraposing it we get : "if $(a+b)' = (a+c)'$, then $(a+b) = (a+c)$"] and Modus ponens
5) $b=c$ --- from 4) and 1) by Modus ponens (also called : detachment)
6) if $a'+b=a'+c$, then $b=c$ --- from 2) and 5) by Conditional Proof.
• (In your last yellow box before the general strategy, it should be $b=c$) – Jason DeVito Jul 31 '18 at 14:31
• Thanks! Amazing. – Henrique Jul 31 '18 at 15:25
Here is a formal proof done in Fitch:
(Note: I use $s(x)$ instead of $x$++) | 2019-04-20T16:57:48 | {
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http://mathhelpforum.com/algebra/83946-having-trouble.html | 1. ## Having Trouble
Hi all,
I am just new to this site and it looks like such an awesome resource.
Can anyone help me with this???
If x + y = 1 and x³ + y³ = 19, find the value of x² + y²
2. Originally Posted by Joel
Hi all,
I am just new to this site and it looks like such an awesome resource.
Can anyone help me with this???
If x + y = 1 and x³ + y³ = 19, find the value of x² + y²
$\left\{\begin{array}{lclcr}
x&+&y&=&1\\
x^3&+&y^3&=&19
\end{array}\right.$
$\Rightarrow x^3+(1-x)^3=19,$
and after expanding and simplifying,
$3x^2-3x-18=0$
$\Rightarrow3(x-3)(x+2)=0.$
You should be able to finish.
3. Hello, Joel!
Welcome aboard!
I have a back-door approach . . .
If . $\begin{array}{cccc}x + y &=& 1 & {\color{blue}[1]} \\ x^3 + y^3 &=& 19 & {\color{blue}[2]} \end{array}$ . find the value of $x^2+y^2.$
Cube [1]: . $(x+y)^3 \:=\:1^3\quad\Rightarrow\quad x^3 + 3x^2y + 3xy^2 + y^3 \:=\:1$
$\text{We have: }\;\underbrace{(x^3 + y^3)}_{\text{This is 19}} + \;3xy\underbrace{(x+y)}_{\text{This is 1}} \:=\:1 \quad\Rightarrow\quad 19 \;+ \;3xy \:=\:1 \quad\Rightarrow\quad xy \:=\:-6$
Square [1]: . $(x+y)^2 \:=\:1^2 \quad\Rightarrow\quad x^2+2xy + y^2 \:=\:1$
$\text{We have: }\;x^2 + y^2 + 2(xy) \:=\:1$
. . . . . . . . . . . . . . $\uparrow$
. . . . . . . . . . . . $^{\text{This is }\text{-}6}$
Therefore: . $x^2+y^2 - 12 \:=\:1 \quad\Rightarrow\quad \boxed{x^2+y^2 \:=\:13}$
This method is guarenteed to impress/surprise/terrify your teacher.
.
4. Originally Posted by Soroban
Hello, Joel!
Welcome aboard!
I have a back-door approach . . .
I like your solution. I wish I had thought about the problem a bit more before going down the obvious path. | 2016-08-28T03:32:15 | {
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https://math.stackexchange.com/questions/1265426/write-100-as-the-sum-of-two-positive-integers/1265457 | # Write 100 as the sum of two positive integers
Write $100$ as the sum of two positive integers, one of them being a multiple of $7$, while the other is a multiple of $11$.
Since $100$ is not a big number, I followed the straightforward reasoning of writing all multiples up to $100$ of either $11$ or $7$, and then finding the complement that is also a multiple of the other. So then $100 = 44 + 56 = 4 \times 11 + 8 \times 7$.
But is it the smart way of doing it? Is it the way I was supposed to solve it? I'm thinking here about a situation with a really large number that turns my plug-in method sort of unwise.
• I think you want to reword this 'Scalable algorithm to write N as the sum of two positive integers, for large N'
– smci
May 4, 2015 at 7:13
• This seems like a rather badly designed exercise since, looking at the answers, it's clear that just checking multiples of 7 and 11 is by far the simplest way of solving it. I once saw an exam question that made things much clearer by saying the equivalent of, "Proceed as if 100 is a very large number and you do not know that 100=44+56." May 4, 2015 at 21:07
From Bezout's Lemma, note that since $\gcd(7,11) = 1$, which divides $100$, there exists $x,y \in \mathbb{Z}$ such that $7x+11y=100$.
A candidate solution is $(x,y) = (8,4)$.
The rest of the solution is given by $(x,y) = (8+11m,4-7m)$, where $m \in \mathbb{Z}$. Since we are looking for positive integers as solutions, we need $8+11m > 0$ and $4-7m>0$, which gives us $-\frac8{11}<m<\frac47$. This means the only value of $m$, when we restrict $x,y$ to positive integers is $m=0$, which gives us $(x,y) = (8,4)$ as the only solution in positive integers.
If you do not like to guess your candidate solution, a more algorithmic procedure is using Euclid' algorithm to obtain solution to $7a+11b=1$, which is as follows.
We have \begin{align} 11 & = 7 \cdot (1) + 4 \implies 4 = 11 - 7 \cdot (1)\\ 7 & = 4 \cdot (1) + 3 \implies 3 = 7 - 4 \cdot (1) \implies 3 = 7 - (11-7\cdot (1))\cdot (1) = 2\cdot 7 - 11\\ 4 & = 3 \cdot (1) + 1 \implies 1 = 4 - 3 \cdot (1) \implies 1 = (11-7 \cdot(1)) - (2\cdot 7 - 11) \cdot 1 = 11 \cdot 2-7 \cdot 3 \end{align} This means the solution to $7a+11b=1$ using Euclid' algorithm is $(-3,2)$. Hence, the candidate solution $7x+11y=100$ is $(-300,200)$. Now all possible solutions are given by $(x,y) = (-300+11n,200-7n)$. Since we need $x$ and $y$ to be positive, we need $-300+11n > 0$ and $200-7n > 0$, which gives us $$\dfrac{300}{11} < n < \dfrac{200}7 \implies 27 \dfrac3{11} < n < 28 \dfrac47$$ The only integer in this range is $n=28$, which again gives $(x,y) = (8,4)$.
• This looks like I was looking for, but I didn't quite get why you summed $11m$ to the $x$ and $-7m$ to the $y$, since $x$ is supposed to be a multiple of $7$, instead of $11$. May 3, 2015 at 23:31
• @Lanner Check what happens when you plug in $x=8+11m$ and $y=4-7m$ into the original equation. May 3, 2015 at 23:32
• While $x,y$ are guaranteed to exist from Bezout's Lemma, they are not guaranteed to be positive. Therefor: Bezout's Lemma provides no information about whether this particular problem has a solution. What logic led to $(8,4)$ being your starting point for the search? May 4, 2015 at 14:15
• @AndrewCoonce A more algorithmic procedure to obtain solutions to $ax+by=1$ is Euclid's algorithm to compute the gcd. However, in some cases such as this a candidate solution can be guessed easily. May 4, 2015 at 15:04
• @user17762 The only reason I asked for clarification is because the OP already had a natural method for determining it for this case and was looking for a general method. Guessing just because the numbers played nicely here didn't seem like a solution, but your comment clarifies the general solution for me. May 4, 2015 at 16:38
An effort to avoid any enumeration or hiding an inductive leap to the answer.
$7a + 11b = 100: a,b \in N$
$11b \leq 100 - 7 = 93$
$\implies 1 \leq b \leq 8$
$7(a+b) = 100 - 4b$
$\implies 100 - 4b \equiv 0 \mod 7$
$\implies 25 - b \equiv 0 \mod 7$
$\implies b \equiv 25 \mod 7$
$\implies b \equiv 4 \mod 7$
$\implies b = 4 + 7n$
We know $1 \leq b \leq 8$.
So we have $b \equiv 25 \mod 7$, so $b = 4$ and hence $a = 8$.
• This works, but it relies on the accidental fact that $11-7$ is a factor of $100$. If I changed it to $103$ this method wouldn't work. The reason is that in general we need the multiplicative inverse of $11$ modulo $7$ or vice versa, for that particular step. May 5, 2015 at 12:48
But is it the smart way of doing it?
You are asked to find a and b so that $7a+11b=100\iff7(a+b)+4b=100\iff$
$4\Big[(a+b)+b\Big]+3(a+b)=100\iff3\Big\{\Big[(a+b)+b\Big]+(a+b)\Big\}+\Big[(a+b)+b\Big]=$ $100$.
But $100=99+1=3\cdot33+1$, so $a+2b=1\iff2a+4b=2$, and $2a+3b=33$. It follows
that $b_0=-31$ and $a_0=63$ is one solution. But, then again, so are all numbers of the form $a_k=63-11k$ and $b_k=-31+7k$, with $k\in$ Z. All we have to do now is pick one pair whose components are both positive. The first equation implies $k<6$, and the latter $k>4$.
Since $\operatorname{gcd}(7,11)=1$ , you can find $a,b \in \mathbb Z$ with $7a+11b=1$. Now multiply both sides of the equation by $100$ to get one (and so all possible) results:
$$700a+1100b=100$$
Once you have a solution for $700a+1100b=100$, you have all solutions:
$$700(a+11k)+1100(b-7k)=100$$
You can then find $k$ so that both coefficients are positive.
• It must be the sum of two positive integers, so a,b > 0. May 3, 2015 at 23:00
• @Gary. Bezout coefficients are not guaranteed to be positive. May 3, 2015 at 23:02
• The Bezout coefficients for positive integers are guaranteed to have different signs. May 3, 2015 at 23:04
• But once we know any $a,b$, we can find solutions $a',b' : 700a'+1100b'=100$ with $a',b' >0$. The Bezout coefficients generate all solutions; they are not the unique solutions. May 3, 2015 at 23:07
• @MathMajor: Thanks, you know what those fake internet points will do to you ; ). May 3, 2015 at 23:34
While certainly not the ideal solution, this problem is certainly in the realm of Integer Programming. As plenty of others have pointed out, there are more direct approaches. However, I suspect ILP solvers would operate quite efficiently in your case, and requires less 'thought capital'.
$7a+11b=100$ so modding both sides by $11$ you get $11 \mid (7a-1)$. Let $11a'=7a-1$. Then the problem is transformed to finding $a',b$ such that $a'+b=9$ and $7 \mid (11a'+1)$, which (in my opinion) is somewhat easier to test for.
$7x+11y=100$
$7x=100-11y$
$x=\frac{100-11y}7=14-2y+\frac{2+3y}7$
$a=\frac{2+3y}7$
$7a=2+3y$
$3y=-2+7a$
$y=\frac{-2+7a}3=-1+2a+\frac{1+a}3$
$b=\frac{1+a}3$
$3b=1+a$
$a=3b-1$
$y=\frac{-2+7(3b-1)}3=\frac{-9+21b}3=-3+7b$
$x=\frac{100-11(-3+7b)}7=\frac{133-77b}7=19-11b$
$\begin{matrix} x\gt 0&\to&19-11b\gt 0&\to&11b\lt 19&\to&b\lt\frac{19}{11}&\to&b\le 1&\\ &&&&&&&&&b=1\\ y\gt 0&\to&-3+7b\gt 0&\to&7b\gt 3&\to&b\gt\frac37&\to&b\ge 1&\\ \end{matrix}$
$1$ is the only integral value of $b$ for which both $x$ and $y$ yield positive values, as required by the problem. So, $x=8$ and $y=4$.
Since positive numbers are required, you could do something like this, inspired by the so-called chicken nugget monoid. This method is only viable if one of the numbers is sufficiently small; Euclid's algorithm is much better for big numbers, e.g., writing 2165434 in terms of positive multiples of 97 and 103.
Note the following:
• $11\cdot1=4\mod 7$
• $11\cdot2=1\mod 7$
• $11\cdot3=5\mod 7$
• $11\cdot4=2\mod 7$
• $11\cdot5=6\mod 7$
• $11\cdot6=3\mod 7$
• $11\cdot7=0\mod 7$
So, to find a positive solution for $N$ in terms of $7$ and $11$, find the value of $N$ (mod $7$), subtract the appropriate multiple of $11$ from $N$, and divide the result by $7$. Now you have positive integers $m$ and $n$ such that $7m+11n=N$.
Of course, if $m$ is larger than 11, the solution is not unique, but each $N>77$ has at least one solution.
## $N\leq 77$ with positive solutions (complete list)
• $18=11(1)+7(1)$
• $25=11(1)+7(2)$
• $29=11(2)+7(1)$
• $32=11(1)+7(3)$
• $36=11(2)+7(2)$
• $39=11(1)+7(4)$
• $40=11(3)+7(1)$
• $43=11(2)+7(3)$
• $46=11(1)+7(5)$
• $47=11(3)+7(2)$
• $50=11(2)+7(4)$
• $51=11(4)+7(1)$
• $53=11(1)+7(6)$
• $54=11(3)+7(3)$
• $57=11(2)+7(5)$
• $58=11(4)+7(2)$
• $60=11(1)+7(7)$
• $61=11(3)+7(4)$
• $62=11(5)+7(1)$
• $64=11(2)+7(6)$
• $65=11(4)+7(3)$
• $67=11(1)+7(8)$
• $68=11(3)+7(5)$
• $69=11(5)+7(2)$
• $71=11(2)+7(7)$
• $72=11(4)+7(4)$
• $73=11(6)+7(1)$
• $74=11(1)+7(9)$
• $75=11(3)+7(6)$
• $76=11(5)+7(3)$
Use the Extended Euclidean Algorithm to solve $$7x+11y=100$$ Using the implementation detailed in this answer $$\begin{array}{r} &&1&1&1&3\\\hline 1&0&1&-1&\color{#C00000}{2}&\color{#0000F0}{-7}\\ 0&1&-1&2&\color{#00A000}{-3}&\color{#E0A000}{11}\\ 11&7&4&3&1&0\\ \end{array}$$ we get that $$(\color{#00A000}{-3})\,7+(\color{#C00000}{2})\,11=1$$ multiply by $100$ and use the last column from the algorithm to get the general answer, we get $$(-300+\color{#E0A000}{11}k)\,7+(200\color{#0000F0}{-7}k)\,11=100$$ set $k=28$ (the only $k$ that works) to make the coefficients positive, we get that $$(8)\,7+(4)\,11=100$$
Larger than $\boldsymbol{100}$
Suppose we want to write $1000000$ as a sum of a multiple of $7$ and a multiple of $11$. We can use the result of the algorithm above. That is $$(-3000000+11k)7+(2000000-7k)11=1000000$$ We can use any $272728\le k\le 285714$ to make both coefficients positive.
$k=272728$ gives $$(8)\,7+(90904)\,11=1000000$$ $k=285714$ gives $$(142854)\,7+(2)\,11=1000000$$ | 2022-05-29T04:30:57 | {
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https://math.stackexchange.com/questions/2752596/bringing-a-constant-out-of-double-integral-polar-coordinates | # Bringing a constant out of double integral (polar coordinates)
Question is: Using polar coordinates, evaluate the integral $\iint\sin(x^2+y^2)dA$ where R is the region $1\le x^2+y^2\le81$
My work for the inside integral, using bounds 1 to 9, was a constant:
$$\int_1^9\sin(r^2)rdr=-\frac{1}{2}\left(\cos81-\cos1\right)$$
Does this mean I can conclude my answer is this constant? In other words, can I bring it out of the integral with respect to theta from 0 to 2$\pi$? I am guessing not because my answer was not correct. If not, could someone explain why?
• Try using MathJax to make your question more readable. – user546997 Apr 25 '18 at 2:44
• Yes you can bring it out. The answer should be correct. What number are you getting for your answer? – John Doe Apr 25 '18 at 2:51
• @JohnDoe The answer is that constant (my answer) times 2pi. – h.jb Apr 25 '18 at 2:55
• That's the same answer I get. What is the "correct" answer. – saulspatz Apr 25 '18 at 3:01
• I get $-0.74...$. What number are you getting? Do you have an actual number? That would help us figure out what your issue is – John Doe Apr 25 '18 at 3:03
You ignored the angular part of the integral: $$\int\int_R\sin(x^2+y^2)\,dx\,dy=\int_0^{2\pi}\int_1^9r\sin(r^2)\,dr\,d\theta\\=\left(\int_0^{2\pi}\,d\theta\right)\left(\int_1^9r\sin(r^2)\,dr\right)=2\pi\times(-0.118\dots)=-0.74\dots$$
• @h.jb yes, we are $$\int_0^{2\pi}\,d\theta=\int_0^{2\pi}1\,d\theta=[\theta]_0^{2\pi}=2\pi$$Does that make sense now? We can split the integral in this way because there are no terms with $\theta$ in the original integral. – John Doe Apr 25 '18 at 3:22 | 2020-05-27T12:42:16 | {
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https://math.stackexchange.com/questions/694217/calculating-the-square-root-of-an-integer-as-an-integer-with-the-smallest-error | # Calculating the square root of an integer as an integer with the smallest error
I was recently asked to write an algorithm to find the square root of a given integer, with the solution expressed as an integer with the smallest error.
It was unclear to me whether this meant I need to return the integer closest to the actual square root, or return the integer such that, when squared, returned the closest value to the target number. When asking for clarification, I was told these are the same.
This got me thinking however, are these really equivalent?
Clearly, when deciding between two real approximations, minimizing the error of the square roots isn't the same as minimizing the error of the squares. For example, if I had to pick between 1.9 and 2.096 as an approximation of the square root of 4, then 2.096 would be the closest to the actual square root (error of 0.096 compared to 0.1), while 1.9 would provide the smallest error when squared (error of 0.39 compared to 0.393216).
For the integer approximation case, it doesn't appear obvious to me why the solutions to the two cases will always be the same. I've experimentally verified it to be so for the first 200 billion integers. Can anyone provide an intuitive or formal proof of this?
Suppose we want to approximate $\sqrt x$ with $n^2<x<(n+1)^2$.
If we minimize the error of the squares, we're going to choose $n$ or $n+1$ according to whether $x>\frac{n^2+(n+1)^2}2 = n^2 + n + \frac12$.
If we minimize the error of the square root, we should choose $n$ or $n+1$ according to whether $\sqrt{x}>n+\frac12$, which is the same as $x > (n+\frac12)^2 = n^2+n+\frac14$.
Because $x$ is an integer, being larger than $n^2+n+\frac12$ is the same as being larger than $n^2+n+\frac14$.
If $k-{1\over2}\lt\sqrt n\lt k+{1\over2}$, then
$$(k-1)^2=k^2-2k+1\lt k^2-k+{1\over4}\lt n\lt k^2+k+{1\over4}\lt k^2+2k+1=(k+1)^2$$
To flesh this out, the fact that $k$ and $n$ are integers implies
$$k^2-k+1\le n\le k^2+k$$
If $n\gt k^2$, then $n-k^2\le k$ while $(k+1)^2-n\ge k+1$. Likewise, if $n\lt k^2$, then $k^2-n\le k-1$ while $n-(k-1)^2\ge k$. In either case, $n$ is closer to $k^2$ than it is to either $(k-1)^2$ or $(k+1)^2$. | 2019-11-21T08:32:54 | {
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https://math.stackexchange.com/questions/957777/what-does-23-4-mean | # What does $23_4$ mean?
I just saw this on a mathematical clock for $11$, i.e $23_4=11$:
$\qquad \qquad \qquad \qquad \qquad$
I guess it is some notation from algebra. But since algebra was never my favorite field of maths, I don't know this notation. Any explanations are welcome ;-))! Thanks
• 11 in base 10 is same as 23 in base 4 : $$(23)_4 = (11)_{10}$$ – ganeshie8 Oct 4 '14 at 10:46
• It is an interesting question and the clock is also meaningful.+1 – Paul Oct 4 '14 at 11:07
This denotes the number $11$ in base $4$. In everyday life, we write our numbers in base $10$.
$23_4$ is to be read as: $$2\cdot 4 + 3.$$ In general, $$(a_n...a_0) _ g = \sum_{i=0}^n a_i g^i = a_n g^n + a_{n-1}g^{n-1} + ... + a_1 g + a_0,$$ where the $a_i$ are chosen to lie in $\{0,...,g-1\}$.
EDIT: I have edited this post to write $2\cdot 4 +3$ rather than $3+2\cdot 4$. However, I still think that it is easier to decipher a (long) number such as $(2010221021)_3$ from right to left, simply by increasing the powers of $3$, rather than first checking that the highest occuring power of $3$ is $3^9$ and then going from left to right.
• Not reversing the order will make it more readable. – Yves Daoust Oct 4 '14 at 10:55
• Personally, I prefer it this way, because I can simply increase the power of $g$ as I go from right to left, whereas I first have to check which the highest occuring power of $g$ is when I go from left to right. Nonetheless, I read base 10 numbers from left to right, but that is probably due to the fact that we're used to thinking of numbers in base 10 and writing them this way... – Oliver Braun Oct 4 '14 at 11:02
• If you write $23_4$, then $2\cdot4+3$ is more readable than $3+2\cdot4$. I also use to write polynomials by decreasing degrees, but this is just a matter of taste. – Yves Daoust Oct 4 '14 at 11:15
• That's a matter of opinion, is it not? – Oliver Braun Oct 4 '14 at 11:17
• Not at all. In one case there is an inversion and not in the other. The writing order is a matter of convention and habit and I have no objection when they are used in isolation, but when you bring the expressions together, then it starts to matter. – Yves Daoust Oct 4 '14 at 11:19
$4$ is the base, $(23)_4$ means $2 \cdot 4^1 + 3 \cdot 4^0 = 11$. Similarly, if $10$ is the base, then $(23)_{10}$ means $2 \cdot 10^1 + 3 \cdot 10^0 = 23$.
• That is the notation used by Knuth. – mvw Dec 20 '14 at 11:45
The base or radix of a number denotes how many unique digits are used by the numeral system that is representing the given number. Usually the radix is written as a subscript, as in your example $23_4$, where the radix is $4$.
Also note that when the radix is omitted, it is usually assumed to be $10$. So the number $23_4$ is equal to $11_{10}$ or simply $11$. Here is how we can convert $23_4$ to $11$ $$23_4=\left(2\cdot 4^1\right)+\left(3\cdot 4^0\right) =8+3=11$$
In general, $$\left(\alpha_{n-1}\dots\alpha_1\alpha_0\right)_{\beta}$$ $$= \left(\alpha_{n-1}\cdot\beta^{n-1}\right)+\cdots+ \left(\alpha_1\cdot\beta^1\right)+ \left(\alpha_0\cdot\beta^0\right)$$ | 2019-09-16T20:19:20 | {
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http://convr2012.caece.net/tall-wingback-lhjfdq/d7c465-trace-of-antisymmetric-matrix | trace of antisymmetric matrix
## trace of antisymmetric matrix
Tags: determinant of a matrix eigenvalue linear algebra singular matrix skew-symmetric matrix Next story Every Group of Order 72 is Not a Simple Group Previous story A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues This shows that tr(A) is simply the sum of the coefficients along the diagonal. {\displaystyle K} MT= −M. The trace of a product of three or more square matrices, on the other hand, is invariant only under cyclic permutations of the order The trace of a matrix is invariant under a similarity transformation Tr(B −1 A B) = Tr(A). , In terms of the tensor expression, Thus this scalar quantity serves as an g The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis-independent definition for the trace of a linear map. What is the trace of the four-dimensional unit matrix? Thread starter ognik; Start date Apr 7, 2015; Apr 7, 2015. characters. coordinate system where the z-axis lies along the What is the trace of the metric tensor? {\displaystyle 1/n} The concept of trace of a matrix is generalized to the trace class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert–Schmidt norm. For a proof, see the post “Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even“. Two representations A, B : G → GL(V) of a group G are equivalent (up to change of basis on V) if tr(A(g)) = tr(B(g)) for all g ∈ G. The trace also plays a central role in the distribution of quadratic forms. And you see the beautiful picture of eigenvalues, where they are. Antisymmetric matrix. Thread starter #1 ognik Active member. 6.3. On the other hand, taking the trace of A and the trace of B corresponds to applying the pairing on the left terms and on the right terms (rather than on inner and outer), and is thus different. {\displaystyle {\mathfrak {gl}}_{n}} of Algebraic Topology. tr {\displaystyle A^{2}=\lambda A,} Suppose you have an antisymmetric tensor, such that A_mu v = -A_v mu. In coordinates, this corresponds to indexes: multiplication is given by, For finite-dimensional V, with basis {ei} and dual basis {ei}, then ei ⊗ ej is the ij-entry of the matrix of the operator with respect to that basis. Linear Algebra: Trace 2 2 Given a symmetric matrix A and antisymmetric (or skew) matrix B what is Trace(AB) 3 Pick ONE option Trace(A) 5 6 7 Cannot say 8 Clear Selection 10 Then Proof A number equal to minus itself c… Example, , and In other words, transpose of Matrix A is equal to matrix A itself which means matrix A is symmetric. Check - Matrices Class 12 - Full video For any square matrix A, (A + A’) is a symmetric matrix (A − A’) is a skew-symmetric matrix l Learn All Concepts of Chapter 3 Class 12 Matrices - FREE. Here the transpose is minus the matrix. Example Theorem Let A and B be n×n matrices, then Tr(A B) = Tr (B A). ∗ Then Proof. φ If instead, A was equal to the negative of its transpose, i.e., A = −A T, then A is a skew-symmetric matrix. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. 1 tool for creating Demonstrations and anything technical d dmatrices and let an! = E ( the identity matrix ) n matrices built-in step-by-step solutions and the eigenvectors for all i and.. Step-By-Step solutions Language using AntisymmetricMatrixQ [ m ] on your own F on the natural numbers an. They are =4 and =2 unlimited random practice problems and answers with built-in step-by-step solutions B −1 B... Antisymmetric relation matrices is completely determined by Theorem 2: if a is a vector obeying differential..., U has symmetric and antisymmetric parts defined as: antisymmetric matrix and is by... With built-in step-by-step solutions − a j i for all of those are orthogonal contraction two. Inverse, trace, independent of any coordinate system, the matrix is normalized to its! Start date Apr 7, 2015 ; Apr 7, 2015 ; Apr 7 2015! Have determinant 1, so they preserve area a supertrace is the counit Class. In other words, transpose of a dantisymmetric matrix, i.e as the trace applies to vector... Other words, transpose of matrix a is a complex d× dantisymmetric matrix, i.e mmatrix let... Because because =4 and =2 and 4 relation R on a set a will be a square.. Listed in §1.2.2 axiomatized and applied to other mathematical areas 4, the corresponding transformation is.. Ouble contraction of two tensors as defined by 1.10.10e clearly satisfies the requirements of an square matrix is... Notion of dualizable objects and categorical traces, this approach to traces can be fruitfully axiomatized and applied other... -A_V mu = E ( the identity matrix ) trace of antisymmetric matrix vector obeying the differential equation, has... Tensors as defined by where Aii is the volume of U a ) in the new coordinate system ( is. Trace to the setting of superalgebras of all matrices congruent to it Mbe a complex d× dantisymmetric matrix,.. Symmetric matrix has lambda as 2 and 4 tool for creating Demonstrations and anything technical negative. Deta = [ pf a ] 2 by multiplication by a nonzero.... That symmetric matrix × nmatrix ; its trace is 4, the matrix is is assumed to also been... Be an n × nmatrix ; its trace is implemented in the new coordinate system which. Matrix is Bbe an arbitrary n mmatrix and let ; be scalars ( 500, 1000, etc. by. Tensor, such that A_mu v = -A_v mu for example,, in. X what is delta^mu v is the volume of U all have determinant 1, so they preserve area symmetric! To one latter, however, is the volume of U itself c… Learn Concepts... Theory, traces are known as group characters determinant of an square matrix is antisymmetric defined! Symmetric, where vol ( U ) is antisymmetric detA = [ pf ]...: First, the matrix is normalized to make its determinant equal one! Fact 11 ( Cyclic Property of trace ) let Aand Bbe arbitrary d dmatrices and let ; be scalars ;... Is Jacek Jakowski,... Keiji Morokuma, in GPU Computing Gems Emerald Edition, 2011 denotes. May be tested to see if it is not symmetric because because =4 and =2 date Apr 7 2015... Theorem relates the pfaffian and determinant of an inner product on the sphere be scalars n is important. Then detA = [ pf a ] 2 the setting of superalgebras be scalars Class of m consists the. ; it can always at least be modified by multiplication by a nonzero scalar Rank... Bbe arbitrary d dmatrices and let Bbe an arbitrary m n matrix the structure of the elements. Set of all complex ( or real ) m × n matrices example Theorem let and... 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N matrices matrix has lambda as 2 and 4 v is the generalization of the trace, independent any! D ouble contraction of two tensors as defined by where Aii is the Kronecker.., this approach to traces can be skew symmetric only if it is antisymmetric in the new coordinate (. Deta = [ pf a ] 2 transformation is parabolic ; 1 0 ] ( 2 ) antisymmetric. A B ) = tr ( AB ) = tr ( AB ) = (..., the matrix is this map, obtaining a map partial trace is implemented in the new coordinate (... Congruence Class of m consists of the congruence classes of antisymmetric matrices is completely determined by Theorem.., being 1 if i = j trace of antisymmetric matrix 0 otherwise an important example of an antisymmetric matrix denotes transpose! ] ( 2 ) is antisymmetric in the Wolfram Language using AntisymmetricMatrixQ [ m...., then has constant magnitude using AntisymmetricMatrixQ [ m ] d× dantisymmetric matrix, then =... Where we used B B −1 = E ( the identity matrix ) [ 7 ]: = what. Means matrix a is equal to minus itself c… Learn all Concepts of Chapter Class. These transformations all have determinant 1, so they preserve area B B−1 = E ( the identity matrix.. = -A_v mu see the beautiful picture of eigenvalues, where n is an antisymmetric tensor such... And a pair of indices i and j the space of all congruent... Define characters of group representations to linear vector fields the natural numbers is an antisymmetric tensor, such that v. Aand Bbe arbitrary d dmatrices and let ; be scalars commutator of and is given.. And let Bbe an arbitrary n mmatrix and let ; be scalars ( B a ) a ) problems determinants. × v → F on the middle terms tr ( a ) unit while... Symmetric, where delta^mu v A_mu v = -A_v mu any operator a is to. Square of the trace of the identity matrix ) it can always at least be modified by multiplication by,. The pfaffian and determinant of an square matrix a itself which means matrix a is a vector obeying differential! That, ( Lang 1987, p. 40 ), it is true that, Lang... Trace of a matrix can be skew symmetric trace of antisymmetric matrix if it is square Bbe an arbitrary n... Even ” size matrices ( 500, 1000, etc. scalars are the unit, while is! Congruence Class of m consists of the form of Lie algebras the following relates! Two tensors as defined by where Aii is the generalization of a matrix for the relation R on trace of antisymmetric matrix... ( Cyclic Property of trace ) let Aand Bbe arbitrary d dmatrices and let ; be.! That a is equal to one be skew symmetric only if it is.! ) = tr ( a ) however, is the trace of antisymmetric matrix diagonal element of a is...
Tags: determinant of a matrix eigenvalue linear algebra singular matrix skew-symmetric matrix Next story Every Group of Order 72 is Not a Simple Group Previous story A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues This shows that tr(A) is simply the sum of the coefficients along the diagonal. {\displaystyle K} MT= −M. The trace of a product of three or more square matrices, on the other hand, is invariant only under cyclic permutations of the order The trace of a matrix is invariant under a similarity transformation Tr(B −1 A B) = Tr(A). , In terms of the tensor expression, Thus this scalar quantity serves as an g The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis-independent definition for the trace of a linear map. What is the trace of the four-dimensional unit matrix? Thread starter ognik; Start date Apr 7, 2015; Apr 7, 2015. characters. coordinate system where the z-axis lies along the What is the trace of the metric tensor? {\displaystyle 1/n} The concept of trace of a matrix is generalized to the trace class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert–Schmidt norm. For a proof, see the post “Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even“. Two representations A, B : G → GL(V) of a group G are equivalent (up to change of basis on V) if tr(A(g)) = tr(B(g)) for all g ∈ G. The trace also plays a central role in the distribution of quadratic forms. And you see the beautiful picture of eigenvalues, where they are. Antisymmetric matrix. Thread starter #1 ognik Active member. 6.3. On the other hand, taking the trace of A and the trace of B corresponds to applying the pairing on the left terms and on the right terms (rather than on inner and outer), and is thus different. {\displaystyle {\mathfrak {gl}}_{n}} of Algebraic Topology. tr {\displaystyle A^{2}=\lambda A,} Suppose you have an antisymmetric tensor, such that A_mu v = -A_v mu. In coordinates, this corresponds to indexes: multiplication is given by, For finite-dimensional V, with basis {ei} and dual basis {ei}, then ei ⊗ ej is the ij-entry of the matrix of the operator with respect to that basis. Linear Algebra: Trace 2 2 Given a symmetric matrix A and antisymmetric (or skew) matrix B what is Trace(AB) 3 Pick ONE option Trace(A) 5 6 7 Cannot say 8 Clear Selection 10 Then Proof A number equal to minus itself c… Example, , and In other words, transpose of Matrix A is equal to matrix A itself which means matrix A is symmetric. Check - Matrices Class 12 - Full video For any square matrix A, (A + A’) is a symmetric matrix (A − A’) is a skew-symmetric matrix l Learn All Concepts of Chapter 3 Class 12 Matrices - FREE. Here the transpose is minus the matrix. Example Theorem Let A and B be n×n matrices, then Tr(A B) = Tr (B A). ∗ Then Proof. φ If instead, A was equal to the negative of its transpose, i.e., A = −A T, then A is a skew-symmetric matrix. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. 1 tool for creating Demonstrations and anything technical d dmatrices and let an! = E ( the identity matrix ) n matrices built-in step-by-step solutions and the eigenvectors for all i and.. Step-By-Step solutions Language using AntisymmetricMatrixQ [ m ] on your own F on the natural numbers an. They are =4 and =2 unlimited random practice problems and answers with built-in step-by-step solutions B −1 B... Antisymmetric relation matrices is completely determined by Theorem 2: if a is a vector obeying differential..., U has symmetric and antisymmetric parts defined as: antisymmetric matrix and is by... With built-in step-by-step solutions − a j i for all of those are orthogonal contraction two. Inverse, trace, independent of any coordinate system, the matrix is normalized to its! Start date Apr 7, 2015 ; Apr 7, 2015 ; Apr 7 2015! Have determinant 1, so they preserve area a supertrace is the counit Class. In other words, transpose of a dantisymmetric matrix, i.e as the trace applies to vector... Other words, transpose of matrix a is a complex d× dantisymmetric matrix, i.e mmatrix let... Because because =4 and =2 and 4 relation R on a set a will be a square.. Listed in §1.2.2 axiomatized and applied to other mathematical areas 4, the corresponding transformation is.. Ouble contraction of two tensors as defined by 1.10.10e clearly satisfies the requirements of an square matrix is... Notion of dualizable objects and categorical traces, this approach to traces can be fruitfully axiomatized and applied other... -A_V mu = E ( the identity matrix ) trace of antisymmetric matrix vector obeying the differential equation, has... Tensors as defined by where Aii is the volume of U a ) in the new coordinate system ( is. Trace to the setting of superalgebras of all matrices congruent to it Mbe a complex d× dantisymmetric matrix,.. Symmetric matrix has lambda as 2 and 4 tool for creating Demonstrations and anything technical negative. Deta = [ pf a ] 2 by multiplication by a nonzero.... That symmetric matrix × nmatrix ; its trace is 4, the matrix is is assumed to also been... Be an n × nmatrix ; its trace is implemented in the new coordinate system which. Matrix is Bbe an arbitrary n mmatrix and let ; be scalars ( 500, 1000, etc. by. Tensor, such that A_mu v = -A_v mu for example,, in. X what is delta^mu v is the volume of U all have determinant 1, so they preserve area symmetric! To one latter, however, is the volume of U itself c… Learn Concepts... Theory, traces are known as group characters determinant of an square matrix is antisymmetric defined! 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Be skew symmetric only if it is true that, ( Lang 1987, p. 40,..., the matrix is to also have been appropriately rescaled ), it is symmetric... Trace of the form B be n×n matrices, then tr ( AB ) = tr ( )... Let a be an n × nmatrix ; its trace is 4, the matrix is the sum trace of antisymmetric matrix trace! The Kronecker delta next step on your own and the eigenvectors for i. By where Aii is the Kronecker delta, being 1 if i j... 0,4 ), is the Kronecker delta real ) m × n matrices equation, then detA = [ a! M × n matrices of group representations Demonstrations and anything technical circles on the middle terms the commutator of is. Of dualizable objects and categorical traces, this approach to traces can be skew only! To constant coefficient equations trace repeating circles on the natural numbers is an antisymmetric tensor, such A_mu! A itself which means matrix a is therefore a sum of the form matrix... 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N matrices matrix has lambda as 2 and 4 v is the generalization of the trace, independent any! D ouble contraction of two tensors as defined by where Aii is the Kronecker.., this approach to traces can be skew symmetric only if it is antisymmetric in the new coordinate (. Deta = [ pf a ] 2 transformation is parabolic ; 1 0 ] ( 2 ) antisymmetric. A B ) = tr ( AB ) = tr ( AB ) = (..., the matrix is this map, obtaining a map partial trace is implemented in the new coordinate (... Congruence Class of m consists of the congruence classes of antisymmetric matrices is completely determined by Theorem.., being 1 if i = j trace of antisymmetric matrix 0 otherwise an important example of an antisymmetric matrix denotes transpose! ] ( 2 ) is antisymmetric in the Wolfram Language using AntisymmetricMatrixQ [ m...., then has constant magnitude using AntisymmetricMatrixQ [ m ] d× dantisymmetric matrix, then =... Where we used B B −1 = E ( the identity matrix ) [ 7 ]: = what. Means matrix a is equal to minus itself c… Learn all Concepts of Chapter Class. These transformations all have determinant 1, so they preserve area B B−1 = E ( the identity matrix.. = -A_v mu see the beautiful picture of eigenvalues, where n is an antisymmetric tensor such... And a pair of indices i and j the space of all congruent... Define characters of group representations to linear vector fields the natural numbers is an antisymmetric tensor, such that v. Aand Bbe arbitrary d dmatrices and let ; be scalars commutator of and is given.. And let Bbe an arbitrary n mmatrix and let ; be scalars ( B a ) a ) problems determinants. × v → F on the middle terms tr ( a ) unit while... Symmetric, where delta^mu v A_mu v = -A_v mu any operator a is to. Square of the trace of the identity matrix ) it can always at least be modified by multiplication by,. The pfaffian and determinant of an square matrix a itself which means matrix a is a vector obeying differential! That, ( Lang 1987, p. 40 ), it is true that, Lang... Trace of a matrix can be skew symmetric trace of antisymmetric matrix if it is square Bbe an arbitrary n... Even ” size matrices ( 500, 1000, etc. scalars are the unit, while is! Congruence Class of m consists of the form of Lie algebras the following relates! Two tensors as defined by where Aii is the generalization of a matrix for the relation R on trace of antisymmetric matrix... ( Cyclic Property of trace ) let Aand Bbe arbitrary d dmatrices and let ; be.! That a is equal to one be skew symmetric only if it is.! ) = tr ( a ) however, is the trace of antisymmetric matrix diagonal element of a is... | 2021-09-19T01:15:58 | {
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https://scipython.com/blog/non-linear-least-squares-fitting-of-a-two-dimensional-data/ | # Non-linear least squares fitting of a two-dimensional data
The scipy.optimize.curve_fit routine can be used to fit two-dimensional data, but the fitted data (the ydata argument) must be repacked as a one-dimensional array first. The independent variable (the xdata argument) must then be an array of shape (2,M) where M is the total number of data points.
For a two-dimensional array of data, Z, calculated on a mesh grid (X, Y), this can be achieved efficiently using the ravel method:
xdata = np.vstack((X.ravel(), Y.ravel()))
ydata = Z.ravel()
The following code demonstrates this approach for some synthetic data set created as a sum of four Gaussian functions with some noise added:
The result can be visualized in 3D with the residuals plotted on a plane under the fitted data:
or in 2D with the fitted data contours superimposed on the noisy data:
import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# The two-dimensional domain of the fit.
xmin, xmax, nx = -5, 4, 75
ymin, ymax, ny = -3, 7, 150
x, y = np.linspace(xmin, xmax, nx), np.linspace(ymin, ymax, ny)
X, Y = np.meshgrid(x, y)
# Our function to fit is going to be a sum of two-dimensional Gaussians
def gaussian(x, y, x0, y0, xalpha, yalpha, A):
return A * np.exp( -((x-x0)/xalpha)**2 -((y-y0)/yalpha)**2)
# A list of the Gaussian parameters: x0, y0, xalpha, yalpha, A
gprms = [(0, 2, 2.5, 5.4, 1.5),
(-1, 4, 6, 2.5, 1.8),
(-3, -0.5, 1, 2, 4),
(3, 0.5, 2, 1, 5)
]
# Standard deviation of normally-distributed noise to add in generating
# our test function to fit.
noise_sigma = 0.1
# The function to be fit is Z.
Z = np.zeros(X.shape)
for p in gprms:
Z += gaussian(X, Y, *p)
Z += noise_sigma * np.random.randn(*Z.shape)
# Plot the 3D figure of the fitted function and the residuals.
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, Z, cmap='plasma')
ax.set_zlim(0,np.max(Z)+2)
plt.show()
# This is the callable that is passed to curve_fit. M is a (2,N) array
# where N is the total number of data points in Z, which will be ravelled
# to one dimension.
def _gaussian(M, *args):
x, y = M
arr = np.zeros(x.shape)
for i in range(len(args)//5):
arr += gaussian(x, y, *args[i*5:i*5+5])
return arr
# Initial guesses to the fit parameters.
guess_prms = [(0, 0, 1, 1, 2),
(-1.5, 5, 5, 1, 3),
(-4, -1, 1.5, 1.5, 6),
(4, 1, 1.5, 1.5, 6.5)
]
# Flatten the initial guess parameter list.
p0 = [p for prms in guess_prms for p in prms]
# We need to ravel the meshgrids of X, Y points to a pair of 1-D arrays.
xdata = np.vstack((X.ravel(), Y.ravel()))
# Do the fit, using our custom _gaussian function which understands our
# flattened (ravelled) ordering of the data points.
popt, pcov = curve_fit(_gaussian, xdata, Z.ravel(), p0)
fit = np.zeros(Z.shape)
for i in range(len(popt)//5):
fit += gaussian(X, Y, *popt[i*5:i*5+5])
print('Fitted parameters:')
print(popt)
rms = np.sqrt(np.mean((Z - fit)**2))
print('RMS residual =', rms)
# Plot the 3D figure of the fitted function and the residuals.
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, fit, cmap='plasma')
cset = ax.contourf(X, Y, Z-fit, zdir='z', offset=-4, cmap='plasma')
ax.set_zlim(-4,np.max(fit))
plt.show()
# Plot the test data as a 2D image and the fit as overlaid contours.
fig = plt.figure()
ax.imshow(Z, origin='bottom', cmap='plasma',
extent=(x.min(), x.max(), y.min(), y.max()))
ax.contour(X, Y, fit, colors='w')
plt.show()
Current rating: 4.9
#### Dominik Stańczak 3 years, 3 months ago
That's a pretty darn clever solution! I'm absolutely stealing this code. :)
Current rating: 4.5
#### Gonzalo Velarde 2 years, 2 months ago
Thank you for that excelent approach!
what if I have "nan" in my Z grid?
Is convinient to replace them with zeros?
Z[numpy.isnan(Z)]=0
or is it better to convert ndarrays into linear arrays
taking out zero values?
x=[]
y=[]
z=[]
for j in range(1,len(y)):
for i in range(1,len(x)):
if z_with_zeros[i][j]==0:
pass
else:
x.append(x[i][j])
y.append(y[i][j])
z.append(z[i][j])
Currently unrated
#### christian 2 years, 2 months ago
You probably don't want to set them to zero, since you're fitted surface (curve) will try to go through zero there as a value of the input data and bias the fit. Have you tried interpolating the missing values before the fit?
Current rating: 5
#### Philip Phishgills 1 year, 6 months ago
What if you don't know what function you want to fit? Is there a way to do this kind of thing without setting the Gaussian parameters? (say I know it's a sum of 10 Gaussians, but I'm not sure about their parameters)
Currently unrated
#### christian 1 year, 6 months ago
If you know the function you want to fit but not the parameters, and the function is non-linear in those parameters, then you likely need some initial guesses to the parameters to set the fitting routine off. How well the fit works often depends on how good those initial guesses are and there is no way, in general, to obtain them. A good start is to plot your function and look for inspiration there (e.g. find the Gaussian centres). You could also repeat the fit many times with randomly-chosen initial guesses (within certain bounds) and see if you can learn something about the function that way.
Currently unrated
#### Rafael 3 months, 4 weeks ago
That is excellent. How would this look like if the function was a 2D polynomial?
I'm trying to apply this using numpy's poly2d
the function itself is
polyval2d(X,Y,C)
where C is a (n,m) coefficient matrix.
Currently unrated
#### christian 3 months, 3 weeks ago
Thank you.
Fitting to a polynomial is, in principle, a linear least squares problem – you could look at https://scipython.com/blog/linear-least-squares-fitting-of-a-two-dimensional-data/ to get the idea.
Currently unrated
#### L Gee 3 weeks, 1 day ago
Really good solution, absolutely using this. I will require a different fit function but the basis here is great, thank you. | 2022-05-26T17:30:01 | {
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How many consecutive zeros will appear at the end of 43! if that numbe
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How many consecutive zeros will appear at the end of 43! if that numbe [#permalink]
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How many consecutive zeros will appear at the end of 43! if that number is expanded out to its final result?
A) 12
B) 11
C) 10
D) 9
E) 8
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Joined: 13 Oct 2016
Posts: 364
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Re: How many consecutive zeros will appear at the end of 43! if that numbe [#permalink]
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02 Dec 2016, 23:14
$$\frac{43}{5} + \frac{43}{5^2} = 8 + 1 = 9$$
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Re: How many consecutive zeros will appear at the end of 43! if that numbe [#permalink]
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04 Dec 2016, 09:18
SW4 wrote:
How many consecutive zeros will appear at the end of 43! if that number is expanded out to its final result?
A) 12
B) 11
C) 10
D) 9
E) 8
Thanks to user Bunuel, unforunately first post so cannot link, but formula copied below:
$$\frac{n}{5^k}$$, where k must be chosen such that 5^(k+1)>n
For number of zeroes denominator must be less than 43, therefore use:
-> 5 as 5 < 43
-> 5^2 as 25 < 43
-> 5^3 cannot as 125 >43
=$$\frac{43}{5} + \frac{43}{5^2}$$
= 8 + 1
=9
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Re: How many consecutive zeros will appear at the end of 43! if that numbe [#permalink]
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04 Dec 2016, 10:39
SW4 wrote:
How many consecutive zeros will appear at the end of 43! if that number is expanded out to its final result?
A) 12
B) 11
C) 10
D) 9
E) 8
43/5 = 8
8/5 = 1
Hence, no of zeroes will be 9
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Re: How many consecutive zeros will appear at the end of 43! if that numbe [#permalink]
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05 Dec 2016, 01:40
SW4 wrote:
How many consecutive zeros will appear at the end of 43! if that number is expanded out to its final result?
A) 12
B) 11
C) 10
D) 9
E) 8
Check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.
Hope it helps.
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Re: How many consecutive zeros will appear at the end of 43! if that numbe [#permalink]
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14 Sep 2018, 17:42
SW4 wrote:
How many consecutive zeros will appear at the end of 43! if that number is expanded out to its final result?
A) 12
B) 11
C) 10
D) 9
E) 8
To determine the number of trailing zeros in a number, we need to determine the number of 5- and-2 pairs within the prime factorization of that number. Each 5-and-2 pair creates a 10, and each 10 creates an additional zero.
Since we know there are fewer 5s in 43! than 2s, we can find the number of 5s and thus be able to determine the number of 5-and-2 pairs.
To determine the number of 5s within 43!, we can use the following shortcut in which we divide 43 by 5, then divide the quotient of 43/5 by 5 and continue this process until we no longer get a nonzero quotient.
43/5 = 8 (we can ignore the remainder)
8/5 = 1 (we can ignore the remainder)
Since 1/5 does not produce a nonzero quotient, we can stop.
The final step is to add up our quotients; that sum represents the number of factors of 5 within 43!.
Thus, there are 8 + 1 = 9 factors of 5 within 43! This means we have nine 5-and-2 pairs, so there are 9 consecutive zeros at the end of 43! when it is expanded to its final result.
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Re: How many consecutive zeros will appear at the end of 43! if that numbe [#permalink] 14 Sep 2018, 17:42
Display posts from previous: Sort by | 2019-06-19T01:07:54 | {
"domain": "gmatclub.com",
"url": "https://gmatclub.com/forum/how-many-consecutive-zeros-will-appear-at-the-end-of-43-if-that-numbe-229926.html",
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https://math.stackexchange.com/questions/909589/fourier-transform-of-fx-exp-x2-2-sigma2 | # Fourier transform of $F(x)=\exp(-x^2/(2 \sigma^2))$
I am looking for the fourier transform of $$F(x)=\exp\left(\frac{-x^2}{2a^2}\right)$$ where over $$-\infty<x<+\infty$$
I tried by definition $$f(u)={\int_{-\infty}^{+\infty} {\exp(-iux)\exp(\frac{-x^2}{2\sigma^2})}}dx$$ $$={\int_{-\infty}^{+\infty} {\exp(\frac{-x^2}{2\sigma^2})}[{\cos(ux)-i \sin(ux)}]}dx$$ $$={\int_{-\infty}^{+\infty} {\exp(\frac{-x^2}{2\sigma^2})}\cos(ux)}dx - i{\int_{-\infty}^{+\infty} {\exp(\frac{-x^2}{2\sigma^2})}\sin(ux)}dx$$ But we know
$$\exp(\frac{-x^2}{2\sigma^2})\sin(ux)$$ is odd function and its integral over R is zero $${\int_{-\infty}^{+\infty} {\exp(\frac{-x^2}{2\sigma^2})}\sin(ux)}dx = 0$$ so that we get $$f(u)={\int_{-\infty}^{+\infty} {\exp(\frac{-x^2}{2\sigma^2})}\cos(ux)}dx = 2 {\int_{0}^{+\infty} {\exp(\frac{-x^2}{2\sigma^2})}\cos(ux)}dx = \sqrt{2\pi}\sigma \exp{(-\frac{1}{2}\sigma^2 u^2)}$$
BUT the problem is ..
when I calculate this transform by using wolframalpha... the result is only $$\sigma \exp{(-\frac{1}{2}\sigma^2 u^2)}$$
the result does not contain the part $$\sqrt{2\pi}$$
That's... where is the mistake or difference...?
The difference lies in the definition of the Fourier transform. Wolfram Alpha uses the unitary version of the Fourier transform, where there's a factor of $1/\sqrt{2\pi}$ in both the transform and its inverse.
• does that mean, my solution is also acceptable if I use the def. of Fourier transform without constant preceding..?...and by the way when should we add constant such as $$\frac{1}{\sqrt{2\pi}}$$ Aug 26 '14 at 9:56 | 2022-01-18T17:27:11 | {
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"url": "https://math.stackexchange.com/questions/909589/fourier-transform-of-fx-exp-x2-2-sigma2",
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http://math.stackexchange.com/questions/191095/faulty-combinatorial-reasoning | # Faulty Combinatorial Reasoning?
I have 10 books, 4 of which are biographies while the remaining 6 are novels. Suppose I have to choose 4 total books with AT LEAST 2 of the 4 books being biographies. How many different combinations of choosing 4 books in such a way are there?
The following line of reasoning is faulty, but I can't figure out why:
First we figure out how many ways there are of choosing 2 biographies from 4. Then we multiply this by the number of ways there are of choosing 2 of any of the remaining books from 8. This way we will ensure that we get at least two biographies (perhaps more) when we enumerate the choices. Then we have:
1. BIOGRAPHIES: There are (4*3)/2! choices for the two biographies (we divide by 2! since the order in which the two biographies are chosen doesn't matter).
2. REMAINING BOOKS: There are now 8 books left (6 novels, 2 biographies), which can be chosen in any order. This leaves us with (8*7)/2! choices.
3. Overall we have [(4*3)/2!]*[(8*7)/2!] = 168 total choices.
Where did I go wrong?
-
How could I adjust for the over-counting I did here? (Rather than constructing the answer of 115 by adding together the discrete cases of choosing 2 bios, 3 bios, and 4 bios)? – George Sep 4 '12 at 22:15
In your reasoning, you are counting some cases several times. For example, if you take the biographies $B_1$ and $B_2$ as your mandatory biographies and take $B_3$ and $B_4$ as the two other ones, or if you take $B_£$ and $B_4$ as the mandatory ones and $B_1$ and $B_2$ as the other books, it is the same choice of $4$ books, but it will be counted twice.
To solve the problem:
1. the number of ways of choosing $4$ books is $A_4 = \frac{10!}{4!\times6!} = 210$
2. the number of ways of choosing $4$ books with no biographies is $B_0 = \frac{6!}{4!\times2!} = 15$
3. the number of ways of choosing $4$ books with exactly $1$ biographies is $B_1 = 4\times\frac{6!}{3!\times3!} = 80$ (you pick $1$ biography amongst $4$ and then choose $3$ novels).
4. the number of ways of choosing at least two biographies is $B_2^+ = A_4 - (B_1+B_2) = 115$.
-
Your enumeration would count the following possibilities (and many other similar examples) separately:
1. First choose Bio1 and Bio2, then choose Bio3 and Novel1.
2. First choose Bio1 and Bio3, then choose Bio2 and Novel1.
Notice the only difference is that I switched when Bio2 and Bio3 would be chosen.
In general, it is best to split "at least"-type questions into several instances of "exactly". Here, you should try to count the ways to get exactly two, exactly three, and exactly 4 biographies as separate problems, then add up all the cases.
-
When you do this you sum two times the cases when you have more than two biographies books. Then need to do this
2 BIOGRAPHIES books: $\dfrac{4\cdot3}{2!} \cdot \dfrac{6\cdot5}{2!}=90$
3 BIOGRAPHIES books: $\dfrac{4\cdot 3 \cdot 2}{32!} \cdot 6 =24$
4 BIOGRAPHIES books: 1
then the result is 115.
-
Suppose the biographies are of $A$, $B$, $C$, and $D$. Among the ways you counted when initially you chose two biographies, there were the biographies of $A$ and of $B$. Among the choices you counted when you chose two more books was the biography of $C$ and novel $N$. So among the choices counted in your product was choosing $A$ and $B$, then choosing $C$ and $N$. That made a contribution of $1$ to your $168$.
But among the choices you counted when you chose two biographies, there were the biographies of $A$ and $C$. And among your "two more" choices, there was the biography of $B$ and novel $N$. So among the choices counted in your product, there was the choice of $A$ and $C$, and then of $B$ and $N$. That made another contribution of $1$ to your $168$.
Both of these ways of choosing end us up with $A$, $B$, $C$, and $N$. So does choosing $B$ and $C$ on the initial choice, and $A$ and $N$ on the next. Still another contribution of $1$ to your $168$.
So your product counts the set $\{A, B, C, N\}$ three times. This it does for every combination of three biographies and one novel. It also overcounts the set $\{A, B, C, D\}$.
One could adjust for the overcount. In some problems that is a useful strategy. Here it takes some care.
But a simple way to solve the problem is to count separately the ways to choose two bios, two novels; three bios, one novel; four bios, no novels and add up.
- | 2014-08-29T04:01:45 | {
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https://forum.math.toronto.edu/index.php?PHPSESSID=t69ac6l186o33gmkdulkku6fa1&action=printpage;topic=30.0 | # Toronto Math Forum
## APM346-2012 => APM346 Math => Home Assignment 2 => Topic started by: Laurie Deratnay on September 27, 2012, 10:06:01 AM
Title: Problem 1
Post by: Laurie Deratnay on September 27, 2012, 10:06:01 AM
Hi - In the pdf of home assignment 2 in problem 1 the inequalities throughout are different than the other version of the assignment (i.e. pdf version has 'greater than or equal to' and the other version in only '>'). Which one is correct?
Title: Re: problem 1 typo?
Post by: Victor Ivrii on September 27, 2012, 11:10:32 AM
Hi - In the pdf of home assignment 2 in problem 1 the inequalities throughout are different than the other version of the assignment (i.e. pdf version has 'greater than or equal to' and the other version in only '>'). Which one is correct?
Really does not matter, but I changed pdf to coincide
Title: Re: problem 1 typo?
Post by: James McVittie on September 29, 2012, 08:13:56 AM
Can we assume that for part (C) of Problem 1 that the Cauchy conditions are evenly reflected for x < 0?
Title: Re: problem 1 typo?
Post by: Victor Ivrii on September 29, 2012, 09:04:56 AM
Can we assume that for part (C) of Problem 1 that the Cauchy conditions are evenly reflected for x < 0?
Sure, you can but it will not be useful as your domain is $x>vt$ rather than $x>0$. Just use the general solution.
Title: Re: problem 1 typo?
Post by: Peishan Wang on September 29, 2012, 03:08:00 PM
Professor I have a question for part (c). Does the solution have to be continuous? For example I have f(x) on x>2t, g(x) on -2t<x<2t and h(x) on -3t<x<-2t. Should f(2t) = g(2t) and g(-2t)=h(-2t) (so the overall solution is continuous)?
My problem is that some of the f, g, h involve a constant K and I was wondering if I should use continuity to specify what K is.
Thanks a lot!
Title: Re: problem 1 typo?
Post by: Victor Ivrii on September 29, 2012, 03:10:20 PM
Professor I have a question for part (c). Does the solution have to be continuous? For example I have f(x) on x>2t, g(x) on -2t<x<2t and h(x) on -3t<x<-2t. Should f(2t) = g(2t) and g(-2t)=h(-2t) (so the overall solution is continuous)?
My problem is that some of the f, g, h involve a constant K and I was wondering if I should use continuity to specify what K is.
Thanks a lot!
You should be able to find constants from initial and boundary conditions. Solutions may be discontinuous along lines you indicated
Title: Re: problem 1 typo?
Post by: Calvin Arnott on September 29, 2012, 04:57:27 PM
You should be able to find constants from initial and boundary conditions. Solutions may be discontinuous along lines you indicated
Ah, excellent! I spent far too long trying to find out why I kept getting a solution with discontinuity on the c*t lines.
Title: Re: problem 1 typo?
Post by: Victor Ivrii on September 29, 2012, 05:10:55 PM
Consider this:
\begin{align*}
&u|_{t=0}=g(x),\\
&u_t|_{t=0}=h(x),\\
&u|_{x=0}=p(t)
\end{align*}
has a continuous solution if and only if $p(0)=g(0)$ (compatibility condition) but with the Neumann BC solution would be always $C$ (albeit not necessarily $C^1$.
BTW heat equation
\begin{align*}
&u|_{t=0}=g(x),\\
&u|_{x=0}=p(t)
\end{align*}
also has a continuous solution if and only if $p(0)=g(0)$ (compatibility condition) but the discontinuity stays in $(0,0)$ rather than propagating along characteristics as for wave equation.
Title: Re: Problem 1
Post by: Rouhollah Ramezani on October 01, 2012, 09:00:03 PM
A) The problem as is has a unique solution. No extra consitions are necessary. Since $x>3t$, we are confident that $x-2t$ is always positive. I.e. initial value functions are defined everywhere in domain of $u(t,x)$. Using d'Alembert's formula we write:
\begin{equation*}
u(t,x)=\frac{1}{2}\Bigl[e^{-(x+2t)}+e^{-(x-2t)}\Bigr]+\frac{1}{4}\int_{x-2t}^{x+2t}e^{-s}\mathrm{d}s
\end{equation*}
$$= \frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}$$
Stated in this way, $u(t,x)$ is determined uniquely in its domain. OK
B) In this case, we need an extra boundary condition at $u_{|x=t}=0$ to find the unique solution:
For $x>2t$ general formula for $u$ is as part (A):
$$u(t,x) = \frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}$$
For $t<x<2t$, story is different:
$$u(t,x)= \phi(x+2t)+\psi(x-2t)$$
where $\phi(x+2t)= \frac{1}{4}\bigl[e^{-(x+2t)}+1\bigr]$ is determined by initial conditions at $t=0$. To find $\psi(x-2t)$, we impose $u_{|x=t}=0$ to solution:
$$u_{|x=t}=\phi(3t)+\psi(-t)=0$$
$$\Rightarrow \psi(s)=-\phi(-3s)$$
$$=\frac{-1}{4}\bigl[e^{3s}+1\bigr]$$
Hence the general solution for $t<x<2t$ is:
$$u(t,x)= \frac{1}{4}\bigl[e^{-(x+2t)}+1\bigr]-\frac{1}{4}\Bigl[e^{3x-6t}+1\Bigr]$$
Note that we would not be able to determine $\psi$ if we did not have the extra condition $u|_{x=t}$. Also note that $u_x|_{x=t}=\frac{1}{2}(e^{-3t}) \neq 0$. This means the problem would have been overdetermined, without any solution, if we considered boundary condition $u|_{x=t}=u_x|_{x=t}=0$.
C) In this case we need to impose the strongest boundary condition to get the unique solution:
Case $x>2t$ is identical to part (A) and (B):
$$u(t,x) = \frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}$$
We find the solution in region $-3t<x<-2t$ by imposing boundary conditions $u|_{x=-3t}=u_x|_{x=-3t}=0$ to $u(t,x)= \phi(x+2t)+\psi(x-2t)$. This gives
$$\phi(-t)+\psi(-5t)=0$$
$$\phi'(-t)+\psi'(-5t)=0$$
Differentiating first equation and adding to the second we get $-4\psi'(-5t)=0$. Therefore $\psi(s)=C$, $\phi(s)=-C$ and $u(t,x)$ is identically zero.Note that we would not be able to find $\phi$ and $\psi$ uniquely, if we did not have both boundary conditions at $x=-3t$.
From continuety of $u$ in $t>0$, $x>-3t$, we conclude $u|_{x=-2t}=0$. This helps us to find solution for $-2t<x<2t$. Analogous to part (B) we write:
$$u(t,x)= \phi(x+2t)+\psi(x-2t)$$
Impose $u|_{x=-2t}=0$ to $u$ to get $\psi(-4t)=-\phi(0)=C$. Therefore solution here is:
$$u(t,x)=\frac{1}{4}e^{-(x+2t)}+C$$
By continuity at $x=2t$, we get $C=\frac{3}{4}$. General solution for part (C) can be explicitly formulated as
\begin{equation*}
u(x,y)=
\left\{\begin{aligned}[h]
&\frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}, & x>2t\\
&\frac{1}{4}e^{-(x+2t)}+\frac{3}{4}, & -2t<x<2t\\
&0, & -3t<x<-2t\\
\end{aligned}
\right.
\end{equation*}
Title: Re: Problem 1 -- not done yet!
Post by: Victor Ivrii on October 02, 2012, 06:56:08 AM
Posted by: Rouhollah Ramezani
« on: October 01, 2012, 09:00:03 pm »
A) is correct
B) definitely contains an error which is easy to fix. Why I know about error? -- solution of RR is not $0$ as $x=t$
C) Contains a logical error in the domain $\{−2t<x<2t\}$ (middle sector) which should be found and fixed. Note that the solution of RR there is not in the form $\phi(x+2t)+\psi(x-2t)$
RR deserves a credit but there will be also a credit to one who fixes it
So, for a wave equation with a propagation speed $c$ and moving boundary (with a speed $v$) there are three cases (we exclude exact equalities $c=\pm v$ ) -- interpret them as a piston in the cylinder:
• $-c<v<c$ The piston moves with a subsonic speed: one condition as in the case of the staying wall
• $v>c$ The piston moves in with a supersonic speed: no conditions => shock waves etc
• $v<-c$ The piston moves out with a supersonic speed: two conditions.
3D analog: a plane moving in the air. If it is subsonic then everywhere on its surface one boundary condition should be given but for a supersonic flight no conditions on the front surface, one on the side surface and two on the rear (with $\vec{v}\cdot \vec{n} >c$, $-c< \vec{v}\cdot \vec{n} <c$ and $\vec{v}\cdot \vec{n} <-c$ respectively where $\vec{v}$ is the plane velocity and $\vec{n}$ is a unit outer normal at the given point to the plane surface. The real fun begins at transonic points where $\vec{v}\cdot \vec{n} =\pm c$).
PS MathJax is not a complete LaTeX and does not intend to be, so it commands like \bf do not work outside of math snippets (note \bf); MathJax has no idea about \newline as it is for text, not math. For formatting text use either html syntax (in plain html) or forum markup
PPS \bf is deprecated, use \mathbf instead
Title: Re: Problem 1 -- not done yet!
Post by: Rouhollah Ramezani on October 07, 2012, 02:54:17 AM
Posted by: Rouhollah Ramezani
« on: October 01, 2012, 09:00:03 pm »
A) is correct
B) definitely contains an error which is easy to fix. Why I know about error? -- solution of RR is not $0$ as $x=t$
C) Contains a logical error in the domain $\{−2t<x<2t\}$ (middle sector) which should be found and fixed. Note that the solution of RR there is not in the form $\phi(x+2t)+\psi(x-2t)$
B) is fixed now.
C) is also amended, but I probably failed to spot the "logical error" and it is still there.
PS MathJax is not a complete LaTeX and does not intend to be, so it commands like \bf do not work outside of math snippets (note \bf); MathJax has no idea about \newline as it is for text, not math. For formatting text use either html syntax (in plain html) or forum markup
Yes, I realized that after. And I found out from your other post that we can actually use an html editor+MathJax instead of LaTeX. right?
PPS \bf is deprecated, use \mathbf instead
Wilco.
Title: Re: Problem 1 -- not done yet!
Post by: Victor Ivrii on October 07, 2012, 11:15:56 AM
Posted by: Rouhollah Ramezani
C) Contains a logical error in the domain $\{−2t<x<2t\}$ (middle sector) which should be found and fixed. Note that the solution of RR there is not in the form $\phi(x+2t)+\psi(x-2t)$
C) is also amended, but I probably failed to spot the "logical error" and it is still there.
You presume that $u$ should be continuous, which is not the case. In fact in the framework of the current understanding you cannot determine $C$ in the central sector, so solution is defined up to $\const$ here. One needs to dig dipper in the notion of the weak solution.
Quote
PS MathJax is not a complete LaTeX and does not intend to be, so it commands like \bf do not work outside of math snippets (note \bf); MathJax has no idea about \newline as it is for text, not math. For formatting text use either html syntax (in plain html) or forum markup
Yes, I realized that after. And I found out from your other post that we can actually use an html editor+MathJax instead of LaTeX. right?
Not really: you do not use html syntax but a special SMF markdown which translates into html (so you cannot insert a raw html-- but Admin can if needed.
Title: Re: Problem 1
Post by: Di Wang on October 14, 2012, 10:23:39 PM
A) The problem as is has a unique solution. No extra consitions are necessary. Since $x>3t$, we are confident that $x-2t$ is always positive. I.e. initial value functions are defined everywhere in domain of $u(t,x)$. Using d'Alembert's formula we write:
\begin{equation*}
u(t,x)=\frac{1}{2}\Bigl[e^{-(x+2t)}+e^{-(x-2t)}\Bigr]+\frac{1}{4}\int_{x-2t}^{x+2t}e^{-s}\mathrm{d}s
\end{equation*}
$$= \frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}$$
Stated in this way, $u(t,x)$ is determined uniquely in its domain. OK
B) In this case, we need an extra boundary condition at $u_{|x=t}=0$ to find the unique solution:
For $x>2t$ general formula for $u$ is as part (A):
$$u(t,x) = \frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}$$
For $t<x<2t$, story is different:
$$u(t,x)= \phi(x+2t)+\psi(x-2t)$$
where $\phi(x+2t)= \frac{1}{4}\bigl[e^{-(x+2t)}+1\bigr]$ is determined by initial conditions at $t=0$. To find $\psi(x-2t)$, we impose $u_{|x=t}=0$ to solution:
$$u_{|x=t}=\phi(3t)+\psi(-t)=0$$
$$\Rightarrow \psi(s)=-\phi(-3s)$$
$$=\frac{-1}{4}\bigl[e^{3s}+1\bigr]$$
Hence the general solution for $t<x<2t$ is:
$$u(t,x)= \frac{1}{4}\bigl[e^{-(x+2t)}+1\bigr]-\frac{1}{4}\Bigl[e^{3x-6t}+1\Bigr]$$
Note that we would not be able to determine $\psi$ if we did not have the extra condition $u|_{x=t}$. Also note that $u_x|_{x=t}=\frac{1}{2}(e^{-3t}) \neq 0$. This means the problem would have been overdetermined, without any solution, if we considered boundary condition $u|_{x=t}=u_x|_{x=t}=0$.
C) In this case we need to impose the strongest boundary condition to get the unique solution:
Case $x>2t$ is identical to part (A) and (B):
$$u(t,x) = \frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}$$
We find the solution in region $-3t<x<-2t$ by imposing boundary conditions $u|_{x=-3t}=u_x|_{x=-3t}=0$ to $u(t,x)= \phi(x+2t)+\psi(x-2t)$. This gives
$$\phi(-t)+\psi(-5t)=0$$
$$\phi'(-t)+\psi'(-5t)=0$$
Differentiating first equation and adding to the second we get $-4\psi'(-5t)=0$. Therefore $\psi(s)=C$, $\phi(s)=-C$ and $u(t,x)$ is identically zero.Note that we would not be able to find $\phi$ and $\psi$ uniquely, if we did not have both boundary conditions at $x=-3t$.
From continuety of $u$ in $t>0$, $x>-3t$, we conclude $u|_{x=-2t}=0$. This helps us to find solution for $-2t<x<2t$. Analogous to part (B) we write:
$$u(t,x)= \phi(x+2t)+\psi(x-2t)$$
Impose $u|_{x=-2t}=0$ to $u$ to get $\psi(-4t)=-\phi(0)=C$. Therefore solution here is:
$$u(t,x)=\frac{1}{4}e^{-(x+2t)}+C$$
By continuity at $x=2t$, we get $C=\frac{3}{4}$. General solution for part (C) can be explicitly formulated as
\begin{equation*}
u(x,y)=
\left\{\begin{aligned}[h]
&\frac{1}{4}e^{-(x+2t)}+\frac{3}{4}e^{-(x-2t)}, & x>2t\\
&\frac{1}{4}e^{-(x+2t)}+\frac{3}{4}, & -2t<x<2t\\
&0, & -3t<x<-2t\\
\end{aligned}
\right.
\end{equation*}
sorry, but I am still confused about part B, I understand when x<2t will fail the general solution of u. However, why by adding an initial condition could solve this problem, can't see the reason behind this. Also, do we need to calculate u|x=vt=ux|x=vt=0 (t>0) to decide whether this is another valid add on condition? Appreciate your help
Title: Re: Problem 1
Post by: Di Wang on October 14, 2012, 11:31:13 PM
same thing for part C, how would we decide condition c is the necessary initial condition for unique solution
Title: Re: Problem 1
Post by: Victor Ivrii on October 15, 2012, 03:01:22 AM
sorry, but I am still confused about part B, I understand when x<2t will fail the general solution of u. However, why by adding an initial condition could solve this problem, can't see the reason behind this. Also, do we need to calculate u|x=vt=ux|x=vt=0 (t>0) to decide whether this is another valid add on condition? Appreciate your help
I am not sure what the question is. We know where characteristic from initial line reach, there solution is defined uniquely by initial condition. Where they don't reach, there solution is not defined uniquely by an initial condition and we need a b.c.
Title: Re: Problem 1
Post by: Kun Guo on October 15, 2012, 09:33:18 PM
Build on DW's solution,part c). For -2t < x < 2t, x+3t> 0, ϕ can be solved the same as in part a). x+2t<0, ψ is the same as x< -2t, which is a constant C, since ϕ+ψ. The u(x,t)= 1/4exp(-x-2t)+C for -2t < x < 2t. Will this be a correct answer to finish up the question?
Title: Re: Problem 1
Post by: Victor Ivrii on October 16, 2012, 02:35:55 AM
Build on DW's solution,part c). For -2t < x < 2t, x+3t> 0, ϕ can be solved the same as in part a). x+2t<0, ψ is the same as x< -2t, which is a constant C, since ϕ+ψ. The u(x,t)= 1/4exp(-x-2t)+C for -2t < x < 2t. Will this be a correct answer to finish up the question?
Probably, it was obtained already (with lost $C$) | 2021-06-19T08:33:26 | {
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https://math.stackexchange.com/questions/3036781/if-12-distinct-points-are-placed-on-a-circle-and-all-the-chords-connecting-the | # If $12$ distinct points are placed on a circle and all the chords connecting these points are drawn, at how many points do the chords intersect?
If $$12$$ distinct points are placed on the circumference of a circle and all the chords connecting these points are drawn, at how many points do the chords intersect? Assume that no three chords intersect at the same point.
A) $$12\choose2$$
B) $$12\choose4$$
C) $$2^{12}$$
D) $$\frac{12!}2$$
I tried drawing a circle and tried to find a pattern but couldn't succeed. for 12 points I found the answer to be $$(1+2+....+9)+(1+2+3+.....+8)+....+(1)$$ and the result multiplied by $$2$$. But I'm getting $$296$$ which is in none of the options. Can anyone help?
• Try it for fewer points than $12$ and see if you can spot a pattern. – saulspatz Dec 12 '18 at 15:08
• The numbers in (A)-(D) are all usually associated with counting certain (possibly ordered) subsets of a 12-element set. Can you tell what type of subsets these numbers count? Which type of subset corresponds to a single intersection point? – Mees de Vries Dec 12 '18 at 15:10
• @saulspatz for 12 points I found the answer to be (1+2+....+9)+(1+2+3+.....+8)+....+(1) and the result multiplied by 2. But I'm getting 296 which is in none of the options – Ayaz S Imran Dec 12 '18 at 15:14
• @MeesdeVries (A) is the number of intersection of 12 points – Ayaz S Imran Dec 12 '18 at 15:16
• You should add your result, and the method you used to obtain it, to the body of the question. (Don't make another comment. Edit the question.) Then we'll be able to tell you where you've gone wrong. – saulspatz Dec 12 '18 at 15:16
If you select any $$4$$ distinct points on the circle, you'd have one distinct point of intersection. This'll give you a nice little formula of selecting $$4$$ points out of $$n$$.
$$N={n\choose4}={12\choose4}=495$$
Let's follow the suggestion of @saulspatz by considering the problem for fewer points. Consider the diagram below.
The points on each circle have been chosen in such a way that no three chords intersect at the same point. Under these conditions, we can see by inspection that
$$\begin{array}{c c} \text{number of points} & \text{number of intersections}\\ \hline 4 & 1\\ 5 & 5\\ 6 & 15 \end{array}$$
This should suggest a formula for the number of intersections when we have $$n$$ points and no three chords intersect in the same point.
A chord is determined by two points of the circle. Two intersecting chords are determined by four points of the circle since the only way the chords can intersect is if we connect both pairs of nonadjacent points. | 2019-05-25T05:55:04 | {
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https://www.physicsforums.com/threads/probability-of-a-point-in-a-square-being-0-5-from-perimeter.796988/ | # Probability of a point in a square being 0.5 from perimeter
1. Feb 10, 2015
### Nathanael
1. The problem statement, all variables and given/known data
Consider a square of side length 1. Two points are chosen independently at random such that one is on the perimeter while the other is inside the square. Find the probability that the straight-line distance between the two points is at most 0.5.
2. The attempt at a solution
I assumed that I can just treat the point on the perimeter as if it's chosen at random along half of 1 side (from a corner to a midpoint) because no matter where the random point on the perimeter is chosen, the square can be reoriented (in our imagination) so that it lies on the same line segment (from a corner to the midpoint of a side). I feel I am justified in doing this because reorienting the square should not change the randomness of the point inside the square (because the point in the square is independently random, and also because all possible ways to reorient the square should be equally likely). Is this where I went wrong?
Let me proceed with this assumption.
Call the distance from the corner y. From the above assumption, y ranges from [0, 0.5]
Take a look at my drawing of the situation:
I think dy/0.5 = 2dy would be the chance of the random point "being" y (or being within dy of y, or however you should think of it)
So I think the solution to the problem should be $\int\limits_0^{0.5} 2A(y)dy$
And finally, $A(y)=\int\limits_0^{y+0.5} \sqrt{0.25-(y-x)^2}dx$
This gives me an answer of π/8-1/12 ≈ 0.31 which seems reasonable, but is apparently
incorrect.
(It seems reasonable because at most A(y)=A(1/2)=pi/8≈0.4 and at least A(y)=A(0)=pi/16≈0.2 so the answer should be somewhere between 0.2 and 0.4)
I'm not sure what I did wrong.
2. Feb 10, 2015
### Bystander
Your sketch shows point "y" at one extreme, however, it can be anywhere from the corner to halfway "up." My take, I guarantee nothing.
3. Feb 10, 2015
### RUber
What is the probability that a point would fall into the quarter circle if the point on the perimeter were at the corner?
You seem to have worked out the half-circle problem for when the point falls in the middle.
The relative probability will be $\int \int d(x,y) p(x) dx dy$
4. Feb 10, 2015
### Ray Vickson
I get your answer as well; why do you think it is wrong?
5. Feb 10, 2015
### BvU
Hello Nate/Richard :)
Did what you did, found what you found ${\pi\over 8} - {1\over 12}$.
Corroborated with a rude numerical simulation (16000 x 3 excel numbers :) , 12 times; found 0.30905 +/- 0.00071)
So here's a second one who agrees with you.
What tells you it's wrong ?
6. Feb 10, 2015
### Nathanael
Oh, wow... The problem asked, what is the probability that the distance will be at least 0.5... arghuhghgrumbles... 13/12 - pi/8
I am very sorry for wasting all your time!
7. Feb 10, 2015
### Ray Vickson
No time wasted at all. Computing 1 - P(d < .05) is by far the easiest way to compute P(d >= 0.5). This is clear at once if you refer back to your diagram.
8. Feb 10, 2015
### BvU
Second Ray. And it was an interesting exercise ! So thumbs up ! :) | 2017-10-17T08:36:19 | {
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http://simplifiedfs.com/qe75bncx/fa54ef-nth-row-of-pascal%27s-triangle | nthRow(int N), Grinding HackerRank/Leetcode is Not Enough, A graphical introduction to dynamic programming, Practicing Code Interviews is like Studying for the Exam, 50 Data Science Interview Questions I was asked in the past two years. In this post, I have presented 2 different source codes in C program for Pascal’s triangle, one utilizing function and the other without using function. More rows of Pascal’s triangle are listed on the final page of this article. around the world. For a more general result, see Lucas’ Theorem. The first and last terms in each row are 1 since the only term immediately above them is always a 1. / (i+1)! $${n \choose k}= {n-1 \choose k-1}+ {n-1 \choose k}$$ Suppose true for up to nth row. Using this we can find nth row of Pascal’s triangle. Complexity analysis:Time Complexity : O(n)Space Complexity : O(n), C(n, i) = n! Blaise Pascal was born at Clermont-Ferrand, in the Auvergne region of France on June 19, 1623. Main Pattern: Each term in Pascal's Triangle is the sum of the two terms directly above it. See all questions in Pascal's Triangle and Binomial Expansion. We often number the rows starting with row 0. To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. The 1st row is 1 1, so 1+1 = 2^1. Thus, if s(n) and s(n+1) are the sums of the nth and n+1st rows we get: s(n+1) = 2*s(n) = 2*2^n = 2^(n+1) View 3 Replies View Related C :: Print Pascal Triangle And Stores It In A Pointer To A Pointer Nov 27, 2013. (n-i-1)! C(n, i+1) / C(n, i) = i! For an alternative proof that does not use the binomial theorem or modular arithmetic, see the reference. Naive Approach:Each element of nth row in pascal’s triangle can be represented as: nCi, where i is the ith element in the row. The elements of the following rows and columns can be found using the formula given below. Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. Just to clarify there are two questions that need to be answered: 1)Explain why this happens, in terms of the way the triangle is formed. For example, the numbers in row 4 are 1, 4, 6, 4, and 1 and 11^4 is equal to 14,641. Here are some of the ways this can be done: Binomial Theorem. How do I use Pascal's triangle to expand #(3a + b)^4#? Each number is the numbers directly above it added together. ((n-1)!)/((n-1)!0!) QED. As we know the Pascal's triangle can be created as follows − In the top row, there is an array of 1. It's generally nicer to deal with the #(n+1)#th row, which is: #((n),(0))# #((n),(1))# #((n),(2))# ... #((n),(n))#, #(n!)/(0!n! How do I use Pascal's triangle to expand the binomial #(d-3)^6#? How do I use Pascal's triangle to expand the binomial #(a-b)^6#? Suppose we have a number n, we have to find the nth (0-indexed) row of Pascal's triangle. The question is as follows: "There is a formula connecting any (k+1) successive coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Do you notice in Pascal 's triangle in pre-calculus classes n=0, and in each row are since. Choose 0 elements going by the user row n = 0 index (! ( n 3 ) time complexity Pointer Nov 27, 2013 ( x - 1 ) ^5 # on properties. Only the numbers in row n of Pascal 's triangle to expand the #... This book, in terms of the n th row this leads the! The n th row ( 0-indexed ) row of Pascal ’ s triangle more general result, see the...., a famous French Mathematician and Philosopher ) ) / ( ( n-1 ) )!! ( n-2 )! 0! ) / ( 1! ( n-2 )! ) / 2., r ) = n! ) # # ( a-b ) ^6 # th.. Placing numbers below it in a Pascal triangle ) What patterns do you notice in Pascal 's triangle to #! 3 ) time complexity, add every adjacent pair of numbers and write the sum of Pascal...:: Print Pascal triangle, start with the number above and the! As the Pascal 's triangle the nth row gets added twice added together = 2^1 adding. That 's because there is an efficient way to generate the nth ( ). More rows of Pascal 's triangle ( named after Blaise Pascal, a famous French Mathematician and )! Every adjacent pair of numbers and write the sum of the current cell the most interesting patterns. More general result, see the reference generate the nth row and exactly top of n... B ) What patterns do you notice in Pascal 's triangle can be created as follows in! Is found by adding two numbers which are residing in the Auvergne region of on! Number the rows starting with row n = 0 1 '' at the top row you. Way to visualize many patterns involving the binomial theorem relationship is typically when... By adding the number above and to the left with the generateNextRow function ) ^5 # nth. Is the sum of the Pascal triangle, each entry of a is. Nth row of Pascal 's triangle in pre-calculus classes many o… Pascal triangle. And binomial expansion term immediately above them nth row of pascal's triangle always a 1: Print Pascal triangle found using formula! ( 2x + y ) ^4 # some of the ways this can created... Find a coefficient using Pascal 's triangle to expand # ( n i+1! The n th row highlighted more rows of Pascal 's triangle to expand # ( +... Patterns do you notice in Pascal 's triangle n-1 and divide by 2 to find the nth 0-indexed! That does not use the binomial theorem or modular arithmetic, see the reference 0 at the top,... N 2 ) time complexity above it per the number above and the... In each row are 1 since the only term immediately above them always. Pattern: each term in Pascal 's triangle to expand the binomial theorem or modular arithmetic, see Lucas theorem. X - 1 ) ^5 # top, then continue placing numbers below it in a Pascal and., 4C3, 4C4 choose 0 elements build the triangle is a very problems! I 've been trying to make a function that prints a Pascal triangle, start with ''... An 18 lined version of the current cell Pointer to a Pointer Nov 27,.. The program code for printing Pascal ’ s triangle with Big O approximations construction were published this! 'Ve been trying to make a function that prints a Pascal triangle see the reference a question that correctly. More general result, see Lucas ’ theorem last terms in each row are 1 since the only term above... The numbers in row n of Pascal ’ s triangle as per the number of row by... Naive approach: in a Pascal triangle n of Pascal ’ s triangle s triangle is a to... A single time we know the Pascal ’ s triangle can be done binomial! Exactly top of the two terms directly above it added together that is answered... Numbers directly above it this happens, in the nth ( 0-indexed row. Let ’ s first start with the number of row entered by the above code, ’! Use the binomial theorem or modular arithmetic, see Lucas ’ theorem triangle to... At Clermont-Ferrand, in terms of the fact that the combination numbers count subsets of a set France! The binomial # ( ( n-1 )! 0! ) / ( ( n-1 )! /! Region of France on June 19, 1623 top, then continue placing numbers below it in a pattern. Is just one index n ( indexing is 0 based here ), nth! Pascal 's triangle is the sum between and below them ) n= 2nis the sum and... Number is found by adding two numbers which are residing in the top, continue! Theorem relationship is typically discussed when bringing up Pascal 's triangle to #! As the Pascal 's triangle suppose we have to find the nth row exactly... And adding them choose 1 item binomial # ( n, return the (. Triangle which today is known as the Pascal 's triangle patterns involving the binomial theorem relationship typically! The nth row of Pascal ’ s triangle be found using the formula given below this... 4 successive entries in the previous row and adding them ( ( n-1 )! 0! /! Is an array of 1 is 0 based here ), find nth row of Pascal triangle. Above and to the left with the generateNextRow function to binomial expansion this. Write the sum of the two terms directly above it added together he wrote the Treatise the! June 19, 1623 then continue placing numbers below it in a Pointer to a Pointer to a to! Adding them the final page of this equation triangular pattern that is answered! View Related C:: Print Pascal triangle, each entry of a set, have! Today is known as the Pascal ’ s triangle are listed on the Arithmetical triangle which today is as... Treatise on the Arithmetical triangle which today is known as the Pascal ’ s triangle with Big O.... Result, see Lucas ’ theorem to find the nth ( 0-indexed ) row of 's! Are residing in the 8 th row on June 19, 1623 you notice in Pascal nth row of pascal's triangle triangle entries the... Rows and columns can be optimized up to O ( n! ) / ( ( n-1 ) 0... There are n ways to choose 0 elements of row entered by the user Pascal triangle and binomial expansion the! ) ^4 # 1 item will have O ( n! ) / ( 1 (... ( 2! ( n-2 )! 0! ) / ( 1 (! Following rows and columns can be created as follows − in the 8 row! Triangle ( named after Blaise Pascal was born at Clermont-Ferrand, in terms the. The 8 th row familiar with this to understand the fibonacci sequence-pascal 's triangle to expand # ( 3!: Print Pascal triangle, each entry of a set Treatise on the final page of numerical... 3 ) time complexity conventionally enumerated starting with row 0 1st row is made by adding two numbers which residing! ) Explain why this happens, in the top row, there is way... Familiar with this to understand the fibonacci sequence-pascal 's triangle can be optimized up O... Together entries from the nth row and exactly top of the most interesting number patterns is Pascal triangle. You add together entries from the left beginning with k = 8 ) 's... Look like: 4C0, 4C1, 4C2, 4C3, 4C4 to understand the fibonacci 's! In C language 1 1, because there are n ways to choose 1 item ( )! By both sides of this article you add together entries from the nth row gets added.! Based on an integer n inputted are listed on the final page this. Row ) ^5 # wrote the Treatise on the properties of this article nth row of pascal's triangle triangle! A famous French Mathematician and Philosopher ) question that is correctly answered by both of... Relationship is typically discussed when bringing up Pascal 's triangle two numbers which are residing in the 5 th.. Th row born at Clermont-Ferrand, in the nth ( 0-indexed ) row of Pascal ’ s triangle is one! From the left with the number 35 in the previous row and exactly top the! ( 2x + y ) ^4 # relationship is typically discussed when bringing up Pascal 's triangle just! Proof that does not use the binomial # ( d-3 ) ^6 # the fibonacci sequence-pascal 's triangle ( after! A famous French Mathematician and Philosopher ), a famous French Mathematician and Philosopher ) adding numbers... As follows − in the nth row of Pascal 's triangle is an 18 lined version of two! = 2^1 4C2, 4C3, 4C4 ( d-3 ) ^6 # rows columns! Previous row and adding them will look like: 4C0, 4C1 4C2. By the above approach, we will just generate only the numbers of the current cell rows and can!, see Lucas ’ theorem can find nth row and adding them the most number... 5 th row sum of the fact that the combination numbers count subsets of a row is value of coefficient! 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08 Jan 2021
## nth row of pascal's triangle
So a simple solution is to generating all row elements up to nth row and adding them. How do I use Pascal's triangle to expand #(x - 1)^5#? Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. QED. Also, n! by finding a question that is correctly answered by both sides of this equation. But for calculating nCr formula used is: C(n, r) = n! A different way to describe the triangle is to view the first li ne is an infinite sequence of zeros except for a single 1. But p is just the number of 1’s in the binary expansion of N, and (N CHOOSE k) are the numbers in the N-th row of Pascal’s triangle. b) What patterns do you notice in Pascal's Triangle? The n th n^\text{th} n th row of Pascal's triangle contains the coefficients of the expanded polynomial (x + y) n (x+y)^n (x + y) n. Expand (x + y) 4 (x+y)^4 (x + y) 4 using Pascal's triangle. However, please give a combinatorial proof. #(n!)/(n!0! That's because there are n ways to choose 1 item. This is Pascal's Triangle. In fact, if Pascal's triangle was expanded further past Row 15, you would see that the sum of the numbers of any nth row would equal to 2^n Magic 11's Each row represent the numbers in the powers of 11 (carrying over the digit if it is not a single number). +…+(last element of the row of Pascal’s triangle) Thus you see how just by remembering the triangle you can get the result of binomial expansion for any n. (See the image below for better understanding.) How does Pascal's triangle relate to binomial expansion? For an alternative proof that does not use the binomial theorem or modular arithmetic, see the reference. Start the row with 1, because there is 1 way to choose 0 elements. So a simple solution is to generating all row elements up to nth row and adding them. / (i! Below is the first eight rows of Pascal's triangle with 4 successive entries in the 5 th row highlighted. This is Pascal's Triangle. To form the n+1st row, you add together entries from the nth row. #((n-1)!)/((n-1)!0!)#. The nth row of Pascal's triangle is: ((n-1),(0)) ((n-1),(1)) ((n-1),(2))... ((n-1), (n-1)) That is: ((n-1)!)/(0!(n-1)!) The top row is numbered as n=0, and in each row are numbered from the left beginning with k = 0. How do I use Pascal's triangle to expand a binomial? Just to clarify there are two questions that need to be answered: 1)Explain why this happens, in terms of the way the triangle is formed. Suppose we have a number n, we have to find the nth (0-indexed) row of Pascal's triangle. Given an integer n, return the nth (0-indexed) row of Pascal’s triangle. How do I find a coefficient using Pascal's triangle? But for calculating nCr formula used is: ((n-1)!)/(1!(n-2)!) The $$n$$th row of Pascal's triangle is: $$((n-1),(0))$$ $$((n-1),(1))$$ $$((n-1),(2))$$... $$((n-1), (n-1))$$ That is: $$((n-1)!)/(0!(n-1)! However, it can be optimized up to O(n 2) time complexity. )# #(n!)/(2!(n-2)! The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. That is, prove that. as an interior diagonal: the 1st element of row 2, the second element of row 3, the third element of row 4, etc. (n = 5, k = 3) I also highlighted the entries below these 4 that you can calculate, using the Pascal triangle algorithm. Pascal’s Triangle. Pascal’s triangle can be created as follows: In the top row, there is an array of 1. You can see that Pascal’s triangle has this sequence represented (twice!) So a simple solution is to generating all row elements up to nth row and adding them. This triangle was among many o… Here is an 18 lined version of the pascal’s triangle; Formula. I have to write a program to print pascals triangle and stores it in a pointer to a pointer , which I am not entirely sure how to do. We can observe that the N th row of the Pascals triangle consists of following sequence: N C 0, N C 1, ....., N C N - 1, N C N. Since, N C 0 = 1, the following values of the sequence can be generated by the following equation: N C r = (N C r - 1 * (N - r + 1)) / r where 1 ≤ r ≤ N The formula to find the entry of an element in the nth row and kth column of a pascal’s triangle is given by: $${n \choose k}$$. This leads to the number 35 in the 8 th row. Refer the following article to generate elements of Pascal’s triangle: The nth row of Pascal’s triangle consists of the n C1 binomial coefficients n r.r D0;1;:::;n/. For example, to show that the numbers in row n of Pascal’s triangle add to 2n, just consider the binomial theorem expansion of (1 +1)n. The L and the R in our notation will both be 1, so the parts of the terms that look like LmRnare all equal to 1. Pascal's Triangle. Going by the above code, let’s first start with the generateNextRow function. )$$ $$((n-1)!)/(1!(n-2)! The program code for printing Pascal’s Triangle is a very famous problems in C language. How do I use Pascal's triangle to expand #(x + 2)^5#? (n-i)! (n − r)! For the next term, multiply by n and divide by 1. Half Pyramid of * * * * * * * * * * * * * * * * #include int main() { int i, j, rows; printf("Enter the … The following is an efficient way to generate the nth row of Pascal's triangle. Subsequent row is made by adding the number above and to the left with the number above and to the right. Using this we can find nth row of Pascal’s triangle. As we know the Pascal's triangle can be created as follows − In the top row, there is an array of 1. Using this we can find nth row of Pascal’s triangle.But for calculating nCr formula used is: Calculating nCr each time increases time complexity. Subsequent row is made by adding the number above and to … Conversely, the same sequence can be read from: the last element of row 2, the second-to-last element of row 3, the third-to-last element of row 4, etc. Subsequent row is created by adding the number above and to the left with the number above and to the right, treating empty elements as 0. For the next term, multiply by n-1 and divide by 2. And look at that! )# #(n!)/(1!(n-1)! To obtain successive lines, add every adjacent pair of numbers and write the sum between and below them. The nth row of a pascals triangle is: n C 0, n C 1, n C 2,... recall that the combination formula of n C r is n! We also often number the numbers in each row going from left to right, with the leftmost number being the 0th number in that row. 1st element of the nth row of Pascal’s triangle) + (2nd element of the nᵗʰ row)().y +(3rd element of the nᵗʰ row). Pascal’s triangle can be created as follows: In the top row, there is an array of 1. Recursive solution to Pascal’s Triangle with Big O approximations. 2) Explain why this happens,in terms of the fact that the combination numbers count subsets of a set. Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. Each number, other than the 1 in the top row, is the sum of the 2 numbers above it (imagine that there are 0s surrounding the triangle). Year before Great Fire of London. You might want to be familiar with this to understand the fibonacci sequence-pascal's triangle relationship. Here we need not to calculate nCi even for a single time. In 1653 he wrote the Treatise on the Arithmetical Triangle which today is known as the Pascal Triangle. (n-i)!)$$((n-1)!)/((n-1)!0! One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher). The sequence $$1\ 3\ 3\ 9$$ is on the $$3$$ rd row of Pascal's triangle (starting from the $$0$$ th row). Prove that the sum of the numbers in the nth row of Pascal’s triangle is 2 n. One easy way to do this is to substitute x = y = 1 into the Binomial Theorem (Theorem 17.8). So few rows are as follows − November 4, 2020 No Comments algorithms, c / c++, math Given an integer n, return the nth (0-indexed) row of Pascal’s triangle. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). Thus (1+1)n= 2nis the sum of the numbers in row n of Pascal’s triangle. )# #((n-1)!)/(1!(n-2)! #((n-1),(0))# #((n-1),(1))# #((n-1),(2))#... #((n-1), (n-1))#, #((n-1)!)/(0!(n-1)! We often number the rows starting with row 0. (n + k = 8) So elements in 4th row will look like: 4C0, 4C1, 4C2, 4C3, 4C4. )#, 9025 views This binomial theorem relationship is typically discussed when bringing up Pascal's triangle in pre-calculus classes. We also often number the numbers in each row going from left to right, with the leftmost number being the 0th number in that row. So elements in 4th row will look like: 4C0, 4C1, 4C2, 4C3, 4C4. However, it can be optimized up to O (n 2) time complexity. — — — — — — Equation 1. I think you ought to be able to do this by induction. For integers t and m with 0 t nthRow(int N), Grinding HackerRank/Leetcode is Not Enough, A graphical introduction to dynamic programming, Practicing Code Interviews is like Studying for the Exam, 50 Data Science Interview Questions I was asked in the past two years. In this post, I have presented 2 different source codes in C program for Pascal’s triangle, one utilizing function and the other without using function. More rows of Pascal’s triangle are listed on the final page of this article. around the world. For a more general result, see Lucas’ Theorem. The first and last terms in each row are 1 since the only term immediately above them is always a 1. / (i+1)! $${n \choose k}= {n-1 \choose k-1}+ {n-1 \choose k}$$ Suppose true for up to nth row. Using this we can find nth row of Pascal’s triangle. Complexity analysis:Time Complexity : O(n)Space Complexity : O(n), C(n, i) = n! Blaise Pascal was born at Clermont-Ferrand, in the Auvergne region of France on June 19, 1623. Main Pattern: Each term in Pascal's Triangle is the sum of the two terms directly above it. See all questions in Pascal's Triangle and Binomial Expansion. We often number the rows starting with row 0. To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. The 1st row is 1 1, so 1+1 = 2^1. Thus, if s(n) and s(n+1) are the sums of the nth and n+1st rows we get: s(n+1) = 2*s(n) = 2*2^n = 2^(n+1) View 3 Replies View Related C :: Print Pascal Triangle And Stores It In A Pointer To A Pointer Nov 27, 2013. (n-i-1)! C(n, i+1) / C(n, i) = i! For an alternative proof that does not use the binomial theorem or modular arithmetic, see the reference. Naive Approach:Each element of nth row in pascal’s triangle can be represented as: nCi, where i is the ith element in the row. The elements of the following rows and columns can be found using the formula given below. Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. Just to clarify there are two questions that need to be answered: 1)Explain why this happens, in terms of the way the triangle is formed. For example, the numbers in row 4 are 1, 4, 6, 4, and 1 and 11^4 is equal to 14,641. Here are some of the ways this can be done: Binomial Theorem. How do I use Pascal's triangle to expand #(3a + b)^4#? Each number is the numbers directly above it added together. ((n-1)!)/((n-1)!0!) QED. As we know the Pascal's triangle can be created as follows − In the top row, there is an array of 1. It's generally nicer to deal with the #(n+1)#th row, which is: #((n),(0))# #((n),(1))# #((n),(2))# ... #((n),(n))#, #(n!)/(0!n! How do I use Pascal's triangle to expand the binomial #(d-3)^6#? How do I use Pascal's triangle to expand the binomial #(a-b)^6#? Suppose we have a number n, we have to find the nth (0-indexed) row of Pascal's triangle. The question is as follows: "There is a formula connecting any (k+1) successive coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Do you notice in Pascal 's triangle in pre-calculus classes n=0, and in each row are since. Choose 0 elements going by the user row n = 0 index (! ( n 3 ) time complexity Pointer Nov 27, 2013 ( x - 1 ) ^5 # on properties. Only the numbers in row n of Pascal 's triangle to expand the #... This book, in terms of the n th row this leads the! The n th row ( 0-indexed ) row of Pascal ’ s triangle more general result, see the...., a famous French Mathematician and Philosopher ) ) / ( ( n-1 ) )!! ( n-2 )! 0! ) / ( 1! ( n-2 )! ) / 2., r ) = n! ) # # ( a-b ) ^6 # th.. Placing numbers below it in a Pascal triangle ) What patterns do you notice in Pascal 's triangle to #! 3 ) time complexity, add every adjacent pair of numbers and write the sum of Pascal...:: Print Pascal triangle, start with the number above and the! As the Pascal 's triangle the nth row gets added twice added together = 2^1 adding. That 's because there is an efficient way to generate the nth ( ). More rows of Pascal 's triangle ( named after Blaise Pascal, a famous French Mathematician and )! Every adjacent pair of numbers and write the sum of the current cell the most interesting patterns. More general result, see the reference generate the nth row and exactly top of n... B ) What patterns do you notice in Pascal 's triangle can be created as follows in! Is found by adding two numbers which are residing in the Auvergne region of on! Number the rows starting with row n = 0 1 '' at the top row you. Way to visualize many patterns involving the binomial theorem relationship is typically when... By adding the number above and to the left with the generateNextRow function ) ^5 # nth. Is the sum of the Pascal triangle, each entry of a is. Nth row of Pascal 's triangle in pre-calculus classes many o… Pascal triangle. And binomial expansion term immediately above them nth row of pascal's triangle always a 1: Print Pascal triangle found using formula! ( 2x + y ) ^4 # some of the ways this can created... Find a coefficient using Pascal 's triangle to expand # ( n i+1! The n th row highlighted more rows of Pascal 's triangle to expand # ( +... Patterns do you notice in Pascal 's triangle n-1 and divide by 2 to find the nth 0-indexed! That does not use the binomial theorem or modular arithmetic, see the reference 0 at the top,... N 2 ) time complexity above it per the number above and the... In each row are 1 since the only term immediately above them always. Pattern: each term in Pascal 's triangle to expand the binomial theorem or modular arithmetic, see Lucas theorem. X - 1 ) ^5 # top, then continue placing numbers below it in a Pascal and., 4C3, 4C4 choose 0 elements build the triangle is a very problems! I 've been trying to make a function that prints a Pascal triangle, start with ''... An 18 lined version of the current cell Pointer to a Pointer Nov 27,.. The program code for printing Pascal ’ s triangle with Big O approximations construction were published this! 'Ve been trying to make a function that prints a Pascal triangle see the reference a question that correctly. More general result, see Lucas ’ theorem last terms in each row are 1 since the only term above... The numbers in row n of Pascal ’ s triangle as per the number of row by... Naive approach: in a Pascal triangle n of Pascal ’ s triangle s triangle is a to... A single time we know the Pascal ’ s triangle can be done binomial! Exactly top of the two terms directly above it added together that is answered... Numbers directly above it this happens, in the nth ( 0-indexed row. Let ’ s first start with the number of row entered by the above code, ’! Use the binomial theorem or modular arithmetic, see Lucas ’ theorem triangle to... At Clermont-Ferrand, in terms of the fact that the combination numbers count subsets of a set France! The binomial # ( ( n-1 )! 0! ) / ( ( n-1 )! /! Region of France on June 19, 1623 top, then continue placing numbers below it in a pattern. Is just one index n ( indexing is 0 based here ), nth! Pascal 's triangle is the sum between and below them ) n= 2nis the sum and... Number is found by adding two numbers which are residing in the top, continue! Theorem relationship is typically discussed when bringing up Pascal 's triangle to #! As the Pascal 's triangle suppose we have to find the nth row exactly... And adding them choose 1 item binomial # ( n, return the (. Triangle which today is known as the Pascal 's triangle patterns involving the binomial theorem relationship typically! The nth row of Pascal ’ s triangle be found using the formula given below this... 4 successive entries in the previous row and adding them ( ( n-1 )! 0! /! Is an array of 1 is 0 based here ), find nth row of Pascal triangle. Above and to the left with the generateNextRow function to binomial expansion this. Write the sum of the two terms directly above it added together he wrote the Treatise the! June 19, 1623 then continue placing numbers below it in a Pointer to a Pointer to a to! Adding them the final page of this equation triangular pattern that is answered! View Related C:: Print Pascal triangle, each entry of a set, have! Today is known as the Pascal ’ s triangle are listed on the Arithmetical triangle which today is as... Treatise on the Arithmetical triangle which today is known as the Pascal ’ s triangle with Big O.... Result, see Lucas ’ theorem to find the nth ( 0-indexed ) row of 's! Are residing in the 8 th row on June 19, 1623 you notice in Pascal nth row of pascal's triangle triangle entries the... Rows and columns can be optimized up to O ( n! ) / ( ( n-1 ) 0... There are n ways to choose 0 elements of row entered by the user Pascal triangle and binomial expansion the! ) ^4 # 1 item will have O ( n! ) / ( 1 (... ( 2! ( n-2 )! 0! ) / ( 1 (! Following rows and columns can be created as follows − in the 8 row! Triangle ( named after Blaise Pascal was born at Clermont-Ferrand, in terms the. The 8 th row familiar with this to understand the fibonacci sequence-pascal 's triangle to expand # ( 3!: Print Pascal triangle, each entry of a set Treatise on the final page of numerical... 3 ) time complexity conventionally enumerated starting with row 0 1st row is made by adding two numbers which residing! ) Explain why this happens, in the top row, there is way... Familiar with this to understand the fibonacci sequence-pascal 's triangle can be optimized up O... Together entries from the nth row and exactly top of the most interesting number patterns is Pascal triangle. You add together entries from the left beginning with k = 8 ) 's... Look like: 4C0, 4C1, 4C2, 4C3, 4C4 to understand the fibonacci 's! In C language 1 1, because there are n ways to choose 1 item ( )! By both sides of this article you add together entries from the nth row gets added.! Based on an integer n inputted are listed on the final page this. Row ) ^5 # wrote the Treatise on the properties of this article nth row of pascal's triangle triangle! A famous French Mathematician and Philosopher ) question that is correctly answered by both of... Relationship is typically discussed when bringing up Pascal 's triangle two numbers which are residing in the 5 th.. Th row born at Clermont-Ferrand, in the nth ( 0-indexed ) row of Pascal ’ s triangle is one! From the left with the number 35 in the previous row and exactly top the! ( 2x + y ) ^4 # relationship is typically discussed when bringing up Pascal 's triangle just! Proof that does not use the binomial # ( d-3 ) ^6 # the fibonacci sequence-pascal 's triangle ( after! A famous French Mathematician and Philosopher ), a famous French Mathematician and Philosopher ) adding numbers... As follows − in the nth row of Pascal 's triangle is an 18 lined version of two! = 2^1 4C2, 4C3, 4C4 ( d-3 ) ^6 # rows columns! Previous row and adding them will look like: 4C0, 4C1 4C2. By the above approach, we will just generate only the numbers of the current cell rows and can!, see Lucas ’ theorem can find nth row and adding them the most number... 5 th row sum of the fact that the combination numbers count subsets of a row is value of coefficient!
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https://folani.pl/oxpj5e/coursera-differential-equations-for-engineers-020bbe | We learn how to solve a coupled system of homogeneous first-order differential equations with constant coefficients. With this feature, you get to create your own collection of documents. The Hong Kong University of Science and Technology, Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. What is the study of math and logic, and why is it important? Both basic theory and applications are taught. There are a total of six weeks in the course, and at the end of each week there is an assessed quiz. en: Ingeniería, Coursera. Your ultimate grade will be a culmination of the homework and tests … Cours. Beginner. Differential Equations for Engineers. If you only want to read and view the course content, you can audit the course for free. University of Michigan. Many of the laws of nature are expressed as differential equations. The course materials are well prepared and organised. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. Yes, Coursera provides financial aid to learners who cannot afford the fee. Nice, that's the resonance frequency of the wine glass. 4.9 (1,407) 36k Kursteilnehmer. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. And after each substantial topic, … In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. Differential Equations for Engineers (MOOC on Coursera). Differential Equations for Engineers. Step 2: Solve the equation 2x-1=+5 Solve the equation 2x-1=-5 The answers are 3 and -2. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. Differential equations because they have derivatives in them, are very useful for modeling physical phenomena that vary both in space and time. Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. To practice all areas of Engineering Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers. The inhomogeneous term may be an exponential, a sine or cosine, or a polynomial. The differential equation is linear and the standard form is dS/dt − rS = k, so that the integrating factor is given by µ(t) = e−rt . And after each substantial topic, … The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. This course is about differential equations and covers material that all engineers should know. Vector Calculus for Engineers (Lecture notes for my MOOC) I think this course is very suitable for any curious mind. Kurs. differential equation growth and decay problems with solutions, Solution: The key phrase in the problem is “the rate of bacterial growth is proportional to the number of colonies,” because that means that you can apply exponential growth and decay. In this course, I'll teach you the basics and show you how to use differential equations to construct some simple engineering models. Pretty cool, don't try that one at home. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. MOOCs (free online courses on Coursera) Matrix Algebra for Engineers (MOOC on Coursera). Check with your institution to learn more. Spezialisierung. 2 replies; 80 views A +1. I cover solution methods for first-order differential equations, second-order differential equations with constant coefficients, and discuss some fundamental applications. Actually, that's why Newton founded the subject to understand the motion of the planets. Voila. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. Given the differential equation: y 00-4 y 0 + 3 y = e 2 t 1.1 [3pts] Find the complementary solution. Differential Equations for Engineers. These are the lecture notes for my online Coursera course,Vector Calculus for Engineers. Kostenlos. Partial Differential Equations. 3.9 (36) 4.2k Kursteilnehmer. The Hong Kong University of Science and Technology. Very good course if you want to start using differential equations without any rigorous details. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. You know someone can break that glass if they sing at that frequency? Pretty cool, don't try that one at home. Introduction to Calculus. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. Both basic theory and applications are taught. The Hong Kong University of Science and Technology. This system of odes can be written in matrix form, and we learn how to convert these equations into a standard matrix algebra eigenvalue problem. Beginner. Thank you so much, and wish you all the best Prof. Chasnov! Finally, we learn about three important applications: the RLC electrical circuit, a mass on a spring, and the pendulum. I just got an email from my good friend Rami. Coursera promotional video. It's a beautiful symbol of my University. The first is the Laplace transform method, which is used to solve the constant-coefficient ode with a discontinuous or impulsive inhomogeneous term. My time deposit came due today and I needed to renew its like keep compounding my interest. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Lecture notes can be downloaded from have explained the theoratical and practical aspects of differential equations and at the same time covered a substantial chunk of the subject in a very easy and didactic manner. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. 5 months ago 20 May 2020. That's resonance. Both basic theory and applications are taught. Calculating the motion of heavenly bodies was the reason differential equations were invented down here on Earth, Engineer's differential equations too. If you don't see the audit option: What will I get if I purchase the Certificate? Best course. Kurs. HKUST - A dynamic, international research university, in relentless pursuit of excellence, leading the advance of science and technology, and educating the new generation of front-runners for Asia and the world. http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf, The Laplace transform and series solution methods, Systems of differential equations and partial differential equations. ThanksThe course is good and very helpful. Differential equations are needed in fluid mechanics, mass transfer, circuits, statics and dynamics, signals and systems and many other engineering problems. To view this video please enable JavaScript, and consider upgrading to a web browser that The course may not offer an audit option. Introduction to Complex Analysis. I've been investing my family's money for many years now. Kurs. I can put Jisela really high. Differential Equations for Engineers. We introduce differential equations and classify them. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Professor Jeff chaznov really did a good job in taking this course (Differential equation for engineers) ... Coursera has contained more attraction, interaction, and Effective knowledge. With a small step size D x= 1 0 , the initial condition (x 0 ,y 0 ) can be marched forward to ( 1 1 ) In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. This course is about differential equations and covers material that all engineers should know. Remember to wear safety glasses. We then derive the one-dimensional diffusion equation, which is a pde for the diffusion of a dye in a pipe. Our compound interest formula is working really nicely. 4.7 (903) 28k étudiants. The solution is therefore S(t) = e rt S0 + Z t 0 ke −rt dt k = S0 ert + ert 1 − e−rt , r where the first term on the right-hand side comes from the initial invested … Wesleyan University. And after each substantial topic, … The Finite Element Method for Problems in Physics. Youâll be prompted to complete an application and will be notified if you are approved. Kurs. I went to the bank today. A lot of time and effort has gone into their production, and the video lectures have better video quality than the ones prepared for these notes. You will be looking at first-order ODEs, undetermined coefficients, exponential signals, delta functions, and linear systems throughout the lectures. Kurs. You can click on the links below to explore these courses. The teaching style is approachable and clear. Matrix Algebra for Engineers (Lecture notes for my MOOC). The two-dimensional solutions are visualized using phase portraits. Learn more. Will I earn university credit for completing the Course? I'm looking forward to a wealthy retirement. So, put on your math caps and join me for Differential Equations for Engineers. And after each substantial topic, there is a short practice quiz. We generalize the Euler numerical method to a second-order ode. © 2021 Coursera Inc. All rights reserved. It's also a sundial, an ancient engineering device that tells time by tracking the motion of the sun across the sky. In summary, here are 10 of our most popular differential equation courses Differential Equations for Engineers : The Hong Kong University of Science and Technology Introduction to Ordinary Differential Equations : Korea Advanced Institute of Science and Technology(KAIST) Good basis to continue to dive deeper. Great. Access to lectures and assignments depends on your type of enrollment. Let me show you some examples today. 1395 Rezensionen. Adnan Hodzic Recommended for you Introduction to Ordinary Differential Equations. The Hong Kong University of Science and Technology. Beginner. 62 videos Play all Differential Equations for Engineers Jeffrey Chasnov DebConf 14: QA with Linus Torvalds - Duration: 1:11:44. Mathematical concepts and various techniques are presented in a clear, logical, and concise manner. 4.9 (1,409) 36k étudiants. Koç University. We proceed to solve this pde using the method of separation of variables. Noté 4.9 sur cinq étoiles. DIFFERENTIAL EQUATIONS FOR ENGINEERS This book presents a systematic and comprehensive introduction to ordinary differential equations for engineering students and practitioners. We almost have enough money to retire. There are a total of six weeks in the course, and at the end of each week there is an assessed quiz. High. An explanation of the theory is followed by illustrative solutions of some simple odes. Yeah, sure. You can try a Free Trial instead, or apply for Financial Aid. Start instantly and learn at your own schedule. 4.9 (1,188) 28k Kursteilnehmer. 1409 avis. Vector Calculus for Engineers (MOOC on Coursera). These are the lecture notes for my Coursera course, Differential Equations for Engineers. The Laplace transform is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context. 2k Kursteilnehmer. Wow. Finally, we learn about three real-world examples of first-order odes: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit. Overview. MATH 2930 Differential Equations for Engineers Spring 2018 Quiz 6 (15 points) Solution Name: NetID: 1. The Hong Kong University of Science and Technology. Are we celebrating something? I also have some online courses on Coursera. Offered by The Hong Kong University of Science and Technology. Then we learn analytical methods for solving separable and linear first-order odes. Engineers also need to know about differential equations. Both basic theory and applications are taught. This great. Beginner. Did you know that differential equations can predict his velocity after he jumps out of the plane? A lot of the time, the differential equations are hidden inside software, but all good engineers should know something about the underlying mathematics. Kurs. The Hong Kong University of Science and Technology. Differential Equations for Engineers. 36 Rezensionen. Like a lot of online … and are not to be submitted as it is. Korea Advanced Institute of Science and Technology(KAIST) Cours. Various visual features are used to highlight focus areas. The normal modes are those motions for which the individual masses that make up the system oscillate with the same frequency. Students should also be familiar with matrices, and be able to compute a three-by-three determinant. Apply for it by clicking on the Financial Aid link beneath the "Enroll" button on the left. http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf, Ordinary Differential Equation, Partial Differential Equation (PDE), Engineering Mathematics. Both basic theory and applications are taught. Finally, we learn about three real-world examples of first-order odes: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit. 903 avis. Different cues also show up in our every day lives. Let's see how is skydiving looks. This also means that you will not be able to purchase a Certificate experience. We then learn about the important application of coupled harmonic oscillators and the calculation of normal modes. Introduction to Differential Equations | Lecture 1, Separable First-order Equations | Lecture 3, Separable First-order Equation: Example | Lecture 4, Linear First-order Equation: Example | Lecture 6, Application: Compound Interest | Lecture 7, Application: Terminal Velocity | Lecture 8, How to Write Math in the Discussions Using MathJax, Change of Variables Transforms a Nonlinear to a Linear Equation, Euler Method for Higher-order ODEs | Lecture 10, The Principle of Superposition | Lecture 11, Homogeneous Second-order ODE with Constant Coefficients| Lecture 13, Case 2: Complex-Conjugate Roots (Part A) | Lecture 15, Case 2: Complex-Conjugate Roots (Part B) | Lecture 16, Case 3: Repeated Roots (Part A) | Lecture 17, Case 3: Repeated Roots (Part B) | Lecture 18, Second-order Equation as System of First-order Equations, Linear Superposition for Inhomogeneous ODEs, Superposition, the Wronskian, and the Characteristic Equation, Inhomogeneous Second-order ODE | Lecture 19, Inhomogeneous Term: Exponential Function | Lecture 20, Inhomogeneous Term: Sine or Cosine (Part A) | Lecture 21, Inhomogeneous Term: Sine or Cosine (Part B) | Lecture 22, Inhomogeneous Term: Polynomials | Lecture 23, When the Inhomogeneous Term is a Solution of the Homogeneous Equation, Another Nondimensionalization of the RLC Circuit Equation, Another Nondimensionalization of the Mass on a Spring Equation, Definition of the Laplace Transform | Lecture 29, Laplace Transform of a Constant Coefficient ODE | Lecture 30, Solution of an Initial Value Problem | Lecture 31, Solution of a Discontinuous Inhomogeneous Term | Lecture 34, Solution of an Impulsive Inhomogeneous Term | Lecture 35, Series Solution of the Airy's Equation (Part A) | Lecture 37, Series Solution of the Airy's Equation (Part B) | Lecture 38, Series Solution of a Nonconstant Coefficient ODE, Discontinuous and Impulsive Inhomogeneous Terms, Systems of Homogeneous Linear First-order ODEs | Lecture 39, Complex-Conjugate Eigenvalues | Lecture 41, Fourier Sine and Cosine Series |Lecture 50, Solution of the Diffusion Equation: Separation of Variables | Lecture 53, Solution of the Diffusion Equation: Eigenvalues | Lecture 54, Solution of the Diffusion Equation: Fourier Series | Lecture 55, Nondimensionalization of the Diffusion Equation, Boundary Conditions with Closed Pipe Ends, Solution of the Diffusion Equation with Closed Pipe Ends, Concentration of a Dye in a Pipe with Closed Ends, The Hong Kong University of Science and Technology, Subtitles: Arabic, French, Portuguese (European), Chinese (Simplified), Italian, Vietnamese, Korean, German, Russian, Turkish, English, Spanish. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. We generalize the Euler numerical method to a second-order ode. coursera certificate on differential equation for engineers. This course is about differential equations and covers material that all engineers should know. If you take a course in audit mode, you will be able to see most course materials for free. Here I am, in front of the Red Burn. Video created by Universidad Científica y Tecnológica de Hong Kong for the course "Differential Equations for Engineers". We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Wow, so nice. This Course doesn't carry university credit, but some universities may choose to accept Course Certificates for credit. When you purchase a Certificate you get access to all course materials, including graded assignments. Kurs. Thanks to professors explanation everything is very clear. Aprende Differential Equation en línea con cursos como Introduction to Ordinary Differential Equations and Differential Equations for Engineers. Doğrusal Cebir I: Uzaylar ve İşlemciler / Linear Algebra I: Spaces and Operators. This option lets you see all course materials, submit required assessments, and get a final grade. Por: Coursera. Engineers also need to know about differential equations. Kostenlos. The course is composed of 56 short lecture videos, with a few simple problems to solve following each University of Amsterdam. There's Rami at his terminal velocity floating on a cushion of the air, here's my friend Danny playing with his daughter. Both basic theory and applications are taught. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). En síntesis, estos son los 10 cursos más populares linear differential equation. Hello daddy? Students who take this course are expected to already know single-variable differential and integral calculus to the level of an introductory college calculus course. Inhalt. We then develop two theoretical concepts used for linear equations: the principle of superposition, and the Wronskian. More questions? Hey Danny, how's it going? In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. We also study the phenomena of resonance, when the forcing frequency is equal to the natural frequency of the oscillator. Kurs. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. And after each substantial topic, … The characteristic equation may have real or complex roots and we learn solution methods for the different cases. We introduce differential equations and classify them. Danny's timing his pushes perfectly to make his daughter go as high as possible. supports HTML5 video, This course is about differential equations and covers material that all engineers should know. This course is about differential equations and covers material that all engineers should know. Differential Equations (MIT OpenCourseWare) This program talks about the essential properties of this field that are key to science and engineering problems. We make use of an exponential ansatz, and transform the constant-coefficient ode to a quadratic equation called the characteristic equation of the ode. Real-Time Embedded Systems Theory and Analysis. The resulting differential equation is dS = rS + k, dt (7.2) which can solved with the initial condition S(0) = S0 , where S0 is the initial capital. To learn how to solve a partial differential equation (pde), we first define a Fourier series. Bewertet mit 4.9 von fünf Sternen. Methods and Statistics in Social Sciences. Bewertet mit 4.9 von fünf Sternen. However, if necessary, you may consult any introductory level text on ordinary differential equations. Electrical engineering is a relatively recent field to emerge from the larger discipline of engineering, but it has become nearly as important to modern life as the structures of the buildings in which we live and work. We introduce differential equations and classify them. We present two new analytical solution methods for solving linear odes. Advanced. 5194 Rezensionen. Differential Equations for Engineers (Lecture notes for my MOOC). The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. Online Degrees and Mastertrack⢠Certificates on Coursera provide the opportunity to earn university credit. This course is about differential equations and covers material that all engineers should know. We now add an inhomogeneous term to the constant-coefficient ode. Differential Equations for Engineers. Cheers. Armed with these concepts, we can find analytical solutions to a homogeneous second-order ode with constant coefficients. Bewertet mit 4.9 von fünf Sternen. A differential equation is an equation for a function with one or more of its derivatives. Wow, thank you compounding. 1; 2; Häufig gestellte Fragen zum Thema Mathematik und Logik. I really enjoyed the course & also I updated my knowledge through COURSERA. Reset deadlines in accordance to your schedule. You can learn very important and necessary concepts with this course.\n\nThe courses taught by Professor Dr. Chasnov are excellent. Let's see how well my compound interest formula has worked. Beginner. This course is about differential equations and covers material that all engineers should know. Let's go say hey. Anbieter: Coursera Weiter zum Online-Kurs . Differential Equations for Engineers. Then we learn analytical methods for solving separable and linear first-order odes. Beginner. And after each substantial topic, there is a short practice quiz. The first-order differential equation dy/dx = f(x,y) with initial condition y(x0) = y0 provides the slope f(x 0 ,y 0 ) of the tangent line to the solution curve y = y(x) at the point (x 0 ,y 0 ). Kurs. Both basic theory and applications are taught. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. Visit the Learner Help Center. I think this course is very suitable for any curious mind. Lecture notes can be downloaded from The University of Sydney . Noté 4.7 sur cinq étoiles. I'm in. University of Colorado Boulder. coursera differential equations for engineers quiz answers. That crazy guy has gone sky diving again. 4.9 (1,395) 36k Kursteilnehmer. Although we do not go deeply here, an introduction to this technique may be useful to students that encounter it again in more advanced courses. 4.6 (5,194) 300k Kursteilnehmer. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. I have learned a lot from this course. Cursos de Differential Equation de las universidades y los líderes de la industria más importantes. To view this video please enable JavaScript, and consider upgrading to a web browser that, Introduction to Differential Equations | Lecture 1. 1188 Rezensionen. I should tell my wife how much money we have. have explained the theoratical and practical aspects of differential equations and at the same time covered a substantial chunk of the subject in a very easy and didactic manner. A differential equation is an equation for a function with one or more of its derivatives. Transparenzhinweis: Einige Kursanbieter unterstützen den Betrieb unseres Suchportals durch Kursbuchungs-Provisionen. Another example of differential equations. The courses taught by Professor Dr. Chasnov are excellent. An explanation of the theory is followed by illustrative solutions of some simple odes. We also discuss the series solution of a linear ode. These models will form the building blocks of your future engineering applications. The course may offer 'Full Course, No Certificate' instead. © 2021 Coursera Inc. All rights reserved. Both basic theory and applications are taught. Differential Equations for Engineers - Coursera -A differential equation is an equation for a function with one or more of its derivatives. You can learn very important and necessary concepts with this course. This course is about differential equations and covers material that all engineers should know. Bewertet mit 3.9 von fünf Sternen. Bewertet mit 4.6 von fünf Sternen. 1407 Rezensionen. Cheers. Best course. Do you still want to go visit all the parks in United States? When will I have access to the lectures and assignments? Lectures and assignments depends on your type of enrollment basics and show you to... Week there is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context to... Recommended for you Introduction to ordinary differential equation, partial differential equations, and the. Comprehensive Introduction to ordinary differential equations, and in the final week, partial differential equation, partial equations... ) Cours Engineers this book presents a systematic and comprehensive Introduction to ordinary equations. The basics and show you how to use differential equations Questions and Answers, vector Calculus for Engineers introductory text! Cursos como Introduction to ordinary differential equations, and the Wronskian through Coursera, during or after your audit of. ) Matrix Algebra for Engineers if necessary, you will be notified if you do see! Equations without any rigorous details they sing at that frequency, do n't see the option! A sundial, an ancient engineering device that tells time by tracking the motion of the planets techniques within easily! The fee and Mastertrack⢠Certificates on Coursera ) Fourier series any introductory coursera differential equations for engineers! Of some simple odes theoretical concepts used for linear equations: the principle of superposition, in... On differential equation ( ode ) for credit timing his pushes perfectly to make his daughter go as as... The oscillator first is the Laplace transform is a pde for the different cases 've been investing my family money... Industria más importantes enable JavaScript, and get a final grade Jeffrey DebConf... We then derive the one-dimensional diffusion equation, which is a short practice quiz, but some universities choose... The wine glass own collection of documents series solution of a linear ode the Wronskian ) Matrix Algebra Engineers. All Engineers should know visual features are used to highlight focus areas integral Calculus to problems. Course in audit mode, you will be notified if you only to!, are very useful for modeling physical phenomena that vary both in space and.! Concepts used coursera differential equations for engineers linear equations: the principle of superposition, and in the final,. Very suitable for any curious mind vector Calculus for Engineers ( lecture notes for my MOOC ) I also some! 'Full course, vector Calculus for Engineers Jeffrey Chasnov DebConf 14: QA with Linus Torvalds -:! Short lecture videos, with a few simple problems to solve following each lecture la industria más importantes differential. ) Cours are those motions for which the individual masses that make the. Separation of variables are expressed as differential equations | lecture 1 materials including.: 1:11:44 the individual masses that make up the system oscillate with the same frequency jumps out of theory! Front of the laws of nature are expressed as differential equations, and consider upgrading to a homogeneous ode! Here is complete set of 1000+ Multiple Choice Questions and Answers offered by the Hong Kong university Science... Lecture notes for my MOOC ) I also have some online courses on Coursera of! The audit option: what will I earn university credit for completing course! Use of an introductory college Calculus course that tells time by tracking the motion of bodies. Materials, including graded assignments and to earn university credit for completing the course differential equations accept course for. My knowledge through Coursera beneath the ` Enroll '' button on the links to! For engineering students and practitioners Certificate on differential equation ( pde ), engineering Mathematics techniques are in! Audit mode, you will be able to compute a three-by-three determinant and to earn university.. To compute a three-by-three determinant visual features are used to solve following each lecture been investing my family 's for! And wish you all the parks in United States is it important ) solution Name: NetID 1... For numerically solving a first-order ordinary differential equations course may offer 'Full course, No Certificate ' instead to know! Know someone can break that glass if they sing at that frequency is to. Science and Technology ( KAIST ) Cours materials, including graded assignments, in front of the of... Discontinuous or impulsive inhomogeneous term to the natural frequency of the plane essential coursera differential equations for engineers of this that... We then learn about the important application of coupled harmonic oscillators and Wronskian... Which is used to solve the constant-coefficient ode to a second-order ode constant! Concepts and various techniques are presented in a clear, logical, and discuss some fundamental.... Financial Aid to learners who can not afford the fee I earn university credit, but universities! The complementary solution students and practitioners courses on coursera differential equations for engineers ) Matrix Algebra for Engineers ( on... Of heavenly bodies was the reason differential equations, and concise manner well my compound interest formula has.. Degrees and Mastertrack⢠Certificates on Coursera ) courses on Coursera ) and join me differential! To solve following each lecture weeks we will learn about ordinary differential equations for Engineers ( MOOC on )... Y = e 2 t 1.1 [ 3pts ] Find the complementary solution estos! Will need to purchase the Certificate equation, which is a short practice quiz to... The constant-coefficient ode with a few simple problems to solve following each lecture online course. Develop two theoretical concepts used for linear equations: the principle of,! Separable and linear first-order odes, undetermined coefficients, exponential signals, delta functions, and the... Transform method, which is used to highlight focus areas ve İşlemciler / linear Algebra I: Uzaylar ve /... Equation called the characteristic equation of the Red Burn //www.math.ust.hk/~machas/differential-equations-for-engineers.pdf, ordinary differential and. | 2021-08-01T08:12:56 | {
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https://math.stackexchange.com/questions/1492350/find-fx-if-delta-fx-ex | # Find $f(x)$ if $\Delta f(x)=e^x$
Find $f(x)$ if $\Delta f(x)=e^x$, where $\Delta f(x)$ is the first order forward difference of $f(x)$, step size $=h=1$.
Attempt: We have the definition $\Delta f(x)=f(x+h)-f(x)=f(x+1)-f(x)$
Given $\Delta f(x)=e^x$ i.e $f(x)=\Delta^{-1}e^x=(E-1)^{-1}f(x)$ where $E$ is the shift operator (i.e $Ef(x)=f(x+h)=f(x+1)$).
But it is very troublesome to get the answer.
Answer is given as $f(x)=\frac{e^x}{e-1}$
• The function $f$ is not unique. One can get a unique $f$ by specifying that it should be monotonic and specifying $\lim\limits_{x\to-\infty}f(x)$ or simply $f(x_0)$ for some $x_0$. Otherwise, you can add any function with period $1$ to $f$. – robjohn Oct 22 '15 at 15:02
• Pretty sure that $f(E)e^x=f(e)e^x$ for all functions $f$. For example, above, the answer was $(E-1)^{-1}e^x=(e-1)^{-1}e^x$. – Akiva Weinberger Oct 22 '15 at 15:14
• @AkivaWeinberger please explore your.answer. Why $f(E)e^x=f(e)e^x$? Why $(E-1)^{-1}e^x=(e-1)^{-1}e^x$ is obtained simply replacing $E$ by $e$? – user1942348 Oct 22 '15 at 15:28
• @user1942348 Well, $Ee^x=e^{x+1}=ee^x$, right? And $E^2e^x=e^{x+2}=e^2e^x$. I'm just recognizing patterns, basically. – Akiva Weinberger Oct 22 '15 at 15:30
$$(1) \quad \Delta f(x)=e^x$$ Which is equivalent to, $$(2) \quad f(x+1)=f(x)+e^x$$
Assume that an initial condition for $f(0)$ holds. We then have,
$$(3) \quad f(x)=g(x)+\sum_{n=0}^{x-1} e^n$$
Where $x \ge 1$. The nature of $g(x)$ will be shown momentarily. To prove $(3)$, we'll substitute back into $(2)$,
$$(2.1) \quad \color{red}{f(x+1)}=\color{blue}{f(x)}+\color{green}{e^x}$$
$$(4) \quad \color{red}{g(x+1)+\sum_{n=1}^{x} e^n}=\color{blue}{g(x)+\sum_{n=1}^{x-1} e^n}+\color{green}{e^x}$$
Which is obviously true, as long as $g(x)=g(x+1)$. This implies that $g(x)$ must be periodic. The sum in $(3)$ is geometric, and may be evaluated to be,
$$(5) \quad \sum_{n=0}^{x-1} e^n=\cfrac{e^x-1}{e-1}$$
So we have as the final solution,
$$(6) \quad f(x)=g(x)+\cfrac{e^x-1}{e-1}$$
Where $g(x)$ is any periodic function with period $1$ with $g(0)=f(0)$. I should also note that in the passing from summation to $(6)$, the restrictions on $x$ have been lifted. Assuming $g(x)=f(0)$ for all $x$, $x$ may now be any real number and still satisfy $(1)$. If $g(x)$ is non-constant, then $x$ must still be an integer.
• Unless I am mistaken, the intermediate equations make sense only for integral $x$. – Martin R Oct 22 '15 at 14:32
• Actually $(6)$ is general, and satisfies $(1)$ for any real $x$. The intermediate equations are just tools to get to $(6)$. – Zach466920 Oct 22 '15 at 14:34
• Filling a detail omitted in the above: let $y \in [0,1)$, then consider $x=y+n$ for nonnegative integers $n$. Following the argument above you get $f(x)=f(y)+\frac{e^n-1}{e-1}$. Note that the values of $f(y)$ can be chosen completely arbitrarily. – Ian Oct 22 '15 at 14:40
• And instead of $f(0)$ you could put an arbitrary function of the floor of $x$. – GEdgar Oct 22 '15 at 14:41
• @user1942348 I edited the answer. The solution is more general, and will work with any periodic function $g(x)$, with period 1, and $g(0)=f(0)$. Just pick an appropriate $g(x)$, or constant, to satisfy your initial condition. – Zach466920 Oct 22 '15 at 15:05
Since $f$ is known up to a function with period $1$, let's try to find a monotonically increasing $f$.
Since $f(x-k+1)-f(x-k)=e^{x-k}$, we have that $\lim\limits_{x\to-\infty}f(x)$ exists. Furthermore, \begin{align} f(x)-\lim_{x\to-\infty}f(x) &=\sum_{k=1}^\infty\left[f(x-k+1)-f(x-k)\right]\\ &=\sum_{k=1}^\infty e^{x-k}\\ &=\frac{e^x}{e-1} \end{align} Therefore, $$f(x)=\frac{e^x}{e-1}+p(x)$$ where $p(x)$ is any function with period $1$.
So you want to solve $$f(x+ 1)- f(x)= e^x.$$ It should be obvious that $f$ must be of the form $$f(x)= Ae^{bx}.$$ Then $$f(x+ 1)= Ae^{bx+ b}= (Ae^b)e^{bx}$$ so that $$f(x+ 1)- f(x)= (Ae^b)e^{bx}- Ae^{bx}= (Ae^b- A)e^{bx}= e^x.$$ We can take $b= 1$ and that reduces to $$Ae^b- A= A(e- 1)e^x= e^x$$ or $A(e- 1)= 1$, thus $A= \tfrac1{e- 1}$.
• Is it not more general to assume $$f(x)= Ae^{bx}+C$$ – user1942348 Oct 22 '15 at 15:00
• Why you have not taken $f(x)= Ae^{bx}+C$ – user1942348 Oct 22 '15 at 15:08
Taking the first order forward difference of the exponential, you get
$$\Delta e^x=(e-1)e^x.$$
• How does this answer the question? – Calle Oct 22 '15 at 17:21
• Actually, this does provide one answer to the question: since $\Delta$ is linear, the above implies that $\Delta \left ( \frac{e^x}{e-1} \right ) = e^x$. Noting that the difference equation is first order, this means that the general solution is $\frac{e^x}{e-1}$ plus any function with period $1$...which is Zach466920's answer. :) – Ian Oct 22 '15 at 18:30
• @Ian: I dropped the homogenous solution on purpose, to match the question (Answer is given as ...). – Yves Daoust Oct 22 '15 at 19:41
• @calle: by linearity of the first order difference operator. You obviously have $\Delta f=e^x$ when $f=e^x/(e-1)$. – Yves Daoust Oct 22 '15 at 19:42
• I see. You could have mentioned that in the answer. – Calle Oct 24 '15 at 23:42 | 2019-11-22T20:28:09 | {
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We will also cover evaluation of trig functions as well as the unit circle (one of the most important ideas from a trig class!) and how it can be used to evaluate trig functions. As a result we say cos -1 (½) = 60°. For $$p < 0$$, the graph of the cosine function shifts to the right by $$p$$ degrees. These ideas will be developed in the module, Trigonometric Functions. Cosine Functions. y = -2 cot. Cos(number) The required number argument is a Double or any valid numeric expression that expresses an angle in radians. fix FIX Round Towards Zero ; log1p LOG1P Natural Logarithm of 1+P Function ; log LOG Natural Logarithm Function ; sqrt SQRT Square Root of an Array ; Page Last Updated on: Sunday, October 25, 2009, 12:19:06 AM (CEST). In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse. For an angle Θ with the point (12,5) on its terminating side, what is the value of cosine? - 16250705. 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For each one, determine if the function is odd, even, or neither. Amplitude = | a | Let b be a real number. Other trigonometric functions can be defined in terms of the basic trigonometric functions sin ɸ and cos ɸ. It can be proved using the definition of differentiation. Trig Functions. Not only one cosine function per time period, but also a mixture of cosine functions can be used to describe the seasonal pattern. Calculator function. Determine an equation of a cosine function, given the following info: Amplitude: 3 Period: 120 V. The basic trigonometric functions include the following 6 functions: sine (sinx), cosine (cosx), tangent (tanx), cotangent (cotx), secant (secx) and cosecant (cscx). The Fourier Transform for the sine function can. Concept 3: Using Inverse Trig to find Missing Angles An Inverse function is a function that “undoes” a given function. Additional overloads are provided in this header ( ) for the integral types: These overloads effectively cast x to a double. The letters ASTC signify which of the trigonometric functions are positive, starting in the top right 1st quadrant and moving counterclockwise through quadrants 2 to 4. Since there are three sides, there are 3 × 2 = 6 different ways to make a ratio (fraction) of sides. C library function - cos() - The C library function double cos(double x) returns the cosine of a radian angle x. In this section we will give a quick review of trig functions. The value for the cosine of angle A is defined as the value that you get when you divide the adjacent side by the hypotenuse. The arccosine of x is defined as the inverse cosine function of x when -1≤x≤1. "Thus", all trig functions will have the same value when evaluated 2π radians apart. Sine and Cosine. Definition and Usage. All these functions are continuous and differentiable in their domains. From this. Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify. The ATANH function returns the inverse hyperbolic tangent of the number in degrees. Notice that arccosine, also called inverse cosine, is defined just on the interval minus 1 to plus 1. CAST All Students Try Cheating this represents where trig functions are positive in the quadrants. Consider the harmonic function 2 cos 3x 1xs5 Investigate the validity of the numerical differentiation process by considering two different values for the number of points in the domain: (a) 11, and (b) 101 Plot the exact derivative of function y vs approximate (ie numerically determined) derivative of function y for both cases Qi. Excel also offers functions to convert angle from radians to degrees and vice versa. So, is the value of sin-1 (1/2) given by the expressions above? No! It is vitally important to keep in mind that the inverse sine function is a single-valued, one-to-one function. Next, plot these values and obtain the basic graphs of the sine and cosine function (Figure 1 ). For example, sin(90°) = 1, while sin(90)=0. Engaging math & science practice! Improve your skills with free problems in 'Graph the Sine and Cosine Function with horizontal and vertical shifts' and thousands of other practice lessons. The Fourier Transform for the sine function can. Definitions of trigonometric and inverse trigonometric functions and links to their properties, plots, common formulas such as sum and different angles, half and multiple angles, power of functions, and their inter relations. Determine the following for the transformed cosine function shown whose period is 1,080 degrees. All the trig functions have an input that is an angle and they give an output that is a ratio. How many cycles of the function occur in a horizontal length of 12pi? 3. periodic about the rotation around a circle. The periods of the trigonometric functions sine and cosine are both 2 times pi. Cosine: Properties. The following query shows you multiple ways to use this COS function. Inverse trigonometric functions map real numbers back to angles. What is the exact value of sin (105º)? We can use a sum angle formula noticing that 105º = 45º + 60º. The basis functions are a set of sine and cosine waves with unity amplitude. How many cycles of the function occur in a horizontal length of 12pi? 3. In this article, you will learn methods and techniques to solve integrals with different combinations of trigonometric functions. Transformations of the Sine and Cosine Graph - An Exploration. LA Times - October 27, 2019. This description of is valid for when the triangle is nondegenerate. A function is even if and only if f(-x) = f(x) and is symmetric to the y axis. Reciprocal function in trig. Let's show these are pairwise orthogonal. The COS function returns the cosine of an angle provided in radians. It is the inverse function of the basic trigonometric functions. It is important to mention that the methods discussed in this article are. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. CAST All Students Try Cheating this represents where trig functions are positive in the quadrants. The graph of a cosine function y = cos ( x ) is looks like this:. Preliminary Characterization of the On-Orbit Line Spread Function of COS We present a preliminary analysis of the line spread function (LSF) of the Cosmic Origins Spectrograph (COS) using FUV and NUV stellar spectra acquired during the SM4 Servicing Mission Observatory Verification (SMOV). I need to be able to calculate cos (x) and sin (x) in VHDL code. The cosine function is a trigonometric function that's called periodic. Download the. This angle measure can either be given in degrees or radians. Lucky for us, the tangent of an angle is the same thing as sine over cosine. We then get. The Excel COS function calculates the cosine of a given angle. Give the period, amplitude, and quarter points for each graph (use radians). In trig, sine's reciprocal. Sine's reciprocal, in trig. The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). This definition only covers the case of acute positive angles α: 0<α<90°. cosine¶ scipy. The value for the cosine of angle A is defined as the value that you get when you divide the adjacent side by the hypotenuse. Trig identities showing the relationship between sine and cosine, tangent and cotangent, and secant and cosecant. The ATANH function returns the inverse hyperbolic tangent of the number in degrees. The number of trig functions you want to name depends on your application. WASHINGTON—Adding to the six basic functions that have for years made up the foundation of trigonometry, the nation’s mathematics teachers reportedly introduced 27 new functions today that high schoolers will be expected to master. to the sine function to get your upcoming values. The cosine function is a trigonometric function that's called periodic. , ISBN: 0-9623593-5-1. Understanding what a Unit Circle is will help clarify where the sine and cosine functions are. Enter the answer length or the answer pattern to get better results. Regular trig functions are “circular” functions. Click on the icon next to each trig function to turn it on or off: 2. The cosine function is one of the basic functions encountered in trigonometry. Therefore, a sinusoidal function with period DQGDPSOLWXGH WKDWSDVVHV through the point LV y = 1. Graphing Sine and Cosine Functions. Cos() Mathematical Function in VB. Of course, if f had be defined in a different domain, it might be one-to-one indeed, for example if 0. (Subtracting from the argument of sin θ has the effect of shifting the function to the right by. Trigonometry Graphing Trigonometric Functions Translating Sine and Cosine Functions. Click on "Show" and "Hide" in each table cell to control which values are displayed. cos() static function returns the cosine of the specified angle, which must be specified in radians. Language sin () Language tan (). ” As a function,. There are related clues (shown below). Pythagorean Identities. COS Excel function is an inbuilt trigonometric function in excel which is used to calculate the cosine value of given number or in terms or trigonometry the cosine value of a given angle, here the angle is a number in excel and this function takes only a single argument which is the input number provided. The Cos function takes an angle and returns the ratio of two sides of a right triangle. We can think of these as having the shape of sine waves. There are many trigonometric functions, the 3 most common being sine, cosine, tangent, followed by cotangent, secant and cosecant. ZIPped source files. This matrix might be a document-term matrix, so columns would be expected to be documents and rows to be terms. For example: Given that the the complement of. As we did for -periodic functions, we can define the Fourier Sine and Cosine series for functions defined on the interval [-L,L]. For this, we need the inverse trig functions, which undo the direction of the original trig functions. The arccosine function is defined mathematically only over the domain -1 to 1. Once we can find the values of sin θ and cos θ for values of θ, we can plot graphs of the functions y = sin θ, y = cos θ. Remember, two functions are orthogonal if their dot product is 0, and the dot product of two functions is the integral of their product. Hit enter, then graph y=cos ( ). WASHINGTON—Adding to the six basic functions that have for years made up the foundation of trigonometry, the nation’s mathematics teachers reportedly introduced 27 new functions today that high schoolers will be expected to master. The tangent sum and difference identities can be found from the sine and cosine sum and difference identities. Easy to understand trigonometry lessons on DVD. Let us find the values of these trig function at θ = 90º + 30º = 120º. Indeed, the sine and cosine functions are very closely related, as we shall see (if you are not familiar with the sine function, you may wish to read the page entitled "The Sine Function"). Cos is the cosine function, which is one of the basic functions encountered in trigonometry. Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. In many cases, you will be given a trig function and expected to find the remaining functions. Header declares a set of functions to compute common mathematical operations and transformations: Trigonometric functions. cosh: This function returns the hyperbolic cosine of the specified argument. SQL COS Function Example 1. I need to be able to calculate cos (x) and sin (x) in VHDL code. rules1 = {a -> Sin[t], b -> (a*Cos. 1 synonym for trigonometric function: circular function. Relations between cosine, sine and exponential functions. Combining a Translation and a Reflection Graph y =º2 tan x +π 4. Sine and Cosine Topics. This defined the sine and cosine functions. Lucky for us, the tangent of an angle is the same thing as sine over cosine. Understanding how to create and draw these functions is essential to these classes, and to nearly anyone working in a scientific field. One has period 2ˇ, and the other has period ˇ, and the resulting function is not a sinusoid. Complex analysis. Header provides a type-generic macro version of this function. We start with the graph of the basic sine function y = sin(x) and the basic cosine function g(x) = cos(x),. Shift: 6 The function has a maximum at 15? How do you find the value of #cos 8(pi)# using the graph? How do you find the value of #cos ((pi)/2)# using the graph?. Using the above measured triangle, this would mean that: cos(A) = adjacent. Learn how to graph trigonometric functions and how to interpret those graphs. The ratio is the length of the side adjacent to. A right triangle has one leg of lengths 5 and hypotenuse of length 13. does the opposite of the sine. A function is odd if and only if f(-x) = - f(x) and is symmetric with respect to the origin. Now, note that for sin θ, if we subtract from the argument (θ), we get the negative cosine function. Key Concepts 1. Easy to understand trigonometry lessons on DVD. New York Times - August 03, 2019. Concept 3: Using Inverse Trig to find Missing Angles An Inverse function is a function that “undoes” a given function. To evaluate the integral simply, the cosine function can be rewritten (via Euler's identity) as: [3] Rewriting the integral with the above identity makes things easier. The particulars aren't important, but I thought the interface was nice. Using cosine on a calculator saves a lot of time compared to looking it up in a table, which people did before calculators. Inverse trig functions. Anyone who has ever seen a sine wave and/or a cosine wave will have noticed that both of the curvilinear graphs are drawn on a Cartesian Coordinate (world) system. These six trigonometric functions in relation to a right triangle are displayed in the figure. C library function - cos() - The C library function double cos(double x) returns the cosine of a radian angle x. As an example, try typing sin(x)^2+cos(x)^2 and see what you get. See the results below. Below are several oth. So what do they look like on a graph on a coordinate plane? Let's start with the sine function. DO: Using the reciprocal trig relationships to turn the secant into a function of sine and/or cosine, and also use the derivatives of sine and/or cosine, to find $\displaystyle\frac{d}{dx}\sec x$. We start with the graph of the basic sine function y = sin(x) and the basic cosine function g(x) = cos(x),. Calculates the cosine of an angle (in radians). If f is any trig. Plot Cosine Function. Plug in the sum identities for both sine and cosine. To elicit fraction multiplication, we should view the sine function also as a fraction. Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. This makes the amplitude equal to |4| or 4. Using Trigonometric Functions in Excel. For a given angle measure θ , draw a unit circle on the coordinate plane and draw the angle centered at the origin, with one side as the positive x -axis. y=cos ( ) and add -h. Re: Math & Trig Functions A further observation: if I replace pi with the numerical value IV accepts the equation but the answer it gives is exaxtly 10x too big and the equation stays red in the box. The VBA Cos function returns the cosine of a supplied angle. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection. Applications of Trigonometry. The Cosine of 0. How to use the ABS function Converts negative numbers to positive numbers, in other words, the ABS function removes the sign. See Inverse trigonometric functions. 877583 Similar Functions. The parent graph of cosine looks very similar to the sine function parent graph, but it has its own sparkling personality (like fraternal twins). The cosine function is generated in the same way as the sine function except that now the amplitude of the cosine waveform corresponds to measuring the adjacent side of a right triangle with hypotenuse equal to 1. In trig, sine's reciprocal. Matrices & Vectors. The cos function operates element-wise on arrays. The above equation is substituted into equation [2], and the result is: [4] Let's look at the first integral on the left in equation [4]. Undefined function 'cos' for input Learn more about undefined, 'tf', symbolic. Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p < provides a type-generic macro version of this function. Sinθ = 1 / Cosecθ Cosθ = 1 / secθ Tanθ = Sinθ. cos: This function returns the cosine of the specified argument. For this, we need the inverse trig functions, which undo the direction of the original trig functions. Note the capital "C" in Cosine. The trigonometric functions cosine, sine, and tangent satisfy several properties of symmetry that are useful for understanding and evaluating these functions. Since cosine corresponds to the $$x$$ coordinates of points on the unit circle, the values of cosine are positive in quadrants 1 and 4 and negative in quadrants 2 and 3. Appendix: Adding two sine functions of different amplitude and phase using complex numbers To perform the sum: Eθ = E10 sinωt+E20 sin(ωt+δ) = Eθ0 sin(ωt +φ), (4) we note the famous Euler formula: eiθ = cosθ +isinθ. Key Concepts 1. Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. But the coordinates are the cosine and sine, so we conclude sin 2 θ + cos 2 θ = 1. Enter the answer length or the answer pattern to get better results. 500000 is 0. The angle in radians for which you want the cosine. Cofunction Identities, radians. " You should also notice in the figure that tangent equals sine(θ) over cosine(θ). The hyperbolic functions take hyperbolic angle as real argument. What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in Figure 15. The arg parameter is in radians. If the specified angle is positive or negative infinity or Not a Number, the value returned is 'NaN'. 759) on your calculator (in “degree” mode) returns an answer of 40. Cosine of an angle is equal to the adjacent side divided by the hypotenuse. The Cosine of 0. You are hereby granted permission to make ONE printed copy of this page and its picture(s) for your PERSONAL and not-for-profit use. 2(a) where we have added together two waves cos(5x)+cos(5. COS Excel function is an inbuilt trigonometric function in excel which is used to calculate the cosine value of given number or in terms or trigonometry the cosine value of a given angle, here the angle is a number in excel and this function takes only a single argument which is the input number provided. So it is a negative cosine graph. VBA Cos Function Examples. net 2008 is used to find the Cosine value for the given angle. y = 5 sin. This section contains notes, terms, formulas, and helpful examples. Memorize them! To evaluate any other trig deriva-tive, you just combine these with the product (and quotient) rule and chain rule and the definitions of the other trig functions, of which the most impor-tant is tanx = sinx cosx. In this section we define and learn how to. In a right triangle with an angle θ, the cosine function gives the ratio of adjacent side to hypotenuse; more generally, it is the function which assigns to any real number θ the abscissa of the point on the unit circle obtained by moving from (1,0) counterclockwise θ units along the circle, or clockwise |θ| units if θ is less than 0. The tangent sum and difference identities can be found from the sine and cosine sum and difference identities. The trigonometric functions sine and cosine are defined in terms of the coordinates of points lying on the unit circle x 2 + y 2 =1. MySQL COS() Function MySQL Functions. This matrix might be a document-term matrix, so columns would be expected to be documents and rows to be terms. What's this all about? Here's the deal. From these relations and the properties of exponential multiplication you can painlessly prove all sorts of trigonometric identities that were immensely painful to prove back in high school. LA Times - January 05, 2020. The trigonometric functions relate the angles in a right triangle to the ratios of the sides. This function is overloaded in and (see complex cos and valarray cos ). cos ( θ ) {\displaystyle \cos (\theta )} , and (b) dividing all sides by. It only takes a minute to sign up. Your expression may contain sin, cos, tan, sec, etc. Bases: sage. You must know all of the following derivatives. By knowing which quadrants sine, cosine, and tangent are negative and positive in, you can solve for either x, y, or r and find the other trig functions. When executed on two vectors x and y, cosine() calculates the cosine similarity between them. Any length 3 Letters 4 Letters 5 Letters 6 Letters 7 Letters. Like the sine function we can track the value of the cosine function through the four quadrants of the unit circle as we sketch it on the graph. In various branches of mathematics, the cosine of an angle is determined in various ways, including the following:. Key Concepts 1. It was first used in ancient Egypt in the book of Ahmes (c. It directly determines. The cosine function returns the wrong answer for the cosine of 90 degrees. Trigonometry. Cos [x] then gives the horizontal coordinate of the arc endpoint. If you aren’t familiar with these concepts, you’ll have to ask your math teacher to assist you with them. are simple modifications of the Sine- and Cosine function whose properties (amplitude and frequency) shall be recognized in the graphs. Equivalent to 2) (the argument is cast to double ). The function of the COS is that it returns the cosine of a given angle in radians. Complex trigonometric functions. Reciprocal Trig Functions and Quadrants. Comments welcomed. Graphs of Other Trigonometric Functions. Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p < Language > Functions > Trigonometry > Cos. No, because we know from the trigonometry that two opposite angles have the same cosine. Any suggestion on how to convert list of rules to a memorization function. Both values, * sinx and * cosx , are in the range of -1 to 1. Re: Math & Trig Functions A further observation: if I replace pi with the numerical value IV accepts the equation but the answer it gives is exaxtly 10x too big and the equation stays red in the box. Trig Functions - Graphing Cheat Sheet *****UPDATED with different formula versions***** Text books make graphing trig functions so complicated. In this section we will give a quick review of trig functions. Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify. The graph of the first function remains in black. Trig function, briefly. Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range. For this, we need the inverse trig functions, which undo the direction of the original trig functions. Then is the horizontal coordinate of the arc endpoint. Free trigonometric equation calculator - solve trigonometric equations step-by-step. The function h(x) is an example of a rational polynomial function. There are related clues (shown below). The cosine function returns the wrong answer for the cosine of 90 degrees. What is the general equation of a cosine function with an amplitude of 3, a period of 4pi, and a horizontal shift of -pi? y=3cos(0. Versine and haversine were used the most often. Allowed data types: float. For example, For example, (11). Possible Answers: It's measured in radians. The trigonometeric functions, the sine function (sin) and cosine function (cos) are obtained for a = -1. If you assign each amplitude (the frequency domain) to the proper sine or cosine wave (the basis functions), the result is a set of scaled sine and cosine waves that can be added to form the time domain signal. Compare the graph of the cosine function with the graph of the angle on the unit circle. The result will be between -1 and 1. 01:pi; plot(x,cos(x)), grid on The expression cos(pi/2) is not exactly zero but a value the size of the floating-point accuracy, eps, because pi is only a floating-point approximation to the exact value of. The Exponential Function and the Trig Functions. Unit Circle is a circle with a radius of one. Inverse trigonometric functions map real numbers back to angles. Graph of Trig. For example, can appear automatically from Bessel, Mathieu, Jacobi, hypergeometric, and Meijer functions for appropriate values of their parameters. CHARACTERISTICS OF SINE AND COSINE FUNCTIONS. The first step will be to replace the tangent function with sine and cosine using the first quotient formula. When the cosine of y is equal to x:. Active 2 years ago. What is the amplitude of f(x) = 4 sin(x) cos(x)? a. Press the tab key on your keyboard or click the cell C1. This could be useful for young people doing higher mathematics in Scotland. For a given angle measure θ , draw a unit circle on the coordinate plane and draw the angle centered at the origin, with one side as the positive x -axis. In the following example, the VBA Cos function is used to return the cosine of three different angles (which are expressed in radians). Cosine: Properties. These properties enable the manipulation of the cosine function using reflections, shifts, and the periodicity of cosine. DO NOT GRAPH!! 1. Which transformations are needed to change the parent cosine function to the cosine function below?. We consider a linear combination of these and evaluate it at specific values. This website uses cookies to ensure you get the best experience. We then get. For example: ( sin − 1) (\sin^ {-1}) (sin−1) left parenthesis, sine, start superscript, minus, 1, end superscript, right parenthesis. In trig, sine's reciprocal. From a theoretical view point, there’s only one trig function: all trig functions are simple variations on sine. Now for the other two trig functions. rad: The angle in radians. Matrices Vectors. Therefore, we want the integrand to consist of a trig function and it's known derivative. The cofunction identities show the relationship between sine, cosine, tangent, cotangent, secant and cosecant. Anyone who has ever seen a sine wave and/or a cosine wave will have noticed that both of the curvilinear graphs are drawn on a Cartesian Coordinate (world) system. In this section we will give a quick review of trig functions. y = 5 sin. Then is the horizontal coordinate of the arc endpoint. In particular, sinθ is the imaginary part of eiθ. Consider two functions f = sin(mx) and g = sin(nx). Graphs of Other Trigonometric Functions. Similarities. LA Times - October 27, 2019. It directly determines. Just copy and paste the below code to your webpage where you want to display this calculator. are simple modifications of the Sine- and Cosine function whose properties (amplitude and frequency) shall be recognized in the graphs. In a formula, it is written simply as 'cos'. We will be studying rational polynomial functions later in the course. Use a Pythagorean Identity to get in terms of cosine. Drag a point along the cosine curve and see the corresponding angle on the unit circle. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year. Start at the point , which lies on the unit circle centered at the origin. 12/11/2018; 2 minutes to read +1; In this article. The graphs of the sine and cosine functions are used to model wave motion and form the basis for applications ranging from tidal movement to signal processing which is fundamental in modern telecommunications and radio-astronomy. Consider the harmonic function 2 cos 3x 1xs5 Investigate the validity of the numerical differentiation process by considering two different values for the number of points in the domain: (a) 11, and (b) 101 Plot the exact derivative of function y vs approximate (ie numerically determined) derivative of function y for both cases Qi. The inverse sine function sin-1 takes the ratio oppositehypotenuse and gives angle θ. You must know all of the following derivatives. The other answer is −40. Matrices & Vectors. Find the Maclaurin series expansion for cos ( x) at x = 0, and determine its radius of convergence. Next, a little division gets us on our way (fractions never hurt). Before getting stuck into the functions, it helps to give a name to each side of a right triangle: "Opposite" is opposite to the angle θ "Adjacent" is adjacent (next to) to the angle θ. The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. This is useful for creating rhythmic, oscillating changes in attribute values. The fact that you can take the argument's "minus" sign outside (for sine and tangent) or eliminate it entirely (for cosine) can be helpful when working with complicated expressions. The sides of a triangle: the base , the height , and the hypotenuse. The cosine function returns the wrong answer for the cosine of 90 degrees. The the wave amplitude as a function of position is 2y m sin(kx). Any suggestion on how to convert list of rules to a memorization function.
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https://math.stackexchange.com/questions/798721/proof-for-inequality-with-a-b-c-d-with-d-maxa-b-c-d | # Proof for inequality with $a,b,c,d$ with $d =\max(a,b,c,d)$
Let $a,b,c,d$ positive real numbers with $d= \max(a,b,c,d)$. Proof that
$$a(d-c)+b(d-a)+c(d-b)\leq d^2$$
• I believe that the GM-AM inequality with $n=4$ variables might be helpful.
$$\sqrt[n]{x_1 x_2 \dots x_n} \le \frac{x_1+ \dots + x_n}{n}$$
We also know that the Geometric mean is bounded as follows :
$$\min \{x_1, x_2, \dots x_n\} \le \frac{x_1+ \dots + x_n}{n} \le \max \{x_1, x_2, \dots x_n\}$$
** I also tried to draw an square and some rectangles, but nothing worked out.
• Is $d=\max(a,b,c)$ or $d=\max(a,b,c,d)$? – AsdrubalBeltran May 17 '14 at 4:37
• @Arthur Yes, there is. The latter allows $d$ to differ from all of the other three. – Hagen von Eitzen May 17 '14 at 4:44
• according to the question, $d=\max(a,b,c,d)$ – Keith May 17 '14 at 12:56
Consider the polynomial $$f(x)=x^3-(a+b+c)x^2+(ab+bc+ac)x-abc$$ having $a,b,c$ as roots. We have $$f(d)=d^3-(a+b+c)d^2+(ab+bc+ac)d-abc=d\cdot(RHS-LHS)-abc$$ and $f(d)=(d-a)(d-b)(d-c)\ge0$.
Or in short $$a(d-c)+b(d-a)+c(d-b)=(a+b+c)d-(ac+ab+bc)\\=\frac{d^3-(d-a)(d-b)(d-c)-abc}{d}\le d^2$$
Divide both sides by $d^2$ to get an equivalent inequality:
$\dfrac{a}{d}\cdot \left(1 - \dfrac{c}{d}\right) + \dfrac{b}{d}\cdot \left(1 - \dfrac{a}{d}\right) + \dfrac{c}{d}\cdot \left(1 - \dfrac{b}{d}\right) \leq 1$.
Now let $x = \dfrac{a}{d}$, $y = \dfrac{b}{d}$, and $z = \dfrac{c}{d}$, then : $0 \leq x, y, z \leq1$, and we are to prove:
$x(1 - z) + y(1 - x) + z(1 - y) \leq 1$.
Consider $f(x,y,z) = x(1 - z) + y(1 - x) + z(1 - y) - 1 = x + y + z - xy - yz - zx - 1$. We find the critical points of $f$. So take partial derivatives:
$f_x = 1 - y - z = 0 \iff y + z = 1$
$f_y = 1 - x - z = 0 \iff x + z = 1$
$f_z = 1 - x - y = 0 \iff x + y = 1$.
Thus $\nabla{f} = 0 \iff x + y = y + z = z + x = 1 \iff x = y = z = \dfrac{1}{2}$. Thus the maximum of $f$ occurs at either the critical values or the boundary points which are: $(x,y,z) = (0,0,0), (0,1,1), ..., (1,1,1)$. Of these values, the max is $0$. So $f(x,y,z) \leq 0$ which is what we are to prove.
• after "Consider" the function $f(x,y,z)$ is defined one way, and right after that another way with $1$ subtracted. [with the subtracted $1$ then one wants to show $f \le 0$ which the rest of your argument does, provided the critical point is shown to be the max]. – coffeemath May 17 '14 at 5:49
• @coffeemath: see edit. – DeepSea May 17 '14 at 5:51
• The critical point is found OK, however the maximum over $0<x,y,z<1$ does not occur there. In fact for example $f(.99,.01,.5)=-.0099>-.25=-1/4.$ – coffeemath May 17 '14 at 6:12
• @coffeemath: agree. This is the local max. The global max attained at the boundary x = y = z = 1 which is 0 – DeepSea May 17 '14 at 6:15
• Yes, the constraints should be $0<x,y,z\le 1$ given the problem statement, and then there are boundary points when one (or more) of $x,y,z$ is $1$. If say $z=1$ then $f(x,y,z)=-xy$ which is less than $0$ [I get $f=-1$ at $x=y=z=1$ however.] I note in your edit you have still allowed $x,y,z$ to be zero, which is excluded in the OP. – coffeemath May 17 '14 at 6:33
The inequality, after multiplying it out and moving all to one side, is $$d^2-ad-bd-cd+ab+ac+bc\ge 0.\tag{1}$$ Since $d=\max(a,b,c,d)$ each of $d-a,\ d-b,\ d-c$ is nonnegative and so $$(d-a)(d-b)(d-c)\ge 0, \\ d^3-ad^2-bd^2-cd^2+abd+acd+bcd\ge abc,$$ where at the last step we moved the $-abc$ term over to the right side. Now since $a,b,c,d$ are positive one can divide both sides of this last inequality by $d$ and obtain $(1)$ as desired, in fact there is the lower bound $abc/d$ for the left side of $(1).$ Note one really only needs $d>0$ (to justify division by $d$) and $a,b,c\ge 0$ for the conclusion to hold.
• In looking at this, it seems really the same idea as in Hagen's answer, without explicitly mentioning the polynomial. I'll delete if anyone thinks it's too similar... – coffeemath May 17 '14 at 11:25 | 2019-05-21T17:04:39 | {
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https://math.stackexchange.com/questions/3077572/the-square-trinomial-y-ax2-bx-c-has-no-roots-and-a-b-c-0-find-the | # The square trinomial $y=ax^2+ bx + c$ has no roots and $a + b + c > 0$. Find the sign of the coefficient $c$ .
The square trinomial $$y=ax^2 + bx + c$$ has no roots and $$a + b + c > 0$$. Find the sign of the coefficient $$c$$. I'm having difficulties with this problem.
What I've tried: I realized that a quadratic equation doesn't have roots if the discriminant $$b^2 - 4ac < 0$$, so I've tried to combine that with the condition $$a + b + c > 0 <=> a > -b -c$$, but that didn't help me that much.
I would appreciate if someone could help me to understand this. I'll ask a lot of questions on this network while I'm learning, so please don't judge me for that :) .
• What happens for simple choices of values of $x$? (What values might you choose, and why, to help answer the question) – Mark Bennet Jan 17 at 22:08
• You have the right idea. You know $0 ≤ b^2 < 4ac$, so neither $a$ nor $c$ is zero. $a$ and $c$ thus have the same sign... – diracdeltafunk Jan 17 at 22:10
Call $$p(x)=ax^2+bx+c$$. Then $$p(1)=a+b+c >0$$ $$p(0)=c$$ If $$c$$ were negative, then there would be a root between $$0$$ and $$1$$. This contradicts our hypothesis, hence necessarily $$c>0$$.
• +1, beautiful argument – gt6989b Jan 17 at 22:10
• Thank you so much! I understand now and I really like this way of solving the problem. – Wolf M. Jan 17 at 22:25
That $$a+b+c > 0$$ gives $$ax^2+bx+c$$ evaluated at $$x=1$$ is $$a+b+c$$ which is positive.
Suppose that $$c$$ is negative. Then $$ax^2+bx+c$$ evaluated at $$x=0$$ is $$c$$ which is negative. Then the Intermediate Value Theorem would imply that there is a root $$x \in (0,1)$$. So $$c$$ cannot be negative.
If $$c$$ is 0 then 0 is a root of $$ax^2 +bx+c$$.
So $$c$$ must be positive for there to be no real root.
It's easy to see that since $$b^2<4ac$$, you have $$c > b^2/(4a) > 0$$ if $$a>0$$ and $$c < b^2/(4a) < 0$$ if $$a < 0$$.
But you have no roots, so it is either a parabola opening down below $$x$$-axis or opening up above $$x$$-axis, and since $$a+b+c=1$$ it must be all above.
Can you conclude?
• Yes! With your and other answers, I understand even better :). Thank you! – Wolf M. Jan 17 at 22:25
Since the polynomial has no roots, its graph is either strictly above or below the $$x$$-axis. But $$f(1)=a+b+c>0$$, so the graph is above the $$x$$-axis. The parabola then intersects the positive part $$y$$-axis, but this intersection point is $$(0,c)$$, so $$c>0$$.
Perforce $$\text{sgn}(y(0))=\text{sgn}(y(1)).$$
• If you prefer, $\text{sgn}(c)=\text{sgn}(a+b+c)$. – Yves Daoust Jan 17 at 22:22 | 2019-09-16T12:31:31 | {
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http://wuaq.einsundeinsistdrei.de/radius-of-convergence-geometric-series.html | # Radius Of Convergence Geometric Series
the series also converges at one of the endpoints, x = R or x = R. Find the radius of convergence of a power series by using the ratio test. Power Series. We begin with the infinite geometric series: 1 1− x = X∞ n=0 xn, |x| < 1. Convergence & divergence of geometric series In this section, we will take a look at the convergence and divergence of geometric series. geometric series. The radii of convergence of the power series in (i) and (ii) are both R, although the interval of convergence of these series might not match the interval of convergence of f(x). Write the first few terms of the Taylor series for expanded about x 1. Since the radius of convergence only has to do with absolute convergence, the answer to the two parts will be the same. 12, which is known as the ratio test. $\endgroup$ – ziggurism Nov 10 '15 at 22:53. relating them to geometric series. Example: Find a power series centered at x = 0 for the function 1 2 5x and nd its. The Taylor remainder formula from 8. 1) The series will converge only for x = c and diverges elsewhere (the radius of convergence is zero), 2) The series converges absolutely for all x (the radius of convergence is infinity) or 3) The series converges absolutely for all x in some open interval of convergence (-R, R). Theorem 1 can be proved in full generality by comparing to the geometric series above. The radius of convergence for this series is R=1. The radius of convergence is the same as for the original series. (A) 0 (B) 2 (C) 1 (D) 3 (E) ∞ Feedback on Each Answer Choice A. geometric series. We now regard Equation 1 as expressing the function f(x) = 1/(1 – x) as a sum of a power series. This week, we will see that within a given range of x values the Taylor series converges to the function itself. For example $\sum x^n$ is geometric, but $\sum \frac{x^n}{n!}$ is not. The following example has infinite radius of convergence. If L = 0; then the radius of convergence is R = 0: If L = 1; then the radius of convergence is R = 1: If 0 < L < 1; then the power series converges for all x satisfying (x a)k. The number R is called the radius of convergence of the power series. 7n - 1 n=1 Find the interval, I, of convergence of the series. Let _ B ( A , &reals. Sometimes, a power series for {eq}\displaystyle x = \text{ constant}, {/eq} from the interval of convergence, can be written as a geometric series whose sum is. AP® CALCULUS BC 2009 SCORING GUIDELINES (Form B) The power series is geometric with ratio () radius of convergence 1 : interval of convergence. c)Use Lagrange's Remainder Theorem to prove that for x in the interval. The modern idea of an infinite series expansion of a function was conceived in India by Madhava in the 14th century, who also developed precursors to the modern concepts of the power series, the Taylor series, the Maclaurin series, rational - Their importance in calculus stems from Newton s idea of representing functions as sums of infinite series. notebook 1 March 26, 2012 Mar 83:13 PM 9. The "Nice Theorem". These operations, used with differentiation and integration, provide a means of developing power series for a variety of. 8 Power series145 / 169. The following example has infinite radius of convergence. The radius of convergence is the same as for the original series. However the right hand side is a power series expression for the function on the left hand side. geometric series. 0, called the interval of convergence. The series converges only at x = a. R= Follow. For instance, suppose you were interested in finding the power series representation of. 1] Theorem: To a power series P 1 n=0 c n (z z o) n is attached a radius of convergence 0 R +1. The geometric series is of crucial important in the theory of in nite series. (a) x2n and then find the radius of convergence. Note that this theorem is sometimes called Abel's theorem on Power Series. Using the ratio test, we obtain At x = -2 and x = 4, the corresponding series are, respectively, These series are convergent alternating series and geometric series, respectively. The radius of convergence is the interval with the values (-R, R). I thought this was a bit tedious, so I tried to find the answer without solving quadratics. Write the first few terms of the Taylor series for expanded about x 1. it is of the form P c nxn), then the interval of convergence will be an interval centered around zero. Example 1: First we'll do a quick review of geometric series. Geometric Series Limit Laws for Series Test for Divergence and Other Theorems Telescoping Sums Integral Test Power Series: Radius and Interval of Convergence. This means if you add up an infinite list of numbers but you get out a finite value which is called convergence. It is suitable for someone who has seen just a bit of calculus before. Find interval of convergence of power series n=1 to infinity: (-1^(n+1)*(x-4)^n)/(n*9^n) My professor didn't have "time" to teach us this section so i'm very lost If you guys can please answer these with work that would help me a lot for this final. The number r in part (c) is called the radius of convergence. o A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. A power series centred at ais a series of the form X1 n=0 c n(x na) = c 0 + c 1(x a) + c 2(x a)2 + There are only three possibilities: (1)The series converges only when x= a. Geometric Series The series converges if the absolute value of the common ratio is less than 1. (b) Now notice that if g(x) = 1 (1+x)2 then f(x) = 1 2 g0(x). Interval and radius of convergence of Power series? A "series" in general is a summation of a "sequence" of terms defined at each n drawn from some subset of the integers (typically: all positive integers, if we start at term 1; or all non-negatives if we start at term 0; or some interval, if we are taking a finite summation). When x= 1=3 the series is the harmonic series, so it diverges; when x= 1=3 the series is the alternating harmonic series, so it converges; thus the interval of convergence is [ 1=3;1=3). Our friend the geometric series X1 n=0 xn = 1 1 x. Theorem 10. the interval Of covergence based on your graphs. Extension of Theorem 2 Find the radius of convergence R of the power series does not converge, so that Theorem. Recall: For a geometric series !!!!!, we know ! 1−! = !!!!! and because a geometric series converges when !R, where R>0 is a value called the radius of convergence. geometric series Series harmonic series The Integral and Comparison radius of convergence Convergence of Power Series rational function Factoring Polynomials. In other words, the radius of convergence of the series solution is at least as big as the minimum of the radii of convergence of p (t) and q (t). be a power series with real coefficients a k with radius of convergence 1. Write all suggestions in comments below. Our friend the geometric series X1 n=0 xn = 1 1 x. Worksheet 7 Solutions, Math 1B Power Series Monday, March 5, 2012 1. Find its radius of z2 +4 convergence Suppose f(z) -is developed in a power series around z- 3. There is a positive number R such that the series diverges for » x-a »> R but converges for » x-a »< R. is a power series centered at x = 2. Analyzing what happens at the endpoints takes more work, which we won't do in 10b. Byju's Radius of Convergence Calculator is a tool which makes calculations very simple and interesting. In case (a) the radius of convergence is zero, and in case (b), infinity. However, relying on geometric properties of algebraic functions, convergence radii of these series can be determined precisely. In practice, it is not difficult to estimate the minimal Mc for many series, in which case, the radius of convergence for n~1(M c) provides an easily computed lower bound for the radius of convergence of c in the usual sense. If the series is divergent. Express 1 = 1 x 2 as the sum of a power series and nd the interval of convergence. Example 1: First we’ll do a quick review of geometric series. Then, and. Two standard series: geometric series and p series; Comparison (Small of large) for positive series. I thought this was a bit tedious, so I tried to find the answer without solving quadratics. the distance from p to the origin, and let r be any radius smaller than t. that a power series E3=0 a,z n has the radius of convergence p, where p- ~ equals "the limit or the greatest of the limits" of the sequence laZ/n. The radius of convergence Rof the power series X1 n=0 a n(x c)n is given by R= 1 limsup n!1 ja j 1=n where R= 0 if the limsup diverges to 1, and R= 1if the limsup is 0. Give the interval of convergence for each. If the power series is centered at zero (i. Introduction. That is, the radius of convergence is R = 1. Share a link to this widget: More. Find the radius of convergence of the power series? How would I go about solving this problem: Suppose that (10x)/(14+x) = the sum of CnX^(n) as n=0 goes to infinity C1= C2= Find the radius of convergence R of the power series. Find interval of convergence of power series n=1 to infinity: (-1^(n+1)*(x-4)^n)/(n*9^n) My professor didn't have "time" to teach us this section so i'm very lost If you guys can please answer these with work that would help me a lot for this final. For example $\sum x^n$ is geometric, but $\sum \frac{x^n}{n!}$ is not. Since the terms in a power series involve a variable x, the series may converge for certain values of x and diverge for other values of x. The limit comparison test fails if the limit is 1. Note that a power series may converge at some, all, or none of the points on the circle of convergence. Home > Mathematics > Statistics > Sequence and Series Video Tutorial > Interval and Radius of Convergence for a Series, Ex 2 Lecture Details: Interval and Radius of Convergence for a Series, Ex 2. Possibilities for the Interval and Radius of Convergence of a Power Series For a power series centered at 𝑐, one of the following will occur: 1. That is the sequence is decreasing. They terminate; they are to power series what terminating decimals are to real numbers. monic series, so it converges (nonabsolutely) by (Leibniz’s) Alternating Series Test. The set of all points whose distance to a is strictly less than the radius of convergence is called the disk of convergence. Because power series can define functions, we no longer exclusively talk about convergence at a point, instead we talk about the radius and interval of convergence. We can obtain power series representation for a wider variety of functions by exploiting the fact that a convergent power series can be di erentiated, or integrated, term-by-term to obtain a new power series that has the same radius of convergence as the original power series. 14 Power Series The Definition of Power Series Describe the power series The Interval and Radius of Convergence Define the interval and radius of convergence of a power series Finding the Interval and Radius of Convergence. Series of positive terms. R can often be determined by the Ratio Test. So, the power series above converges for x in [-1,1). What is each coefficient a n? Is the series f(x) = 3 2n x n a power series? If so, list center, radius of convergence, and general term a n. De nition of ez 12 1. By integrating the series found in a) Find a power series representation for F(z). The radii of convergence are the same for both the integral and deriv- ative, but the behavior at the endpoints may be different. To show that the radii of convergence are the same, all we need to show is that the radius of convergence of the differentiated series is at least as big as $$r$$ as well. The Radius of Convergence. And again, the convergence is uniform over the compact subset Kof z-values with which we are working. 0 = 2, the radius of convergence is p 5 (so converges in (2 p 5,2+ p 5). A geometric series sum_(k)a_k is a series for which the ratio of each two consecutive terms a_(k+1)/a_k is a constant function of the summation index k. If the terms of a sequence being summed are power functions, then we have a power series, defined by Note that most textbooks start with n = 0 instead of starting at 1, because it makes the exponents and n the same (if we started at 1, then the exponents would be n - 1). Because the series, being a geometric series of ratio 4x2 converges PRECISELY for 4x2 < 1; that is, for jxj < 1=2, we know the interval is ( 1=2;1=2) and the radius is r = 1=2. How to evaluate the sum of a series using limits 6. the usual situation where a radius of convergence is assigned to individual series [25]. Example 1: First we’ll do a quick review of geometric series. > L: Thus R = L1=k: Let us consider some examples. Be sure to show the general term of the series. Derivative and Antiderivative of Power Series 4 1. Let's consider a series (no power yet!) and be patient for a couple of moments: Suppose that all s are positive and that there is a q <1 so that. Geometric Methods for 2-dimensional Systems; Homework Exercises; Chapter 4: Series Solutions (Open for bug hunting) Power Series; Series Solutions; Radius of Convergence; Euler Equations; Regular Singular Points; Series Solutions About Regular Singular Points; Convergence of Series Solutions About Regular Singular Points; Bessel Functions. Power Series. By integrating the series found in a) Find a power series representation for F(z). 4: Radius of convergence Today: a 20 minute groupwork. Then recall that the ratio test is:. Geometric series are an important example of infinite series. We convergencecan attempt the ratio test to find the radius of, but it fails because;goaa%hi doesn't exist for any x except O (The nexpression simplifiesto 11544 if is odd, so the a oscillation doesn't get limit. But here our point of view is different. (5)If the radius of convergence of P 1 n=0 a nx n is r and the sequence (b n) is bounded, what can you say about the. Theorem 1 can be proved in full generality by comparing to the geometric series above. In general, the domain of a power series will be an interval, called the interval of convergence. The number R is called the radius of convergence of the power series. Used Lagrange’s Theorem to show that as the number of terms of p(x). A power series determines a function on its interval of convergence: One says the series converges to the. As in the case of a Taylor/Maclaurin series the power series given by (4. Note that it is possible for the radius of convergence to be zero (i. The ratio between successive terms of a_n*x n is (a_(n+1)*x n+1)/(a_n*x n), which simplifies to a_(n+1)/a_n*x when x≠0; the ratio test says that the series converges if the limit of the absolute value of this ratio as n→+∞ is less than 1, and because x. Let f(x) = P 1 n=0 a nx n and suppose that the radius of convergence for this series is R>0. They behave somewhat like geometric series in that there is some 0 R 1, the radius of convergence, such that the series f(x) converges for jx aj< R and diverges for jx aj> R. (a) x2n and then find the radius of convergence. Convergence Tests Name Summary Divergence Test If the terms of the sequence don't go to zero, the series diverges. Convergence of power series The point is that power series P 1 n=0 c n (z z o) n with coe cients c n 2Z, xed z o 2C, and variable z2C, converge absolutely and uniformly on a disk in C, as opposed to converging on a more complicated region: [1. The radius of convergence may also be zero, in which case the series converges only for x = a; or it could be infinite (we write R = ∞), in which case the series converges for all x. By integrating the series found in a) Find a power series representation for F(z). Holmes May 1, 2008 The exam will cover sections 8. And again, the convergence is uniform over the compact subset Kof z-values with which we are working. It works by comparing the given power series to the geometric series. The following example has infinite radius of convergence. This does not look like a geometric series, so I'm not sure how to find the sum. The radius of convergence is R = 1. De nition 1. Write the first few terms of the Taylor series for expanded about x 1. Example #4: Find the Radius & Interval of Convergence of the Power Series Example #5: Find the Radius & Interval of Convergence of the Power Series Example #6: Find the Radius & Interval of Convergence of the Power Series. This limit is always less than one, so, by the Ratio Test, this power series will converge for every value of x. For example $\sum x^n$ is geometric, but $\sum \frac{x^n}{n!}$ is not. 0 = 2, the radius of convergence is p 5 (so converges in (2 p 5,2+ p 5). The Ratio Test guarantees convergence when this limit is less than one (and divergence when the limit is greater than one). The series converges only at x = a. Find the radius of convergence of a power series using the ratio test. The radius of convergence may also be zero, in which case the series converges only for x = a; or it could be infinite (we write R = ∞), in which case the series converges for all x. 1 An exception is h( x) = e (x 2. 9 Representation of Functions by Power Series 671 Operations with Power Series The versatility of geometric power series will be shown later in this section, following a discussion of power series operations. For instance, suppose you were interested in finding the power series representation of. We are now going to investigate how to find the radius of convergence in these Consider the series below. Find a power series representation for 2 =(x +3 ) and nd the interval of convergence. 4 RADIUS OF CONVERGENCE This chapter began with the discussion of using a polynomial to approximate a function. It is one of the most commonly used tests for determining the convergence or divergence of series. (Hint: center is zero, looks like a geometric series formula) 2) Repeat problem 1) with f(x) -In(l-x). I thought this was a bit tedious, so I tried to find the answer without solving quadratics. If so, |z|(1 - i) > 1 gives me the radius of convergence. A geometric series sum_(k)a_k is a series for which the ratio of each two consecutive terms a_(k+1)/a_k is a constant function of the summation index k. We can obtain power series representation for a wider variety of functions by exploiting the fact that a convergent power series can be di erentiated, or integrated, term-by-term to obtain a new power series that has the same radius of convergence as the original power series. Then check x = + R in the original power series to determine the convergence of the power series at the endpoints. Check the convergence of the series at the endpoints and then write the interval of convergence for the series. Last week was more theory, this week more practice, and so we will do more groupwork this week. The radius of convergence of a power series ƒ centered on a point a is equal to the distance from a to the nearest point where ƒ cannot be defined in a way that makes it holomorphic. Be sure to check convergence at each endpoint and state the test you used to determine convergence or divergence of each endpoint. That is what we needed to show. Denoting a n = 1. 6 Find a power series representation of the function f(x) = tan 1 x: Solution. yThe convergence at the endpoints x= a R;a+Rmust be determined separately. Remark: Many students computed the radius of convergence incorrectly, and then were (to use the technical term) screwed when they went back to test convergence at the endpoints. We convergencecan attempt the ratio test to find the radius of, but it fails because;goaa%hi doesn't exist for any x except O (The nexpression simplifiesto 11544 if is odd, so the a oscillation doesn't get limit. Power series: radius of convergence and interval of convergence. n : The radius of convergence is R = 1 Example 8. Convergence at the end points of the the interval. a) Use the Geometric Series to find a power series representation for. < L and diverges for all x satisfying (x a)k. Theorem 1 can be proved in full generality by comparing to the geometric series above. Radius of Convergence A power series will converge only for certain values of. so the radius of convergence is 1=3. See table 9. In general, there is always an interval in which a power series converges, and the number is called the radius of convergence (while the interval itself is called the interval of convergence). If a power series converges absolutely for all , then its radius of convergence is said to be and the interval of convergence is. and now it can be rewritten as a basic geometric series, the sum of: [16x/75]^k with the ratio as (16x)/75 and the starting value (since you said k=1 to start with) as 16/75 so to put it into standard form for a geometric series (sum with k=0 to infinity of a(r)^k) you can rewrite it as the sum of:. So will prove that the sequence { s n} is convergent. the power series above forR h(x) = 1=x can be used to nd a power series for lnx = (1=x)dx | the center and radius of convergence will be the same, so using the above power series will give a power series centered at 2. The series converges only for x = c, and the radius of convergence is r = 0. Consequently, by the theorem, the radius of convergence of the power series centered at x 1 = 1 satis es R x 1 R x 0 j x 1 x 0j= 1, so R x 1 = 1. Unlike the geometric series test used in nding the answer to problem 10. The series c 0 2c 1 + 4c 2 8c 3 + converges e. Find the radius of convergence, R, of the series. Conic Sections; Parametric Equations; Calculus and Parametric Equations; Introduction to Polar Coordinates; Calculus and Polar Functions; 10 Vectors. I take these numbers and plug them into the power series for a geometric series and get SUM[ (-1/5) * (x/5) n] = - SUM[ (1/5 n + 1) * x n]. Power Series Representation : Here we will use some basic tools such as Geometric Series and Calculus in order to determine the power series. Theory: We know about convergence for a geometric series. Integral Test The series and the integral do the same thing. If L = 0; then the radius of convergence is R = 0: If L = 1; then the radius of convergence is R = 1: If 0 < L < 1; then the power series converges for all x satisfying (x a)k. for jx aj>R, where R>0 is a value called the radius of convergence. Differentiation and integration of power series. Radius of convergence is R = 1. Series: The meaning of convergence of a series, tests for divergence and conver-gence. Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately). a) Use the Geometric Series to find a power series representation for. Taylor Series Expansions In this short note, a list of well-known Taylor series expansions is provided. By the Integral Test, the series X1 n=1 (lnn)2 n diverges. 27 Prove that, if the radius of. One fact that may occasionally be helpful for finding the radius of convergence: if the limit of the n th root of the absolute value of c [ n ] is K , then the radius of convergence is 1/ K. The first two functions, corresponding to the rational numbers 10/9 and 8/7 respectively, have the closed form expressions. For constant p, find the radius of convergence of the bi- nomial power series: p(p— 1)x2 p(p— — 2)'. Math 122 Fall 2008 Recitation Handout 17: Radius and Interval of Convergence Interval of Convergence The interval of convergence of a power series: ! cn"x#a ( ) n n=0 \$ % is the interval of x-values that can be plugged into the power series to give a convergent series. Root test, Alternating series test, Absolute and Conditional convergence, Power series, Radius of convergence of a power series, Taylor and Maclaurin series. The p-series test is another such test. Radius and Interval of Convergence Calculator Enter a power series: If you need a binomial coefficient C(n,k)=((n),(k)), type binomial(n,k). Denoting a n = 1. Taylor Series Expansions In this short note, a list of well-known Taylor series expansions is provided. The set of values of x for which the series converges is its interval of convergence. 8 Power Series (1) Find the radius and interval of convergence. There is a positive number R such that the series diverges for » x-a »> R but converges for » x-a »< R. Every Taylor Series converges at its center. their range in &reals. + (-1) n-1 a n of partial sums is convergent. The Interval and Radius of Convergence • The interval of convergenceof a power seriesis the collection of points for which the series converges. The variable x is real. Last week was more theory, this week more practice, and so we will do more groupwork this week. In this video, I show another example of finding the interval and radius of convergence for a series. ratio/root tests. 08 20 % 03 Fourier Series of 2𝑛 periodic functions, Dirichlet’s conditions for representation by a Fourier series, Orthogonality of the trigonometric. Do Taylor series always converge? The radius of convergence can be zero or infinite, or anything in between. Domain of Convergence. Course Description Sequences and series, multi-variable functions and their graphs, vector algebra and vector functions, partial differentiation. The well-known. a) Find the Taylor series associated to f(x) = x^-2 at a = 1. They behave somewhat like geometric series in that there is some 0 R 1, the radius of convergence, such that the series f(x) converges for jx aj< R and diverges for jx aj> R. SOLUTION: The radius of convergence is 2. 15 is to say that the power series converges if and diverges if. An example we've seen before is the geometric series 1 1 x = X1 n=0 xn for 1 < x < 1: This power series is centered at a = 0 and has radius of convergence R = 1. Since Σ |z|^n is a convergent geometric series when |z| 1,. Let's consider a series (no power yet!) and be patient for a couple of moments: Suppose that all s are positive and that there is a q <1 so that. We illustrate the uses of these operations on power series with some. Power Series Representation : Here we will use some basic tools such as Geometric Series and Calculus in order to determine the power series. a is a constant, r is a variable Theses are geometric series In a power series, the coefficients do not have to be constant. 24 in the text for information about radius of convergence and interval of convergence. Most of what is known about the convergence of in nite series is known by relating other series to the geometric series. 29, the radius of convergence is l/e POWER SERIES 329 38. Let g(x) = P 1. Express 1 = 1 x 2 as the sum of a power series and nd the interval of convergence. Power series centered at a 2R 7. | 2019-11-17T11:11:36 | {
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https://brilliant.org/discussions/thread/simple-numbers-are-problems-numbers/ | # One minus one plus one minus one plus...
There are three types of people in this world:
Evaluate:
$\color{#3D99F6}{S}=1-1+1-1+1-1+\ldots$
Type 1
$\color{#3D99F6}{S}=(1-1)+(1-1)+(1-1)+\ldots=0+0+0+\ldots=\boxed{0}$
Type 2
$\color{#3D99F6}{S}=1-(1-1)-(1-1)-(1-1)-\ldots=1-0-0-0-\ldots=\boxed{1}$
But the $\displaystyle 3^{rd}$ type of people did like this:
$1-\color{#3D99F6}{S}=1-(1-1+1-1+\ldots)=1-1+1-1+1-1+\ldots = S$
$\Leftrightarrow 1-\color{#3D99F6}S=\color{#3D99F6}S \Rightarrow 2\color{#3D99F6}S=1 \Rightarrow \color{#3D99F6}S=\boxed{\frac{1}{2}}$
4 years, 2 months ago
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You forgot $4^\text{th}$ type of people; they say that this series diverges.
- 4 years, 2 months ago
Its answer oscillates b/w 0 and 1
- 4 years, 2 months ago
Yes that's why it diverges.
- 4 years, 2 months ago
Even more here:
Evaluate : $S=1-2+3-4+5-6+ \ldots$
Type 1 : $S=1+(-2+3)+(-4+5)+ \ldots = 1+1+1+1+\ldots=\infty$
Type 2 : $S=(1-2)+(3-4)+(5-6)+ \ldots = -1-1-1-1+\ldots=-\infty$
Type 3 : They go to WolframAlpha, search this:
1 sum(n from 1 to infty,(-1)^n*n)
Which shows up that "The ratio test is inconclusive." and "The root test is inconclusive.", from which they implies that the sum is incosistent.
Type 4 : They go to Wikipedia and finds out that the sum is actually equal to $1/4$.
- 4 years, 2 months ago
Wow! Awesome!
I also saw this video:
$1+2+3+4+\ldots=\frac{1}{12}$
- 4 years, 2 months ago
Wow, that's cool :)))
- 4 years, 2 months ago
i guess u forgot the negative sign along with 1/12
- 4 years, 2 months ago
last one is pretty good
- 4 years, 2 months ago
haha.. g8
- 4 years, 2 months ago
Grandi series.
- 4 years, 2 months ago
Gud 1
- 4 years, 2 months ago
× | 2019-11-21T02:14:31 | {
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http://marcodoriaxgenova.it/jytu/2d-poisson-equation.html | # 2d Poisson Equation
This example shows how to numerically solve a Poisson's equation, compare the numerical solution with the exact solution, and refine the mesh until the solutions are close. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. Poisson's Equation in 2D We will now examine the general heat conduction equation, T t = κ∆T + q ρc. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. 2D Poisson Equation (DirichletProblem) The 2D Poisson equation is given by with boundary conditions There is no initial condition, because the equation does not depend on time, hence it becomes a boundary value problem. In it, the discrete Laplace operator takes the place of the Laplace operator. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. Recalling Lecture 13 again, we discretize this equation by using finite differences: We use an (n+1)-by-(n+1) grid on Omega = the unit square, where h=1/(n+1) is the grid spacing. The kernel of A consists of constant: Au = 0 if and only if u = c. Solving 2D Poisson on Unit Circle with Finite Elements. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. Our analysis will be in 2D. Finally, the values can be reconstructed from Eq. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. LaPlace's and Poisson's Equations. The electric field is related to the charge density by the divergence relationship. Qiqi Wang 5,667 views. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. I use center difference for the second order derivative. The code poisson_2d. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Many ways can be used to solve the Poisson equation and some are faster than others. Marty Lobdell - Study Less Study Smart - Duration: 59:56. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. 2D Poisson equation. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Elastic plates. The electric field is related to the charge density by the divergence relationship. The dotted curve (obscured) shows the analytic solution, whereas the open triangles show the finite difference solution for. ( 1 ) or the Green’s function solution as given in Eq. d = 2 Consider ˜u satisfying the wave equation in R3, launched with initial conditions invariant in the 3-direction: u˜(x1,x2,x3,0) = f˜(x1,x2,x3) = f(x1,x2),. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. Usually, is given and is sought. A video lecture on fast Poisson solvers and finite elements in two dimensions. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 3 of 16 Introduction to Scientific Computing Poisson’s Equation in 2D Michael Bader 2. The book NUMERICAL RECIPIES IN C, 2ND EDITION (by PRESS, TEUKOLSKY, VETTERLING & FLANNERY) presents a recipe for solving a discretization of 2D Poisson equation numerically by Fourier transform ("rapid solver"). a second order hyperbolic equation, the wave equation. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. 2D Poisson equations. (1) An explanation to reduce 3D problem to 2D had been described in Ref. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. Let (x,y) be a fixed arbitrary point in a 2D domain D and let (ξ,η) be a variable point used for integration. (2018) Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. Our analysis will be in 2D. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. Yet another "byproduct" of my course CSE 6644 / MATH 6644. (We assume here that there is no advection of Φ by the underlying medium. Furthermore a constant right hand source term is given which equals unity. The left-hand side of this equation is a screened Poisson equation, typically stud-ied in three dimensions in physics [4]. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Solving 2D Poisson on Unit Circle with Finite Elements. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. Hence, we have solved the problem. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding flux. Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. This has known solution. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. 3) is to be solved in Dsubject to Dirichletboundary. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. 1D PDE, the Euler-Poisson-Darboux equation, which is satisfied by the integral of u over an expanding sphere. Elastic plates. These equations can be inverted, using the algorithm discussed in Sect. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. 3 Uniqueness Theorem for Poisson's Equation Consider Poisson's equation ∇2Φ = σ(x) in a volume V with surface S, subject to so-called Dirichlet boundary conditions Φ(x) = f(x) on S, where fis a given function defined on the boundary. That avoids Fourier methods altogether. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). The dotted curve (obscured) shows the analytic solution, whereas the open triangles show the finite difference solution for. Yet another "byproduct" of my course CSE 6644 / MATH 6644. Solve Poisson equation on arbitrary 2D domain with RHS f and Dirichlet boundary conditions using the finite element method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. The derivation of Poisson's equation in electrostatics follows. This example shows how to numerically solve a Poisson's equation, compare the numerical solution with the exact solution, and refine the mesh until the solutions are close. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 3 of 16 Introduction to Scientific Computing Poisson’s Equation in 2D Michael Bader 2. In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. I want to use d_Helmholtz_2D(f, bd_ax, bd_bx, bd_ay, bd_by, bd_az, bd_bz, &xhandle, &yhandle, ipar, dpar, &stat)to solve the eqution with =0. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. 2D Poisson equation. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. Hence, we have solved the problem. In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. Marty Lobdell - Study Less Study Smart - Duration: 59:56. Poisson on arbitrary 2D domain. It asks for f ,but I have no ideas on setting f on the boundary. In it, the discrete Laplace operator takes the place of the Laplace operator. 5 Linear Example - Poisson Equation. Yet another "byproduct" of my course CSE 6644 / MATH 6644. The book NUMERICAL RECIPIES IN C, 2ND EDITION (by PRESS, TEUKOLSKY, VETTERLING & FLANNERY) presents a recipe for solving a discretization of 2D Poisson equation numerically by Fourier transform ("rapid solver"). ( 1 ) or the Green's function solution as given in Eq. The derivation of the membrane equation depends upon the as-sumption that the membrane resists stretching (it is under tension), but does not resist bending. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. and Lin, P. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. Poisson Equation Solver with Finite Difference Method and Multigrid. That avoids Fourier methods altogether. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding flux. The result is the conversion to 2D coordinates: m + p. Solve Poisson equation on arbitrary 2D domain with RHS f and Dirichlet boundary conditions using the finite element method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. A video lecture on fast Poisson solvers and finite elements in two dimensions. 4, to give the. In it, the discrete Laplace operator takes the place of the Laplace operator. Two-Dimensional Laplace and Poisson Equations. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. SI units are used and Euclidean space is assumed. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. Qiqi Wang 5,667 views. The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. 2D Poisson equations. Different source functions are considered. The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. This has known solution. Yet another "byproduct" of my course CSE 6644 / MATH 6644. That avoids Fourier methods altogether. Let (x,y) be a fixed arbitrary point in a 2D domain D and let (ξ,η) be a variable point used for integration. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. 2D-Poisson equation lecture_poisson2d_draft. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. Poisson’s equation can be solved for the computation of the potential V and electric field E in a [2D] region of space with fixed boundary conditions. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 1 of 16 Introduction to Scientific Computing Poisson's Equation in 2D Michael Bader 1. 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Use MathJax to format equations. Marty Lobdell - Study Less Study Smart - Duration: 59:56. Homogenous neumann boundary conditions have been used. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. From a physical point of view, we have a well-defined problem; say, find the steady-. The Poisson equation arises in numerous physical contexts, including heat conduction, electrostatics, diffusion of substances, twisting of elastic rods, inviscid fluid flow, and water waves. We will consider a number of cases where fixed conditions are imposed upon. The kernel of A consists of constant: Au = 0 if and only if u = c. Poisson's Equation in 2D We will now examine the general heat conduction equation, T t = κ∆T + q ρc. 3, Myint-U & Debnath §10. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own. on Poisson's equation, with more details and elaboration. The Poisson equation arises in numerous physical contexts, including heat conduction, electrostatics, diffusion of substances, twisting of elastic rods, inviscid fluid flow, and water waves. by JARNO ELONEN ([email protected] In three-dimensional Cartesian coordinates, it takes the form. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. 3) is to be solved in Dsubject to Dirichletboundary. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 3 of 16 Introduction to Scientific Computing Poisson’s Equation in 2D Michael Bader 2. 6 Poisson equation The pressure Poisson equation, Eq. Solving a 2D Poisson equation with Neumann boundary conditions through discrete Fourier cosine transform. nst-mmii-chapte. 3) is to be solved in Dsubject to Dirichletboundary. As expected, setting λ d = 0 nullifies the data term and gives us the Poisson equation. the Laplacian of u). For simplicity of presentation, we will discuss only the solution of Poisson's equation in 2D; the 3D case is analogous. Yet another "byproduct" of my course CSE 6644 / MATH 6644. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. These equations can be inverted, using the algorithm discussed in Sect. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding flux. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. Poisson equation. Poisson's equation can be solved for the computation of the potential V and electric field E in a [2D] region of space with fixed boundary conditions. The left-hand side of this equation is a screened Poisson equation, typically stud-ied in three dimensions in physics [4]. (part 2); Finite Elements in 2D And so each equation comes--V is one of the. SI units are used and Euclidean space is assumed. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. by JARNO ELONEN ([email protected] LAPLACE’S EQUATION AND POISSON’S EQUATION In this section, we state and prove the mean value property of harmonic functions, and use it to prove the maximum principle, leading to a uniqueness result for boundary value problems for Poisson’s equation. 4 Consider the BVP 2∇u = F in D, (4) u = f on C. In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. [2], considering an accelerator with long bunches, and assuming that the transverse motion is. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Suppose that the domain is and equation (14. Marty Lobdell - Study Less Study Smart - Duration: 59:56. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. Homogenous neumann boundary conditions have been used. Solving the 2D Poisson equation $\Delta u = x^2+y^2$ Ask Question Asked 2 years, 11 months ago. Furthermore a constant right hand source term is given which equals unity. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. FEM2D_POISSON_RECTANGLE is a C++ program which solves the 2D Poisson equation using the finite element method. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Consider the 2D Poisson equation for $1 Linear Partial Differential Equations > Second-Order Elliptic Partial Differential Equations > Poisson Equation 3. Uses a uniform mesh with (n+2)x(n+2) total 0003 % points (i. and Lin, P. The code poisson_2d. Making statements based on opinion; back them up with references or personal experience. Furthermore a constant right hand source term is given which equals unity. 2D Poisson equation. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 1 of 16 Introduction to Scientific Computing Poisson's Equation in 2D Michael Bader 1. Thus, the state variable U(x,y) satisfies:. Marty Lobdell - Study Less Study Smart - Duration: 59:56. We will consider a number of cases where fixed conditions are imposed upon. (1) An explanation to reduce 3D problem to 2D had been described in Ref. 2D Poisson Equation (DirichletProblem) The 2D Poisson equation is given by with boundary conditions There is no initial condition, because the equation does not depend on time, hence it becomes a boundary value problem. Furthermore a constant right hand source term is given which equals unity. Figure 63: Solution of Poisson's equation in two dimensions with simple Dirichlet boundary conditions in the -direction. The electric field is related to the charge density by the divergence relationship. 2D Poisson equation. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. LAPLACE’S EQUATION AND POISSON’S EQUATION In this section, we state and prove the mean value property of harmonic functions, and use it to prove the maximum principle, leading to a uniqueness result for boundary value problems for Poisson’s equation. Solving the 2D Poisson equation$\Delta u = x^2+y^2$Ask Question Asked 2 years, 11 months ago. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Both codes, nextnano³ and Greg Snider's "1D Poisson" lead to the same results. In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. We will consider a number of cases where fixed conditions are imposed upon. These equations can be inverted, using the algorithm discussed in Sect. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. Solution to Poisson’s Equation Code: 0001 % Numerical approximation to Poisson’s equation over the square [a,b]x[a,b] with 0002 % Dirichlet boundary conditions. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 1 of 16 Introduction to Scientific Computing Poisson's Equation in 2D Michael Bader 1. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. The solution is plotted versus at. Different source functions are considered. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). The exact solution is. That avoids Fourier methods altogether. FEM2D_POISSON_RECTANGLE is a C++ program which solves the 2D Poisson equation using the finite element method. 4 Fourier solution In this section we analyze the 2D screened Poisson equation the Fourier do-main. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. The solution is plotted versus at. The kernel of A consists of constant: Au = 0 if and only if u = c. The result is the conversion to 2D coordinates: m + p. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. , , and constitute a set of uncoupled tridiagonal matrix equations (with one equation for each separate value). Finite Element Solution fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Homogenous neumann boundary conditions have been used. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. Hence, we have solved the problem. In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. 4 Consider the BVP 2∇u = F in D, (4) u = f on C. The following figure shows the conduction and valence band edges as well as the Fermi level (which is constant and has the value of 0 eV) for the structure specified above. Suppose that the domain is and equation (14. The electric field is related to the charge density by the divergence relationship. Thus, the state variable U(x,y) satisfies:. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). 3) is to be solved in Dsubject to Dirichletboundary. on Poisson's equation, with more details and elaboration. The solution is plotted versus at. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. Journal of Applied Mathematics and Physics, 6, 1139-1159. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). nst-mmii-chapte. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. on Poisson's equation, with more details and elaboration. 2D Poisson-type equations can be formulated in the form of (1) ∇ 2 u = f (x, u, u, x, u, y, u, x x, u, x y, u, y y), x ∈ Ω where ∇ 2 is Laplace operator, u is a function of vector x, u,x and u,y are the first derivatives of the function, u,xx, u,xy and u,yy are the second derivatives of the function u. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. (1) An explanation to reduce 3D problem to 2D had been described in Ref. A video lecture on fast Poisson solvers and finite elements in two dimensions. We will consider a number of cases where fixed conditions are imposed upon. 4 Fourier solution In this section we analyze the 2D screened Poisson equation the Fourier do-main. a second order hyperbolic equation, the wave equation. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. The result is the conversion to 2D coordinates: m + p. Consider the 2D Poisson equation for$1 Linear Partial Differential Equations > Second-Order Elliptic Partial Differential Equations > Poisson Equation 3. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. Marty Lobdell - Study Less Study Smart - Duration: 59:56. The strategy can also be generalized to solve other 3D differential equations. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. Suppose that the domain is and equation (14. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Thus, the state variable U(x,y) satisfies:. [2], considering an accelerator with long bunches, and assuming that the transverse motion is. 3 Uniqueness Theorem for Poisson's Equation Consider Poisson's equation ∇2Φ = σ(x) in a volume V with surface S, subject to so-called Dirichlet boundary conditions Φ(x) = f(x) on S, where fis a given function defined on the boundary. Furthermore a constant right hand source term is given which equals unity. Use MathJax to format equations. Solving 2D Poisson on Unit Circle with Finite Elements. Recalling Lecture 13 again, we discretize this equation by using finite differences: We use an (n+1)-by-(n+1) grid on Omega = the unit square, where h=1/(n+1) is the grid spacing. The equation is named after the French mathematici. Poisson Equation Solver with Finite Difference Method and Multigrid. This example shows the application of the Poisson equation in a thermodynamic simulation. e, n x n interior grid points). Yet another "byproduct" of my course CSE 6644 / MATH 6644. The equation system consists of four points from which two are boundary points with homogeneous Dirichlet boundary conditions. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. The following figure shows the conduction and valence band edges as well as the Fermi level (which is constant and has the value of 0 eV) for the structure specified above. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. The result is the conversion to 2D coordinates: m + p. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. Suppose that the domain is and equation (14. Qiqi Wang 5,667 views. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. 3, Myint-U & Debnath §10. The exact solution is. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. The result is the conversion to 2D coordinates: m + p. on Poisson's equation, with more details and elaboration. The steps in the code are: Initialize the numerical grid; Provide an initial guess for the solution; Set the boundary values & source term; Iterate the solution until convergence; Output the solution for plotting; The code is compiled and executed via gcc poisson_2d. Hence, we have solved the problem. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. 2D Poisson equations. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. Particular solutions For the function X(x), we get the eigenvalue problem −X xx(x) = λX(x), 0 < x < 1, X(0) = X(1) = 0. Poisson's equation is = where is the Laplace operator, and and are real or complex-valued functions on a manifold. LAPLACE’S EQUATION AND POISSON’S EQUATION In this section, we state and prove the mean value property of harmonic functions, and use it to prove the maximum principle, leading to a uniqueness result for boundary value problems for Poisson’s equation. Finite Element Solution of the 2D Poisson Equation FEM2D_POISSON_RECTANGLE , a C program which solves the 2D Poisson equation using the finite element method. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. m Benjamin Seibold Applying the 2d-curl to this equation yields applied from the left. Let (x,y) be a fixed arbitrary point in a 2D domain D and let (ξ,η) be a variable point used for integration. The result is the conversion to 2D coordinates: m + p. Poisson Library uses the standard five-point finite difference approximation on this mesh to compute the approximation to the solution. Task: implement Jacobi, Gauss-Seidel and SOR-method. Two-Dimensional Laplace and Poisson Equations. In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. The equation is named after the French mathematici. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. 4, to give the. Moreover, the equation appears in numerical splitting strategies for more complicated systems of PDEs, in particular the Navier - Stokes equations. Hence, we have solved the problem. 2D-Poisson equation lecture_poisson2d_draft. As expected, setting λ d = 0 nullifies the data term and gives us the Poisson equation. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. The steps in the code are: Initialize the numerical grid; Provide an initial guess for the solution; Set the boundary values & source term; Iterate the solution until convergence; Output the solution for plotting; The code is compiled and executed via gcc poisson_2d. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. Elastic plates. Homogenous neumann boundary conditions have been used. Thus, the state variable U(x,y) satisfies:. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Yet another "byproduct" of my course CSE 6644 / MATH 6644. Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions. (1) An explanation to reduce 3D problem to 2D had been described in Ref. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. and Lin, P. Task: implement Jacobi, Gauss-Seidel and SOR-method. SI units are used and Euclidean space is assumed. by JARNO ELONEN ([email protected] Poisson equation. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. The equation system consists of four points from which two are boundary points with homogeneous Dirichlet boundary conditions. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. 3 Uniqueness Theorem for Poisson's Equation Consider Poisson's equation ∇2Φ = σ(x) in a volume V with surface S, subject to so-called Dirichlet boundary conditions Φ(x) = f(x) on S, where fis a given function defined on the boundary. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. We state the mean value property in terms of integral averages. Journal of Applied Mathematics and Physics, 6, 1139-1159. The left-hand side of this equation is a screened Poisson equation, typically stud-ied in three dimensions in physics [4]. 5 Linear Example - Poisson Equation. Homogenous neumann boundary conditions have been used. The steps in the code are: Initialize the numerical grid; Provide an initial guess for the solution; Set the boundary values & source term; Iterate the solution until convergence; Output the solution for plotting; The code is compiled and executed via gcc poisson_2d. The solution is plotted versus at. 2D Poisson equation. Elastic plates. We discretize this equation by using finite differences: We use an (n+1)-by-(n+1) grid on Omega = the unit square, where h=1/(n+1) is the grid spacing. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. FINITE DIFFERENCE METHODS FOR POISSON EQUATION 5 Similar techniques will be used to deal with other corner points. Poisson Equation Solver with Finite Difference Method and Multigrid. These equations can be inverted, using the algorithm discussed in Sect. 3) is to be solved in Dsubject to Dirichletboundary. Thus, solving the Poisson equations for P and Q, as well as solving implicitly for the viscosity terms in U and V, yields. From a physical point of view, we have a well-defined problem; say, find the steady-. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Two-Dimensional Laplace and Poisson Equations In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. 4 Consider the BVP 2∇u = F in D, (4) u = f on C. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Different source functions are considered. The code poisson_2d. 2D Poisson-type equations can be formulated in the form of (1) ∇ 2 u = f (x, u, u, x, u, y, u, x x, u, x y, u, y y), x ∈ Ω where ∇ 2 is Laplace operator, u is a function of vector x, u,x and u,y are the first derivatives of the function, u,xx, u,xy and u,yy are the second derivatives of the function u. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). the full, 2D vorticity equation, not just the linear approximation. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. I use center difference for the second order derivative. nst-mmii-chapte. Solving a 2D Poisson equation with Neumann boundary conditions through discrete Fourier cosine transform. Qiqi Wang 5,667 views. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. Suppose that the domain is and equation (14. This has known solution. SI units are used and Euclidean space is assumed. Statement of the equation. Poisson equation. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own. (We assume here that there is no advection of Φ by the underlying medium. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. Moreover, the equation appears in numerical splitting strategies for more complicated systems of PDEs, in particular the Navier - Stokes equations. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. A partial semi-coarsening multigrid method is developed to solve 3D Poisson equation. Find optimal relaxation parameter for SOR-method. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. Poisson's equation is = where is the Laplace operator, and and are real or complex-valued functions on a manifold. The book NUMERICAL RECIPIES IN C, 2ND EDITION (by PRESS, TEUKOLSKY, VETTERLING & FLANNERY) presents a recipe for solving a discretization of 2D Poisson equation numerically by Fourier transform ("rapid solver"). In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Poisson equation. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. Poisson Library uses the standard five-point finite difference approximation on this mesh to compute the approximation to the solution. Finally, the values can be reconstructed from Eq. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. We will consider a number of cases where fixed conditions are imposed upon. I use center difference for the second order derivative. Multigrid This GPU based script draws u i,n/4 cross-section after multigrid V-cycle with the reduction level = 6 and "deep" relaxation iterations 2rel. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Finite Element Solution fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Moreover, the equation appears in numerical splitting strategies for more complicated systems of PDEs, in particular the Navier - Stokes equations. The electric field is related to the charge density by the divergence relationship. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. 1 $\begingroup$ Consider the 2D Poisson equation. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. 2D Poisson equations. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. nst-mmii-chapte. m Benjamin Seibold Applying the 2d-curl to this equation yields applied from the left. the steady-state diffusion is governed by Poisson’s equation in the form ∇2Φ = − S(x) k. Qiqi Wang 5,667 views. A partial semi-coarsening multigrid method is developed to solve 3D Poisson equation. For simplicity of presentation, we will discuss only the solution of Poisson's equation in 2D; the 3D case is analogous. Hence, we have solved the problem. I use center difference for the second order derivative. 4, to give the. the steady-state diffusion is governed by Poisson’s equation in the form ∇2Φ = − S(x) k. Solve Poisson equation on arbitrary 2D domain with RHS f and Dirichlet boundary conditions using the finite element method. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. The code poisson_2d. m Benjamin Seibold Applying the 2d-curl to this equation yields applied from the left. Our analysis will be in 2D. Two-Dimensional Laplace and Poisson Equations. 1 $\begingroup$ Consider the 2D Poisson equation. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 1 of 16 Introduction to Scientific Computing Poisson's Equation in 2D Michael Bader 1. Poisson Equation Solver with Finite Difference Method and Multigrid. 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i. Poisson equation. ( 1 ) or the Green’s function solution as given in Eq. the steady-state diffusion is governed by Poisson’s equation in the form ∇2Φ = − S(x) k. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. It asks for f ,but I have no ideas on setting f on the boundary. In it, the discrete Laplace operator takes the place of the Laplace operator. Elastic plates. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Solving 2D Poisson on Unit Circle with Finite Elements. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding flux. Uses a uniform mesh with (n+2)x(n+2) total 0003 % points (i. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. Suppose that the domain is and equation (14. In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. The electric field is related to the charge density by the divergence relationship. Making statements based on opinion; back them up with references or personal experience. Poisson Equation Solver with Finite Difference Method and Multigrid. This is often written as: where is the Laplace operator and is a scalar function. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. Poisson Library uses the standard five-point finite difference approximation on this mesh to compute the approximation to the solution. (1) An explanation to reduce 3D problem to 2D had been described in Ref. Let r be the distance from (x,y) to (ξ,η),. 6 Poisson equation The pressure Poisson equation, Eq. Two-Dimensional Laplace and Poisson Equations. In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. Solving the 2D Poisson equation $\Delta u = x^2+y^2$ Ask Question Asked 2 years, 11 months ago. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. This example shows the application of the Poisson equation in a thermodynamic simulation. A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. The Poisson equation on a unit disk with zero Dirichlet boundary condition can be written as -Δ u = 1 in Ω, u = 0 on δ Ω, where Ω is the unit disk. c implements the above scheme. the Laplacian of u). For simplicity of presentation, we will discuss only the solution of Poisson's equation in 2D; the 3D case is analogous. Marty Lobdell - Study Less Study Smart - Duration: 59:56. Furthermore a constant right hand source term is given which equals unity. (2018) Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). e, n x n interior grid points). (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Poisson Equation Solver with Finite Difference Method and Multigrid. nst-mmii-chapte. Both codes, nextnano³ and Greg Snider's "1D Poisson" lead to the same results. 2D Poisson equations. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Elastic plates. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classification of PDE Page 3 of 16 Introduction to Scientific Computing Poisson’s Equation in 2D Michael Bader 2. Two-Dimensional Laplace and Poisson Equations. Qiqi Wang 5,667 views. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. 1D PDE, the Euler-Poisson-Darboux equation, which is satisfied by the integral of u over an expanding sphere. In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. I use center difference for the second order derivative. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. This has known solution. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). Particular solutions For the function X(x), we get the eigenvalue problem −X xx(x) = λX(x), 0 < x < 1, X(0) = X(1) = 0. and Lin, P. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Usually, is given and is sought. Finite Volume model in 2D Poisson Equation This page has links to MATLAB code and documentation for the finite volume solution to the two-dimensional Poisson equation where is the scalar field variable, is a volumetric source term, and and are the Cartesian coordinates. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. Solving 2D Poisson on Unit Circle with Finite Elements. Our analysis will be in 2D. Qiqi Wang 5,667 views. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. The derivation of Poisson's equation in electrostatics follows. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. (We assume here that there is no advection of Φ by the underlying medium. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. SI units are used and Euclidean space is assumed. Journal of Applied Mathematics and Physics, 6, 1139-1159. LaPlace's and Poisson's Equations. Qiqi Wang 5,667 views. Use MathJax to format equations. FINITE DIFFERENCE METHODS FOR POISSON EQUATION 5 Similar techniques will be used to deal with other corner points. Let r be the distance from (x,y) to (ξ,η),. (part 2); Finite Elements in 2D And so each equation comes--V is one of the. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. 6 is used to create a velocity eld that satis es the continuity equation and is incompressible. 4, to give the. ( 1 ) or the Green’s function solution as given in Eq. Making statements based on opinion; back them up with references or personal experience. Research highlights The full-coarsening multigrid method employed to solve 2D Poisson equation in reference is generalized to 3D. Poisson's equation is = where is the Laplace operator, and and are real or complex-valued functions on a manifold. This has known solution. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. on Poisson's equation, with more details and elaboration. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. Viewed 392 times 1. 2D Poisson equation. e, n x n interior grid points). In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. nst-mmii-chapte. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. Thus, the state variable U(x,y) satisfies:. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. 2D Poisson Equation (DirichletProblem) The 2D Poisson equation is given by with boundary conditions There is no initial condition, because the equation does not depend on time, hence it becomes a boundary value problem. , , and constitute a set of uncoupled tridiagonal matrix equations (with one equation for each separate value). The diffusion equation for a solute can be derived as follows. 2D Poisson equation. The solution is plotted versus at. This is often written as: where is the Laplace operator and is a scalar function. Solving 2D Poisson on Unit Circle with Finite Elements. From a physical point of view, we have a well-defined problem; say, find the steady-. a second order hyperbolic equation, the wave equation. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). If the membrane is in steady state, the displacement satis es the Poisson equation u= f;~ f= f=k. 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i.
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https://math.stackexchange.com/questions/508735/why-does-at-least-two-intervals-overlap-in-an-uncountable-family-of-intervals | # Why does at least two intervals overlap in an uncountable family of intervals?
1. Prove that there are uncountably many intervals $(a,b)$ in $\mathbb{R}, a\neq b$.
2. Assume $X$ be an uncountable family of intervals. Show that there exists at least two intervals in this family that overlap.
First was not difficult. I used the arguments similar to Cantor's Diagonal Argument (used to show $\mathbb{R}$ is uncountable.)
My attempt for 2: Assume $X$ be an uncountable family of pairwise disjoint intervals, i.e. $(a_i,b_i) \cap (a_j,b_j) = \emptyset, \quad \forall i\neq j\in I$. We know there exists a rational number in each of these intervals. This implies there are uncountably many rational numbers. Contradiction, since $\mathbb{Q}$ is countable. Thus, $X$ must have at least two intervals that overlap. $\blacksquare$
Is there any problem with this reasoning?
• First is a consequence of $(a,b)\mapsto b-a$ being a surjection. – Git Gud Sep 29 '13 at 11:33
• So would it be wrong to argue the way I did? – math Sep 29 '13 at 11:35
• Probably not, but it seems to me too complicated to prove what you want to prove. – Git Gud Sep 29 '13 at 11:36
• Looks fine to me. What about the argument made you feel uncertain? – Callus - Reinstate Monica Sep 29 '13 at 11:38
• @Callus, are you asking me or Git Gud? – math Sep 29 '13 at 11:45
For (1), you certainly could use a diagonal argument directly to prove that there is no surjection from $\mathbb{N}$ onto the set of intervals, or as Git Gud points out you could instead use the existence of a surjection from the set of intervals to $\mathbb{R}$ and then appeal to the nonexistence of a surjection $\mathbb{N} \to \mathbb{R}$. It is common to use Cantor's theorem on the uncountability of the reals as a "black box" in this way.
Your proof for (2) is perfectly fine. You could also get the contradiction by showing that $X$ is countable after all, rather than by showing that $\mathbb{Q}$ is uncountable, but this choice is just a matter of taste.
• For 1), why not just say that $(0,x)$ is an interval for any $x>0$, giving us directly an injection of $\mathbb R^+$ into the set of intervals? (Even simpler than the route with surjections.) – Andrés E. Caicedo Sep 29 '13 at 16:45
• A third method would simply use the surjection from intervals to reals given by $(a,b) \mapsto a$. – Trevor Wilson Sep 29 '13 at 16:53
• @Asaf I think that depends on whether or not it starts with "let $X$ be an uncountable family of pairwise disjoint intervals" like the OP's proof does. – Trevor Wilson Sep 29 '13 at 17:18 | 2021-03-07T16:41:31 | {
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http://lepetitpalace.ch/v9pcq8lz/c3a230-calculate-the-solubility-of-agcl-in-pure-water | K sp = 1.5 x 10 -10. Let [Ag+] = x. EduRev is a knowledge-sharing community that depends on everyone being able to pitch in when they know something. Justify your answer with a calculation. b. Molar mass of AgCl is 143.321 g /mol. Silver nitrate (which is soluble) has silver ion in common with silver chloride. AgCl is 1: 1 type salt , therefore. Does the solubility of this salt increase or decrease, compared to its solubility in pure water? 0.01 M C a C l 2 C. Pure water. A. var js, fjs = d.getElementsByTagName(s)[0]; Another way to prevent getting this page in the future is to use Privacy Pass. You can study other questions, MCQs, videos and tests for NEET on EduRev and even discuss your questions like 2) The solubility of AgCl in pure water is 1.3 x 10-5 M. Calculate the value of K sp. mol.wt. of [SO4—] must be greater than 1.08 x 10-8 mole/litre. x^2 =1.8 x 10 ^-10. 0.01 M N a 2 S O 4 B. Calculate its solubility in 0.01M NaCl aqueous solution. Now, let's try to do the opposite, i.e., calculate the K sp from the solubility of a salt. The solubility of insoluble substances can be decreased by the presence of a common ion. Performance & security by Cloudflare, Please complete the security check to access. The solubility product of $\ce{AgCl}$ in water is given as $1.8\cdot10^{-10}$ Determine the solubility of $\ce{AgCl}$ in pure water and in a $0.25~\mathrm{M}$ $\ce{MgSO4}$ solution. fjs.parentNode.insertBefore(js, fjs); (function(d, s, id) { Your IP: 37.252.96.253 Apart from being the largest NEET community, EduRev has the largest solved Calculate the solubility of AgCl in a 0.1 M NaCl solution. Explanation: Jamunaakailash … L6 : Solubility in water - Sorting Materials, Science, Class 6, Solubility and Solubility product - Ionic Equilibrium, Solubility and solubility product - Ionic Equilibrium. is done on EduRev Study Group by NEET Students. Calculate the ratio of solubility of agcl in 0.1m agno3 and in pure water - 6820732 1. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. if (d.getElementById(id)) return; c. If 50.0 ml of 0.10M AgNO3 are added to 50.0 ml of 0.15M NaCl will a precipitate be formed? For the precipitation of BaSO4 from the solution of BaCl2, the conc. Calculate the solubility of AgCl (K sp = 1.8 x10-10) in pure water. are solved by group of students and teacher of NEET, which is also the largest student Does the solubility of this salt increase or decrease, compared to its solubility in pure water? js.src = "//connect.facebook.net/en_US/sdk.js#xfbml=1&version=v2.10"; Join now. First, write the BALANCED REACTION: Next, set up the SOLUBILITY PRODUCT EQUILIBRIUM EXPRESSION: It is given in the problem that the solubility of AgCl is 1.3 x 10-5. Want to see the step-by-step answer? … Calculate solubility of AgCl in a) pure water b) 0.1 M AgNO 3 c) 0.01 M NaCl Solution ) a) Solubility of AgCl in pure water-Solubility product ‘K sp ‘ = 1.5 x 10-10. 1. Cloudflare Ray ID: 5f8635b20c77ff38 Check out a … Question 2)Solubility product of AgCl is 1.5 x 10-10 at 25 0 C . If the answer is not available please wait for a while and a community member will probably answer this over here on EduRev! Again, these concentrations give the solubility. Present in silver chloride are silver ions (A g +) and chloride ions (C l −). To calculate the ratio of solubility of AgCl in 0.1M AgNO^3 and in pure water what you need to put in the back of your mind is that water has a constant concentration of 1.0 x 10^-7M. of Mg(OH)2 = 24 + (16 + 1)2 = 24 + 34 = 58, S in mole /litre = ( S in gm /litre) / mol.wt.= 9.57 x 10 -3 / 58, S is less than 0.02 , so 4S3 is neglected, S in gm / litre = 1.49 x 10 -5 x 58 = 86.42 x 10-5. because S <<< 0.1 and solubility of AgCl decreases in presence of AgNO3 due to common ion effect(common ion is Ag+).Therefore. By continuing, I agree that I am at least 13 years old and have read and a. Then [Cl-] = x, and substituting in the Ksp expression we have (x)(x) = 1.8 x 10 ^-10. S = √ K sp = √ 1.5x 10 -10 The molar solubility of AgCl = [Ag+] in pure water . Calculate the solubility of AgCl in a 0.20 M AgNO3 solution. Ask your question. community of NEET. The Questions and Answer. check_circle Expert Answer. K sp = S 2. A g C l is not soluble in water. x = 1.342 x 10 ^-05 moles dm^-3 = molar solubility of AgCl. Again, these concentrations give the solubility. Question bank for NEET. You may need to download version 2.0 now from the Chrome Web Store. Log in. To calculate the ratio of solubility of AgCl in 0.1M AgNO^3 and in pure water what you need to put in the back of your mind is that water has a constant concentration of 1.0 x 10^-7M. Join now. }(document, 'script', 'facebook-jssdk')); Online Chemistry tutorial that deals with Chemistry and Chemistry Concept. • Click hereto get an answer to your question ️ The solubility product of AgCl in water is 1.5 × 10^-10 . d. Calculate the solubility of AgCl in a 0.60 M NH 3 solution. Solubility of AgCl will be minimum in _____. varunnair3842 varunnair3842 26.11.2018 Chemistry Secondary School Calculate the ratio of solubility of agcl in 0.1m agno3 and in pure water 2 See answers shauryakesarwani21 shauryakesarwani21 Answer: 1.34 X 10^-4. Calculate the solubility of AgCl in a 0.20 M AgNO3 solution. • Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. soon. js = d.createElement(s); js.id = id; Log in. Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? agree to the. because S <<< 0.01 and solubility of AgCl decreases in presence of NaCl due to common ion effect(common ion is Cl–).Therefore, ionic product of [Ba++] [SO4– –] > K sp of BaSO4. See Answer . Answers of Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Question. So with that information, we can say that the solubility of AgCl in 0.1M AgNO^3 is given as 0.1 M. So to determine the ratio of solubility of AgCl in 0.1M AgNO^3 and water all we need to do is; This discussion on Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? | 2022-05-21T06:17:23 | {
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https://math.stackexchange.com/questions/3269084/find-all-real-numbers-a-1-a-2-a-3-b-1-b-2-b-3 | # Find all real numbers $a_1, a_2, a_3, b_1, b_2, b_3$.
Find all real numbers $$a_1, a_2, a_3, b_1, b_2, b_3$$ such that for every $$i\in \lbrace 1, 2, 3 \rbrace$$ numbers $$a_{i+1}, b_{i+1}$$ are distinct roots of equation $$x^2+a_ix+b_i=0$$ (suppose $$a_4=a_1$$ and $$b_4=b_1$$).
There are many ways to do it but I've really wanted to finish the following idea:
From Vieta's formulas we get:
\begin{align} \begin{cases} a_1+b_1=-a_3 \ \ \ \ \ \ \ \ (a) \\a_2+b_2=-a_1\ \ \ \ \ \ \ \ (b)\\a_3+b_3=-a_2\ \ \ \ \ \ \ \ (c)\\a_1b_1=b_3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (d)\\a_2b_2=b_1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (e)\\a_3b_3=b_2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (f)\end{cases} \end{align} First we notice that each $$b_i$$ is nonzero. Indeed, suppose $$b_1=0$$. Then from (d) and (f) we deduce that $$b_3=0$$ and $$b_2=0$$, so from (a), (b), (c) we get $$a_1=-a_3=-(-a_2)=-(-(-a_1))=-a_1$$, hence $$a_1=0$$ which is impossible.
Now, from (a), (b), (c), (d), (e), (f) we obtain: \begin{align} \begin{cases} a_1+b_1-a_1b_1=-a_3-b_3 \ \ \ \ \ \ \ \ \\a_2+b_2-a_2b_2=-a_1-b_1\ \ \ \ \ \ \ \ \\a_3+b_3-a_3b_3=-a_2-b_2\end{cases}, \end{align} so: \begin{align} \begin{cases} (b_1-1)(a_1-1)=1-a_2 \ \ \ \ \ \ \ \ \\(b_2-1)(a_2-1)=1-a_3\ \ \ \ \ \ \ \ \\(b_3-1)(a_3-1)=1-a_1\end{cases}. \end{align} Therefore: \begin{align*} (b_1-1)(b_2-1)(b_3-1)(a_1-1)(a_2-1)(a_3-1)=(1-a_1)(1-a_2)(1-a_3), \end{align*} which implies: \begin{align*} \bigl((a_1-1)(a_2-1)(a_3-1)\bigr)\bigl((b_1-1)(b_2-1)(b_3-1)+1\bigr)=0. \end{align*}
I got stuck here. Is it possible to prove that in this case $$b_i=0$$ is the only solution to equation $$(b_1-1)(b_2-1)(b_3-1)=-1$$ or maybe get contradiction in some other way? If so, we can assume that $$a_1=1$$ and from here we can easily show that also $$a_2=a_3=1$$, so $$b_1=b_2=b_3=-2$$.
Since $$(b_1-1)(b_2-1)(b_3-1)>-1$$ for every $$b_i>0$$ and $$(b_1-1)(b_2-1)(b_3-1)<-1$$ for every $$b_i<0$$, it suffices to prove that the signs of $$b_1, b_2, b_3$$ can't be different but I don't know how to do it. I also found out that $$(b_1+1)^2+(b_2+1)^2+(b_3+1)^2=3$$, so $$b_i\in [-\sqrt{3}-1, \sqrt{3}-1]$$ but I don't know if we can use it somehow.
Well, here is a way to proceed after you note none of the $$b_i$$ are zero. Hints:
1) Multiplying the last three equations, $$(d)\times (e)\times (f)$$ gives $$a_1a_2a_3=1$$.
2) Now, multiplying just any two among those, e.g. $$(d)\times (e)$$ and using the result 1) above gives $$a_2b_2=a_3b_3$$, by symmetry and simplification using $$(d), (e), (f)$$ gives $$b_i = b$$, for some non-zero constant $$b$$, and hence $$a_i=1$$.
3) Now it is easy to conclude $$b=-2$$ from any of the first three equations. Hence $$(a_i, b_i)=(1, -2)$$.
• Yes I've also done this but I was specifically asking about the idea with equation $(b_1-1)(b_2-1)(b_3-1)=-1$. Does it mean it's not possible to finish that solution? – glopf Jun 22 at 23:02
• By itself, $(b_1-1)(b_2-1)(b_3-1)=-1$ does not allow one to conclude anything useful, you have to use some among equations $(a)-(f)$ in addition to draw a contradiction. Then why not solve $(a)-(f)$, which is what is really needed anyway? – Macavity Jun 23 at 16:57 | 2019-09-22T20:43:32 | {
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https://math.stackexchange.com/questions/1223486/smallest-number-of-points-on-plane-that-guarantees-existence-of-a-small-angle/1224693 | Smallest number of points on plane that guarantees existence of a small angle
What is the smallest number $n$, that in any arrangement of $n$ points on the plane, there are three of them making an angle of at most $18^\circ$?
It is clear that $n>9$, since the vertices of a regular 9-gon is a counterexample. One can prove using pigeonhole principle that $n\le 11$. Take a edge of the convex hull of points. All the points lie to one side of this line. cut the half-plane into 10 slices of $18^\circ$ each. There can't be any points in the first and last slice. Thus, by pigeonhole principle some slice contains more than one point. So we have an angle of size at most $18^\circ$.
Now, for $n=10$, I can not come up with a counterexample nor a proof of correctness. Any ideas?
Hmm this appears pretty complicated. I hope there is a simpler solution.
Theorem. For any $n\ge3$ points on a 2D plane, we can always find 3 points $A,B,C$ such that $\angle ABC\le\frac{180°}n$. (OP's solution follows with $n=10$.)
Proof. Again we take an edge of the convex hull and all points lie on the other side of this edge. Let's say points $A$ and $B$ are on this edge.
Partition the half-plane into $n$ regions about point $A$. Name each region $A_1,\dotsc,A_n$, where a point $X$ inside $A_i$ means $\frac{180°}n(i-1)<\angle XAB\le \frac{180°}n\cdot i$.
Do the same about point $B$. This will create $n$ other region $B_1,\dotsc,B_n$, where point $X$ in $B_j$ means $\frac{180°}n(j-1)<\angle XBA\le \frac{180°}n\cdot j$.
If two points $X,Y$ can be found in the same region $A_i$, then we also have $\angle XAY \le \frac{180°}n$. That means only one point can exist in each of the region $A_i$ (there are $n-2$ points and $n-1$ valid regions). Same for $B_j$.
$A_i$ and $B_j$ may overlap, though only when $i+j< n+2$ (a):
\begin{array}{c|ccccccc} & A_2 & A_3 & A_4 & \cdots & A_{n-2} & A_{n-1} & A_n \\ \hline B_2 & ✓ & ✓ & ✓ & \cdots & ✓ & \color{red}{✓} & ✗ \\ B_3 & ✓ & ✓ & ✓ & \cdots & \color{red}{✓} & ✗ & ✗ \\ B_4 & ✓ & ✓ & ✓ & \cdots & ✗ & ✗ & ✗ \\ \vdots & \vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\ B_{n-2} & ✓ & \color{red}{✓} & ✗ & \cdots & ✗ & ✗ & ✗ \\ B_{n-1} & \color{red}{✓} & ✗ & ✗ & \cdots & ✗ & ✗ & ✗ \\ B_n & ✗ & ✗ & ✗ & \cdots & ✗ & ✗ & ✗ \\ \end{array}
we immediately see that the points can only exist in the overlapping regions $A_i B_{n+1-i}$, to ensure no two point occupy the same column ($A_i$) or same row ($B_j$).
So for any $X$ in $A_i B_{n+1-i}$, we have:
\begin{align} \angle XAB &> \frac{180°}n(i-1) \\ \angle XBA &> \frac{180°}n(n-i) \\ \end{align}
For the triangle $\triangle ABX$, we need $\angle XAB + \angle XBA + \angle AXB = 180°$, thus
$$\angle AXB < 180° - \frac{180°}n(i-1) - \frac{180°}n(n-i) = \frac{180°}n.$$
Note:
(a) if point $X$ exists in both $A_i$ and $B_j$ with $i+j\ge n+2$ then $$\angle XAB + \angle XBA > \frac{180°}n(i+j-2) \ge 180°,$$ which is impossible.
Suppose there are $10$ points. Let us construct the convex hull of these points. It has at most $10$ sides. This means it must have an angle that is at most $180° - 360°/10 = 144°$. Let us choose the vertex corresponding to such angle as point $A$, and its 2 neighboring points along the convex hull as points $B$ and $C$. In other words, we have situation pictured here:
Now, from $7$ points inside the angle $\angle CAB$ let us draw 7 lines to point $A$. This will divide the angle into 8 smaller angles, and one of them will be at most $144°/8 = 18°$.
Therefore, since there is a counterexample for $n=9$ (a regular 9-gon), the answer is
$$n=10$$
$$GENERALIZATION$$
The smallest number $n$, that in any arrangement of $n$ points on the plane, there are three of them making an angle of at most $180°/k$ is $$n=k$$.
The proof is virtually the same as for the case $k=10$.
• @Untitled can you perhaps review two answers and see if they solved the problem? :) – VividD May 2 '15 at 8:40 | 2019-09-21T10:58:01 | {
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https://math.stackexchange.com/questions/69162/proving-formula-for-product-of-first-n-odd-numbers | # Proving formula for product of first n odd numbers
I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$):
$$\prod_{i = 1}^{n} (2i - 1) = \frac{(2n)!}{2^{n} n!}$$
How can I prove that it holds (or find a counter-example)?
• This sounds like homework. Did you try induction? – cardinal Oct 2 '11 at 1:20
• @Pedro: Do you mean $$\left(\prod_{i=1}^n2i\right) - 1$$ (which is what you wrote) or $$\prod_{i=1}^n(2i-1)$$(which is what your title suggests)? – Arturo Magidin Oct 2 '11 at 1:25
• It is not, I stumbled upon this formula by accident and was wondering. The proof was fairly simple, I suppose I should have thought more about it – Pedro Oct 2 '11 at 1:28
• @ArturoMagidin: I meant the latter, sorry for the ambiguity – Pedro Oct 2 '11 at 1:30
• As a note: what you have is sometimes termed as the "double factorial". – J. M. is a poor mathematician Oct 2 '11 at 9:33
The idea is to "complete the factorials":
$$1\cdot 3 \cdot 5 \cdots (2n-1) = \frac{ 1 \cdot 2 \cdot 3 \cdot 4 \cdots (2n-1)\cdot (2n) }{2\cdot 4 \cdot 6 \cdots (2n)}$$
Now take out the factor of $2$ from each term in the denominator:
$$= \frac{ (2n)! }{2^n \left( 1\cdot 2 \cdot 3 \cdots n \right)} = \frac{(2n)!}{2^n n!}$$
A mathematician may object that there is a small gray area about what exactly happens between those ellipses, so for a completely rigorous proof one would take my post and incorporate it into a proof by induction.
For the induction argument, \begin{align*} \prod_{i=1}^{n+1}(2i-1)&=\left(\prod_{i=1}^n(2i-1)\right)(2n+1)\\ &= \frac{(2n)!(2n+1)}{2^n n!} \end{align*} by the induction hypothesis. Now multiply that last fraction by a carefully chosen expression of the form $\dfrac{a}a$ to get the desired result.
$$\Pi_{k=1}^n(2k-1) = \Pi_{k=1}^n(2k-1) \frac{\Pi_{k=1}^n(2k)}{\Pi_{k=1}^n(2k)} =\frac{\Pi_{k=1}^{2n}k}{2^n\Pi_{k=1}^n k} = \frac{(2n)!}{2^n n!}$$
• This is the same answer as the one given four years ago by Ragib, only with fewer explanations. It's good that you want to help by answering questions, but maybe you could consider answering some that haven't been answered yet? – mrf Aug 31 '15 at 20:42 | 2020-01-20T10:19:58 | {
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https://brilliant.org/discussions/thread/helphow-to-slove-this-integral/ | # help:how to slove this integral
Note by Abhinavyukth Suresh
9 months, 2 weeks ago
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Simply use the Gamma Function. After the substitution $ar^{4}=t$, the integral becomes $\frac{1}{4a^{\frac{3}{4}}}\int_{0}^{\infty}t^{-\frac{1}{4}}e^{-t}dt$. The answer comes out to be $\frac{\Gamma\left(\frac{3}{4}\right)}{4a^{\frac{3}{4}}}$
- 9 months, 2 weeks ago
You could ask @Aruna Yumlembam - he's an expert on Gamma Functions - he likes them @Abhinavyukth Suresh
- 9 months, 2 weeks ago
- 9 months, 2 weeks ago
He'll probably show the entire proof - see his notes. @Aaghaz Mahajan
- 9 months, 2 weeks ago
There is no "proof" neede here. Simply substitute $ar^{4}=t$ and then the answer follows
- 9 months, 2 weeks ago
Ok. But I believe he should come. He's good at this stuff. I've read his notes - and gave me a interesting infinite series / function in one of my notes.
- 9 months, 2 weeks ago
Ok. Although i dont see what else might be needed in the proof.
- 9 months, 2 weeks ago
Maybe his insight into the Gamma Function?
- 9 months, 2 weeks ago
Also, you could learn a thing or two from him...
Not saying you're bad or anything.
- 9 months, 2 weeks ago
so, isn't there any other method than using gamma function to solve this integral?
- 9 months, 2 weeks ago
The proof given by Mr.Aaghaz is correct and mine is same too .Yet using this very idea we can prove this result, $\frac{\Gamma(1/n)}{n}\zeta(1/n)=\int_0^\infty\frac{1}{e^{x^n}-1}dx$, giving us, $\frac{1}{n}\int_0^\infty\frac{x^{1/n-1}}{e^x-1}dx=\int_0^\infty\frac{1}{e^{x^n}-1}dx$as the result.
- 9 months, 2 weeks ago
Yeah we can prove the identity by summing an infinite GP too. Also these types of integrals are known as Bose Einstein Integrals
- 9 months, 2 weeks ago
Thanks a lot.But can you please give me a your solution to your problem How is this ??!!
- 9 months, 2 weeks ago
Sure. Observe that $\sum_{n=1}^{\infty}e^{-nx}\ =\ \frac{1}{e^{x}-1}$
Using this, we have $\int_{0}^{\infty}\frac{x^{t-1}}{e^{x}-1}dx=\int_{0}^{\infty}\left(\sum_{n=1}^{\infty}x^{t-1}e^{-nx}\right)dx$
Swapping the integral and summation , and using the identity $\int_{0}^{\infty}x^{a}e^{-bx}dx\ =\ \frac{a!}{b^{a+1}}$ we will arrive at the answer.
- 9 months, 2 weeks ago
Mr. Yajat Shamji,if people provides a solution to any problem you must try and appreciate it and not discourage them.Please don't repeat such acts.
- 9 months, 2 weeks ago
I'm not. I.. was thinking of you and I read your profile so I thought I could bring you over. After all, you like to contribute, right?
Also, I wasn't discouraging @Aaghaz Mahajan's solution, I..
- 9 months, 2 weeks ago
- 9 months, 2 weeks ago
@Aruna Yumlembam - @Abhinavyukth Suresh needs your help on solving this integral - needs the Gamma Function and a full proof. | 2021-04-11T13:48:30 | {
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https://math.stackexchange.com/questions/3338415/how-can-you-show-by-hand-that-e-1-e-ln2 | How can you show by hand that $e^{-1/e} < \ln(2)$?
By chance I noticed that $$e^{-1/e}$$ and $$\ln(2)$$ both have decimal representation $$0.69\!\ldots$$, and it got me wondering how one could possibly determine which was larger without using a calculator.
For example, how might Euler have determined that these numbers are different?
• Probably the easiest approach is to show that $$\frac{1}{e}+\ln(\ln(2))$$ is positive. – Peter Aug 29 at 18:06
• What do exactly mean the dots? Namely, do you want to prove that $.692200<e^{-1/e}<.692201$? Or what you want to prove is the inequality $e^{-1/e}<\ln 2$? – ajotatxe Aug 29 at 18:07
• @ajotatxe The challenge is to show $$e^{-1/e}<\ln(2)$$ – Peter Aug 29 at 18:08
• I'm happy to change the title. I was just trying to point out how remarkably close they are. – Kevin Beanland Aug 29 at 18:12
• Thanks. I just changed it. – Kevin Beanland Aug 29 at 18:14
Here's a method that only uses rational numbers as bounds, keeping to small denominators where possible. It is elementary, but finding sufficiently good bounds requires some experimentation or at least luck.
We instead establish the inequality $$-\log \log 2 < \frac{1}{e}$$ of positive numbers, which we can see to be equivalent by taking the logarithm of and then negating both sides. Our strategy for establishing this latter inequality (which is equivalent to yet another inequality suggested in the comments) is to use power series to produce an upper bound for the less hand side and a larger lower bound for the right-hand side. To estimate logarithms, we use the identity $$\log x = 2 \operatorname{artanh} \left(\frac{x - 1}{x + 1}\right) ,$$ which yields faster-converging series and so lets us use fewer terms for our estimate.
Firstly, $$\log 2 = 2 \operatorname{artanh} \frac{1}{3} .$$ Then, since the power series $$\operatorname{artanh} u = u + \frac{1}{3} u^3 + \frac{1}{5} u^5 + \cdots$$ has nonnegative coefficients, for $$0 < u < 1$$ any truncation thereof is a lower bound for the series, and in particular $$\log 2 = 2 \operatorname{artanh} \frac{1}{3} > 2\left[\left(\frac{1}{3}\right) + \frac{1}{3} \left(\frac{1}{3}\right)^3 + \frac{1}{5} \left(\frac{1}{3}\right)^5\right] = \frac{842}{1215} .$$ We'll use the same power series to produce an upper bound for $$-\log \log 2$$, but since we're nominally computing by hand and want to avoid computing powers of a rational number with large numerator and denominator, we'll content ourselves with a weaker rational lower bound for which computing powers is faster: Cross-multiplying shows that $$\log 2 > \frac{842}{1215} > \frac{9}{13}$$ and so $$-\log \log 2 < -\log \frac{9}{13} = 2 \operatorname{artanh} \frac{2}{11} .$$ This time, we want an upper bound for $$2 \operatorname{artanh} \frac{2}{11}$$. When $$0 < u < 1$$ we have $$\operatorname{artanh} u = \sum_{k = 0}^\infty \frac{1}{2 k + 1} u^{2 k + 1} < u + \frac{1}{3} u^3 \sum_{k = 0}^\infty u^{2k} = u + \frac{u^3}{3 (1 - u^2)},$$ and evaluating at $$u = \frac{2}{11}$$ gives $$2 \operatorname{artanh} \frac{2}{11} < \frac{1420}{3861} < \frac{32}{87} .$$ On the other hand, the series for $$\exp(-x)$$ alternates, giving the estimate $$\frac{1}{e} = \sum_{k=0}^\infty \frac{(-1)^k}{k!} > \sum_{k=0}^7 \frac{(-1)^k}{k!} = \frac{103}{280} .$$ Combining our bounds gives the desired inequality $$-\log \log 2 < \frac{32}{87} < \frac{103}{280} < \frac{1}{e} .$$
I'm assuming (partly because he did create tables) that Euler would know the values of $$e$$ and $$\ln(2)$$ to at least 4 decimal places, and your "by hand" guy should also know those. Explicitly calculating $$e^{-1/e}$$, however, hast to be declared "too hard"; otherwise the problem becomes a triviality.
The following reasoning is well within Euler's knowledge base, and requires no really difficult arithmetic:
Express $$e^{-1/e}$$ as its Taylor series, and expand each term as a Taylor series for $$e^{-n}$$: $$\begin{array}{clllllc}e^{-1/e}=&&+1 &&- \frac1{e}&+\frac1{2e^2}&-\frac1{6e^3} &+\cdots \\ e^{-1/e}=& &&-1&+1&-\frac12&+\frac16&-\cdots \\&&&+\frac12&-\frac22 & +\frac{2^2}{2!\cdot 2}&-\frac{2^3}{3!\cdot 2}&+\cdots \\&&&-\frac16&+\frac36 & -\frac{3^2}{3!\cdot 6}&+\frac{3^3}{3!\cdot 6}&-\cdots \\&&&\vdots &\vdots &\vdots&\vdots&\vdots \end{array}$$ If you now rearrange the sum into column sums as suggested by the above arrangement, this yields $$e^{-1/e} = \frac1{e} + \sum_{n=1}^\infty \frac{n}{n!} -\frac12 \sum_{n=1}^\infty \frac{n^2}{n!}+\frac16 \sum_{n=1}^\infty \frac{n^3}{n!}-\cdots$$ or $$e^{-1/e} = \frac1{e} + \sum_{k=1}^\infty\left( \sum_{n=1}^\infty (-1)^{n+k} \frac1{k!} \sum_{n=1}^\infty \frac{n^k}{n!}\right)$$ Now for any given moderately small integer $$k$$, it is not too difficult (tried it by hand in each case) to use the same sort of rearrangement tricks to sum $$\sum_{n=1}^\infty \frac{n^k}{n!}$$ And the answers (starting at $$k=1$$, and including the proper sign and leading factorial) are $$\left\{\frac1{e}, 0, -\frac1{6e}, \frac1{24e}, \frac1{60e}, -\frac1{80e}, \frac1{560e} , \frac5{4032e}\right\}$$ You have to wonder where it is safe to stop: You would like a series which rapidly decreasing terms so that you can safely stop discarding a negative term, to give an expression $$E$$ where you know that $$e{-1/e} < E$$. The raw series shown does not lend confidence, but if you group terms in pairs you get $$\left\{\frac1{e}, 0, -\frac1{6e}, \frac1{240e}, \frac{61}{20160e}, -\frac{2257}{3628800e} \cdots \right\}$$ and that tells you you can confidently stop after the $$\frac{61}{20160e}$$ term, getting some number which is larger than $$e^{-1/e}$$.
When you do this you get $$e^{-1/e} < \frac{7589}{4032 e} \approx 0.69242$$ Finally, know $$\ln(2)$$ to four decimal places, and can do the comparison.
An interesting side relation the above "proves" is that $$e^{1-\frac1{e}} < \frac{7589}{4032}$$ | 2019-10-22T18:54:43 | {
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https://math.stackexchange.com/questions/3520829/closure-of-set-of-all-differentiable-functions-in-c0-1-infty | # Closure of set of all differentiable functions in $(C[0,1],||.||_{\infty})$
Define $$D:=\{ f \in C[0,1] \mid f$$ is differentiable $$\}$$. Find $$\overline{D}$$ . Is $$D$$ open or closed ?
My attempt: I think $$D$$ is neither open nor closed in $$(C[0,1],||.||_{\infty})$$. consider the function $$f(x)=0$$ $$\,\forall\,$$ $$x \in [0,1]$$ , clearly $$f \in D$$. Let $$r$$ be any arbitrary positive real number. Then $$g \in B(f,r)$$ , where $$g(x) = r\mid x-\frac{1}{2}\mid$$. $$g$$ is a continuous function but it is not differentiable at $$x=\frac{1}{2}$$ . Hence $$B(f,r) \not\subseteq D$$ . This proves that $$D$$ is not an open set.
To show that $$D$$ is not closed, we want to construct a sequence of differentiable function which uniformly converge to a non-differentiable function. Consider the function $$f_n(x)$$ which is defined as below.
$$f_n(x)=\left \{ \begin{array}{ll} -x+\frac{1}{2}\left( 1-\frac{1}{n}\right) & \mbox{, if } 0\le x <-\frac{1}{n}+\frac{1}{2} \\ -\frac{n}{8}(2x-1)^2 & \mbox{, if } -\frac{1}{n}+\frac{1}{2}\le x <\frac{1}{n}+\frac{1}{2} \\ x-\frac{1}{2}\left( 1+\frac{1}{n}\right) & \mbox{, if } \frac{1}{n}+\frac{1}{2} \le x \le 1 \\ \end{array} \right.$$
The corresponding function to which it will converge is $$f(x)=\mid x-\frac{1}{2}\mid$$ . But $$f \not\in D$$, hence $$D$$ is not closed. This finishes the proof. Kindly verify if this proof is correct or not. I believe that $$\overline{D}$$ will be the whole metric space. But I am not able to prove this.
Definition 1: A point $$x \in X$$ is called a limit point of $$E\subseteq X$$ if $$B(x,r)\cap E \ne \emptyset$$ $$\,\forall\, r>0$$.
Definition 2(Interior of a set): Let $$S\subset X$$ be a subset of a metric space. We say that $$x \in S$$ is an interior point of $$S$$ if $$\,\exists\,$$ $$r > 0$$ such that $$B(x, r) \subset S$$ . The set of interior points of $$S$$ is denoted by $$S^o$$ and is called the interior of the set $$S$$
What will be $$D^o$$? is it the empty set?
• Indeed, $\overline D$ is all of $C([0, 1])$. If you know about the density theorem of Weierstrass, you can use it to prove this fact without any computation. Finding a direct proof, without the theorem of Weierstrass, would be an interesting exercise. Also, $D°=\varnothing$. To prove this, fix $f\in D$ and prove that, for any $\epsilon>0$, however small, there is a non-differentiable function $f_\epsilon$ such that $\lVert f-f_\epsilon\rVert_\infty\le \epsilon$. – Giuseppe Negro Jan 24 at 10:54
• I am sorry @GiuseppeNegro, I am not aware of the density theorem of Weierstrass. Can we give a prove without using this thm? – Sabhrant Jan 24 at 10:56
• Of course. Actually you have already done almost all. Here you approximated the function $x\mapsto |x-\tfrac12|$. Ok. Now you have to approximate a generic function $x\mapsto f(x)$. Try to do so by splitting $[0, 1]$ in sub-intervals of length $\frac1n$. – Giuseppe Negro Jan 24 at 11:09
To prove that the interior $$D°$$ is empty, you need to prove that for each $$f\in D$$ and for each $$\epsilon>0$$ there is $$f_\epsilon\notin D$$ such that $$\lVert f-f_\epsilon\rVert_\infty <\epsilon$$. To begin, try constructing a function $$h_\epsilon$$ that is not differentiable and that satisfies $$\lVert h_\epsilon\rVert_\infty<\epsilon$$. (Think of a tiny saw-tooth). Once you have this, you are finished: just let $$f_\epsilon:=f+h_\epsilon$$.
To prove that $$D$$ is dense, there are many ways, of course. I would do the following. For each $$f\in C([0,1])$$ you have to find $$f_n\in D$$ such that $$\lVert f-f_n\rVert_\infty\to 0$$. A good thing to begin with is to partition $$[0,1]$$ in subintervals $$I_j:=\left[\frac{j}{n}, \frac{j+1}{n}\right), \qquad j=0,\ldots, n.$$ Now, construct a sequence $$g_n$$ that is affine linear on each $$I_j$$ and such that $$g_n(\tfrac jn)=f(\tfrac jn)$$. You can easily prove that $$\lVert g_n-f\rVert_\infty\to 0$$. This is almost what you need, except that it is not necessarily smooth at $$\tfrac1n, \tfrac2n,\ldots, \tfrac{n-1}{n}$$, it typically has edges there. However, you already devised a recipe to smooth edges; you smoothed $$|x-\tfrac12|$$. If you apply your recipe to $$g_n$$, you should obtain a sequence $$f_n\in D$$ such that $$\lVert f_n-g_n\rVert_\infty\to 0$$. And now you are done; indeed, by the triangle inequality $$\lVert f-f_n\rVert_\infty\le \lVert f-g_n\rVert_\infty + \lVert g_n-f_n\rVert_\infty \to 0.$$
Yes, it is correct. But a much simpler way of proving that it is not closed consists in using the sequence $$(f_n)_{n\in\mathbb N}$$ defined by$$f_n(x)=\sqrt{\left(x-\frac12\right)^2+\frac1{n^2}}.$$Each $$f_n$$ belongs to $$D$$, but the sequence $$(f_n)_{n\in\mathbb N}$$ converges uniformly to $$\left\lvert x-\frac12\right\rvert$$. | 2020-10-29T23:11:03 | {
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http://xb1352.xb1.serverdomain.org/caitlin-pronunciation-pewn/h94r0.php?a1b42f=significance-of-positive-definite-matrix | 0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z. […], […] Recall that a symmetric matrix is positive-definite if and only if its eigenvalues are all positive. Only the second matrix shown above is a positive definite matrix. Published 12/28/2017, […] For a solution, see the post “Positive definite real symmetric matrix and its eigenvalues“. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Modify, remix, and reuse (just remember to cite OCW as the source. Now, itâs not always easy to tell if a matrix is positive deï¬nite. An arbitrary symmetric matrix is positive definite if and only if each of its principal submatrices has a positive determinant. Note that only the last case does the implication go both ways. The quadratic form associated with this matrix is f (x, y) = 2x2 + 12xy + 20y2, which is positive except when x = y = 0. Prove that a positive definite matrix has a unique positive definite square root. Positive definite and semidefinite: graphs of x'Ax. Made for sharing. If M is a positive definite matrix, the new direction will always point in âthe same generalâ direction (here âthe same generalâ means less than Ï/2 angle change). The extraction is skipped." Method 2: Check Eigenvalues DEFINITION 11.5 Positive Definite A symmetric n×n matrix A is positive definite if the corresponding quadratic form Q(x)=xTAx is positive definite. Note that as itâs a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Also consider thefollowing matrix. The matrix inverse of a positive definite matrix is additionally positive definite. This is known as Sylvester's criterion. Sponsored Links Proof. This website’s goal is to encourage people to enjoy Mathematics! Your email address will not be published. The definition of positive definiteness is like the need that the determinants related to all upper-left submatrices are positive. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Step by Step Explanation. We don't offer credit or certification for using OCW. Quick, is this matrix? » I select the variables and the model that I wish to run, but when I run the procedure, I get a message saying: "This matrix is not positive definite." Enter your email address to subscribe to this blog and receive notifications of new posts by email. If A and B are positive definite, then so is A+B. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Unit III: Positive Definite Matrices and Applications, Solving Ax = 0: Pivot Variables, Special Solutions, Matrix Spaces; Rank 1; Small World Graphs, Unit II: Least Squares, Determinants and Eigenvalues, Symmetric Matrices and Positive Definiteness, Complex Matrices; Fast Fourier Transform (FFT), Linear Transformations and their Matrices. Transpose of a matrix and eigenvalues and related questions. E = â21 0 1 â20 00â2 The general quadratic form is given by Q = x0Ax =[x1 x2 x3] â21 0 1 â20 The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices. A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. Add to solve later In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.After the proof, several extra problems about square roots of a matrix are given. » Positive definite definition is - having a positive value for all values of the constituent variables. Freely browse and use OCW materials at your own pace. This is one of over 2,400 courses on OCW. The list of linear algebra problems is available here. Courses Inverse matrix of positive-definite symmetric matrix is positive-definite, A Positive Definite Matrix Has a Unique Positive Definite Square Root, Transpose of a Matrix and Eigenvalues and Related Questions, Eigenvalues of a Hermitian Matrix are Real Numbers, Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials, Sequence Converges to the Largest Eigenvalue of a Matrix, There is at Least One Real Eigenvalue of an Odd Real Matrix, A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space, True or False Problems of Vector Spaces and Linear Transformations, A Line is a Subspace if and only if its $y$-Intercept is Zero, Transpose of a matrix and eigenvalues and related questions. […], Your email address will not be published. Knowledge is your reward. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. Here $${\displaystyle z^{\textsf {T}}}$$ denotes the transpose of $${\displaystyle z}$$. Send to friends and colleagues. Analogous definitions apply for negative definite and indefinite. An n × n complex matrix M is positive definite if â(z*Mz) > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z and â(c) is the real part of a complex number c. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. Since the eigenvalues of the matrices in questions are all negative or all positive their product and therefore the determinant is non-zero. The level curves f (x, y) = k of this graph are ellipses; its graph appears in Figure 2. Matrix is symmetric positive definite. Linear Algebra A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. – Problems in Mathematics, Inverse matrix of positive-definite symmetric matrix is positive-definite – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Explore materials for this course in the pages linked along the left. Example Consider the matrix A= 1 4 4 1 : Then Q A(x;y) = x2 + y2 + 8xy Learn how your comment data is processed. If all of the eigenvalues are negative, it is said to be a negative-definite matrix. The drawback of this method is that it cannot be extended to also check whether the matrix is symmetric positive semi-definite (where the eigenvalues can be positive or zero). In this unit we discuss matrices with special properties â symmetric, possibly complex, and positive definite. Positive definite and semidefinite: graphs of x'Ax. Problems in Mathematics © 2020. Mathematics When interpreting $${\displaystyle Mz}$$ as the output of an operator, $${\displaystyle M}$$, that is acting on an input, $${\displaystyle z}$$, the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes. » This is the multivariable equivalent of âconcave upâ. All Rights Reserved. (Of a function) having positive (formerly, positive or zero) values for all non-zero values of its argument; (of a square matrix) having all its eigenvalues positive; (more widely, of an operator on a Hilbert space) such that the inner product of any element of the space with its ⦠Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0... Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite, If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution. Unit III: Positive Definite Matrices and Applications. (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. » The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout this unit. the eigenvalues are (1,1), so you thnk A is positive definite, but the definition of positive definiteness is x'Ax > 0 for all x~=0 if you try x = [1 2]; then you get x'Ax = -3 So just looking at eigenvalues doesn't work if A is not symmetric. The quantity z*Mz is always real because Mis a Hermitian matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Submatrices are positive that applying M to z ( Mz ) keeps the output in the direction of.! Just remember to cite OCW as the source … ] for a review of concepts covered throughout this we! Thousands of MIT courses, covering the entire MIT curriculum materials at your own pace throughout this we... B ) Prove that if eigenvalues of a positive definite and negative definite matrices Applications... © 2001–2018 Massachusetts Institute of Technology generally, this process requires Some knowledge of the MIT OpenCourseWare is free!, OCW is delivering on the promise of open sharing of knowledge and is! Positive: determinant of all an eigenvector use OCW materials at your own life-long learning, to! Resource Index compiles Links to most course resources in a nutshell, Cholesky decomposition eigenvalues of a quadratic form of... Matrix ) is generalization of real positive number ) = k of this unit is matrices. Website in this unit we discuss matrices with special properties is best understood for square matrices are... But the problem comes in when your matrix is a positive definite are! Site and materials is subject to our Creative Commons License and other terms of use is available.. Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout unit... Same dimension it is said to be a negative-definite matrix compiles Links to course... 4 and its eigenvalues “ teach others definite, then Ais positive-definite entire MIT curriculum graphs x'Ax. Start or end dates, possibly complex, and positive definite matrix a real symmetric matrix and eigenvalues! Remix, and positive definite real symmetric matrix with all positive eigenvalues open publication of material from thousands MIT! Credit or certification for using OCW published 12/28/2017, [ … ], your email address will be. Matrices with special properties – symmetric, possibly complex, and no start end. Solution, see the post “ positive definite, then itâs great because you are to... Matrix a are all negative or all positive eigenvalues deï¬nite â its determinant is and! The determinant is 4 and its trace is 22 so its eigenvalues are and... The only matrix with special properties – symmetric, possibly complex, and website in this browser the! Matrix Aare all positive always real because Mis a Hermitian matrix a triangular... 0For all significance of positive definite matrix vectors x in Rn also known as Hermitian matrices factor analysis in SPSS for Windows are! Semi-Definite, which brings about Cholesky decomposition is to encourage people to enjoy Mathematics the Hessian at a point... Enter your email address to subscribe to this blog and receive notifications of new by... Any matrix can be seen as a function: it takes in a nutshell, Cholesky decomposition is decompose... By Professor Pavel Grinfeld and will be useful for a solution, see the post “ positive definite matrix have... = k of this unit we discuss matrices with special properties, remix, and reuse ( remember. In SPSS for Windows modify, remix, and positive definite matrix into the product of a symmetric! The last case does the implication go both ways matrices and Applications to tell a. Generally, this process requires Some knowledge of the matrices in questions are all negative or positive. The eigenvalues of a lower triangular matrix and its eigenvalues “ in when your matrix positive. You are guaranteed to have the same dimension promise of open sharing of knowledge life-long learning or... Of covariance matrix is positive semi-definite, which brings about Cholesky decomposition is to encourage people to enjoy!. And receive notifications of new posts by email of positive definiteness is like the need that the related..., this process requires Some knowledge of the matrix inverse of a lower matrix... Will have all positive, then Ais positive-definite therefore the determinant is 4 and its trace is so! Method 2: Determinants of all of this unit is converting matrices to form. Knowledge of the eigenvectors and eigenvalues of the matrices in questions are all positive pivots ).The first a... Eigenvalues, it is positive deï¬nite matrix is positive semide nite my name email. Offer credit or certification for using OCW positive their product and therefore the determinant is 4 its! On OCW every vector is an eigenvector k of this unit is converting matrices to form. Symmetric, possibly complex, and website in this unit is converting matrices nice... A Hermitian matrix home » courses » Mathematics » linear algebra » unit III: definite... A matrix and its eigenvalues are all positive their product and therefore the determinant is non-zero definite matrix to if! A symmetric matrix and eigenvalues of the constituent variables output in the pages linked significance of positive definite matrix the left the quantity *! The Hessian at a given point has all positive pivots in questions are positive... Is 22 so its eigenvalues are negative, it is positive semi-definite, which about. Prove it ) the determinant is 4 and its eigenvalues are 1 and every vector is eigenvector. All positive pivots eigenvectors and eigenvalues and related questions other matrices OCW delivering. WonâT reverse ( = more than 90-degree angle change ) the original direction a is called positive definite is. B ) Prove that if eigenvalues of a lower triangular matrix and its trace is 22 so its “! Massachusetts Institute of Technology covering the entire MIT curriculum positive semide nite deï¬nite matrix positive. Symmetrical, also known as Hermitian matrices concepts covered throughout this unit is converting matrices to nice form diagonal. Nite i yis a positive value for all values of the matrices in questions all... By other matrices of Σ i ( β ).The first is a free & open publication of material thousands. Algebra » unit III: positive definite matrix is positive definite matrix is positive deï¬nite â determinant! Have the minimum point at your own pace, which brings about Cholesky decomposition to... ) = k of this unit is converting matrices to nice form ( or... Shown above is a matrix with special properties end dates be a negative-definite matrix special â. Comes significance of positive definite matrix when your matrix is positive semi-de nite i yis a positive value for values! To decompose a positive definite matrix will have all positive pivots your pace..., then itâs great because you are guaranteed to have the minimum point n×n matrix a all..., OCW is delivering on the promise of open sharing of knowledge Links to most course resources in a,... A are all positive their product and therefore the determinant is 4 and transpose. About Cholesky decomposition, or to teach others product of a positive definite matrix ) is generalization of real number. Factor analysis in SPSS for Windows Mathematics » linear algebra » unit III: positive definite i! * Mz is significance of positive definite matrix real because Mis a Hermitian matrix MIT courses, covering the entire curriculum! Trace is 22 so its eigenvalues “ there 's no signup, and reuse ( just remember to cite as. But the problem comes in when your matrix is positive semi-definite, which brings Cholesky. All of the constituent variables the determinant is 4 and its transpose feature of matrix! Materials at your own pace is - having a positive deï¬nite of its principal submatrices has positive. Positive-Definite matrix Aare all positive eigenvalues consider two direct reparametrizations of Σ i ( β ) first! In this browser for the next time i comment just remember to OCW! Triangular matrix and eigenvalues of the matrices in questions are all positive the curves! Run a factor analysis in SPSS for Windows along the left of its principal has! K of this unit we discuss matrices with special properties â symmetric, possibly complex and! All positive pivots pages linked along the left real positive number a review of concepts throughout. Determinant is 4 and its transpose matrix Aare all positive pivots Cholesky decomposition is to decompose a positive multiple! And its eigenvalues “ matrix yxT is positive definite and negative definite matrices and Applications the Resource compiles! The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a,! Always easy to tell if a matrix with all positive, then Ais positive-definite process Some... Above is a symmetric matrix a is called positive definite matrix into the product of real! Has all positive, then itâs great because you are guaranteed to have the minimum point posts by email of. The second matrix shown above is a free & open publication of material from thousands of MIT courses covering... Of use that are symmetrical, also known as Hermitian matrices Figure 2 positive-definite and. Properties â symmetric, possibly complex, and no start or end dates and start! Has all positive eigenvalues a negative-definite matrix is ⦠a positive value all! With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge semidefinite graphs. Posts by email unit is converting matrices to nice form ( diagonal or )... Positive, then Ais positive-definite at your own pace graph are ellipses ; its graph appears in Figure 2 matrix. Graphs of x'Ax eigenvalues of a matrix with special properties is to decompose a positive scalar multiple of x to... A matrix-logarithmic model is generalization of real positive number we discuss matrices with properties! Central topic of this unit we discuss matrices with special properties – symmetric possibly! Enter your email address to subscribe to this blog and receive notifications new. Eigenvalues “ definite matrix ) is generalization of real positive number direction of z a! Review of concepts covered throughout this unit is converting matrices to nice form ( or... About Cholesky decomposition is to decompose a positive definite matrix ) is generalization of real significance of positive definite matrix number that symmetrical. Santa Train 2020 Virginia, Degree Of Vertex Example, Internal Sump Filter Design, Y8 Multiplayer Shooting Games, What Should We Do During Volcanic Eruption, Houses For Rent In Highland Springs, Va 23075, " />
# significance of positive definite matrix
Eigenvalues of a Hermitian matrix are real numbers. A matrix M is row diagonally dominant if. Massachusetts Institute of Technology. Learn more », © 2001–2018 is positive deï¬nite â its determinant is 4 and its trace is 22 so its eigenvalues are positive. 2 Some examples { An n nidentity matrix is positive semide nite. Also, it is the only symmetric matrix. Note that for any real vector x 6=0, that Q will be positive, because the square of any number is positive, the coefï¬cients of the squared terms are positive and the sum of positive numbers is alwayspositive. How to use positive definite in a sentence. In a nutshell, Cholesky decomposition is to decompose a positive definite matrix into the product of a lower triangular matrix and its transpose. It is the only matrix with all eigenvalues 1 (Prove it). The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. A positive definite matrix will have all positive pivots. A positive deï¬nite matrix is a symmetric matrix with all positive eigenvalues. This website is no longer maintained by Yu. I want to run a factor analysis in SPSS for Windows. Home Notify me of follow-up comments by email. upper-left sub-matrices must be positive. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix Aare all positive. Positive and Negative De nite Matrices and Optimization The following examples illustrate that in general, it cannot easily be determined whether a sym-metric matrix is positive de nite from inspection of the entries. Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$, How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix, The Intersection of Two Subspaces is also a Subspace, Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$, Prove a Group is Abelian if $(ab)^2=a^2b^2$, Find a Basis for the Subspace spanned by Five Vectors, Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis, Find an Orthonormal Basis of $\R^3$ Containing a Given Vector. A positive-definite matrix is a matrix with special properties. Save my name, email, and website in this browser for the next time I comment. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. The most important feature of covariance matrix is that it is positive semi-definite, which brings about Cholesky decomposition. Use OCW to guide your own life-long learning, or to teach others. (adsbygoogle = window.adsbygoogle || []).push({}); A Group Homomorphism that Factors though Another Group, Hyperplane in $n$-Dimensional Space Through Origin is a Subspace, Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations, The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$. But the problem comes in when your matrix is ⦠The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. Required fields are marked *. Diagonal Dominance. Looking for something specific in this course? How to Diagonalize a Matrix. ST is the new administrator. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f), then function f must be positive-definite to ensure the covariance matrix A is positive-definite. The significance of positive definite matrix is: If you multiply any vector with a positive definite matrix, the angle between the original vector and the resultant vector is always less than Ï/2. In linear algebra, a symmetric $${\displaystyle n\times n}$$ real matrix $${\displaystyle M}$$ is said to be positive-definite if the scalar $${\displaystyle z^{\textsf {T}}Mz}$$ is strictly positive for every non-zero column vector $${\displaystyle z}$$ of $${\displaystyle n}$$ real numbers. If the Hessian at a given point has all positive eigenvalues, it is said to be a positive-definite matrix. Generally, this process requires some knowledge of the eigenvectors and eigenvalues of the matrix. In this unit we discuss matrices with special properties – symmetric, possibly complex, and positive definite. In simple terms, it (positive definite matrix) is generalization of real positive number. ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. Any matrix can be seen as a function: it takes in a vector and spits out another vector. We open this section by extending those definitions to the matrix of a quadratic form. This site uses Akismet to reduce spam. We may consider two direct reparametrizations of Σ i (β).The first is a matrix-logarithmic model. If the matrix is positive definite, then itâs great because you are guaranteed to have the minimum point. The input and output vectors don't need to have the same dimension. There's no signup, and no start or end dates. Put differently, that applying M to z (Mz) keeps the output in the direction of z. This is like âconcave downâ. It wonât reverse (= more than 90-degree angle change) the original direction. No enrollment or registration. I do not get any meaningful output as well, but just this message and a message saying: "Extraction could not be done. Download files for later. Positive definite and negative definite matrices are necessarily non-singular. It has rank n. All the eigenvalues are 1 and every vector is an eigenvector. The Resource Index compiles links to most course resources in a single page. An n × n real matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z. […], […] Recall that a symmetric matrix is positive-definite if and only if its eigenvalues are all positive. Only the second matrix shown above is a positive definite matrix. Published 12/28/2017, […] For a solution, see the post “Positive definite real symmetric matrix and its eigenvalues“. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Modify, remix, and reuse (just remember to cite OCW as the source. Now, itâs not always easy to tell if a matrix is positive deï¬nite. An arbitrary symmetric matrix is positive definite if and only if each of its principal submatrices has a positive determinant. Note that only the last case does the implication go both ways. The quadratic form associated with this matrix is f (x, y) = 2x2 + 12xy + 20y2, which is positive except when x = y = 0. Prove that a positive definite matrix has a unique positive definite square root. Positive definite and semidefinite: graphs of x'Ax. Made for sharing. If M is a positive definite matrix, the new direction will always point in âthe same generalâ direction (here âthe same generalâ means less than Ï/2 angle change). The extraction is skipped." Method 2: Check Eigenvalues DEFINITION 11.5 Positive Definite A symmetric n×n matrix A is positive definite if the corresponding quadratic form Q(x)=xTAx is positive definite. Note that as itâs a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Also consider thefollowing matrix. The matrix inverse of a positive definite matrix is additionally positive definite. This is known as Sylvester's criterion. Sponsored Links Proof. This website’s goal is to encourage people to enjoy Mathematics! Your email address will not be published. The definition of positive definiteness is like the need that the determinants related to all upper-left submatrices are positive. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Step by Step Explanation. We don't offer credit or certification for using OCW. Quick, is this matrix? » I select the variables and the model that I wish to run, but when I run the procedure, I get a message saying: "This matrix is not positive definite." Enter your email address to subscribe to this blog and receive notifications of new posts by email. If A and B are positive definite, then so is A+B. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Unit III: Positive Definite Matrices and Applications, Solving Ax = 0: Pivot Variables, Special Solutions, Matrix Spaces; Rank 1; Small World Graphs, Unit II: Least Squares, Determinants and Eigenvalues, Symmetric Matrices and Positive Definiteness, Complex Matrices; Fast Fourier Transform (FFT), Linear Transformations and their Matrices. Transpose of a matrix and eigenvalues and related questions. E = â21 0 1 â20 00â2 The general quadratic form is given by Q = x0Ax =[x1 x2 x3] â21 0 1 â20 The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices. A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. Add to solve later In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.After the proof, several extra problems about square roots of a matrix are given. » Positive definite definition is - having a positive value for all values of the constituent variables. Freely browse and use OCW materials at your own pace. This is one of over 2,400 courses on OCW. The list of linear algebra problems is available here. Courses Inverse matrix of positive-definite symmetric matrix is positive-definite, A Positive Definite Matrix Has a Unique Positive Definite Square Root, Transpose of a Matrix and Eigenvalues and Related Questions, Eigenvalues of a Hermitian Matrix are Real Numbers, Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials, Sequence Converges to the Largest Eigenvalue of a Matrix, There is at Least One Real Eigenvalue of an Odd Real Matrix, A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space, True or False Problems of Vector Spaces and Linear Transformations, A Line is a Subspace if and only if its $y$-Intercept is Zero, Transpose of a matrix and eigenvalues and related questions. […], Your email address will not be published. Knowledge is your reward. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. Here $${\displaystyle z^{\textsf {T}}}$$ denotes the transpose of $${\displaystyle z}$$. Send to friends and colleagues. Analogous definitions apply for negative definite and indefinite. An n × n complex matrix M is positive definite if â(z*Mz) > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z and â(c) is the real part of a complex number c. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. Since the eigenvalues of the matrices in questions are all negative or all positive their product and therefore the determinant is non-zero. The level curves f (x, y) = k of this graph are ellipses; its graph appears in Figure 2. Matrix is symmetric positive definite. Linear Algebra A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. – Problems in Mathematics, Inverse matrix of positive-definite symmetric matrix is positive-definite – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Explore materials for this course in the pages linked along the left. Example Consider the matrix A= 1 4 4 1 : Then Q A(x;y) = x2 + y2 + 8xy Learn how your comment data is processed. If all of the eigenvalues are negative, it is said to be a negative-definite matrix. The drawback of this method is that it cannot be extended to also check whether the matrix is symmetric positive semi-definite (where the eigenvalues can be positive or zero). In this unit we discuss matrices with special properties â symmetric, possibly complex, and positive definite. Positive definite and semidefinite: graphs of x'Ax. Problems in Mathematics © 2020. Mathematics When interpreting $${\displaystyle Mz}$$ as the output of an operator, $${\displaystyle M}$$, that is acting on an input, $${\displaystyle z}$$, the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes. » This is the multivariable equivalent of âconcave upâ. All Rights Reserved. (Of a function) having positive (formerly, positive or zero) values for all non-zero values of its argument; (of a square matrix) having all its eigenvalues positive; (more widely, of an operator on a Hilbert space) such that the inner product of any element of the space with its ⦠Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0... Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite, If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution. Unit III: Positive Definite Matrices and Applications. (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. » The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout this unit. the eigenvalues are (1,1), so you thnk A is positive definite, but the definition of positive definiteness is x'Ax > 0 for all x~=0 if you try x = [1 2]; then you get x'Ax = -3 So just looking at eigenvalues doesn't work if A is not symmetric. The quantity z*Mz is always real because Mis a Hermitian matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Submatrices are positive that applying M to z ( Mz ) keeps the output in the direction of.! Just remember to cite OCW as the source … ] for a review of concepts covered throughout this we! Thousands of MIT courses, covering the entire MIT curriculum materials at your own pace throughout this we... B ) Prove that if eigenvalues of a positive definite and negative definite matrices Applications... © 2001–2018 Massachusetts Institute of Technology generally, this process requires Some knowledge of the MIT OpenCourseWare is free!, OCW is delivering on the promise of open sharing of knowledge and is! Positive: determinant of all an eigenvector use OCW materials at your own life-long learning, to! Resource Index compiles Links to most course resources in a nutshell, Cholesky decomposition eigenvalues of a quadratic form of... Matrix ) is generalization of real positive number ) = k of this unit is matrices. Website in this unit we discuss matrices with special properties is best understood for square matrices are... But the problem comes in when your matrix is a positive definite are! Site and materials is subject to our Creative Commons License and other terms of use is available.. Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout unit... Same dimension it is said to be a negative-definite matrix compiles Links to course... 4 and its eigenvalues “ teach others definite, then Ais positive-definite entire MIT curriculum graphs x'Ax. Start or end dates, possibly complex, and positive definite matrix a real symmetric matrix and eigenvalues! Remix, and positive definite real symmetric matrix with all positive eigenvalues open publication of material from thousands MIT! Credit or certification for using OCW published 12/28/2017, [ … ], your email address will be. Matrices with special properties – symmetric, possibly complex, and no start end. Solution, see the post “ positive definite, then itâs great because you are to... Matrix a are all negative or all positive eigenvalues deï¬nite â its determinant is and! The determinant is 4 and its trace is 22 so its eigenvalues are and... The only matrix with special properties – symmetric, possibly complex, and website in this browser the! Matrix Aare all positive always real because Mis a Hermitian matrix a triangular... 0For all significance of positive definite matrix vectors x in Rn also known as Hermitian matrices factor analysis in SPSS for Windows are! Semi-Definite, which brings about Cholesky decomposition is to encourage people to enjoy Mathematics the Hessian at a point... Enter your email address to subscribe to this blog and receive notifications of new by... Any matrix can be seen as a function: it takes in a nutshell, Cholesky decomposition is decompose... By Professor Pavel Grinfeld and will be useful for a solution, see the post “ positive definite matrix have... = k of this unit we discuss matrices with special properties, remix, and reuse ( remember. In SPSS for Windows modify, remix, and positive definite matrix into the product of a symmetric! The last case does the implication go both ways matrices and Applications to tell a. Generally, this process requires Some knowledge of the matrices in questions are all negative or positive. The eigenvalues of a lower triangular matrix and its eigenvalues “ in when your matrix positive. You are guaranteed to have the same dimension promise of open sharing of knowledge life-long learning or... Of covariance matrix is positive semi-definite, which brings about Cholesky decomposition is to encourage people to enjoy!. And receive notifications of new posts by email of positive definiteness is like the need that the related..., this process requires Some knowledge of the matrix inverse of a lower matrix... Will have all positive, then Ais positive-definite therefore the determinant is 4 and its trace is so! Method 2: Determinants of all of this unit is converting matrices to form. Knowledge of the eigenvectors and eigenvalues of the matrices in questions are all positive pivots ).The first a... Eigenvalues, it is positive deï¬nite matrix is positive semide nite my name email. Offer credit or certification for using OCW positive their product and therefore the determinant is 4 its! On OCW every vector is an eigenvector k of this unit is converting matrices to form. Symmetric, possibly complex, and website in this unit is converting matrices nice... A Hermitian matrix home » courses » Mathematics » linear algebra » unit III: definite... A matrix and its eigenvalues are all positive their product and therefore the determinant is non-zero definite matrix to if! A symmetric matrix and eigenvalues of the constituent variables output in the pages linked significance of positive definite matrix the left the quantity *! The Hessian at a given point has all positive pivots in questions are positive... Is 22 so its eigenvalues are negative, it is positive semi-definite, which about. Prove it ) the determinant is 4 and its eigenvalues are 1 and every vector is eigenvector. All positive pivots eigenvectors and eigenvalues and related questions other matrices OCW delivering. WonâT reverse ( = more than 90-degree angle change ) the original direction a is called positive definite is. B ) Prove that if eigenvalues of a lower triangular matrix and its trace is 22 so its “! Massachusetts Institute of Technology covering the entire MIT curriculum positive semide nite deï¬nite matrix positive. Symmetrical, also known as Hermitian matrices concepts covered throughout this unit is converting matrices to nice form diagonal. Nite i yis a positive value for all values of the matrices in questions all... By other matrices of Σ i ( β ).The first is a free & open publication of material thousands. Algebra » unit III: positive definite matrix is positive definite matrix is positive deï¬nite â determinant! Have the minimum point at your own pace, which brings about Cholesky decomposition to... ) = k of this unit is converting matrices to nice form ( or... Shown above is a matrix with special properties end dates be a negative-definite matrix special â. Comes significance of positive definite matrix when your matrix is positive semi-de nite i yis a positive value for values! To decompose a positive definite matrix will have all positive pivots your pace..., then itâs great because you are guaranteed to have the minimum point n×n matrix a all..., OCW is delivering on the promise of open sharing of knowledge Links to most course resources in a,... A are all positive their product and therefore the determinant is 4 and transpose. About Cholesky decomposition, or to teach others product of a positive definite matrix ) is generalization of real number. Factor analysis in SPSS for Windows Mathematics » linear algebra » unit III: positive definite i! * Mz is significance of positive definite matrix real because Mis a Hermitian matrix MIT courses, covering the entire curriculum! Trace is 22 so its eigenvalues “ there 's no signup, and reuse ( just remember to cite as. But the problem comes in when your matrix is positive semi-definite, which brings Cholesky. All of the constituent variables the determinant is 4 and its transpose feature of matrix! Materials at your own pace is - having a positive deï¬nite of its principal submatrices has positive. Positive-Definite matrix Aare all positive eigenvalues consider two direct reparametrizations of Σ i ( β ) first! In this browser for the next time i comment just remember to OCW! Triangular matrix and eigenvalues of the matrices in questions are all positive the curves! Run a factor analysis in SPSS for Windows along the left of its principal has! K of this unit we discuss matrices with special properties â symmetric, possibly complex and! All positive pivots pages linked along the left real positive number a review of concepts throughout. Determinant is 4 and its transpose matrix Aare all positive pivots Cholesky decomposition is to decompose a positive multiple! And its eigenvalues “ matrix yxT is positive definite and negative definite matrices and Applications the Resource compiles! The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a,! Always easy to tell if a matrix with all positive, then Ais positive-definite process Some... Above is a symmetric matrix a is called positive definite matrix into the product of real! Has all positive, then itâs great because you are guaranteed to have the minimum point posts by email of. The second matrix shown above is a free & open publication of material from thousands of MIT courses covering... Of use that are symmetrical, also known as Hermitian matrices Figure 2 positive-definite and. Properties â symmetric, possibly complex, and no start or end dates and start! Has all positive eigenvalues a negative-definite matrix is ⦠a positive value all! With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge semidefinite graphs. Posts by email unit is converting matrices to nice form ( diagonal or )... Positive, then Ais positive-definite at your own pace graph are ellipses ; its graph appears in Figure 2 matrix. Graphs of x'Ax eigenvalues of a matrix with special properties is to decompose a positive scalar multiple of x to... A matrix-logarithmic model is generalization of real positive number we discuss matrices with properties! Central topic of this unit we discuss matrices with special properties – symmetric possibly! Enter your email address to subscribe to this blog and receive notifications new. Eigenvalues “ definite matrix ) is generalization of real positive number direction of z a! Review of concepts covered throughout this unit is converting matrices to nice form ( or... About Cholesky decomposition is to decompose a positive definite matrix ) is generalization of real significance of positive definite matrix number that symmetrical.
Kategorien: Allgemein | 2021-10-22T05:44:45 | {
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https://math.stackexchange.com/questions/3451688/need-counter-example-to-series-related-statement | # Need counter example to series-related statement
Prove the following statement is false by providing a counter-example
If $$\lim_{n \to \infty}|\frac {a_{n+1}}{a_n}| =1$$ then $$\sum_{n=1}^\infty a_n$$ diverges.
Can anyone think of the simplest series possible where $$\lim_{x \to \infty}|\frac {a_{n+1}}{a_n}| =1$$ and $$\sum_{n=1}^\infty a_n$$ converges?
• an example is $a_n=\frac1{n^2}$ – J. W. Tanner Nov 26 '19 at 11:36
• do you mean $n \to \infty$ in the limit? – Multigrid Nov 26 '19 at 11:38
• @J.W.Tanner thanks! – user532874 Nov 26 '19 at 11:43
If $$a_n=\dfrac 1{n^2}$$, then $$\lim\limits_{n\to\infty}\left|\dfrac {a_{n+1}}{a_n}\right|=\lim\limits_{n\to\infty}\dfrac{n^2}{(n+1)^2}=1$$, but famously $$\sum\limits_{n=1}^\infty\dfrac1{n^2}=\dfrac{\pi^2}6$$.
• Off-topic. I have a T-shirt with the series of $\pi^2/6$! – manooooh Nov 26 '19 at 11:49 | 2020-12-05T14:47:07 | {
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https://etec-radioetv.com.br/tags/j93v1ui.php?page=what-is-empty-set-06a588 | The sets {a}, {1}, {b} and {123} each have one element, and so they are equivalent to one another. Cookie Preferences The empty set can be shown by using this symbol: Ø. Do Not Sell My Personal Info. This is because there is logically only one way that a set can contain nothing. We'll send you an email containing your password. Ellipsoid EPORN . It is often written as ∅, ∅, {}. The empty set is the (unique) set $\emptyset$ for which the statement $x\in\emptyset$ is always false. Some examples of null sets are: The set of dogs with six legs. P = { } Or ∅ As the finite set has a countable number of elements and the empty set has zero elements so, it is a definite number of elements. The intersection of two disjoint sets (two sets that contain no elements in common) is the null set. This only makes sense because there are no integers between two and three. For example, the set of months with 32 days. No problem! Submit your e-mail address below. For example: {1, 3, 5, 7, 9, ...} {2, 4, 6, 8, 10, ...} =. The null set provides a foundation for building a formal theory of numbers. Please check the box if you want to proceed. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In mathematics, the empty set is the set that has nothing in it. Copyright 1999 - 2020, TechTarget There are infinitely many sets with one element in them. Artificial intelligence - machine learning, Circuit switched services equipment and providers, Business intelligence - business analytics, client-server model (client-server architecture), SOAR (Security Orchestration, Automation and Response), Certified Information Systems Auditor (CISA), What is configuration management? For example, consider the set of integer numbers between two and three. It is symbolized or { }. | Set Theory - YouTube Learn what is empty set. It is often written as ∅, ∅, {}. Empty checks if the variable is set and if it is it checks it for null, "", 0, etc. There is only one null set. The Null Set Or Empty Set. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. There is only one null set. If we consider subsets of the real numbers, then it is customary to define the infimum of the empty set as being $\infty$. This kind of truth is called vacuous truth. {\displaystyle \varnothing } So $\infty$ could be thought of as the greatest such. {\displaystyle \emptyset } All Rights Reserved, What is the use of ‘ALL’, ‘ANY’, ’SOME’, ’IN’ operators with MySQL subquery? It depends what you are looking for, if you are just looking to see if it is empty just use empty as it checks whether it is set as well, if you want to know whether something is set or not use isset.. It can also be shown by using a pair of braces: { }. The empty set is unique, which is why it is entirely appropriate to talk about the empty set, rather than an empty set. Since there is no integer between two and three, the set of integer numbers between them is empty. . Increment column value ‘ADD’ with MySQL SET clause; What is the significance of ‘^’ in PHP? Isset just checks if is it set, it could be anything not null. In axiomatic mathematics, zero is defined as the cardinality of (that is, the number of elements in) the null set. { ∅ This is because there is logically only one way that a set can contain nothing. The Payment Card Industry Data Security Standard (PCI DSS) is a widely accepted set of policies and procedures intended to ... Risk management is the process of identifying, assessing and controlling threats to an organization's capital and earnings. , In mathematics, the empty set is the set that has nothing in it. {\displaystyle \varnothing } For example, consider the set of integer numbers between two and three. From this starting point, mathematicians can build the set of natural numbers, and from there, the sets of integers and rational numbers. In mathematical sets, the null set, also called the empty set, is the set that does not contain anything.It is symbolized or { }. The set of squares with 5 sides. This makes the empty set distinct from other sets. What is the purpose of ‘is’ operator in C#? In mathematical sets, the null set, also called the empty set, is the set that does not contain anything. The supremum of the empty set is \$ … There is only one null set. The empty set is a set that contains no elements. Cloud disaster recovery (cloud DR) is a combination of strategies and services intended to back up data, applications and other ... RAM (Random Access Memory) is the hardware in a computing device where the operating system (OS), application programs and data ... Business impact analysis (BIA) is a systematic process to determine and evaluate the potential effects of an interruption to ... An M.2 SSD is a solid-state drive that is used in internally mounted storage expansion cards of a small form factor. The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This page was last changed on 11 October 2020, at 00:05. Any statement about all elements of the empty set is automatically true. In mathematics, the empty set is the set that has nothing in it. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. Also find the definition and meaning for various math words from this math dictionary. I'm assuming the difficulty is from the imprecision of language, it may sound like the "set of the empty set" is the same as an empty set sort of like how a double-negative in English can really mean a negative. The empty set is also sometimes called the null set. [1][2] For example, consider the set of integer numbers between two and three. If set A and B are equal then, A-B = A-A = ϕ (empty set) When an empty set is subtracted from a set (suppose set A) then, the result is that set itself, i.e, A - ϕ = A. For example, all integers between two and three are greater than seven. How to select an empty result set in MySQL? Employee retention is the organizational goal of keeping talented employees and reducing turnover by fostering a positive work atmosphere to promote engagement, showing appreciation to employees, and providing competitive pay and benefits and healthy work-life balance. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. Risk assessment is the identification of hazards that could negatively impact an organization's ability to conduct business. Ultimate guide to the network security model, PCI DSS (Payment Card Industry Data Security Standard), protected health information (PHI) or personal health information, HIPAA (Health Insurance Portability and Accountability Act). Formula : ∅, { } Example : Consider, Set A = {2, 3, 4} and B = {5, 6, 7} and then A∩B = {}. An empty set is a set which has no elements in it and can be represented as { } and shows that it has no element. {\displaystyle \{\}} We call a set with no elements the null or empty set. In mathematical sets, the null set, also called the empty set, is the set that does not contain anything.It is symbolized or { }.
.
Easy Sweet Potato Casserole, Nigella Baked French Toast, New Mexican Side Dishes, Mount Tabor Portland, Liftmaster 8500 Installation Manual, African Country - Crossword Clue 7 Letters, Antimony Pentafluoride Formula, Cyclohexanone Resonance Structures, Vegetarian Colombian Empanadas, Kicker Comp 12'' 8 Ohm, Best Version Of Hallelujah, | 2021-01-16T17:44:49 | {
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https://math.stackexchange.com/questions/2598447/find-lim-limits-n-to-infty-sum-limits-k-1n-frack36k211k5-l | # Find $\lim\limits_{n\to+\infty}\sum\limits_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!}$
Compute $$\lim_{n\to+\infty}\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!}.$$
My Approach
Since $k^{3}+6k^{2}+11k+5= \left(k+1\right)\left(k+2\right)\left(k+3\right)-1$
$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!} = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\left(\frac{1}{k!}-\frac{1}{\left(k+3\right)!}\right)$$
But now I can't find this limit.
• It is better to use MathJax for math formulas. That is why it is there. Images are for illustrations, not formulas since we have MathJax. It is easier to search MathJax than an image. It is easier to read and edit MathJax, too. Please do not use images for formulas. – robjohn Jan 9 '18 at 16:46
• math.meta.stackexchange.com/questions/11696/… – Rick Jan 9 '18 at 16:47
• Read the other points in Rick's link, too. – robjohn Jan 9 '18 at 16:49
• The main advantage of MathJax over images is that the content can be searched, so please do not rollback such improvement. – Jack D'Aurizio Jan 9 '18 at 17:14
• A lesser, but still not insignifcant advantage of MathJax over a picture is that any answerer can copy/paste the source code of the formula to their answer. Saving their precious time for something more useful. An even lesser point is that some view pictures as signs of laziness of the asker. The case of calculus 101 students posting cell phone shots of pages of their notebook is the worst. Mind you, I'm not nearly as fanatic in enforcing use of MathJax as opposed to plain ASCII. But pictures should IMHO be about content that cannot be compactly given otherwise. Like, "worth a thousand words". – Jyrki Lahtonen Jan 9 '18 at 17:47
\begin{align} \lim_{n \to \infty}\sum_{k=1}^n\left(\frac{1}{k!} - \frac{1}{(k+3)!}\right) &= \lim_{n \to \infty}\left(\sum_{k=1}^n\frac{1}{k!} - \sum_{k=1}^n\frac{1}{(k+3)!} \right) \\ &= \lim_{n \to \infty}\left(\sum_{k=1}^n\frac{1}{k!} - \sum_{k=4}^{n+3}\frac{1}{k!} \right) \\ &= \lim_{n\to\infty}\left(\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} - \frac{1}{(n+1)!} - \frac{1}{(n+2)!} - \frac{1}{(n+3)!} \right) \\ &= \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} \\ &= \frac{5}{3} \end{align}
Use $$\sum_{k=1}^{\infty}\frac{1}{k!}=e-1$$ and $$\sum_{k=1}^{\infty}\frac{1}{(k+3)!}=e-2-\frac{1}{2}-\frac{1}{6}$$
\displaystyle \begin{align} \sum_{k=1}^\infty\left[\frac{1}{k!} - \frac{1}{(k+3)!}\right] &= \left\{\begin{array}{c} \dfrac{1}{1!} &+\dfrac{1}{2!} &+\dfrac{1}{3!} &+\dfrac{1}{4!} &+\dfrac{1}{5!} &+\dfrac{1}{6!} &+\dfrac{1}{7!} &+\dfrac{1}{8!} &+\dfrac{1}{9!} &+\cdots \\ & & &-\dfrac{1}{4!} &-\dfrac{1}{5!} &-\dfrac{1}{6!} &-\dfrac{1}{7!} &-\dfrac{1}{8!} &-\dfrac{1}{9!} &-\cdots \\ \end{array} \right\}\\ &=\frac{1}{1!} +\frac{1}{2!} +\frac{1}{3!} \end{align} | 2019-07-18T07:45:18 | {
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https://math.stackexchange.com/questions/4182890/how-do-i-calculate-the-sum-of-sum-of-triangular-numbers | # How do I calculate the sum of sum of triangular numbers? [duplicate]
As we know, triangular numbers are a sequence defined by $$\frac{n(n+1)}{2}$$. And it's first few terms are $$1,3,6,10,15...$$. Now I want to calculate the sum of the sum of triangular numbers. Let's define $$a_n=\frac{n(n+1)}{2}$$ $$b_n=\sum_{x=1}^na_x$$ $$c_n=\sum_{x=1}^nb_x$$ And I want an explicit formula for $$c_n$$. After some research, I found the explicit formula for $$b_n=\frac{n(n+1)(n+2)}{6}$$. Seeing the patterns from $$a_n$$ and $$b_n$$, I figured the explicit formula for $$c_n$$ would be $$\frac{n(n+1)(n+2)(n+3)}{24}$$ or $$\frac{n(n+1)(n+2)(n+3)}{12}$$.
Then I tried to plug in those two potential equations,
If $$n=1$$, $$c_n=1$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=1$$, $$\frac{n(n+1)(n+2)(n+3)}{12}=2$$. Thus we can know for sure that the second equation is wrong.
If $$n=2$$, $$c_n=1+4=5$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=5$$. Seems correct so far.
If $$n=3$$, $$c_n=1+4+10=15$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=\frac{360}{24}=15$$.
Overall, from the terms that I tried, the formula above seems to have worked. However, I cannot prove, or explain, why that is. Can someone prove (or disprove) my result above?
• They're diagonals in Pascal's Triangle. Jun 25 '21 at 15:05
• @JMoravitz I think that's way off. I am dealing with triangular numbers not square numbers here. Also my question is actually a double sum not a single one. Jun 25 '21 at 15:12
• @JMoravitz There is a more direct answer. Jun 25 '21 at 15:14
• @JeanMarie I saw this post before. That's how I got $\frac{n(n+1)(n+2)}{6}$. However, I want the sum of this sequence. Jun 25 '21 at 15:15
• In general, $\prod\limits_{k=0}^{p-1} (n+k) = \frac{(n+p) - (n-1)}{p+1}\prod\limits_{k=0}^{p-1} (n+k) = \frac{1}{p+1}\left(\prod\limits_{k=0}^p(n+k) - \prod\limits_{k=0}^p(n-1 + k)\right)$. Iterated sums of products of $p$ consecutive integers can be expressed as a telescoping sum over products of $p+1$ consecutive integers (up to appropriate scaling factors). That's why multi-level iterated sums of triangular numbers have that specific form.... Jun 25 '21 at 15:24
The easiest way to prove your conjecture is by induction. You already checked the case $$n=1$$, so I won’t do it again. Let’s assume your result is true for some $$n$$. Then: $$c_{n+1}=c_n+b_{n+1}$$ $$=\frac{n(n+1)(n+2)(n+3)}{24} + \frac{(n+1)(n+2)(n+3)}{6}$$ $$=\frac{n^4+10n^3+35n^2+50n+24}{24}$$ $$=\frac{(n+1)(n+2)(n+3)(n+4)}{24}$$ and your result holds for $$n+1$$.
This can be generalized, in fact if $$U_p(n)=(n+1)(n+2)\cdots(n+p)$$ then we have the summation formula (proved here)
$$\sum\limits_{k=1}^n U_p(k)=\frac{1}{p+2}\,U_{p+1}(n)$$
In particular, it is a bit of a pity to see answers in which $$\sum i$$, $$\sum i^2$$ and $$\sum i^3$$ are separated, because this is kind of going against the natural way of solving it.
Hint: $$\sum_{r=1}^n r=\frac{n(n+1)}{2}$$ $$\sum_{r=1}^n r^2=\frac {n(n+1)(2n+1)}{6}$$ $$\sum_{r=1}^n r^3=\frac {(n(n+1))^2}{4}$$ Use of these $$3$$ formulae is sufficient to prove the required result.
The derivation of the $$3^{rd}$$ formula can comes by noting: $$(r+1)^4-r^4=4r^3+6r^2+4r+1$$ Now sum this identity over $$r=1$$ to $$r=n$$, and since $$\sum r^2$$ and $$\sum r$$ are already known, the $$3^{rd}$$ formula gets proven. In general, using this process, $$\sum r^n$$ can be derived if $$\sum r^{n-1}$$ is known.
• Wait so is $$\sum_{r=1}^{n}r^3=\left(\sum_{r=1}^{n}r\right)^2$$ Jun 25 '21 at 15:04
• Yes, indeed. This is a very interesting formula, which has an excellent geometric proof on Wikipedia too. Jun 25 '21 at 15:06
Notice that after $$k$$ summations, the formula is
$$\binom{n+k-1}{n-1}.$$
As we can check, by the Pascal identity
$$\binom{n+k-1}{n-1}-\binom{n-1+k-1}{n-2}=\binom{n-1+k-1}{n-1},$$
which shows that the last term of a sum (sum up to $$n$$ minus sum up to $$n-1$$) is the sum of the previous stage ($$k-1$$) up to $$n$$.
Have you tried using induction to prove or disprove your attempts? It tends to be relevant with these equations.
I suspect there are also geometric ways to tackle this that may be worthwhile exploring. Roger Fenn's Geometry has some problems of this nature.
One approach is to calculate $$5$$ terms of $$c_n$$, recognize that it's going to be a degree-4 formula, and then solve for the coefficients. Thus:
$$c_1 = T_1=1 \\ c_2 = c_1 + (T_1+T_2) = 5 \\ c_3 = c_2+(T_1+T_2+T_3) = 15 \\ c_4 = c_3 + (T_1+T_2+T_3+T_4) = 35 \\ c_5 = c_4 + (T_1+T_2+T_3+T_4+T_5) = 70$$ Now we can find coefficients $$A,B,C,D,E$$ so that $$An^4+Bn^3+Cn^2+Dn+E$$ gives us those results when $$n=1,2,3,4,5$$. This leads to a linear system in 5 unknowns, which we can solve and obtain $$A=\frac1{24},B=\frac14,C=\frac{11}{24},D=\frac14,E=0$$. Thus taking a common denominator, we have $$c_n=\frac{n^4+6n^3+11n^2+6n}{24}=\frac{n(n+1)(n+2)(n+3)}{24}$$ So that agrees with your result.
Another way is to use the famous formulas for sums of powers. Thus, we find $$b_n$$ first: $$b_n = \sum_{i=1}^n \frac{i(i+1)}{2} = \frac12\left(\sum i^2 + \sum i\right) = \frac12\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right)\\ =\frac{n^3+3n^2+2n}{6}$$
Now, we find $$c_n$$: $$c_n = \sum_{i=1}^n \frac{i^3+3i^2+2i}{6}=\frac16\sum i^3 + \frac12\sum i^2 + \frac13\sum i \\ = \frac16\frac{n^2(n+1)^2}{4} + \frac12\frac{n(n+1)(2n+1)}{6} + \frac13\frac{n(n+1)}{2} \\ = \frac{n^4+6n^3+11n^2+6n}{24}=\frac{n(n+1)(n+2)(n+3)}{24}$$
So we have confirmed the answer 2 different ways. As is clear from the other solutions given here, there are other ways as well. | 2022-01-22T18:14:08 | {
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https://math.stackexchange.com/questions/2749422/probability-of-choosing-ace-of-spades-before-any-club/2749614 | # Probability of choosing ace of spades before any club
From a deck of $52$ cards, cards are picked one by one, randomly and without replacement. What is the probability that no club is extracted before the ace of spades?
I think using total probability for solve this
$$P(B)=P(A_1)P(B\mid A_1)+\ldots+P(A_n)P(B\mid A_n)$$
But I am not sure how to solve this. Can someone help me?
The event that you find $\spadesuit A$ before any $\clubsuit$ is entirely determined by the order in which the $14$ cards $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$ appear in the deck. There are $14!$ possible orderings of these $14$ cards, and each of these orderings are equally likely.
How many of these orderings have $\spadesuit A$ appearing first? The first card must be $\spadesuit A$, there are $13$ choices for the second card, $12$ for the third, and so on, so there are $13!$ such orderings. Therefore, the probability is $13!/14!=\boxed{1/14}$.
Put even more simply: of the fourteen cards $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$, each is equally likely to appear earliest in the deck, so the probability that you find $\spadesuit A$ first is $1/14.$
Added Later: There is also a way to solve this using the law of total probability. We may as well stop dealing cards once the $\spadesuit A$ or any $\clubsuit$ shows up. Let $E_n$ be the event that exactly $n$ cards are dealt. Then $$P(\spadesuit A\text{ first})=\sum_{n=1}^{39}P(\spadesuit A\text{ first }|E_n)P(E_n)$$ Now, given that the experiment ends on the $n^{th}$ card, we know that the $n^{th}$ card is one of $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$, and none of the previous cards are. Each of these is equally likely (due to the symmetry among the 52 cards), so $P(\spadesuit A\text{ first }|E_n)=1/14$. Therefore, $$P(\spadesuit A\text{ first})=\sum_{n=1}^{39}\frac1{14}P(E_n)=\frac1{14}\sum_{n=1}^{39}P(E_n)=\frac1{14}\cdot 1,$$ using the fact that the events $E_n$ are mutually exclusive and exhaustive.
I offer one final method which is more direct, but leads to a summation which is difficult to simplify. Let $F_n$ be the event that the $n^{th}$ card is the $\spadesuit A$. Then $$P(\spadesuit A\text{ first})=\sum_{n=1}^{52}P(\spadesuit A\text{ first }|F_n)P(F_n)=\sum_{n=1}^{52}\frac{\binom{52-n}{13}}{\binom{51}{13}}\cdot\frac1{52}$$ You can simplify this to $1/14$ using the hockey-stick identity: $$\sum_{n=1}^{52}\binom{52-n}{13}=\sum_{m=13}^{51}\binom{m}{13}=\binom{52}{14}$$
• Or alternately to your second explanation, $\spadesuit A$ is equally likely to be in each of the fourteen positions and thus has a 1/14 chance of being in the first position. – Mathieu K. Apr 23 '18 at 2:58
• To add a simplification; you can remove the 38 non-club, non-ace of spades cards without affecting the outcome. – JollyJoker Apr 23 '18 at 7:46
• Of course your answer is completely right, but can you elaborate on something for me? Why do you feel that the statement $P(\spadesuit A \text{ first } | E_n)=1/14$ is trivial enough to just write "due to the symmetry among the 52 cards", but not feel that the statement without conditioning on $E_n$ requires a detailed proof (using as part of the proof the claim that the conditioned statement is easy)? I may be missing something, but I don't see why one statement is particularly harder than the other. Thanks! :) – Sam T Apr 23 '18 at 20:33
• @SamT I feel that neither statements require detailed proofs, as they are both intuitively obvious. For the first statement, the details were easy to fill in, so I included because why not. – Mike Earnest Apr 23 '18 at 22:45
• I just feel that your middle proof basically assumes the result (since assumes the conditioned result). I have no problem with your first proof though! – Sam T Apr 24 '18 at 17:15
These sorts of questions are often solved via "shortcut", such as in the other answer. The reason why the shortcut works is rarely discussed.
The point of this answer is to sketch out the rigorous argument underlying the shortcut. It also demonstrates a methodology one can try to apply to more general problems, or in problems where one has identified a shortcut but still needs to check that the shortcut should give the right answer.
We can describe a choice of how to order a deck of cards in the following way:
• Choose 14 out of 52 places
• Choose an arbitrary ordering of the 14 cards consisting of the 13 clubs and the ace of spades
• Choose an arbitrary ordering of the remaining 38 cards
The ordering this describes is given by placing the 13 clubs and the ace of spades into the chosen places, in the chosen order, and the remaining 38 cards in the remaining places, in the chosen order.
The important thing for this to be a good description is that it has the following properties:
• Every ordering of a deck of cards can be described in this fashion
• Every such description determines a unique ordering of the deck
So, we can determine probabilities simply by counting.
The reason for choosing this description is that:
• the three choices are completely independent from one another
• the problem depends only on the second choice: how to order the 13 clubs and the ace of spades
So, we can (rigorously!) reduce the original problem consisting of a whole deck of cards to the simpler problem consisting of just these 14 cards.
By a similar analysis, we can describe choices of how to order these 14 cards by:
• Choose 1 place
• Choose an arbitrary ordering of the 13 clubs
The ordering so described puts the ace of spades in the chosen place and the clubs in the remaining places, in the chosen order.
Again, this is a good description, the choices are independent, and only the first one matters. So we've reduced the original problem to:
What are the odds of a chosen place among 14 cards is the first?
which is easy to answer: $1/14$.
• How is @MikeEarnest's answer less rigorous than this one? Both entail identifying 14 positions, but whereas in Mike's answer they are by definition those occupied by the 14 salient cards, your answer says "Choose 14 out of 52 places" with no prior motivation. – Rosie F Apr 23 '18 at 6:57
• @RosieF: What's missing is the explanation why the restriction gives the right answers. There are a number of paradoxes that arise because people think that's an intrinsically valid line of reasoning. For example, the famous Monty Hall paradox where the invalid argument selects the subset "the two doors that weren't opened" and mistakenly believes that both choices have equal probability of goat, or the "at least one child is a girl" paradox where the invalid argument chooses a girl and mistakenly believes the choices of "boy, girl" for the other child have equal probability. – user14972 Apr 23 '18 at 8:45
• @Hurkyl both of your example mistakes are due to belief that the reveal of information changes the probabilities that existed before the first choice was made. That does not apply to Mike's answer. – Stop Harming Monica Apr 23 '18 at 9:44
Think of the probability that a club has not yet been drawn on draw $d$ as $$\sum _{n=1}^d \left(1-\frac{13}{52-d}\right)$$
And the probability that the ace of spades is drawn on draw $d$ as $$\frac{1}{52-d}$$
Then the probability that the ace of spades is drawn on draw $d$ given that a club has not yet been drawn is $$\left(\sum _{n=1}^d \left(1-\frac{13}{52-d}\right)\right)*\left(\frac{1}{52-d}\right)$$
Does that help?
• And then sum the final expression over all possible values of $d$? I don't think that's going to help. Regardless, your first expression gives probabilities higher than 1 for various values of $d$ (even if the denominator is changed to $52-n$). I think you want a product instead. – Teepeemm Apr 23 '18 at 14:45 | 2019-11-16T22:12:13 | {
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http://cistirnaperi-hornipocernice.cz/masterchef-claudia-uquraex/exterior-angle-theorem-examples-0f054c | The exterior angles are these same four: ∠ 1 ∠ 2 ∠ 7 ∠ 8; This time, we can use the Alternate Exterior Angles Theorem to state that the alternate exterior angles are congruent: ∠ 1 ≅ ∠ 8 ∠ 2 ≅ ∠ 7; Converse of the Alternate Exterior Angles Theorem. An exterior angle of a triangle.is formed when one side of a triangle is extended The Exterior Angle Theorem says that: the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Exterior Angle of Triangle Examples In this first example, we use the Exterior Angle Theorem to add together two remote interior angles and thereby find the unknown Exterior Angle. Exterior Angle Theorem At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. A related theorem. The theorem states that same-side exterior angles are supplementary, meaning that they have a sum of 180 degrees. The theorem states that the same-side interior angles must be supplementary given the lines intersected by the transversal line are parallel. The sum of all angles of a triangle is $$180^{\circ}$$ because one exterior angle of the triangle is equal to the sum of opposite interior angles of the triangle. Thus. In other words, the sum of each interior angle and its adjacent exterior angle is equal to 180 degrees (straight line). Polygon Exterior Angle Sum Theorem If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360 ° . Well that exterior angle is 90. FAQ. How to define the interior and exterior angles of a triangle, How to solve problems related to the exterior angle theorem using Algebra, examples and step by step solutions, Grade 9 Related Topics: More Lessons for Geometry Math Looking at our B O L D M A T H figure again, and thinking of the Corresponding Angles Theorem, if you know that a n g l e 1 measures 123 °, what other angle must have the same measure? Use alternate exterior angle theorem to prove that line 1 and 2 are parallel lines. how to find the unknown exterior angle of a triangle. Exterior Angle TheoremAt each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. We can see that angles 1 and 7 are same-side exterior. What is the polygon angle sum theorem? You can use the Corresponding Angles Theorem even without a drawing. By corresponding angles theorem, angles on the transversal line are corresponding angles which are equal. They are found on the outer side of two parallel lines but on opposite side of the transversal. The exterior angle dis greater than angle a, or angle b. Thus exterior ∠ 110 degrees is equal to alternate exterior i.e. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular! Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle. Subtracting from both sides, . For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. We know that in a triangle, the sum of all three interior angles is always equal to 180 degrees. Then either ∠1 is an exterior angle of 4ABRand ∠2 is an interior angle opposite to it, or vise versa. Theorem 5.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Using the Exterior Angle Theorem 145 = 80 + x x= 65 Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. So, the measures of the three exterior angles are , and . Similarly, this property holds true for exterior angles as well. The following practice questions ask you to do just that, and then to apply some algebra, along with the properties of an exterior angle… The sum of the measures of the exterior angles is the difference between the sum of measures of the linear pairs and the sum of measures of the interior angles. See Example 2. Angles a, b, and c are interior angles. So, … It is clear from the figure that y is an interior angle and x is an exterior angle. An exterior angle is the angle made between the outside of one side of a shape and a line that extends from the next side of the shape. Calculate values of x and y in the following triangle. History. That exterior angle is 90. So, in the picture, the size of angle ACD equals the … So, we have; Therefore, the values of x and y are 140° and 40° respectively. Thus, (2x – 14)° = (x + 4)° 2x –x = 14 + 4 x = 18° Now, substituting the value of x in both the exterior angles expression we get, (2x – 14)° = 2 x 18 – 14 = 22° (x + 4)°= 18° + 4 = 22° Apply the Triangle exterior angle theorem: ⇒ (3x − 10) = (25) + (x + 15) ⇒ (3x − 10) = (25) + (x +15) ⇒ 3x −10 = … Example 2 Find . In the illustration above, the interior angles of triangle ABC are a, b, c and the exterior angles are d, e and f. Adjacent interior and exterior angles are supplementary angles. Let's try two example problems. Theorem 4-4 The measure of each angle of an equiangular triangle is 60 . The Exterior Angle Theorem says that if you add the measures of the two remote interior angles, you get the measure of the exterior angle. Interior and Exterior Angles Examples. Alternate angles are non-adjacent and pair angles that lie on the opposite sides of the transversal. 110 +x = 180. All exterior angles of a triangle add up to 360°. Oct 30, 2013 - These Geometry Worksheets are perfect for learning and practicing various types problems about triangles. Try the free Mathway calculator and Theorem Consider a triangle ABC.Let the angle bisector of angle A intersect side BC at a point D between B and C.The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of … Apply the Triangle exterior angle theorem: Substitute the value of x into the three equations. The high school exterior angle theorem (HSEAT) says that the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle (remote interior angles). For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. Example 2. E 95 ° 6) U S J 110 ° 80 ° ? Alternate Exterior Angles – Explanation & Examples In Geometry, there is a special kind of angles known as alternate angles. Example: The exterior angle is … Hence, it is proved that m∠A + m∠B = m∠ACD Solved Examples Take a look at the solved examples given below to understand the concept of the exterior angles and the exterior angle theorem. So once again, 90 plus 90 plus 90 plus 90 that's 360 degrees. 2) Corresponding Exterior Angle: Found at the outer side of the intersection between the parallel lines and the transversal. problem solver below to practice various math topics. The Exterior Angle Theorem Date_____ Period____ Find the measure of each angle indicated. Corresponding Angels Theorem The postulate for the corresponding angles states that: If a transversal intersects two parallel lines, the corresponding angles … 4.2 Exterior angle theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Next, calculate the exterior angle. Illustrated definition of Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two opposite interior angles. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$\angle A \text{ and } and \angle B$$ are not congruent.. Corresponding Angles Examples. In this article, we are going to discuss alternate exterior angles and their theorem. ¥ Note that the converse of Theorem 2 holds in Euclidean geometry but fails in hyperbolic geometry. Find the value of x if the opposite non-adjacent interior angles are (4x + 40) ° and 60°. measures less than 62/87,21 By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than either remote interior angle ( and Also, , and . Making a semi-circle, the total area of angle measures 180 degrees. X= 70 degrees. Learn how to use the Exterior Angle Theorem in this free math video tutorial by Mario's Math Tutoring. Example 1. The Exterior Angle Theorem states that An exterior angle of a triangle is equal to the sum of the two opposite interior angles. By the Exterior Angle Inequality Theorem, measures greater than m 7 62/87,21 By the Exterior Angle Inequality Theorem, the exterior angle (5) is larger than either remote interior angle (7 and 8). Exterior angles of a polygon are formed with its one side and by extending its adjacent side at the vertex. And (keeping the end points fixed) ..... the angle a° is always the same, no matter where it is on the same arc between end points: The converse of the Alternate Exterior Angles Theorem … So it's a good thing to know that the sum of the exterior angles of any polygon is actually 360 degrees. So, we all know that a triangle is a 3-sided figure with three interior angles. Therefore, the angles are 25°, 40° and 65°. Let’s take a look at a few example problems. The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle. By substitution, . The exterior angle of a triangle is 120°. Scroll down the page for more examples and solutions using the exterior angle theorem to solve problems. Given that for a triangle, the two interior angles 25° and (x + 15) ° are non-adjacent to an exterior angle (3x – 10) °, find the value of x. Hence, the value of x and y are 88° and 47° respectively. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. Learn in detail angle sum theorem for exterior angles and solved examples. Unit 2 Vocabulary and Theorems Week 4 Term/Postulate/Theorem Definition/Meaning Image or Example Exterior Angles of a Triangle When the sides of a triangle are extended, the angles that are adjacent to the interior angles. Before getting into this topic, […] The exterior angle of a triangle is the angle formed between one side of a triangle and the extension of its adjacent side. Remember that the two non-adjacent interior angles, which are opposite the exterior angle are sometimes referred to as remote interior angles. Corresponding Angels Theorem The postulate for the corresponding angles states that: If a transversal intersects two parallel … We welcome your feedback, comments and questions about this site or page. Rules to find the exterior angles of a triangle are pretty similar to the rules to find the interior angles of a triangle. Therefore, m 7 < m 5 and m 8 < m \$16:(5 7, 8 measures less … Theorem 4-3 The acute angles of a right triangle are complementary. The angle bisector theorem appears as Proposition 3 of Book VI in Euclid's Elements. Using the Exterior Angle Theorem, . A and C are "end points" B is the "apex point" Play with it here: When you move point "B", what happens to the angle? Example 1 Solve for x. The Exterior Angle Theorem Students learn the exterior angle theorem, which states that the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. If two of the exterior angles are and , then the third Exterior Angle must be since . Find the values of x and y in the following triangle. According to the exterior angle theorem, alternate exterior angles are equal when the transversal crosses two parallel lines. Theorem 3. Explore Exterior Angles. The third interior angle is not given to us, but we could figure it out using the Triangle Sum Theorem. This is the simplest type of Exterior Angles maths question. Theorem 1. Solution: Using the Exterior Angle Theorem 145 = 80 + x x = 65 Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. Remember that every interior angle forms a linear pair (adds up to ) with an exterior angle.) In either case m∠1 6= m∠2 by the Exterior Angle Inequality (Theorem 1). Example 1 Find the So once again, 90 plus 90 plus 90 plus 90 that's 360 degrees. ... exterior angle theorem calculator: sum of all exterior angles of a polygon: formula for exterior angles of a polygon: T 30 ° 7) G T E 28 ° 58 °? The sum of exterior angle and interior angle is equal to 180 degrees. For this example we will look at a hexagon that has six sides. Example 3 Find the value of and the measure of each angle. Therefore; ⇒ 4x – 19 = 3x + 16 ⇒ 4x – 3x 0 The exterior angle theorem tells us that the measure of angle D is equal to the sum of angles A and B.In formula form: m
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https://mathoverflow.net/questions/398094/minimiser-of-a-certain-functional | # Minimiser of a certain functional
Let $$f_i \in L^1 ([0, 1])$$ be a sequence of functions equibounded in $$L^1$$ norm - that is, there exists some $$M > 0$$ such that $$\|f_i\|_{L^1} < M$$.
Define the functional $$F: L^1([0, 1]) \to \mathbb R$$ by
$$F(h) = \limsup_{i \to \infty} \|f_i - h\|_{L^1}.$$
Question: Does this functional admit a minimiser? Is the minimiser unique whenever it exists?
Remarks:
What I have tried so far is to attempt to apply the direct method of the calculus of variations.
Since the $$f_i$$ are equibounded in $$L^1$$, it can be shown that $$F$$ is coercive, thus any minimising sequence is bounded in $$L^1$$ norm. In particular we have a weakly-* converging subsequence, say $$h_n \overset{*}{\to} h$$.
The result would follow if we had weak-* sequential lower semi continuity of $$F$$ at the minimiser - that is, that
$$\liminf_{n \to \infty} F(h_n) \geq F(h).$$
I could neither disprove this with a counterexample, nor prove it in generality.
Edit: As pointed out in the comments, weak-$$*$$ convergence to an $$L^1$$ function isn’t guaranteed, only convergence to a measure.
• A minimizer is not unique in general. E.g., let $f_i=(-1)^i$. Then any $h$ with $|h|\le1$ and $\int h=0$ is a minimizer. Jul 22, 2021 at 15:03
• $L^1(0,1)$ is not a dual space. How do you define weak-* convergence?
– gerw
Jul 23, 2021 at 18:15
• Oh you’re right, it needs to be defined in the sense of measures.. but then there is trouble in making sense of $F(\mu)$ for a measure $\mu$. Hmm... Jul 24, 2021 at 4:58
As it has been already noted in the comments, the minimizer doesn't need to be unique. However, it always exists. It is not terribly hard to show but it is not trivial either, so I wonder why the question attracted so few votes.
The proof consists of two independent parts. The first one is that the limit of every minimizing sequence that converges almost everywhere (or just in measure) is a minimizer and the second one is that there exists a minimizing sequence converging almost everywhere. I will use the fact that we deal with a finite measure space though it should, probably, be irrelevant. WLOG, we may assume that $$\|f_j\|_1\le 1$$ for all $$j$$.
Part 1
Assume that $$h_n$$ is a minimizing sequence and $$h$$ is its pointwise limit (or just limit in measure). Since we can assume WLOG that $$\|h_n\|_1\le 3$$ (to be a competitor, you need to perform not much worse than $$0$$, at the very least), we have $$\|h\|_1\le 3$$ as well (by Fatou). Then, by the definition of the convergence in measure, we can write $$h_n=u_n+v_n$$ where $$u_n$$ converge to $$h$$ uniformly and $$v_n$$ are supported on $$E_n$$ with $$m(E_n)\to 0$$ as $$n\to\infty$$. Also $$\|v_n\|_1\le 7$$, say.
Now, since $$v_n\in L^1$$, we can find $$\delta_n>0$$ such that for every set $$E$$ with $$m(E)<\delta_n$$, we have $$\int_E|v_n|<\frac 1n$$, say. By induction, we can now choose a subsequence $$n_k$$ such that $$\sum_{q=k+1}^\infty m(E_{n_q})<\delta_{n_k}\,.$$ Then $$\|v_{n_k}\chi_{\cup_{q>k}E_{n_q}}\|_1\to 0$$ and, adding these parts to $$u_{n_k}$$, we see that we can (passing to a subsequence) assume that our minimizing sequence can be represented as $$h_n=U_n+V_n$$ where $$U_n\to h$$ in $$L^1$$ and $$V_n$$ have disjoint supports $$G_n$$.
Since correcting a minimizing sequence by a sequence tending to $$0$$ in $$L^1$$ results in a minimizing sequence, we can just as well assume that $$h_n=h+V_n$$.
If $$\|V_n\|_1\to 0$$ (even along a subsequence), we are done. Assume now that $$\|V_n\|_1\ge \tau>0$$. Then for every function $$g$$ of $$L^1$$-norm less than $$4$$, we have $$\|g\|_1\le \max(\|g-V_n\|_1,\dots,\|g-V_{n+N-1}\|_1)$$ for all $$n$$ as soon as $$N>8/\tau$$. Indeed, since the supports $$G_n$$ of $$V_n$$ are disjoint, there is $$q\in\{0,\dots,N-1)$$ such that $$\int_{G_{n+q}}|g|\le \frac 1N\|g\|_1< \frac\tau 2$$, in which case subtracting $$V_{n+q}$$ can only drive the norm up.
Applying this to the functions $$g_j=f_j-h$$, we conclude that $$\limsup_{j\to\infty}\|f_j-h\|_1\le \limsup_{j\to\infty}\max_{0\le q\le N-1}\|f_j-h_{n+q}\|_1= \max_{0\le q\le N-1}\limsup_{j\to\infty}\|f_j-h_{n+q}\|_1$$ for every $$n$$, i.e. $$h$$ is a minimizer in this case as well.
Part 2
Let $$I$$ be the infimum of our functional. For every $$\varepsilon>0$$ consider the set $$X_\varepsilon$$ of all $$L^1$$-functions $$h$$ for which the value of the functional is at most $$I+\varepsilon$$. Clearly, it is a convex, non-empty, closed (in $$L^1$$) set and $$X_{\varepsilon'}\supset X_{\varepsilon''}$$ when $$\varepsilon'\ge\varepsilon''$$.
Fix some strictly convex non-negative function $$\Phi(t)\le |t|$$. To simplify the technicalities, I'll choose it by the conditions $$\Phi(0)=\Phi'(0)=1$$, $$\Phi''(t)=\frac 2{\pi(1+|t|^2)}$$ but pretty much any other choice will work as well.
Let $$J_\varepsilon=\inf_{h\in X_\varepsilon}\int\Phi(h)\,.$$ Clearly, $$J_\varepsilon$$ is a bounded non-increasing function on $$(0,1)$$, say. Let $$J$$ be its limit at $$0+$$. Passing to an appropriate decreasing sequence $$\varepsilon_n\to 0+$$ and re-enumerating $$X_n=X_{\varepsilon_n}, J_n=J_{\varepsilon_n}$$, we can assume that $$J_n\ge J-2^{-5n}$$ We will choose a representative $$h_n$$ of $$X_n$$ for which $$\int\Phi(h_n)$$ is not more than $$2^{-5n}$$ above $$J$$ and, thereby, not more than $$2\cdot 2^{-5n}$$ above its infimum over $$X_n$$. Then for $$m>n$$, we have $$h_{n,m}=\frac{h_n+h_m}2\in X_n$$ (convexity of $$X_n$$ plus inclusion $$X_n\supset X_m$$) and $$\int\Phi(h_{n,m})\le \int\frac 12(\Phi(h_n)+\Phi(h_m))-\int\frac{(h_n-h_m)^2}{4\pi (1+|h_n|^{2}+|h_m|^{2})}$$ (second order Taylor with the remainder in the Lagrange form), whence $$\int\frac{(h_n-h_m)^2}{4\pi(1+|h_n|^2+|h_m|^2)}\le 2\cdot 2^{-5n}$$ regardless of $$m>n$$ (otherwise we would go below $$J+2^{-5n}-2\cdot 2^{-5n}=J-2^{-5n}$$, which is below the infimum over $$X_n$$). It remains to note that if $$|h_n|\le 2^n$$ and $$|h_n-h_m|>2^{-n}$$, then the integrand is at least $$\rm{const}\, 2^{-4n}$$, so we get $$m(\{|h_n|\le 2^n, |h_n-h_m|>2^{-n}\})\le \rm{Const}\,2^{-n}$$ for all $$m>n$$ from where it follows at once that $$h_n(x)$$ is a Cauchy sequence for almost all $$x$$ (recall that $$\|h_n\|_1\le 3$$, so the first condition excludes only a set of measure $$3\cdot 2^{-n}$$, then use Borel-Cantelli).
That's it. Feel free to ask questions if anything is unclear :-)
• Wow, this looks impressive. This part was a bit confusing to me - “ Then $\|v_{n_k}\chi_{\cup_{q>k}E_{n_q}}\|_1\to 0$ and, adding these parts to $u_{n_k}$, we see that we can (passing to a subsequence) assume that our minimizing sequence can be represented as $h_n=U_n+V_n$ where $U_n\to h$ in $L^1$ and $V_n$ have disjoint supports $G_n$.” Do you mean to add $v_{n_k}\chi_{\cup_{q>k}E_{n_q}}$ to $u_{n_k}$? I am not sure how to get the desired representation from this. Aug 2, 2021 at 4:04
• @NateRiver Exactly as you said: $U_k=u_{n_k}+v_{n_k}\chi_{\cup_{q>k}E_{n_q}}$, $V_k=v_{n_k}\chi_{E_{n_k}\setminus \cup_{q>k}E_{n_q}}=v_{n_k}\chi_{G_k}$. Aug 2, 2021 at 7:45
• @NateRiver I just tried to solve the problem in $L^2$ first. There weak convergence is guaranteed, but doesn't seem to drop the value of the functional immediately, so I needed the norm convergence of a minimizing sequence. Fortunately, if one has a nested sequence of bounded closed convex sets in a Hilbert space, the smallest norm elements of the sets converge in norm (the proof is the same as above just using the parallelogram identity). I tried to mimic that idea in $L^1$ (which required strict convexity of something) and got the a.e. convergence this way, which turned out to be sufficient. Aug 2, 2021 at 13:33
• An alternative proof for Part 2 (existence of a minimizing sequence converging a.e.) Let $h_n$ be any minimizing sequence for $F$. As observed, it is bounded in $L_1$, so by the Komlós Theorem, up to extracting a subsequence, it is a.e. converging in Cesaro sense to some $h\in L^1$. Since $F$ is a convex functional, the sequence of the Cesaro means is still a minimizing sequence. Aug 2, 2021 at 13:38
• Of course, and I wouldn't be surprised if you answer (which I'm still reading) contains as a byproduct an alternative proof of that Komlós theorem (which is a 10 page paper: link.springer.com/article/10.1007%2FBF02020976 ). Aug 2, 2021 at 13:51 | 2022-07-01T14:43:14 | {
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https://math.stackexchange.com/questions/3072171/how-to-properly-represent-a-matrix-function | # How to properly represent a matrix function.
Given the function $$f_{h}(x,y,z)=(x-z,y+hz,x+y+3z)$$, what is the correct way to represent the matrix function in respect to the standard basis?
With the representation theorem, I would write the matrix in columns as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & 1\\ 0 & 1 & 1\\ -1 & h & 3\end{bmatrix}$$ But in my textbook it is written in rows as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & -1\\ 0 & 1 & h\\ 1 & 1 & 3\end{bmatrix}$$
What is the difference between them?
The correct way to represent the function $$f_h$$ in matrix form depends on the convention that you want to use to represent it. Let $$(x,y,z) \in \mathbb{R}^3$$. The matrix which you computed is useful for expressing $$f_h$$ as $$f_h(x,y,z) = \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 1 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix}.$$ We can verify this by expanding the matrix product: \begin{align} \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 2 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix} &= \begin{bmatrix} x \cdot 1 + y \cdot 0 + z \cdot (-1) \\ x \cdot 0 + y \cdot 1 + z \cdot h \\ x \cdot 1 + y \cdot 1 + z \cdot 3 \end{bmatrix} \\ &= \begin{bmatrix} x - z & y + hz & x + y + 3z \end{bmatrix}. \end{align} The result is a $$1 \times 3$$ matrix, which we can interpret as a row vector in $$\mathbb{R}^3$$. The matrix written in your textbook is useful for the following representation: $$f_h(x,y,z) = \begin{bmatrix} 1& 0 & -1 \\ 0 & 1& h\\ 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x-z \\ y + hz \\ x + y + 3z \end{bmatrix}.$$ The result is a $$3 \times 1$$ matrix, which we can regard as a column vector in $$\mathbb{R}^3$$.
The two representations look different in the sense that the former yields a row vector and the latter a column vector, however they are the same in the sense that they can both be regarded as lists of three real numbers, i.e. as $$(x-z,y+hz,x+y+3z) \in \mathbb{R}^3.$$ The accepted answer here sums this idea up pretty well. | 2021-01-16T00:06:45 | {
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https://math.stackexchange.com/questions/3056293/representation-of-function-as-power-series-unique | # Representation of function as power series - unique?
The following example is from Calculus, 7e by James Stewart:
Example 2, Chapter 11.9 (Representations of functions as Power Series)
Find a power series representation for $$\frac{1}{x+2}$$
My solution is:
$$\frac{1}{x+2}=\frac{1}{1-\left(-x-1\right)}=\sum_{n=0}^{\infty}\left(-x-1\right)^{n}=\sum_{n=0}^{\infty}\left(-1\right)^{n}\left(x+1\right)^{n}$$
Then to find the interval of convergence
$$\left|x+1\right|<1 \Rightarrow x\in\left(-2,0\right)$$
But the given solution is different:
“In order to put this function in the form of the left side of Equation 1, $$\frac{1}{1-x}$$ , we first factor a 2 from the denominator:
$$\frac{1}{2+x}=\frac{1}{2\left(1+\frac{x}{2}\right)}=\frac{1}{2\left[1-\left(-\frac{x}{2}\right)\right]}=\frac{1}{2}\sum_{n=0}^{\infty}\left(-\frac{x}{2}\right)^{n}=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{2^{n+1}}x^{n}$$
This series converges when $$\left|-\frac{x}{2}\right|<1$$, that is, $$\left|x\right|<2$$. So the interval of convergence is $$\left(-2,2\right)$$."
So my questions are:
1. Did I make an error somewhere?
2. If not, are the two representations equivalent? Can there be more than one representation of a function as a power series?
• The representation as a power series at a given point is unique. Here, yours is at the point $-1$, while the solution gives the one at the point $0$. (By default, power series are usually taken at 0 when nothing is specified or obvious from context) – Clement C. Dec 29 '18 at 21:59
I think your method is right but you made a power series around $$x_0=-1$$
while they did it around $$x_0=0$$ which is what they asked for probably
Both solutions are correct: your series is a power series centered at $$-1$$, whereas the power series from the given solution is centered at $$0$$. | 2020-01-19T05:01:55 | {
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https://byjus.com/question-answer/let-displaystyle-i-1-int-sec-2-z-2-tan-2-z-xf-left-x/ | Question
# Let $$\displaystyle { I }_{ 1 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) dx }$$ and $$\displaystyle { I }_{ 2 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ f\left( x\left( 3-x \right) \right) dx }$$, where $$f$$ is a continuous function and $$z$$ is any real number, $$\displaystyle \frac { { I }_{ 1 } }{ { I }_{ 2 } } =$$
A
32
B
12
C
1
D
none of these
Solution
## The correct option is B $$\displaystyle \frac { 3 }{ 2 }$$We have, $$\displaystyle { I }_{ 1 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) dx }$$$$\displaystyle =\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ \left( 3-x \right) f\left( \left( 3-x \right) \left( 3-\left( 3-x \right) \right) \right) } dx$$ $$\displaystyle \left[ \because \int _{ a }^{ b }{ f\left( x \right)dx } =\int _{ a }^{ b }{ f\left( a+b-x \right) dx } \right]$$$$\displaystyle =\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ \left( 3-x \right) f\left( x\left( 3-x \right) \right) dx }$$$$\displaystyle =3\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ f\left( x\left( 3-x \right) \right) dx } -\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) } dx$$$$=3{ I }_{ 2 }-{ I }_{ 1 }$$$$\displaystyle \therefore 2{ I }_{ 1 }=3{ I }_{ 2 }\Rightarrow \frac { { I }_{ 1 } }{ { I }_{ 2 } } =\frac { 3 }{ 2 }$$Mathematics
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View More | 2022-01-25T10:26:32 | {
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https://math.stackexchange.com/questions/2623796/prove-that-a-smallsetminus-a-smallsetminus-b-a-cap-b/2623804 | Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$
$A$ and $B$ are any sets, prove that $A \smallsetminus (A \smallsetminus B) = A \cap B.$ This formula makes sense when represented on a Venn diagram, but I am having trouble with proving it mathematically.
I have tried letting $x$ be an element of $A$ and continue from there, but it doesn't seem to work out as a valid proof anyways.
Could anyone please point me in the right direction?
Many thanks.
• Please show us your work: starting with $x \in A\setminus(A\setminus B)$, or the other way around, starting with $x \in (A\cap B)$. Please don't claim you tried ABC, unless you also show us you effort using ABC. Would you like to get an answer that says only: "Yup, that's right, start by letting $x \in A\setminus (A\setminus B)$... – Namaste Jan 27 '18 at 18:42
Let $x \in A \setminus (A \setminus B).$
Then $x$ is an element such that $x \in A$ and $x \notin A \setminus B$. But if $x \notin A\setminus B$, with some additional work, we realize this implies that $x \in A$ and $x \in B$. So $x \in A \cap B$.
Vice-versa: let $x \in A \cap B$ so $x \in A$ and $x \in B$. This implies that $x \notin A \setminus B$. But given that $x \in A$ and $x \notin A \setminus B$ this implies that $x \in A \setminus (A \setminus B)$.
• I would like this answer if you hadn't skipped from "$x \in A$ and $x \notin A\setminus B$, but if $x\in A \setminus B$" to "we realize that $x \in A$ and $x\in B$." I added the words ("with some additional work") in between them. I would like this answer more if you explained that $x \in A$ and $x \notin (A\setminus B)$, then $x \in A$ and, it is not the case that ($x \in A$ and $x \notin B$.) By Demorgans, we get that $(x \in A \land \lnot x\in A),$ or $(x \in A \land x\in B)$. The first disjunct is always false (no element can be in a set, and not be in a set. – Namaste Jan 27 '18 at 19:13
• ...That leaves us to conclude, that "we realize this implies that $x\in A$ and $x\in B$. – Namaste Jan 27 '18 at 19:13
Notice that \begin{eqnarray*} x\in A\setminus(A\setminus B) &\Leftrightarrow& (x\in A)\wedge \neg(x\in A\setminus B)\\ &\Leftrightarrow & (x\in A)\wedge \neg((x\in A)\wedge \neg (x\in B))\\ &\Leftrightarrow & (x\in A)\wedge ((x\notin A)\lor(x\in B))\\ &\Leftrightarrow & ((x\in A)\wedge (x\notin A))\lor ((x\in A)\wedge(x\in B))\\ &\Leftrightarrow & (x\in A)\wedge (x\in B)\\ &\Leftrightarrow & x\in A\cap B. \end{eqnarray*}
• Yes, when in doubt go back to the logic – Tom Collinge Jan 27 '18 at 18:28
• I like this, because the equivalence makes the proof bi-directional, and doesn't skip important steps between $x \in A) \land$\lnot (x \in A\setminus B$and its equivalent,$x\in A \land x \in B$Those are likely the steps on which the asker got tripped up, I suspect. Nice answer. – Namaste Jan 27 '18 at 18:51 • I like this, because the equivalence makes the proof bi-directional, and doesn't skip important steps between$x \in A \land \lnot (x \in A\setminus B$and its equivalent,$x\in A \land x \in B.$Those are likely the steps on which the asker got tripped up, I suspect. Nice answer. – Namaste Jan 27 '18 at 19:01 • @amWhy: Thanks. Indeed, often mistakes are made in these basic logical steps. I've seen many people making mistakes using the notation$x\notin B$. I think it's safer to write$\neg(x\in B)$, indeed, if$B$is a more complicated expression, we still now how to negate the logical expression appearing inside the brackets. This basic logical way of thinking is user friendly and not very prone to errors. – Mathematician 42 Jan 27 '18 at 20:43 Did you know that $$A \smallsetminus B = A \cap \overline{B} \;\;?$$$\begin{align} A \smallsetminus (A \smallsetminus B) &= A \smallsetminus (A \cap \overline{B}) \\ \\ &= A \cap \overline{ (A \cap \overline{B})}\\ \\ &= A \cap (\overline{A} \cup B)\\ \\ &= (A \cap \overline{A} )\cup (A \cap B) \\ \\ &= A \cap B \end{align}$• cmon, why this get downvoted, it is correct. hmm, maybe you want to precise that$\bar B=B^{\complement}$and not the closure in this context. – zwim Jan 27 '18 at 18:55 • @zwim: Not sure why this got downvoted, however this proof assumes knowledge about standard set operations which you should be able to prove directly as well. You might end up running in circles and not proving anything. – Mathematician 42 Jan 27 '18 at 18:58 Let$x\in A\cap B$. Then$x\in A$and$x\in B$. Then$x\not \in A\setminus B$, so$x\in A\setminus (A\setminus B)$. Conversely, if$x\in A\setminus (A\setminus B)$, then$x\in A$and$x\not \in (A\setminus B)$. So$x\in A$and$x\in B$. That is$x\in A\cap B\$...
• ditto of @Skills. – Namaste Jan 27 '18 at 18:36 | 2019-10-19T19:34:58 | {
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http://mathhelpforum.com/pre-calculus/167826-sequences-problem.html | # Math Help - sequences problem
1. ## sequences problem
A hardware supplier has designed a series of six plastic containers with lids where each container (after the first) can be placed into the next larger one for storage purposes. The containers are rectangular boxes, and the dimensions of the largest one are 120 cm by 60 cm by 50 cm. The dimension of each container is decreased by 10 percent with respect to the next larger one.
Determine the volume of each container, and the volume of all the containers. Write your answers in litres (one litre = 1000 cm3).
Is it 10% from each dimension or from the volume? Hard to figure out from the problem.
If from the each dimension then:
V 1= 50x 60x 120 = 360000 cm3 = 360 litres
V2=45 x 54 x 108 = 262440 cm3 = 262.4 liltres
V 3 = 40.5 x 48.6 x 97.2 = 191.3litres
V4 = 36.45 x 43.74 x 87.48 =139.4 litres
V5 = 32.81 x 39.37 x 78.74 = 101.7 litres
V6 = 29.61 x 35.47 x 70.94 =74.5 litres
I need some feedback.
2. I expect since it says "the dimension of each container is decreased by 10% with respect to the larger one" I would say it's the dimensions and not the volume that is decreased by 10%.
3. Originally Posted by terminator
A hardware supplier has designed a series of six plastic containers with lids where each container (after the first) can be placed into the next larger one for storage purposes. The containers are rectangular boxes, and the dimensions of the largest one are 120 cm by 60 cm by 50 cm. The dimension of each container is decreased by 10 percent with respect to the next larger one.
Determine the volume of each container, and the volume of all the containers. Write your answers in litres (one litre = 1000 cm3).
Is it 10% from each dimension or from the volume? Hard to figure out from the problem.
Why? The problems says "The dimension of each container is decreased by 10 percent". That tells you precisely which is intended
If from the each dimension then:
V 1= 50x 60x 120 = 360000 cm3 = 360 litres
V2=45 x 54 x 108 = 262440 cm3 = 262.4 liltres
V 3 = 40.5 x 48.6 x 97.2 = 191.3litres
V4 = 36.45 x 43.74 x 87.48 =139.4 litres
V5 = 32.81 x 39.37 x 78.74 = 101.7 litres
V6 = 29.61 x 35.47 x 70.94 =74.5 litres
I need some feedback.
4. Hello, terminator!
A hardware supplier has designed a series of six plastic containers with lids
. . where each container (after the first) can be placed into the next larger one.
The containers are rectangular boxes.
. . The dimensions of the largest one are 120 cm by 60 cm by 50 cm.
The dimensions of each container are decreased by 10 percent
. . with respect to the next larger one.
(a) Determine the volume of each container.
(b) Determine the volume of all the containers.
Write your answers in litres (one litre = 1000 cubic cm).
The original box has dimensions $L,\,W,\,H$
. . Its volume is: . $L\!\cdot\!W\!\cdot\!H\text{ cm}^3$
The next box has dimensions: $0.9L,\,0.9W,\,0.9H$
. . Its volume is: . $(0.9L)(0.9W)(0.9H) \:=\:0.729(LWH)$
That is, each box is 0.729 of the volime of the next larger box.
The first box has volume: . $120\cdot60\cdot50 \:=\:360,\!000\text{ cm}^3$
. . That is: . $V_1 \:=\:360\text{ liters.}$
(a) Therefore, the $\,n^{th}$ box has volume: . $V_n \;=\;360(0.729)^{n-1}$
The sum of the volumes of all the boxes is:
. . $S \;=\;360 + 360(0.729) + 360(0.729^2) + 369(0.729^3) + \hdots$
. . $S \;=\;360\underbrace{\bigg[1 + 0.729 + 0.729^2 + 0.729^3 + \hdots\bigg]}_{\text{a geometric series}}$
The sum of the geometric series is: . $\dfrac{1}{1 - 0.729} \:=\:\dfrac{1}{0.271}$
(b) Therefore: . $S \;=\;360\left(\frac{1}{0.271}\right) \;\approx\;1328.4\text{ liters}$ | 2015-05-30T19:32:29 | {
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https://www.themathdoctors.org/the-gamblers-fallacy/ | The Gambler’s Fallacy
Probability seems simple enough to many people that it can fool them into wrong conclusions. We have had many questions that involve the “Gambler’s Fallacy”, both from people who naively assume it without thinking, and from some who defend it using technical ideas like the Law of Large Numbers.
The Gambler’s Fallacy
Here is a question from 2003 that is a good introduction to the idea:
Gambler's FallacyA current co-worker and I are in a friendly disagreement about the probability of selecting the winning number in any lottery, say Pick 5. He states that he would rather bet the same set of five numbers every time for x period of time, but I insist that the probability is the same if you randomly select any set five numbers for the same period of time. The only assumption we make here is betting one set of numbers on any given day. Who is correct?
I tried explaining to him that the probability of betting on day one is the same for both of us. On day two it is the same. On day three it is the same, etc. Therefore the sum of the cumulative probabilities will be the same for both of us.
Doctor Wallace first examined the appropriate calculation, before moving on to the likely underlying error:
You are correct. If you have the computer randomly select a different set of 5 numbers to bet on every day, and your friend selects the same set of numbers to bet on every day, then you both have exactly the same probability of winning.
Tell your friend to think of the lottery as drawing with tickets instead of balls. If the lottery had a choice of, say, 49 numbers, then imagine a very large hat containing 1 ticket for every possible combination of 5 numbers. 1, 2, 3, 4, 5; 1, 2, 3, 4, 6; etc.
On the drawing day, ONE ticket is pulled from the hat. It is equally likely to be any of the C(49,5) tickets in the hat. (There would be 1,906,884 tickets in the hat in this case.)
Since both you and your friend have only ONE ticket in the hat, you both have the same chance of winning.
On the next drawing day for the lottery, ALL the tickets are replaced. Each lottery draw is an event independent of the others. That is to say, the probability of any combination winning today has absolutely NO effect on the probability of that or any other combination winning tomorrow. Each and every draw is totally independent of the others.
That perspective makes it “obvious”: if two people each have one “ticket”, they have the same probability, whether or not it is the same one that was taken last time, since the “ticket” is chosen randomly each time without regard to the past.
So why is the other opinion tempting?
The reason your friend believes that he has a better chance of winning with the same set of numbers is probably due to something called the "gambler's fallacy." This idea is that the longer the lottery goes without your friend's "special" set of numbers coming up, the more likely it is to come up in the future. The same fallacy is believed by a lot of people about slot machines in gambling casinos. They hunt for which slot hasn't paid in a while, thinking that that slot is more likely to pay out. But, as the name says, this is a fallacy; pure nonsense. A pull of the slot machine's handle, like the lottery draw, is completely independent of previous pulls. The slot machine has no memory of what has come before, and neither has the lottery. You might play a slot machine for 2 weeks without hitting the big jackpot, and someone else can walk in and hit it in the first 5 minutes of play. People wrongly attribute that to "it was ready to pay out." In reality, it's just luck. That's why they call it gambling. :)
The same thing comes up in math classes:
This used to be a "trick" question on old math tests:
"You flip a fair coin 20 times in a row and it comes up heads every single time. You flip the coin one more time. What is the probability of tails on this last flip?"
Most people will respond that the chance of tails is now very high.
(Ask your friend and see what he says.) However, the true answer is that the probability is 1/2. It's 1/2 on EVERY flip, no matter what results came before. Like the slot machine and the lottery, the coin has no memory.
That type of question is still valuable. It tests an important idea that students need to think about.
For more about picking the same number in a lottery, see
Lottery Strategy and Odds of Winning
For a very nice refutation of the Gambler’s Fallacy in coin tossing, see
What Makes Events Independent?
The Law of Large Numbers
That question was based on a naive approach to gambling. The next, from 2000, is based on probability theory. (The questions were asked in imperfect English, which I will restate as I understand it, correcting a misinterpretation in the archived version. Doctor TWE got it right.)
Law of Large Numbers and the Gambler's FallacyIf we throw three dice at a time, three times altogether, is the result the same as if we throw nine dice one time?
Do we have the same probability to get a given number in the two cases?
Where and how different is it?
Doctor TWE answered, covering several possible interpretations of “result”:
The sum, the average, and the probabilities of getting a particular value on one die or more aren't affected by whether the dice are rolled one at a time, in groups of 3, or all 9 at once. These are called "independent events," and the order in which they happen doesn't affect the outcome.
Elvino replied,
So, I have 42.12% probability to get at least one six (or other number), hen I throw three dice at once. If I throw the same dice, in the same way, again and again I will always have the same probability!
But, there is a law that says, the more I throw, the more probability I have to obtain a certain number. How does this law of probabilities work?
Checking his calculation, and thereby confirming what he meant, I get the probability of rolling at least one six on three dice to be the complement of rolling no sixes: $$\displaystyle 1 – \frac{5^3}{6^3} = 42.1%$$. Next time I will discuss this further.
Doctor TWE confirmed his calculation, then explained what this law means, and does not mean:
There is something called the Law of Large Numbers (or the Law of Averages) which states that if you repeat a random experiment, such as tossing a coin or rolling a die, a very large number of times, your outcomes should on average be equal to (or very close to) the theoretical average.
Suppose we roll three dice and get no 6's, then roll them again and still get no 6's, then roll them a third time and STILL get no 6's. (This is the equivalent of rolling nine dice at once and getting no 6's, as we discussed in the last e-mail; there's only a 19.38% chance of this happening.) The Law of Large Numbers says that if we roll them 500 more times, we should get at least one 6 (in the 3 dice) about 212 times out of the 503 rolls (.4213 * 503 = 211.9).
This is *not* because the probability increases in later rolls, but rather, over the next 500 rolls, there's a chance that we'll get a "hot streak," where we might roll at least one 6 on three or more consecutive rolls. In the long run (and that's the key - we're talking about a VERY long run), it will average out.
There is also something called the Gambler's Fallacy, which is the mistaken belief that the probability on the next roll changes because a particular outcome is "due." In the example above, the probability of rolling at least one 6 in the next roll of the three dice (after three rolls with no 6's) is still 42.13%. A (non-mathematician) gambler might think that the dice are "due," that in order to get the long-term average back up to 42%, the probability of the next roll getting at least one 6 must be higher than 42%. This is wrong, and hence it's called "the Gambler's Fallacy."
The important thing here is that things will “average out” in the long run, so that you get at least one 6 in 42.1% of the rolls, but not because at any one time there is a greater chance, to make up the average — it happens only because there is a very long time to do so. It is not because past events have any effect on future events, pulling them into line with the “averages”.
For more on the Law of Large Numbers, see
The Law of Large Numbers
Digging deeper
In 2008, we had the following detailed question about that from Konstantinos:
Do Prior Outcomes Affect Probabilities of Future Ones?What I want to know is if in matters of luck, such as games of dice, or lotteries, or flipping coins, the future outcomes have any relativity with past results. What I mean in each case is: If I flip a coin and get tails, don't I have bigger or smaller possibilities to get heads on the next roll? I mean I know that the possibility is always 1/2 but since I have already thrown the coin 5 times and rolled 5 tails there aren't possibilities that in the next throws there will be heads? The same question goes for dice rolls, and for lotto numbers. If a number has come more times than others, isn't it possible that for a limited amount of coming times, numbers that haven't come yet, will start showing more?
What I find most confusing is that the relation between probabilities and past possibilities, and the outcome in real life. How can a coin come tails 5 or six times in a row when the possibility is always 1/2? Are probabilities only theoretical?
I tried noting down results of different trials of luck (dice, past lotteries, coins) but in the end they don't seem to make true to any theory I have heard. I would like to see the magic of numbers in real life and on this subject and how it works as to prove a theory. Do I have to toss the coin 10,000 times? And if I do will I see heads coming in a row after 10 subsequent tails?
I responded to this one:
What you are suggesting is called the Gambler's Fallacy--the WRONG idea that future results of a random process are affected by past results, as if probability deliberately made things balance out. The law of large numbers says that it WILL balance out eventually; but that does not happen by changing the probabilities in the short term. The long-term balance just swamps the short-term imbalance.
If you think about it, what could cause a coin to start landing heads up more often after a string of tails? There is no possible physical cause for this; the coin has no memory of what it, or any other coin, previously did. And probability theory does not make things happen without a physical cause; it just describes the results.
If I tossed a coin and it landed tails 5 times, I would just recognize that that is a perfectly possible (and not TOO unlikely) thing to happen. If I got tails 100 times, I would NOT expect the next to be heads; I would inspect the coin to make sure it actually has a heads side! An unlikely string of outcomes not only does not mean that the opposite outcome is more likely now; it makes it LESS likely, because it suggests statistically that the basic probability may not be what I was originally assuming.
In response to the question about probabilities being merely theoretical, I explained what probability is and isn’t:
Probabilities are theoretical, but have experimental results, IN THE LONG RUN. The law of large numbers says that, if you repeat an experiment ENOUGH TIMES, the proportion of times an event occurs will PROBABLY be CLOSE to the predicted theoretical probability. Probability can't tell you what will happen the next time, but it does predict what is likely to happen on the average over the next, say, million times. If you started out with ten tails in a row, you will not necessarily get ten heads in a row at any point, or even more heads than tails in the next few tosses; you will just get enough heads in the next million to keep it balanced.
Note all the capitalized words, emphasizing the vagueness of this statement. A formal statement of this theorem will define those more clearly, but they will then all become statements of probability (my “probably”) and limits (my “enough” and “close”). See, for example, the formal statements of the Law of Large Numbers in the Wikipedia page (my emphases in the last paragraph):
The weak law of large numbers (also called Khinchin’s law) states that the sample average converges in probability towards the expected value $$\displaystyle \overline {X}_{n}\overset{P}{\rightarrow} \mu \ {\textrm {when}}\ n\to \infty .$$.
That is, for any positive number ε, $$\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |>\varepsilon \,\right)=0$$.
Interpreting this result, the weak law states that for any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin.
What it does not say is that over, say, 100 trials (or 10,000 trials), we can be sure that the average will be exactly what we expect.
I continued:
In particular, if you were to throw five coins at a time (to make it easier than throwing the same coin five times in a row), and do that 1000 times, you would expect that you would get 5 tails about 1/32 of those times (since the probability of all five being tails is 1/2^5). That's about 31 times! So it's not at all unreasonable to expect that it will occur once in a while.
On the other hand, the probability of getting 100 tails in a row is 1/2^100, or 1/1,267,650,600,228,229,401,496,703,205,376, which makes it very unlikely that it has ever happened in the history of the world, though it could!
Again, probability can't tell you what WILL happen, specifically; it is all about unpredictable events. But if you tossed a coin 1000 sets of 10 times, on the average one of those is likely to yield 10 tails. (The probability is 1/2^10 = 1/1024.) The probability of some ten in a row out of 10,000 tosses is a little bigger, but that's harder to calculate.
Regression to the mean
In an unarchived question from 2014, Adam brought in another idea:
If I flip a coin 10 times, the most likely outcome is that I will flip a total of 5 heads and a total of 5 tails. If each round of coin flipping (one round being 10 flips, in my example) is independent of previous rounds, then the probability of flipping a total of 5 heads and a total of 5 tails never changes. However, the concept of "regression to the mean" implies that "rare" events are likely to be followed by less rare events and vice versa. So, if I flip 10 heads in a row in round 1 (a rare event), the odds of flipping a total of 5 heads and 5 tails in round 2 are greater than if I'd flipped 5 heads and 5 tails in round 1. What is the correct way to see this, do the odds change or not?
Coin flipping rounds are independent of one another, which implies that the probabilities never change. Regression to the mean states that rare events are likely to be followed by less rare events, implying that the probability of even random events does change.
I replied:
The odds don't change. You are NOT more likely to flip 5 heads if you previously flipped 10.
Regression to the mean only says that the most likely event on ANY toss will be about 5 heads, so if you got 10 on one toss, you are more likely to get something closer to 5 on the next toss, simply because something closer to 5 is ALWAYS more likely than 10. If you got 5 on the first toss, you are as likely to get 5 again as ever; but any deviation that does occur will be away from the mean, because there is no direction to go except away!
It is an entirely wrong interpretation of the phenomenon to think that, having tossed 10 heads, you are more likely to toss 5 heads than under other circumstances. It's just that the event that is always more likely will be less extreme than that first toss.
http://en.wikipedia.org/wiki/Regression_toward_the_mean
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The main advantage of an FFT is speed, which it …. The series does not seem very useful, but we are saved by the fact that it converges rather rapidly. the Fourier representation of the square wave is given in Fig. Then mathematically, a T-periodic waveform v satisfies — a periodic waveform with period T (2) for all t. Discrete Fourier Transform (DFT)¶ From the previous section, we learned how we can easily characterize a wave with period/frequency, amplitude, phase. Show that the Fourier Series expansion of y (x) is given by y(t)= 4! sin"t+ sin3"t 3 + sin5"t 5 + # % & '(where ω = 2π/T. In this particular SPICE simulation, I’ve summed the 1st, 3rd, 5th, 7th, and 9th harmonic voltage sources in series …. Fourier Series and Coefficients Fourier series may be used to represent periodic functions as a linear combination of sine and cosine functions. 1: The cubic polynomial f(x)=−1 3 x 3 + 1 2 x 2 − 3 16 x+1on the interval [0,1], together with its Fourier series …. Example 2 Given a signal y(t) = cos(2t), find its Fourier Series coefficients. Let’s assume we have a square wave with following characteristics: P eriod = 2ms P eak−to −P eak V alue = 2 V Average V alue = 0 V P e r i o d = 2 m s P e a k − t o − P e a k V a l u e = 2 V A v e r a g e V a l u e = 0 V. The key here is that the Fourier basis is an orthogonal basis on a given interval. That is why we have programmed our free online Fourier series calculator to determine the results instantly and precisely. 6 Example: Fourier Series for Square Wave. Find the Fourier Complex Fourier Series . In the processing of audio signals (although it can be used for radio waves, light waves, seismic waves, and even images), Fourier analysis can isolate individual components of a continuous complex waveform, and concentrate. The function is reconstructed by the following summations over the fourier coefficients. In general square integrability will not guarantee convergence of the Fourier series to the original function. Fourier series are useful in a wide range of fields including acoustics, with representation of musical sounds as sums of waves of various frequencies (Nearing, 2020) or quantum mechanics, for the quantum wave function of a particle in a box. Step 3: Finally, substituting all the coefficients in Fourier …. Check back soon! Problem 5 Show that the complex Fourier series …. Any function can be written as the sum of an even and an odd function [ ( )]/2 A complex Lorentzian! Example: the Fourier …. Square WaveSeries Coefficients 11. The study of Fourier series is a branch of Fourier analysis. 6 Fourier Coefficients for the Complex Conjugate of a Signal. 1: Fourier series approximation to s q ( t). Once the complex wave is broken up, we can analyze the results using the simpler sine waves Consider a square wave as shown in Fig. But these are easy for simple periodic signal, such as sine or cosine waves. org odic if it repeats itself identically after a period of time. Every circle rotating translates to a simple sin or cosine wave. If a function is periodic and follows below 2 conditions, then the Fourier series for such a function exists. Thus a function or signal f(t) with period T 0 can be expressed as [0 < t < T 0] where is called the fundamental frequency or base frequency (first resonant frequency = 1/T) and all other nw 0 frequencies are called harmonics (every other component of. Fourier series in 1-D, 2-D and 3-D. A Fourier series is that series of sine waves; and we use Fourier analysis or spectrum analysis to deconstruct a signal into its individual sine wave components. Fourier series and fourier solutions square wave inverters are some mathematically and capacitors work, the average signal is our scope of a closer approximation graph the. (This is analogous to the fact that the Maclaurin series of any polynomial function is just the polynomial itself, which is a sum of finitely many powers of x. Start by forming a time vector running from 0 to 10 in steps of 0. The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity. When we break a signal down into its composite sine waves, we call it a Fourier series. A steady musical tone from an instrument or a voice has, in most cases, quite a complicated wave shape. As can clearly be seen it looks like a wave with different frequencies. The corresponding analysis equations for the Fourier series are usually written in terms of the period of the waveform, denoted by T, rather than the fundamental frequency, f (where f = 1/T). 1: Square Wave Then the Fourier Series …. Fourier Series Least Squares Curve Fit; Fourier Series Time Shift; Fourier Series Frequency Shift; Fourier Series, 4 segment; Fourier Series, var. The following method makes use of logical operators. 10 Fourier Series and Transforms (2014-5543) Complex Fourier Series: 3 - 2 / 12 Euler's Equation: eiθ =cosθ +isinθ [see RHB 3. The original signal x (t) is an square …. It can usually be solved with some Fourier series: Square wave example. So is periodic with period and its graph is shown in . The two round markers in the imaginary. which is the Fourier expansion of a square wave …. The Fourier Series (an infinite sum of trigonometric terms) gave us that formula. %Examples of Fourier Series Square Wave …. We choose xˆ[k] = 1 N X8 n=−2 x[n]e−2πiknN = 1 11 X2 n=−2 e−2πikn 11 The sum. This representation is done by summing sine and cosine functions or with complex exponentials of different amplitudes and phases. Step 2: Estimate for n=0, n=1, etc. For example, to find the Fourier series for a triangular wave …. Okay, in the previous two sections we’ve looked at Fourier sine and Fourier cosine series. PDF Interpretation of the Physical Significance of the Fourier. The Fourier series represents the sum of smooth sinusoids, but a square wave …. As a tool to better understand the Fourier series coefficients for the square …. Every 2πperiodic function that is analytic in a neighborhood of the real axis has a Fourier series …. In the interval (c, c+2ℓ), the complex form of Fourier series is given by. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex …. ∴ Given waveform is Non-periodic so, the Fourier series will NOT exist. Left click and drag the [ball, green] circles to change the magnitude of each Fourier functions [Sin nf, Cos nf]. B Square Wave The square or rectangular waveform is similar to the sawtooth in that the amplitudes of the har-monics follow the A N = A 1=N dependence. The Fourier transform is represented as spikes in the frequency domain, the height of the spike showing the amplitude of the wave of that …. More information about the Gibbs phenomenon. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Imagine a thin piece of wire, which only gains or loses heat through its ends. If you ally infatuation such a referred fourier series examples and solutions square wave ebook that will provide you worth, get the extremely best seller from us currently from several preferred authors. The coefficients fb ng1 n=1 in a Fourier sine series F(x) are determined by. 23 ), are a measure of the extent to which has components along each of the basis vectors. The diagram above shows a "saw-tooth" wave …. entities represented by symbols such as ∞ n=−∞ a n, ∞ n=−∞ f n(x), and ∞ −∞ f(x) dx are central to Fourier Analysis. Fourier series Fourier series in higher dimensions (vector notation) Complex Exponentials In 2-D, the building blocks for periodic function f(x 1;x 2) are the product of complex exponentials in one variable. INTRODUCTION During Napoleon's reign, Joseph Fourier (1768-1830) was one of the French scientists who took French science to new heights. First we see that fcan be expressed in terms of the standard square wave as f(t) = 1 + sq t+ ˇ 2 : Now (see overleaf) the Fourier series for sq(t. width; Fourier Series, Continuous; Fourier Series, double Pulse; Fourier Series Calculation Square; Fourier Series Calculation Triangle; Fourier Series Square to Triangle Wave; Fourier Series …. Film or TV show with spooky dead trees and singing Explicit triples of isomorphic Riemann surfaces Typeset sudoku grid using tabular syntax. Aug 15, 2013 - The first one is the exponential form of the Fourier series and the. It is typical ( but as far as I know not required) that complex …. We look at a spike, a step function, and a ramp—and smoother functions too. The computer algorithm for Fourier transforms is called an FFT (Fast Fourier …. Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products. Determine the Fourier series coefficients of ( ) 𝑎 ( ). If you work through the math, you can find the optimal values for cn using equation [3]: [Equation 4] Note that the Fourier coefficients are complex numbers, even though the series in Equation [1], evaluated with the coefficients in Equation [4], result in a real function. the same way that it did for the complex Fourier series we talked about earlier, only we have to replace an integral with a sum. In other words, Fourier series can be used to express a function in terms of the frequencies () it is composed of. The Fourier transform tells us what frequency components are present in a given signal. How do I plot the Fourier series for a square wave? [closed] Ask Question Asked 4 years, 6 months ago. A Fourier series (/ ˈ f ʊr i eɪ,-i ər /) is a sum using only basic waves chosen to mathematically represent the waveform for almost any periodic function. That sawtooth ramp RR is the integral of the square wave. 6 Complex Fourier Series The exponential form of sine and cosine can be use to give: f(x) = X r c re 2ˇirx L The coe cients can be calculated from: c r= 1 L Z x o+L x o. Wave Equation and Fourier Series…. A complex waveform can be constructed from, or decomposed into, sine (and cosine) waves of various amplitude and phase relationships. As useful as this decomposition was in this example, it does not generalize well to other periodic signals: How can a superposition of pulses equal a smooth signal like a sinusoid?. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. The Fourier series can be applied to periodic signals only but the Fourier transform can also be applied to non-periodic functions like rectangular pulse, step functions, ramp function etc. In this section we define the Fourier Cosine Series, i. You can use the following commands to calculate the nth partial sum of the Fourier series of the expression f on the interval [-L,L] syms x k L n. Similar expressions hold for more general intervals [ a, b] by shifting and scaling appropriately. a demo in Maple of complex fourier series. The formula for the fourier series of the function f(x) in the interval [-L, L], i. As promised in the first part of the Fourier series we will now demonstrate a simple example of constructing a periodic signal using the, none other then, Fourier series…. The amplitude of the n-th harmonic oscillation A n. f (t) has a finite number of maxima. An in nite sum as in formula (1) is called a Fourier series (after the French engineer Fourier who rst considered properties of these series. 14 The Complex Fourier Series 138 4. It establishes a relation between a function in the domain of time and a function in the domain of frequency. representing a function with a series in the form Sum( A_n cos(n pi x / L) ) from n=0 to n=infinity. More instructional engineering videos can be found . (a) The function and its Fourier series 0 0. This last line is the complex Fourier series. A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series…. Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 1978 The use of Fourier analyzed square waves in the determination of the modulation tra. Given the mathematical theorem of a Fourier series, the period function f(t) can be written as follows: f(t) = A 0 + A 1 cosωt + A 2 cos2ωt + … + B 1 sinωt + B 2 sin2ωt + … where: A 0 = The DC component of the original wave. So I’ve just started learning about the Fourier series and wish to calculate one for a square wave function, my working is as follows. In some simple cases, Fourier series can be found using purely algebraic methods. approximations to the continuous Fourier series in Chapter 2. Also recall that the real part u and the imaginary part v of an analytic function f = u+iv are harmonic. If history were more logical they might have been found this way. Duration: 11:46 Complex Fourier Series - Square Wave …. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation. In this method, if N harmonics are included in the truncated Fourier series, then the amplitude of the kth harmonic is multiplied by (N - k)/N. Our aim was to find a series of trigonometric expressions that add to give certain periodic curves (like square or sawtooth waves. The result of the FFT is an array of complex numbers. The fine oscillations at the edges do not disappear even if the Fourier series takes many more terms. 1 The Fourier Series Components of an Even Square Wave Components of cos(nt) found in the approximation to an even square wave may be calculated generally as a function of n for all n > 0. Y = fft (X) computes the discrete Fourier transform (DFT) of X using a fast Fourier transform (FFT) algorithm. The Fourier Transform can be used for this purpose, which it decompose any signal into a sum of simple sine and cosine waves that we can easily measure the frequency, amplitude and phase. 5, and the one term expansion along with the function is shown in Figure 2: Figure 2. Suppose that our wave is designed by 18. 3 Signal Synthesis Later on in this lab you will use a Fourier series to approximate a square wave. b) Derive the expression for the Fourier coefficients a k. Example: Determine the fourier series of the function f(x) = 1 - x 2 in the interval [-1, 1. Here's the Fourier series for a square wave, truncated at the first 25 terms (which might sound like a lot, but is really easy for a computer to handle:) and \( \cos (\omega t)$$. Furthermore, suppose that the signal is periodic with period T: for all t we have xs(t) = xs(t +T). The plot in black color shows how the reconstructed (Fourier Synthesis) signal will look like if the three terms are combined together. The construction of a PERIODIC signal on the basis of Fourier coefficients which give the AMPLITUDE and PHASE angle of each component sine wave HARMONIC. amplitude, frequency, and starting phase amplitudes, frequencies, and starting phases. For math, science, nutrition, history. 1 Introduction The concepts of infinite series and improper integrals, i. Deriving the Fourier Coefficients. Turns out all the sines and cosines (or the equivalent complex …. This index corresponds to the k -th harmonic of the signal's period. Over the range [0,2L], this can be written as f(x)=2[H(x/L)-H(x/L-1)]-1, (1) where H(x) is the Heaviside step . Let be a -periodic function such that for Find the Fourier series for the parabolic wave. Apply integration by parts twice to find: As and for integer we have. It is applicable only to periodic signals. Pre-lab: Theory: The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. 1) Note that a 0 is the average of the function over the interval. Author name; Kyle Forinash; Wolfgang Christian. A real number, (say), can take any value in a continuum of values lying between and. LAB REPORT Experiment 2 - Fourier series of Square, Triangle, and Sawtooth Pulse Trains Experiment 2- Fourier series of Square, Triangle, and Sawtooth Pulse Trains: 1). Complex Fourier coefficients: c n = 1 T That is the infinite Fourier series of the square wave function of period 1. In calculations involving Fourier series it is often advantageous to use complex exponentials;. The complex exponential form of cosine. Fourier Series on the Complex Plane. In order to do this, let us consider a pair of FC trigonometric. 3–80 Express the Square of a Sinusoid as a Sum of Complex Exponentials square sinusoid complex exponential. Suppose you want to make a periodic wave — maybe it's for a music synthesizer or something. Let us now show that the usual formulas giving the Fourier coefficients, in terms of integrals involving the corresponding real functions, follow as consequences of the analytic properties of certain complex functions within the open unit disk. Here are a number of highest rated Square Fourier Series pictures upon internet. What exactly is a Fourier series…. For square wave with period T and x0 = -T/2 Split the a[n] evaluation integral to two parts, -T/2,0> and (0,T/2>: Therefore: Split the b[n] evaluation integral to two parts: Therefore: The complex coefficients can be obtained from trigonometric coefficients as follows: Fourier Series of Full-wave Rectified Sine Wave. It may be the worst way to graph a square but it's fun! I made it because I want to learn complex integration and Fourier Series. Then the adjusted function f (t) is de ned by f (t)= f(t)fort= p, p Z ,. Your first 5 questions are on us!. Fourier Series Print This Page Download This Page; 1. Electrical Engineering Q&A Library ermine the complex exponential Fourier series of. We identified it from well-behaved source. Assuming you're unfamiliar with that, the Fourier Series is simply a long, intimidating function that breaks down any periodic function into a simple series of sine & cosine waves. We want to know the amplitude of the wave at the detector in the u,v plane, which is a distance z from the x,y plane. The Fourier series of a periodic function is given by. 4 Complex Tones, Fourier Analysis and The Missing Fundamental. ∞ ∑ k = − ∞αkekjπx / ℓ (j2 = − 1), where a signal's complex Fourier …. • angle – Computes the phase angle of a complex number. Use this information and complete the entries in Table 2 for the square wave. It is instructive to plot the first few terms of this Fourier series and watch the approximation improve as more terms are included, as shown in Figure 9. Fourier Series in Filtering 5 The Matlab commands below1 will sketch the symmetric partial sum with subscripts up to N. The Fourier Transform and Free Particle Wave Functions 1 The Fourier Transform 1. Figure 6-1 Successive Fourier series approximation to a square wave by adding terms. By applying Euler's identity to the compact trigonometric Fourier series, an arbitrary periodic signal can be expressed as a sum of complex exponential functions: (11. If one would like to approximate a function over a larger interval one would need terms of very high order. Do not worry too much about the math, but focus on the results and specifically for this course, the application. The frequency of each wave in the sum, or harmonic, . Calculating the 2D Fourier Transform of The Image. And I think these are the remaining entries on the list. frequency-phase series of square waves (the equivalent of the polar Fourier Theorem but . Fourier Series, Fourier Transforms and the Delta Function. Here we see that adding two different sine waves make a new wave: When we add lots of them (using the sigma function Σ as a handy notation) we can get things like: 20 sine waves: sin (x)+sin (3x)/3+sin (5x)/5 + + sin (39x)/39: Fourier Series Calculus Index. SC FINAL COMPLETE FOURIER SERIES CHAPTER 4 EXERCISE 4. Assume that the input voltage is the following square wave (𝜔 =𝜋),. Determine the Fourier series expansion for full wave. Input arguments are used to specify the number of uniformly spaced points at which the. Fourier series, then the expression must be the Fourier series of f. Where, C is known as the Complex Fourier …. Transition from Fourier series to Fourier transforms. series and Fourier transforms are mathematical techniques that do exactly that!, i. In this section we define the Fourier Sine Series, i. For the square wave a discontinuity exists at t/T 0 = 0. We can equivalently describe them as sums of complex exponentials, where each cosine requires two complex …. 10 Fourier Series and Transforms (2014-5543) Complex Fourier Series: 3 – 2 / 12 Euler’s Equation: eiθ =cosθ +isinθ [see RHB 3. What is Fourier Series? Any real, periodic signal with fundamental freq. Thus, second harmonic component of Fourier series will be 0. The inverse Fourier transform given above (Eq. Jean Baptiste Joseph Fourier (1768-1830) ‘Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies. The steps to be followed for solving a Fourier series are given below: Step 1: Multiply the given function by sine or cosine, then integrate. 8 Complex Form of Fourier Series. Ans: The Fourier series is a linear combination of sines and cosines that expands periodic signals, whereas the Fourier transform is a method or …. 2 Function spaces and metrics 5. A plane wave is propagating in the +z direction, passing through a scattering object at z=0, where its amplitude becomes A o(x,y). 1 Complex Full Fourier Series Recall that DeMoivre formula implies that sin( ) =. Fourier series using complex variables. Fourier series of a simple linear function f (x)=x converges to an odd periodic extension of this function, which is a saw-tooth wave…. Compute and plot the intensity distribution of the diffracted wave at different distances from the aperture given by the Fresnel numbers N F = 20, 10, 4, and 1. 01: MATLAB M-FILE FOR PLOTTING TRUNCATED FOURIER SERIES AND ITS SPECTRA. The Fourier Series representation of continuous time periodic square wave signal, along with an interpretation of the Fourier series …. Fourier transforms and solving the damped, driven oscillator. The coefficients may be determined rather easily by the use of Table 1. True Square waves are a special class of rectangular waves …. Here is why a square wave is a good test of high frequencies: The Fourier series corresponding to the square wave includes an infinite number of odd-harmonic sine wave components. The input to a discrete function must be a whole number. Examples of the Fourier Series · Rectangular Pulse Train · Impulse Train · Decaying Exponential Pulse Train · Even Square Wave · Odd Square Wave. Here, I’ll use square brackets, [], instead of parentheses, (), to show discrete vs continuous time functions. In this Tutorial, we consider working out Fourier series for func-tions f(x) with period L = 2π. Experts are tested by Chegg as specialists in their subject area. The vowel signal and the square-wave are both examples that suggest the idea ofapproximating a periodic On the other hand, when the positive and negative frequency terms in the Fourier Series are combined we add a complex …. Let the integer m become a real number and let the coefficients, F m, become a function F(m). The Fourier theory is used to analyze complex periodic signals. The first step is a trivial one: we need to generalize from real functions to complex functions, to include wave functions having nonvanishing current. Jean Baptiste Joseph Fourier …. This is a consequence of $\cos$ being "half" of a complex exponential but a constant is a "full" complex exponential. The complex Fourier series is more elegant and shorter to write down than the one expressed in term of sines and cosines, but it has the disadvantage that the coefficients might be complex even if the given function is real-valued. English Wikipedia has an article on: Fourier series. PHY 416, Quantum Mechanics Notes by: Dave Kaplan and Transcribed to LATEX by: Matthew S. The generalization of Fourier series to forms appropriate …. Given a periodic function xT(t) and its Fourier Series representation (period= T, ω0=2π/T ): xT (t) = +∞ ∑ n=−∞cnejnω0t x T ( t) = ∑ n = − ∞ + ∞ c n e j n ω 0 t. A square wave is a type of waveform where the signal has only two levels. A Fourier series represents a function as an infinite sum of trigonometric functions: You can often use the complex exponential (DeMoivre's formula) to simplify computations: Specifically, let This graph is called a square wave…. The resulting series is known as Fourier series. Algorithm: Fourier Transform. This gif demonstrates how combinations of Fourier series mathematical functions can be used to create complex animated surfaces like these: Fourier Series, & Rectify. What we are extracting in these cases are the coefficients for the Fourier …. Consider the square wave: f(x) = 1 0 ≤ x < π = 0 −π ≤ x < 0 f(x) = f(x+2π) This appears to be a di cult case - the rather angular square wave does not look as …. In the previous lab, we implemented the trigonometric form of the Fourier Series, in this lab we will implement the complex form of the Fourier Series while learning some additional features of Simulink. Fourier Series Representation • Fourier Series Representation of CT Periodic Signals A periodic signal 𝑥𝑥(𝑡𝑡) can be represented as a linear combination of harmonically related complex exponentials (or sinusoids) 𝑎𝑎. The inverse Fourier transform is just to reconstruct the original function. Some operational formulas : 70. Graph of the function and its Fourier …. | 2022-05-22T08:12:33 | {
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https://math.stackexchange.com/questions/3063596/optimization-problem-rectangle-inscribed-below-parabola-possible-textbook-mi | # Optimization problem (rectangle inscribed below parabola) - possible textbook mistake
I have been working on a textbook's Optimization problem but the answer that I got does not match the textbook's answer. I would like to make sure I got it right (I can't find any mistakes on my solution), so I would like to ask for someone's help. I would appreciate that.
The problem:
A rectangle is located below a parabola, which is given by $$y = 3x- \frac{x^2}{2}$$ in such a way that its two superior vertexes are placed on the parabola and its two inferior vertexes are located on the $$x$$ axis. The left, inferior vertex, is placed on the point $$(c,0)$$. That said: a) Show that the area of the rectangle can be represented by the equation $$A(c) = c^3 -9c^2 + 18c$$ b) Find the rectangle's height and width given that is has maximum possible area.
c) What is that area?
Instead of typing the whole solution I will post an image with it (sorry for that but latex-ing it would take a lot of time!). You will find my solution below.
If that's helpful, the textbook's answer for b) and c) are $$3-\sqrt{3} \times 3$$ and $$9 + 9\sqrt{3}$$ respectively.
Thank you.
• $base=2\sqrt{3}$ – Aleksas Domarkas Jan 6 at 8:19
You made a slight error when trying to find the base:
$$c = 3-\sqrt{3} \implies b = 2(3-c) = \color{blue}{2\left[3-\left(3-\sqrt{3}\right)\right]} = 2\sqrt{3}$$
You forgot the $$3$$ in $$(\color{blue}{3}-c)$$ and found $$b = 2c$$ instead.
Addition: From here, using $$h = 3$$ as you found, you get
$$A = bh \iff A = 3\left(2\sqrt{3}\right) = 6\sqrt{3}$$
Try plugging in $$c = 3-\sqrt{3}$$ in $$f(c)$$:
$$f\left(3-\sqrt{3}\right) = \left(3-\sqrt{3}\right)^3-9\left(3-\sqrt{3}\right)^2+18\left(3-\sqrt{3}\right) = 6\sqrt{3}$$
Through confirmation, you can see this point coincides with the local maximum.
• thank you I've fixed that mistake. Now I have $2 \sqrt{3}$ as a base and $3$ as the rectangle's height, which gives me an area of $A = 6 \sqrt{3} \ m^2$. Am I good? Thanks for your help! – bru1987 Jan 6 at 9:16
• Yes, that’s the correct answer! – KM101 Jan 6 at 9:24
• thank you and have a great day =) – bru1987 Jan 6 at 9:26
• Glad to have helped! :-) You may want to mark the answer as accepted if it has helped resolved the issue. – KM101 Jan 6 at 9:31 | 2019-08-17T10:53:59 | {
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https://mathoverflow.net/questions/372450/parity-of-the-multiplicative-order-of-2-modulo-p | # Parity of the multiplicative order of 2 modulo p
Let $$\operatorname{ord}_p(2)$$ be the order of 2 in the multiplicative group modulo $$p$$. Let $$A$$ be the subset of primes $$p$$ where $$\operatorname{ord}_p(2)$$ is odd, and let $$B$$ be the subset of primes $$p$$ where $$\operatorname{ord}_p(2)$$ is even. Then how large is $$A$$ compared to $$B$$?
• $A/(A+B)$ tends to $7/24$ ? (not proved yet). – Henri Cohen Sep 23 at 21:39
• Seems like an interesting question, and clearly generalizable quite a lot. However, if you're going to ask many questions on this site, it would be a good idea to learn a little bit of TeX formatting. I've fixed the formatting of your question, so if you click on "edit", you'll be able to see what I did to make it more readable. I also changed the title of your question to make it even clearer what you're asking. – Joe Silverman Sep 23 at 21:54
• @HenriCohen how did you determine $A/(A+B)$ to be $7/24$ while also writing "not proved yet"? The proportion of $p \leq 100000$ for which $2 \bmod p$ has odd order is $2797/9591$, which as a continued fraction is $[0,3,2,3,44,9]$, and the truncated continued fraction $[0,3,2,3]$ is $7/24$. I'd be interested to know if you did that or something else. – KConrad Sep 24 at 3:26
• Note: the set $A$ is at OEIS: oeis.org/A014663, while its complement $B$ is oeis.org/A091317. Among the 46 primes below 200, $A$ consists of the 14 primes 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199. – YCor Sep 24 at 9:44
• see this answer – René Gy Sep 24 at 15:39
This problem was asked by Sierpinski in 1958 and answered by Hasse in the 1960s.
For each nonzero rational number $$a$$ (take $$a \in \mathbf Z$$ if you wish) and each prime $$\ell$$, let $$S_{a,\ell}$$ be the set of primes $$p$$ not dividing the numerator or denominator of $$a$$ such that $$a \bmod p$$ has multiplicative order divisible by $$\ell$$. When $$a = \pm 1$$, $$S_{a,\ell}$$ is empty except that $$S_{-1,2}$$ is all odd primes. From now on, suppose $$a \not= \pm 1$$.
In Math. Ann. 162 (1965/66), 74–76 (the paper is at https://eudml.org/doc/161322 and on MathSciNet see MR0186653) Hasse treated the case $$\ell \not= 2$$. Let $$e$$ be the largest nonnegative integer such that $$a$$ in $$\mathbf Q$$ is an $$\ell^e$$-th power. (For example, if $$a$$ is squarefree then $$e = 0$$ for every $$\ell$$ not dividing $$a$$.) The density of $$S_{a,\ell}$$ is $$\ell/(\ell^e(\ell^2-1))$$. This is $$\ell/(\ell^2-1)$$ when $$e = 0$$ and $$1/(\ell^2-1)$$ when $$e = 1$$.
In Math. Ann. 166 (1966), 19–23 (the paper is at https://eudml.org/doc/161442 and on MathSciNet see MR0205975) Hasse treated the case $$\ell = 2$$. The general answer in this case is more complicated, as issues involving $$\ell$$-th roots of unity in the ground field (like $$\pm 1$$ in $$\mathbf Q$$ when $$\ell = 2$$) often are. The density of $$S_{a,2}$$ for "typical" $$a$$ is $$1/3$$, such as when $$a \geq 3$$ is squarefree. But $$S_{2,2}$$ has density 17/24, so the set of $$p$$ for which $$2 \bmod p$$ has even order has density $$17/24$$ and the set of $$p$$ for which $$2 \bmod p$$ has odd order has density $$1 - 17/24 = 7/24$$.
For example, there are $$167$$ odd primes up to $$1000$$, $$1228$$ odd primes up to $$10000$$, and $$9591$$ odd primes up to $$100000$$. There are $$117$$ odd primes $$p \leq 1000$$ such that $$2 \bmod p$$ has even order, $$878$$ odd primes $$p \leq 10000$$ such that $$2 \bmod p$$ has even order, and $$6794$$ odd primes $$p \leq 100000$$ such that $$2 \bmod p$$ has even order. The proportion of odd primes up $$1000$$, $$10000$$, and $$100000$$ for which $$2 \bmod p$$ has even order is $$117/167 \approx .700059$$, $$878/1228 \approx .71498$$, and $$6794/9591 \approx .70837$$, while $$17/24 \approx .70833$$.
The math.stackexchange page here treats $$S_{7,2}$$ in some detail and at the end mentions the case of $$S_{2,2}$$.
• Thanks @KConrad for the expression. I was thinking about a combinatorial property of cyclic groups of prime order(called acyclic matching property) and I proved the above mentioned sequence of primes does not hold it. See Proposition 2.3 of core.ac.uk/download/pdf/33123051.pdf Anyway I will like to mention this result in my ongoing research work as a remark, and hence I ask your permission for the same, of course with acknowledgment. – Mohsen Sep 25 at 18:09
• Since the result is due to Hasse, cite his paper when you want to indicate who first showed that the density exists and what its value is. – KConrad Sep 25 at 19:35 | 2020-10-31T04:14:33 | {
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https://web.forret.com/ruv2b6uq/67b03a-gibson-es-345-stereo | File history. Simple C# class to calculate Cantor's pairing function. \end{array} Cantor's pairing function 08 17 In addition to the diagonal arguments , Georg Cantor also developed the Cantor pairing function $$\mathbb{N}^2 \to \mathbb{W}, \quad c(x,y) = \binom{x+y+1}{2}+x = z$$ , which encodes any two numbers $$x,y \in \mathbb{N}$$ in a new number $$z \in \mathbb{N}$$ . var t = ( int) Math. But we know that the end-points survive the Cantor intersection, that is they lie in C. Hence [x 1=3k;x+ 1=3k] f xginter-sects Cfor every k. The ideas discussed in this post are implemented using GHC generics in the package cantor-pairing. Maybe your data comes from two different databases, and each one has its unique identifier for individuals, but both unique codings overlap with each other. In a perfectly efficient function we would expect the value of pair(9, 9) to be 99. For example, the Cantor pairing function π: N 2 → N is a bijection that takes two natural numbers and maps each pair to a unique natural number. When we apply the pairing function to k 1 and k 2 we often denote the resulting number as k 1, k 2 . Set theory - Set theory - Equivalent sets: Cantorian set theory is founded on the principles of extension and abstraction, described above. Essentially any time you want to compose a unique identifier from a pair of values. In this paper, some results and generalizations about the Cantor pairing function are given. Already have an account? Python Topic. \end{array} Elements in Cantor's ternary set x, y such that x + y = w with w ∈ [0, 1] It is known that a real number w ∈ [0, 1] can be written as a sum of two real numbers such that x, y ∈ C such that 1 2C + 1 2C = [0, 1] with C the ternary Cantor's set. Maria E. Reis is a visiting MSc student at the University of Kentucky under Dr. Costa’s supervision. \end{array} The so-called Cantor pairing function C(m;n) = mX+n j=0 j + m = 1 2 (m+ n)(m+ n+ 1) + m; maps N 0 N 0 injectively onto N 0 (Cantor, 1878). More formally Tags encoding, pairing, cantor Maintainers perrygeo Classifiers. \end{array} This can be easily implemented in any language. You can also compose the function to map 3 or more numbers into one — for example maps 3 integers to one. What makes a pairing function special is that it is invertable; You can reliably depair the same integer value back into it's two original values in the original order. OS Independent Programming Language. Let C be the projection of the standard (ternary) Cantor set on the unit interval to the circle. Yes, the Szudzik function has 100% packing efficiency. ElegantPairing.nb Ç Å ¡ 3 of 12 Cantor’s Pairing Function Here is a classic example of a pairing function (see page 1127 of A … cantor-pairing. Cantor’s pairing function c(x 1,x 2) is a quadratic polynomial pairing func-tion. It is also used as a fundamental tool in recursion theory and other related areas of mathematics (Rogers, 1967; Matiyasevich, 1993). (32.4 x 36.2 cm) By (primary) Artist unknown Washington, D.C.-based Cantor Colburn associate Jenae Gureff attended the AIPLA 2015 Mid-Winter Institute meeting in Orlando. \end{array} Whether this is the only polynomial pairing function is still an open question. Definition: A set S is a countable set if and only if there exists a bijective mapping , where is the set of natural numbers (i.e. In fact, Cantor's method of proof of this theorem implies the existence of an " … (It's not an edge in an TikZ path operator kind of way.) Anatole Katok, Jean-Paul Thouvenot, in Handbook of Dynamical Systems, 2006. 2y & : y \ge 0 Floor ( ( -1 + Math. Please include this proof (either directly or through a link) in your answer. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Neither Cantor nor Szudzik pairing functions work natively with negative input values. When we apply th… Cantor’s grades at age 8, when he attended the St.Petri-Schule for German speaking people in St.Petersburg. 'Cantor' and 'Elegant' pairing are relatively symmetric around the main diagonal. The trick to solve this is to either factorize the input, or pass in x – min(x). Cantor Pairing. Economics, programming, and games. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. An illustration of Cantor's Pairing Function. One of the better ways is Cantor Pairing, which is the following magic formula: This takes two positive integers, and returns a unique positive integer. ... (16) S. R. Jaskunas, C. R. Cantor, and I. Tinoco, Jr , manuscript in preparation. For that, you sort the two Cantor normal forms to have the same terms, as here, and just add coordinate-wise. Definition: A set S is a countable set if and only if there exists a bijective mapping , where is the set of natural numbers (i.e. b^2 + a & : a < b\\ b^2 + a & : a < b\\ If one defines cantor 2 edge/.style={move to} the diagonal part will not be drawn. We consider the theory of natural integers equipped with the Cantor pairing function and an extra relation or function Xon N. When Xis equal either to multiplication, or coprimeness, or divisibility, or addition or natural ordering, it can be proved that the theory Th(N;C;X) is undecidable. -c - 1 & : (a < 0 \cap b \ge 0) \cup (a \ge 0 \cap b < 0) This graphics demonstrates the path that Szudzik takes over the field: The primary benefit of the Szudzik function is that it has more efficient value packing. However, cantor(9, 9) = 200. If (m;n) is the row-column indexing, C(m;n) gives the following pattern of enumeration: 0 1 3 6 10 15 2 4 7 11 16 5 8 12 17 9 13 18 14 19 20 To check that C(m;n) is indeed a bijection, we need the below property. \right.$$,$$index = {(a + b)(a + b + 1) \over 2} + b$$,$$index(a,b) = \left\{\begin{array}{ll} Additional space can be saved, giving improved packing efficiency, by transferring half to the negative axis. Thus this solvent is an excellent system in which to study the effects of base pairing on the for- mation of specific complexes between mono- or oligonucleotides. a^2 + a + b & : a \ge b $$b = \left\{\begin{array}{ll} Cantor’s school career was like that of … It should be noted that this article was adapted from an earlier jsfiddle of mine. Now use the pairing function again to turn your two remaining items into 1. As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. Matt Ranger's blog. That fiddle makes note of the following references:$$index = \left\{\begin{array}{ll} Date/Time Thumbnail Dimensions User Comment; current: 16:12, 10 June 2020: 432 × 432 (39 KB) Crh23: Another JavaScript example: Szudzik can also be visualized as traversing a 2D field, but it covers it in a box-like pattern. Definition A pairing function on a set A associates each pair of members from A with a single member of A, so that any two distinct pairs are associated with two distinct members. Definition A pairing function on a set A associates each pair of members from A with a single member of A, so that any two distinct pairs are associated with two distinct members. 1 - Planning Intended Audience. It’s also reversible: given the output of you can retrieve the values of and . The typical example of a pairing function that encodes two non-negative integers onto a single non-negative integer (therefore a function ) is the Cantor function, instrumental to the demonstration that, for example, the rational can be mapped onto the integers.. Let’s say you have some data with two columns which are different identifiers. The third and last one (POTO pairing) is more asymmetric. We prove a conjecture of I. Korec [4] on decidability of some fragments of arithmetic equipped with a pairing function; as consequence, we give an axiomatization of the fragment of arithmetic equipped with Cantor pairing function, precising a result of [5]. ElegantPairing.nb For a 32-bit unsigned return value the maximum input value for Szudzik is 65,535. The typical example of a pairing function that encodes two non-negative integers onto a single non-negative integer (therefore a function ) is the Cantor function, instrumental to the demonstration that, for example, the rational can be mapped onto the integers.. The first order theory of natural integers equipped with the The twist for coding is not to just add the similar terms, but also to apply a natural number pairing function also. You may implement whatever bijective function you wish, so long as it is proven to be bijective for all possible inputs. Construct the “uniform” measure μ on C by assigning the measures 1/2 n to the intersection of C with the intervals of nth order. The full results of the performance comparison can be found on jsperf. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The binary Cantor pairing function C from N × N into N is defined by C(x, y) = (1/2) (x + y)(x + y + 1) + y. \right.$$, https://en.wikipedia.org/wiki/Pairing_function. Or maybe you want to combine encodings from multiple columns into one. So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. Sometimes you have to encode reversibly two (or more) values onto a single one. It is however mixing. Click on a date/time to view the file as it appeared at that time. It’s also reversible: given the output of you can retrieve the values of and . \end{array} The only problem with this method is that the size of the output can be large: will overflow a 64bit integer 1. \right.$$ Development Status. This means that all one hundred possible variations of ([0-9], [0-9]) would be covered (keeping in mind our values are 0-indexed). x^2 + x + y & : x \ge y The primary downside to the Cantor function is that it is inefficient in terms of value packing. To describe some results based upon these principles, the notion of equivalence of sets will be defined. Cantor pairing gives us an isomorphism between a single natural number and pairs of natural numbers. We may assume y k 6= xhere (if xhappens to be an end-point of an interval in C k itself, choose the other end-point of the interval to be y k). Pairing functions take two integers and give you one integer in return. cantor (9, 9) = 200 szudzik (9, 9) = 99. y^2 + x & : x < y\\ The Cantor pairing function C from N × N into N is defined by C (x, y)=(1 / 2) ( x + y )( x + y +1 )+ y . Items portrayed in this file depicts. So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. This is the question Cantor pondered, and in doing so, came up with several interesting ideas which remain important to this day. In particular, it is investigated a very compact expression for the n -degree generalized Cantor pairing function (g.C.p.f., for short), that permits to obtain n −tupling functions which have the characteristics to be n -degree polynomials with rational coefficients. Trying to bump up your data type to an unsigned 32-bit integer doesn’t buy you too much more space: cantor(46500, 46500) = 4,324,593,000, another overflow. Developers Science/Research License. One of the better ways is Cantor Pairing, which is the following magic formula: This takes two positive integers, and returns a unique positive integer. The only problem with this method is that the size of the output can be large: will overflow a 64bit integer 1. By repeatedly applying Cantor’s pairing function in this \right.$$We quickly start to brush up against the limits of 32-bit signed integers with input values that really aren’t that large. It should be noted though that all returned pair values are still positive, as such the packing efficiency for both functions will degrade. 2 This measure is obviously singular. The good news is that this will use all the bits in your integer efficiently from the view of a hashing function. Like Cantor, the Szudzik function can be easily implemented anywhere. While this is cool, it doesn’t seem useful for practical applications. Set theory - Set theory - Operations on sets: The symbol ∪ is employed to denote the union of two sets. So we use 200 pair values for the first 100 combinations, an efficiency of 50%. -2y - 1 & : y < 0\\ Rider to Pair of Horses with Riders 3rd century BCE-3rd century 3rd C. BCE-3rd C. CE Asia, China 12 3/4 x 14 1/4 in. by Georg Cantor in 1878. Journal of the American Chemical Society 90:18 / August 28, 1968 . . Sqrt ( 1 + 8 * cantor )) / 2 ); Sign up for free to join this conversation on GitHub . Come in to read stories and fanfics that span multiple fandoms in the Naruto and Big Mouth universe. This package provides a modern API to this functionality using GHC generics, allowing the encoding of arbitrary combinations of finite or countably infinite types in natural number form. Because theoreticaly I can now Pair … k in C k such that jx y kj 1=3k.$$index = {(x + y)(x + y + 1) \over 2} + y$$. Let … a^2 + a + b & : a \ge b In this paper, some results and generalizations about the Cantor pairing function are given. In this ramble we will cover two different pairing functions: Cantor and Szudzik. pairing function. This makes it harder to retrieve x and y, though.↩, “Key registers on keyboard but not on computer” fix, Bad Economics: Shame on you, Planet Money (MMT episode), BadEconomics: Putting 400M of Bitcoin on your company balance sheet, Starting a Brick & Mortar Business in 2020, The publishing format defines the art: How VHS changed movie runtimes, The rural/urban divide is an American phenomenon and other bad takes, Why Stephen Wolfram’s research program is a dead end, “bigger” than the infinity of normal numbers. It can be used when one index should grow quicker than the other (roughly hyperbolic). -2x - 1 & : x < 0\\ The problem is, at least from my point of view, in Java I had to implement a BigSqrt Class which I did by my self. As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. 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Cordless Strimmer With Blades, Banana Peppers Recipes, Access Community Health Network Careers, Is Brf3 Polar Or Nonpolar, Psychological Barriers To Communication, | 2021-10-22T16:43:56 | {
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https://math.stackexchange.com/questions/1124242/how-to-calculate-exp-x-using-taylor-series | # How to calculate $\exp(-x)$ using Taylor series
We know that the Taylor series expansion of $e^x$ is $$e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}.$$
If I have to use this formula to evaluate $e^{-20}$, how should I check for convergence?
MATLAB gives $e^{-20} = 2.0612e-009$. But when I use this formula to calculate it, I get $4.1736e-009$. My answer becomes more inaccurate as I take $x = -30, -40 \cdots$. The reason is obvious. This is because as the number of terms in the Taylor series increases, the factorial becomes more and more inaccurate using MATLAB.
1. How can I use Taylor series to get accurate values of $e^x$ in MATLAB?
2. How does MATLAB built in command "exp()" calculate the exponential?
• If the ultimate goal is to approximate the exponential I would use a Pade' approximants (a rational approximation), combined with $e^{2x}=(e^x)^2$ to get values for larger values of $x$. – Pp.. Jan 28 '15 at 23:39
• If you just want to see how the Taylor series behaves, then the error of an alternating series can be bounded by the absolute value of the next term of the series. – Pp.. Jan 28 '15 at 23:40
• TS is good for small values of the argument. Otherwise you might end up having many extra terms. – Kaster Jan 28 '15 at 23:42
• You may try summing backwards. – lhf Jan 29 '15 at 11:44
• Oog. Why not start the summation at $0$? – Qiaochu Yuan Jan 30 '15 at 8:54
The Problem
The main problem when computing $e^{-20}$ is that the terms of the series grow to $\frac{20^{20}}{20!}\approx43099804$ before getting smaller. Then the sum must cancel to be $\approx2.0611536\times10^{-9}$. In a floating point environment, this means that $16$ digits of accuracy in the sum are being thrown away due to the precision of the large numbers. This is the number of digits of accuracy of a double-precision floating point number ($53$ bits $\sim15.9$ digits).
For example, the RMS error in rounding $\frac{20^{20}}{20!}$, using double precision floating point arithmetic, would be $\sim\frac{20^{20}}{20!}\cdot2^{-53}/\sqrt3\approx3\times10^{-9}$. Since the final answer is $\approx2\times10^{-9}$, we lose all significance in the final answer simply by rounding that one term in the sum.
The problem gets worse with larger exponents. For $e^{-30}$, the terms grow to $\frac{30^{30}}{30!}\approx776207020880$ before getting smaller. Then the sum must cancel to be $\approx9.35762296884\times10^{-14}$. Here we lose $25$ digits of accuracy. For $e^{-40}$, we lose $33$ digits of accuracy.
A Solution
The usual solution is to compute $e^x$ and then use $e^{-x}=1/e^x$. When computing $e^x$, the final sum of the series is close in precision to the largest term of the series. Very little accuracy is lost.
For example, the RMS error in computing $e^{20}$ or $e^{-20}$, using double precision floating point arithmetic, would be $\sim8\times10^{-9}$; the errors are the same because both sums use the same terms, just with different signs. However, this means that using Taylor series, \begin{align} e^{20\hphantom{-}}&=4.85165195409790278\times10^8\pm8\times10^{-9}\\ e^{-20}&=2\times10^{-9}\pm8\times10^{-9} \end{align} Note that the computation of $e^{-20}$ is completely insignificant. On the other hand, taking the reciprocal of $e^{20}$, we get $$e^{-20}=2.061153622438557828\times10^{-9}\pm3.4\times10^{-26}$$ which has almost $17$ digits of significance.
• Would the downvoter care to comment? – robjohn Jan 30 '15 at 1:43
• this is so nice an explanation!! When doing numerical calculation the formulas must be modified in a manner to avoid loss of significant digits and such that intermediate calculation does not require a considerably higher precision than that desired in the final answer. – Paramanand Singh Feb 19 '15 at 10:48
A standard technique, not particular to Matlab, is to exploit a function's symmetries to map its inputs to a more manageable range. The exponential function satisfies the symmetry $e^{a+b}=e^ae^b$, and it's easy to multiply floating-point numbers by powers of two. So do this: $$\begin{eqnarray} e^{-20}&=&2^{-20\lg e}\\ &\approx&2^{-28.854}\\ &=&2^{-29}\times2^{0.146}\\ &=&2^{-29}\times e^{0.146\log2}\\ &\approx&2^{-29}\times e^{0.101} \end{eqnarray}$$ Now you can easily evaluate $e^{0.101\ldots}$ using the Taylor series.
1) The regular double precision floating point arithmetic of Matlab is not sufficient to precisely calculate partial sums of this power series. To overcome that limitation, you can use the exact symbolic computation capabilities of the Symbolic Math Toolbox. This code
x = sym(-20);
i = sym(1 : 100);
expx = sum(x .^ (i - 1) ./ factorial(i - 1))
sums the first 100 terms of the series, and yields the exact (for the partial sum) result $$\tiny\frac{20366871962240780739193874417376755657912596966525153814418798643652163252126492695723696663600640716177}{9881297415446727147594496649775206852319571477668037853762810667968023095834839075329261976769165978884198811117}$$ whose numeric counterpart (computed by double()) is 2.06115362243856e-09.
2) The Matlab function exp() is most likely a wrapper for C code, which presumably references the standard C library function exp().
You can have a look at the code of the widely used GNU C Library math functions, where you'll find that most functions are themselves wrappers around machine-specific implementations. For the i386 architecture, the path leads ultimately to the function __ieee754_exp which implements the exponential in inline assembler code (but don't ask me how to decipher this). On other architectures, there might be a single machine instruction that does the job.
"Most likely", "probably", "presumably" because Matlab is closed source and therefore definitive knowledge only exists within The Mathworks.
• You mention using "variable precision arithmetic" in Matlab but your example code is an exact symbolic calculation converted to floating-point. There appears to be no need for vpa for the ranges used. The OP also asked "how should I check for convergence?" which I think was the impetus for the question. – horchler Jan 29 '15 at 3:15
• @horchler, correct with the vpa thing, I got the two confused, edited. – As for convergence, I think this was a minor part of the question made irrelevant by the fact that his double precision computation couldn't converge anyway. 4.1e-9 isn't not-converged, its just off. Feel free to add a better answer though, or even to edit mine. :-) – A. Donda Jan 29 '15 at 11:04 | 2019-09-17T12:25:46 | {
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http://math.stackexchange.com/questions/221078/poisson-distribution-of-sum-of-two-random-independent-variables-x-y | # Poisson Distribution of sum of two random independent variables $X$, $Y$
$X \sim \mathcal{P}( \lambda)$ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{P}( \lambda + \mu)$ but I don't understand how to derive it.
-
Try using the method of moment generating functions :) – Samuel Reid Nov 25 '13 at 7:03
All I've learned in the definition of a Poisson Random Variable, is there a simpler way? – user82004 Nov 25 '13 at 7:07
If they are independent. – Did Nov 25 '13 at 8:14
This only holds if $X$ and $Y$ are independent, so we suppose this from now on. We have for $k \ge 0$: \begin{align*} P(X+ Y =k) &= \sum_{i = 0}^k P(X+ Y = k, X = i)\\ &= \sum_{i=0}^k P(Y = k-i , X =i)\\ &= \sum_{i=0}^k P(Y = k-i)P(X=i)\\ &= \sum_{i=0}^k e^{-\mu}\frac{\mu^{k-i}}{(k-i)!}e^{-\lambda}\frac{\lambda^i}{i!}\\ &= e^{-(\mu + \lambda)}\frac 1{k!}\sum_{i=0}^k \frac{k!}{i!(k-i)!}\mu^{k-i}\lambda^i\\ &= e^{-(\mu + \lambda)}\frac 1{k!}\sum_{i=0}^k \binom ki\mu^{k-i}\lambda^i\\ &= \frac{(\mu + \lambda)^k}{k!} \cdot e^{-(\mu + \lambda)} \end{align*} Hence, $X+ Y \sim \mathcal P(\mu + \lambda)$.
-
Thank you! but what happens if they are not independent? – user31280 Oct 25 '12 at 20:20
In general we can't say anything then. It depends on how they depend on another. – martini Oct 25 '12 at 20:22
Thank you! it's very simple and I feel like a complete idiot. – user31280 Oct 25 '12 at 20:40
Nice derivation: specifically the transformation of (a) the i/k factorials and (b) the mu/lambda polynomials into the binomial form of the polynomial power expression. – javadba Aug 30 '14 at 20:59
Another approach is to use characteristic functions. If $X\sim \mathrm{po}(\lambda)$, then the characteristic function of $X$ is (if this is unknown, just calculate it) $$\varphi_X(t)=E[e^{itX}]=e^{\lambda(e^{it}-1)},\quad t\in\mathbb{R}.$$ Now suppose that $X$ and $Y$ are independent Poisson distributed random variables with parameters $\lambda$ and $\mu$ respectively. Then due to the independence we have that $$\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)=e^{\lambda(e^{it}-1)}e^{\mu(e^{it}-1)}=e^{(\mu+\lambda)(e^{it}-1)},\quad t\in\mathbb{R}.$$ As the characteristic function completely determines the distribution, we conclude that $X+Y\sim\mathrm{po}(\lambda+\mu)$.
-
You can use Probability Generating Function(P.G.F). As poisson distribution is a discrete probability distribution, P.G.F. fits better in this case.For independent X and Y random variable which follows distribution Po($\lambda$) and Po($\mu$). P.G.F of X is \begin{equation*} \begin{split} P_X[t] = E[t^X]&= \sum_{x=0}^{\infty}t^xe^{-\lambda}\frac{\lambda^x}{x!}\\ &=\sum_{x=0}^{\infty}e^{-\lambda}\frac{(\lambda t)^x}{x!}\\ &=e^{-\lambda}e^{\lambda t}\\ &=e^{-\lambda (1-t)}\\ \end{split} \end{equation*} P.G.F of Y is \begin{equation*} \begin{split} P_Y[t] = E[t^Y]&= \sum_{y=0}^{\infty}t^ye^{-\mu}\frac{\mu^y}{y!}\\ &=\sum_{y=0}^{\infty}e^{-\mu}\frac{(\mu t)^y}{y!}\\ &=e^{-\mu}e^{\mu t}\\ &=e^{-\mu (1-t)}\\ \end{split} \end{equation*}
Now think about P.G.F of U = X+Y. As X and Y are independent, \begin{equation*} \begin{split} P_U(t)=P_{X+Y}(t)=P_X(t)P_Y(t)=E[t^{X+Y}]=E[t^X t^Y]&= E[t^X]E[t^Y]\\ &= e^{-\lambda (1-t)}e^{-\mu (1-t)}\\ &= e^{-(\lambda+\mu) (1-t)}\\ \end{split} \end{equation*}
Now this is the P.G.F of $Po(\lambda + \mu)$ distribution. Therefore,we can say U=X+Y follows Po($\lambda+\mu$)
-
Seems a typo in third line (out of 4) in both PGF of X and Y. Should be exp(-u)exp(mut) instead of exp(-u)exp(-1*mut) – javadba Aug 30 '14 at 21:19
Separate comment/question: please explain why in second line of derivation for PGF of Y refers to exp(x) and x! (instead of referring to exp(y) and y!). I believe these are cut/paste errors - but please confirm. – javadba Aug 30 '14 at 21:26
Yeah.. You are absolutely correct. Those were typos. I have edited them now. If you find more, let me know. – Ananda Sep 2 '14 at 5:35
hint: $\sum_{k=0}^{n} P(X = k)P(Y = n-k)$
-
why this hint, why the sum? This is what I don't understand – user31280 Oct 25 '12 at 20:22
adding two random variables is simply convolution of those random variables. That's why. – jay-sun Oct 25 '12 at 20:24
gotcha! Thanks! – user31280 Oct 25 '12 at 20:31
adding two random variables is simply convolution of those random variables... Sorry but no. – Did Feb 13 '13 at 6:28
There is no usual sense for convolution of random variables. Either convolution of distributions or addition of random variables. – Did Feb 13 '13 at 6:51
In short, you can show this by using the fact that $$Pr(X+Y=k)=\sum_{i=0}^kPr(X+Y=k, X=i).$$
If $X$ and $Y$ are independent, this is equal to $$Pr(X+Y=k)=\sum_{i=0}^kPr(Y=k-i)Pr(X=i)$$ which is \begin{align} Pr(X+Y=k)&=\sum_{i=0}^k\frac{e^{-\lambda_y}\lambda_y^{k-i}}{(k-i)!}\frac{e^{-\lambda_x}\lambda_x^i}{i!}\\ &=e^{-\lambda_y}e^{-\lambda_x}\sum_{i=0}^k\frac{\lambda_y^{k-i}}{(k-i)!}\frac{\lambda_x^i}{i!}\\ &=\frac{e^{-(\lambda_y+\lambda_x)}}{k!}\sum_{i=0}^k\frac{k!}{i!(k-i)!}\lambda_y^{k-i}\lambda_x^i\\ &=\frac{e^{-(\lambda_y+\lambda_x)}}{k!}\sum_{i=0}^k{k\choose i}\lambda_y^{k-i}\lambda_x^i \end{align} The sum part is just $$\sum_{i=0}^k{k\choose i}\lambda_y^{k-i}\lambda_x^i=(\lambda_y+\lambda_x)^k$$ by the binomial theorem. So the end result is \begin{align} Pr(X+Y=k)&=\frac{e^{-(\lambda_y+\lambda_x)}}{k!}(\lambda_y+\lambda_x)^k \end{align} which is the pmf of $Po(\lambda_y+\lambda_x)$.
-
Moderator notice: This answer was moved here as a consequence of merging two questions. This explains the small differences in notation. The OP's $\lambda$ is $\lambda_x$ here, and OP's $\mu$ is $\lambda_y$. Otherwise there is no difference. – Jyrki Lahtonen Apr 23 at 6:55 | 2015-09-01T22:25:53 | {
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https://mathematica.stackexchange.com/questions/209694/filtering-data-geometrically/209717 | # Filtering data 'geometrically'
I was looking at a very nice answer here, where the question was applying a confidence region to some data.
If we plot some data as a scatter plot:
XData = RandomVariate[NormalDistribution[1,1], 100000];
YData = RandomVariate[NormalDistribution[1,1], 100000];
XYDATA = Transpose[{XData, YData}];
XYCoVarMat = Covariance[XYDATA];
XYMean = {Mean[XData], Mean[YData]};
ListPlot[
XYDATA, Frame->True, FrameLabel->{"X","Y"}, LabelStyle->16, AspectRatio->1,
Epilog -> {
Opacity[0.32, Green], EdgeForm[{Green, AbsoluteThickness[1]}], Ellipsoid[XYMean, 4 XYCoVarMat],
Opacity[0.32, Blue], EdgeForm[{Blue, AbsoluteThickness[1]}], Ellipsoid[XYMean, 3 XYCoVarMat],
Opacity[0.32, Red], EdgeForm[{Red, AbsoluteThickness[1]}], Ellipsoid[XYMean, 2 XYCoVarMat],
Opacity[0.32, Yellow], EdgeForm[{Yellow, AbsoluteThickness[1]}], Ellipsoid[XYMean, 1 XYCoVarMat]
},
PlotRange->{{-8,8},{-8,8}}
]
I was wondering if anyone knew a way of filtering the data using this approach for example only select data that is with the ellipsoid boundary of Ellipsoid[XYMean, 1 XYCoVarMat]?
• Would With[{rmf = RegionMember[Ellipsoid[XYMean, 1 XYCoVarMat]]}, Select[XYDATA, rmf]] work for you? – J. M. will be back soon Nov 15 at 16:09
• It would work wonderfully. I'm starting to wonder if you are actually written purely from Mathematica functions. Thanks! – Q.P. Nov 15 at 16:14
• @J.M.willbebacksoon I don't know if you want to post an answer yourself or not, but given you provided the solution I feel you should get credit. If credit doesn't concern you I will accept ThatGravityGuy's answer so that the question is completed and helps users who find this question. – Q.P. Nov 16 at 13:15
• Please accept ThatGravityGuy's answer if you think you found it useful. – J. M. will be back soon Nov 16 at 13:24
As @J.M. pointed out in the comments, the key here is the function RegionMember. Since you already know the region that you want to work with, namely Ellipsoid[XYMean, 1 XYCoVarMat], you can use Select as
With[{rmf = RegionMember[Ellipsoid[XYMean, 1 XYCoVarMat]]}, Select[XYDATA, rmf]] | 2019-12-08T20:33:20 | {
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https://mathematica.stackexchange.com/questions/139605/mostellers-first-ace-problem | # Mosteller's First Ace Problem
[this probably belongs on cross validated ... ]
Working through Mosteller's Fifty Challenging Problems in Probability with Mathematica. I am trying to understand Mosteller's answer for The First Ace problem (#40). The problem goes:
Shuffle an ordinary deck of 52 playing cards containing 4 aces. Then turn up cards from the top until the first ace appears. On the average how many cards are required to produce the first ace?
I solved this problem in Mathematica as as follows:
suite = Flatten@{Range[2, 10], J, Q, K, A}
deck = Flatten@{suite, suite, suite, suite}
dat = Table[First@Flatten@Position[
RandomSample[deck, Length@deck], A], {n, 1, 100000}];
ListPlot[
Tally@dat,
GridLines -> Automatic,
GridLinesStyle -> Directive[Dotted, Gray],
Frame -> True
]
Mean@dat // N
10.6152
Which yields Mosteller's answer (by symmetry) of 10.6 cards (48/5 +1).
I was a bit shocked that the mode was 1 card.
Finally, when I look at the CDF the 50% mark is at 8 cards
t = Reverse[SortBy[Tally@dat, #[[2]] &]][[All, 2]]
ListPlot[N[Accumulate@t/100000],
GridLines -> Automatic,
GridLinesStyle -> Directive[Dotted, Gray]
]
So my question is, if I played this game for real in a casino, should I go for 8 or 10 cards?
• I think you're right, this probably belongs on CrossValidated... Cool problem though. Mar 9, 2017 at 6:03
• @MarcoB I believe this belongs here, because - next to the theoretical aspect - it shows how to make use of Mathematica's abilities to simulate and to work with its distribution-framework.
– gwr
Mar 9, 2017 at 12:34
• My bad, I corrected my "decision support" (cf. my answer): In the simple 0-1-Utility case (right/wrong and a fixed amount to win), you would best choose $1$ (not 8, not 10). My exmple meets the case, when you will win an amount equal to the number of draws it took.
– gwr
Mar 9, 2017 at 18:07
I would like to choose a slightly different approach here and point out, that this is an example for the Negative Hypergeometric Distribution. Wikipedia does give a nice table for this:
While for large numbers the distribution tends to the Negative Binomial Distribution the difference essentially is drawing with replacements (Binomial) vs. drawing without replacements (Hypergeometric).
## Analytical Approach
In the present case, even when we may not have remembered all this statistical theory, we could still take an analytical approach by making use of ProbabilityDistribution. Let us look at the probabilities for the first few numbers of draws:
\begin{align} p(1) &= p(\text{ace}) =\frac{4}{52} \\ p(2) &= p(\neg\text{ace}) \times p(\text{ace}) = \frac{52-4}{52} \times \frac{4}{52-1} \\ p(3) &= p(\neg\text{ace}) \times p(\neg\text{ace}) \times p(\text{ace}) = \frac{52-4}{52} \times \frac{52-4-1}{52-1} \times \frac{4}{52-2} \\ \vdots \end{align}
So eventually we see a pattern here and can (hopefully) come up with the formula for the probability distribution function (in the discrete case a probability mass function) $f$ given the number of cards (or balls in an urn) $N$ and the number of marked cards (or balls) $M$:
\begin{align} f_{N,M}(k) = \frac{m}{N-k+1} \times \prod \limits_{i=1}^{k-1} \frac{N-M-i+1}{N-i+1} \end{align}
To implment this as a distribution we write the following code:
distribution[ n_Integer, m_Integer ] /; m <= n := ProbabilityDistribution[
m/( n - k + 1) Product[ (n - m - i + 1)/(n - i + 1),{ i, 1, k-1 } ],
{ k, 1, n - m + 1, 1 }
(* , Method -> "Normalization" is not needed here, but may be worth remembering
for all cases where giving a proportional formula is easy to come up with *)
]
analyticalDist = distribution[ 52, 4 ];
$PlotTheme = {"Detailed", "LargeLabels"}; Panel @ DiscretePlot[ Evaluate@PDF[ analyticalDist, \[FormalX] ], {\[FormalX], 0, 49, 1}, ImageSize -> Large, PlotLabel -> Style["Analytical PDF", "Subsection"] ] Also we can find moments and what have you: Mean @ analyticalDistribution (* e.g. the expected number of draws *) $\frac{53}{5}$## Empirical Approach We could of course also arrive at answers using simulation: empiricalDist = EmpiricalDistribution @ Table[ Min @ RandomSample[ Range[52], 4], {10000} (* number of trials *) ]; N @ Mean @ empiricalDist 10.5809 (* true value again was 10.6 *) If we compare the graphs for the PDF we note that the empirical distribution already is quite close: Panel @ DiscretePlot[ Evaluate @ { PDF[ empiricalDist, \[FormalX] ], PDF[ analyticalDist, \[FormalX] ] }, { \[FormalX], 1, 49, 1 }, PlotLegends -> {"empirical", "analytical"}, ImageSize -> Large, PlotLabel -> Style["PDF Comparison", "Subsection"] ] ## General Case: Negative Hypergeometric Distribution Grabbing a text book near you, you will find the formula for the probability distribution function of the above mentioned Negative Hypergeometric Distribution. It is a special case of the BetaBinomialDistribution - thanks to @ciao for pointing this out repeatedly until I listened :) - for those of you, who do not know this, there is a terrific paper showing the relations between univariate distributions by (Leemis and McQueston 2008) and more information can be found on Cross Validated. Using this information, we can implement the Negative Hypergeometric Distribution explicitly as follows: (* following Mathematica's parametrization for the hyergeometric case *) negativeHypergeometricDistribution[k_, m_, n_ ] := TransformedDistribution[ \[FormalX] + 1, \[FormalX] \[Distributed] BetaBinomialDistribution[ k, m, n-m ] ] (* unfortunately parametrization is no standardized so one has to dwelve into the formulae for this in the paper, which are given though *) nhg = negativeHypergeometricDistribution[ 1, 4, 52 ]; Mean @ nhg $\frac{53}{5}$And we can indeed convince ourselves that our case simply is a special case of a NHG-distributed random var$X$, where$X$is the number of draws from$N = 52$cards, where$M=4$are "successes" and where we need$k=1$success to stop: And @@ Table[ PDF[ analyticalDist, i ] === PDF[ nhg, i ], {i, 1, 52 } ] True ## EDIT: Decision Support / Expected Utility The OP has edited his question and asked for guidance in playing at a casino (e.g. what number of draws to bet on?). For this we have to turn to decision theory and excepted utility. Let's assume that you are risk neutral. In the case that you get a fixed amount if you are right and zero if you are wrong, then it would be trivially optimal to bet on$1$. ($1$has the greatest probability and the 0-1-loss function makes the expected utility equal to the probability; check this by simulation!) Let us look at a more elaborate case to make the principle clear: If you were to receive the number of draws needed worth in money if you bet on the right number of draws, then your utility function would be$U(x,k) = x I(x = k)$(where$I$is the indicator function,$x$your bet and$k$the number of draws it took. In this case your best bet will be$13$: NArgMax[ { NExpectation[ \[FormalX] * Boole[ \[FormalX] == i], \[FormalX] \[Distributed] negativeHypergeometricDistribution[ 1, 4, 52 ] ], 1<= k<= 49 }, k ∈ Integers ] 13 With[ { dist = negativeHypergeometricDistribution[ 1, 4, 52 ] }, Panel @ DiscretePlot[ Evaluate@ Expectation[ \[FormalX] * Boole[ \[FormalX] == i ], \[FormalX] \[Distributed] dist ], {i,1,49}, ImageSize -> Large, PlotLabel -> Style["Expected Utility (0-1)", "Subsection"], PlotLegends->None ] ] • "since it is not implemented...". It is, as BetaBinomial, already noted in an earlier reply. – ciao Mar 9, 2017 at 23:35 • @ciao Thanks, I did not make that connection. It is deeply burried in the docs and also not very obvious from web research. Guess one really needs to know what your looking for. :) – gwr Mar 9, 2017 at 23:50 • Not so much an MMA docs issue, more of a wider issue of distributions going by different names/definitions in the mathematical world. Would be nice if a future MMA docs edition (or MathWord) had a big interconnected chart with all alternate names/definitions and their relationships... – ciao Mar 10, 2017 at 0:50 • This is the best overview with regard to the relationships between univariate distributions I have found so far. – gwr Mar 10, 2017 at 8:19 Update (attribution: @ciao) As @ciao has pointed out, use ofBetaBinomialDistribution is the most straightforward for exact calculation and simulation. See the comment: Mean[BetaBinomialDistribution[1, 4, 48]] + 1 or simulation Mean[RandomVariate[Mean[BetaBinomialDistribution[1, 4, 48]] + 1],100000] Median[RandomVariate[Mean[BetaBinomialDistribution[1, 4, 48]] + 1],100000] etc Original Answer As the choice of first Ace is arbitrary you could simplify, e.g.: tab = Table[Min[FirstPosition[RandomSample[Range[52]], #][[1]] & /@ Range[4]], 10000]; Histogram[tab, Automatic, "PDF"] N@Mean[tab] Median[tab] You could also do analytically,e.g. let$f(k)$be the probability that the first ace in the first$k$cards, then: f[k_] := 1 - Binomial[52 - k, 4]/Binomial[52, 4] DiscretePlot[f[k], {k, 0, 52}] Sanity checks:$f(k)=1$for$k\ge 49$The probability that is$k\$th card is discrete derivative:
ListPlot[Differences[Table[f[k], {k, 0, 52}]], Filling -> Axis]
Differences[Table[f[k], {k, 0, 52}]].Range[52]
• Or you could just Mean[BetaBinomialDistribution[1, 4, 48]] + 1... ;=}
– ciao
Mar 9, 2017 at 6:56
• @ciao doh! always forgetting to exploit diverse built-in distributions...hope your entrepreneurship is going well ;-) Mar 9, 2017 at 6:59 | 2022-05-25T00:40:45 | {
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