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• Yes this is the answer, as @Unreasonable Sin has comment, that's a Kolmogorov randomness definition, and that definition is really great, I have been reading Gregory Chaitin AIT en.wikipedia.org/wiki/Algorithmic_information_theory and is the deeper thing I have read about randomness – Hernán Eche Jul 1 '11 at 19:43
The randomness of a sequence, to use @Listing's definition, can be quantified by the entropy rate.
If the order of the numbers does not matter, you can use a statistical test for randomness.
I recommend reading Chapter 3.5 of Knuth, Seminumerical Algorithms (this book is Volume 2 of The Art Of Computer Programming). In fact, I recommend reading everything Knuth has ever written, but this chapter in particular, titled What is a Random Sequence, will help you clarify your concept of randomness. | {
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# Plotting complex Sine
I've got another plotting problem. I want to plot Sin[z] where z is complex. So, I've tried the following:
Plot3D[ Sin[ x + I y], {x, -1, 1}, {y, -1, 1}]
I wanted to see how the sine function looks like on the unit circle. But... I get no output. Am I doing something wrong or is the kernel stuck?
• You can plot Re[Sin[x + I*y]] and Im[Sin[x + I*y]] separately. – b.gates.you.know.what Jun 15 '12 at 15:35
• Possible duplicate of Plotting Complex Quantity Functions – Jens Aug 7 '12 at 16:36
• – Jens Aug 7 '12 at 16:37
• @Jens It's not a duplicate of those posts since the OP asks for plot of sine on the unit circle, although the issue is quite similar. – Artes Aug 7 '12 at 17:18
• @Artes I see this as a special case of the linked question, but since you answered both, I'll go with your judgement here. – Jens Aug 7 '12 at 17:27
Well, you have to treat the real and imaginary parts separately. You can't really have a complex $z$ value in these plots. Here's one way to visualize complex sine:
Table[Plot3D[f[Sin[x + I y]], {x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y, z}, x^2 + y^2 < 1]], {f, {Re, Im,
Abs}}] // GraphicsRow
One further visualization aid would be to color these functions by the argument (adapting a scheme by Roman Maeder):
Table[Plot3D[f[Sin[x + I y]], {x, -5, 5}, {y, -5, 5},
ColorFunction -> Function[{x, y, z}, Hue[(Pi + If[z == 0, 0, Arg[Sin[x + I y]]])/(2Pi)]],
ColorFunctionScaling -> False, PlotLabel -> TraditionalForm[f[Sin[z]]],
RegionFunction -> Function[{x, y, z}, x^2 + y^2 < 25]],
{f, {Re, Im, Abs}}] // GraphicsRow
As Artes notes, one could have used ParametricPlot3D[] so that you are directly working with polar coordinates: | {
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Table[ParametricPlot3D[{r Cos[t], r Sin[t], f[Sin[r Exp[I t]]]},
{r, 0, 5}, {t, -Pi, Pi}, BoxRatios -> OptionValue[Plot3D, BoxRatios],
ColorFunction -> Function[{x, y, z}, Hue[(Pi + If[z == 0, 0, Arg[Sin[x + I y]]])/(2 Pi)]],
ColorFunctionScaling -> False, PlotLabel -> TraditionalForm[f[Sin[z]]]],
{f, {Re, Im, Abs}}] // GraphicsRow
Yet another visualization possibility:
Table[ParametricPlot3D[{r Cos[t], r Sin[t], f[Sin[r Exp[I t]]]},
{r, 0, 5}, {t, -Pi, Pi}, BoxRatios -> OptionValue[Plot3D, BoxRatios],
ColorFunction -> Function[{x, y, z}, Hue[(Pi + If[z == 0, 0, Arg[Sin[x + I y]]])/(2Pi)]],
ColorFunctionScaling -> False, MeshFunctions -> (#4 &),
MeshShading -> {Automatic, None}, MeshStyle -> Transparent,
PlotLabel -> TraditionalForm[f[Sin[z]]], PlotRange -> All],
{f, {Re, Im, Abs}}] // GraphicsRow
• wow, that looks great. Thank you! – Chris Jun 15 '12 at 15:43
You can plot in 3 dimensions only real and/or imaginary parts of a function. One can make use of Plot3D, but since there was a question how the sine function looks like on the unit circle, first I demonstrate usage of ParametricPlot3D and later I'll show a few of many possible uses of Plot3D.
When we'd like to use ParametricPlot3D, then instead of parametrizing complex numbers like x + I y we would rather parametrize them like r * Exp[ I u], where r is a radius of a circle and u is a polar angle. On a unit circle this reduces to Exp[ I u].
ParametricPlot3D[
{ { Cos[u], Sin[u], Re @ Sin[Exp[I u]]},
{ Cos[u], Sin[u], Im @ Sin[Exp[I u]]}}, {u, 0, 2 Pi},
PlotStyle -> {{Thick, Darker @ Green}, {Thick, Darker @ Orange}}, BoxRatios -> Automatic]
It would be easier to realize the structure of the graph of Sine, rotating ParametricPlot3D around z axis. Thus we define the following functions : | {
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F1[t_] :=
Graphics3D[
Rotate[
ParametricPlot3D[ Table[{r Cos[u], r Sin[u], Re @ Sin[r Exp[I u]]}, {r, 0.1, 1, 0.1}],
{u, 0, 2 Pi}, PlotStyle -> Thick,
ColorFunction -> (ColorData["DeepSeaColors"][#3] &),
BoxRatios -> Automatic, Axes -> False, Boxed -> False][[1]],
2 Pi t, {0, 0, 1}], Boxed -> False]
F2[t_] :=
Graphics3D[
Rotate[
ParametricPlot3D[ Table[{r Cos[u], r Sin[u], Im @ Sin[r Exp[I*(u)]]}, {r, 0.1, 1, 0.1}],
{u, 0, 2 Pi}, PlotStyle -> Thick,
ColorFunction -> (ColorData["Rainbow"][#3] &),
BoxRatios -> Automatic, Axes -> False, Boxed -> False][[1]],
2 Pi t, {0, 0, 1}], Boxed -> False]
now we can animate rotation around z-axis :
Animate[
Show[{ F1[t], F2[t],
ParametricPlot3D[{{Cos[v], Sin[v], -1},
{Cos[v], Sin[v], 0},
{Cos[v], Sin[v], 1} }, {v, 0, 2 Pi},
PlotStyle -> {Dashed, Dashed, Dashed}, BoxRatios -> Automatic,
Axes -> False, Boxed -> False]},
ViewPoint -> {Pi, Pi/2, 1/2}],
{t, 0, 1}, DefaultDuration -> 15]
The "deepseacolors" and "rainbow" families of curves are respectively parametric 3D - plots of real and imaginary parts of Sine over circles of radius r in the complex plane and the view point rotates around z - axis. The dashed circles are unit circles in planes {x, y} for z in {-1, 0 , 1}. Here the rotation is surplus but still advantageous for the sake of comprehensible visualization.
Now we provide static 3-D plots of Sine in the complex plane.
GraphicsRow[{
Plot3D[Re@Sin[x + I*y], {x, -2 Pi, 2 Pi}, {y, -2 Pi, 2 Pi}, ClippingStyle -> None],
Plot3D[Im@Sin[x + I*y], {x, -2 Pi, 2 Pi}, {y, -2 Pi, 2 Pi}, ClippingStyle -> None]}]
I extended the range of the plot to {x, -2 Pi, 2 Pi} and {y, -2 Pi, 2 Pi} since in your former case there was nothing interesting to see.
To compare with a familiar pattern of the graph of Sine let's restrict the range of the imaginary part of the variable, e.g.
GraphicsRow[{ Plot3D[ Re @ Sin[x + I*y], {x, -2 Pi, 2 Pi}, {y, -0.3 Pi, 0.3 Pi}],
Plot3D[ Im @ Sin[x + I*y], {x, -2 Pi, 2 Pi}, {y, -0.3 Pi, 0.3 Pi}]}] | {
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or to get the equal scale for all dimensions
GraphicsRow[
{Plot3D[ Re @ Sin[x + I*y], {x, -2 Pi, 2 Pi}, {y, -0.4 Pi, 0.4 Pi},
BoxRatios -> Automatic, PlotLabel -> "Real part"],
Plot3D[ Im @ Sin[x + I*y], {x, -2 Pi, 2 Pi}, {y, -0.4 Pi, 0.4 Pi},
BoxRatios -> Automatic, PlotLabel -> "Imaginary part"] },
PlotLabel -> "Graphs of Sine"]
and if you prefer the both parts of Sine in the complex plane in one plot :
Plot3D[{ Re @ Sin[x + I*y], Im @ Sin[x + I*y]},
{x, -2 Pi, 2 Pi}, {y, -0.4 Pi, 0.4 Pi},
Mesh -> {5, 3}, BoxRatios -> Automatic,
PlotStyle -> {{Opacity[0.35], Lighter[Green, 0.5]},
{Opacity[0.7], Lighter[Blue, 0.7]} } ]
• For a different visualization, on the Riemann sphere, see the paper "Visualizing Complex Functions with the Presentations Application," The Mathematica Journal, vol. 11 #2 (2009), by David J. M. Park and me. Available in CDF and PDF here. (To re-evaluate most code, and to run the dynamic examples, you'll need a copy of Park's Presentations application.) – murray Aug 7 '12 at 16:57
• @murray Thank you for pointing out interesting references. I've seen that paper. – Artes Aug 7 '12 at 17:01 | {
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# Prove that if $M_{1} \subseteq M_{2}$, then inf$(M_{2})\leq$ inf$(M_{1})\leq$ sup$(M_{1})\leq$ sup$(M_{2})$
Prove that if $M_{1} \subseteq M_{2}$, then inf$(M_{2})\leq$ inf$(M_{1})\leq$ sup$(M_{1})\leq$ sup$(M_{2})$
My attempt:
$M_{1} \subseteq M_{2} \implies \forall m\in{M_{1}}$, $m\in{M_{2}}$
$\implies$inf$(M_{2}) \leq$ inf$(M_{1})$ and also that sup$(M_{1})\leq$ sup$(M_{2})$
Additionally, by the definition of supremum we know that inf$(M_{1})\leq$ sup$(M_{1})$
Together we have, inf$(M_{2})\leq$ inf$(M_{1})\leq$ sup$(M_{1})\leq$ sup$(M_{2})$
$\therefore$ If $M_{1} \subseteq M_{2}$, then inf$(M_{2})\leq$ inf$(M_{1})\leq$ sup$(M_{1})\leq$ sup$(M_{2})$
I made a few jumps that I am not sure you can take (line 1 to 2). Is this proof valid? Anything I can change? Thanks!
• Hmm, I think your jumps are too big. How does all m in both M_1 being in M_2 imply the inf M_2 is less or equal to the inf of M_1? How do we know from the definition of sup that inf <= sup? That's not actually part of the definition. I sympathize, as this real does seem trivial and obvious and thus irritatingly difficult to prove. But you I don't think your prove has actually done anything except restate what is to be proven and declared they are obvious. (Which to be fair they sort of are.) Jul 12 '17 at 17:40
There seems to be nothing wrong with your proof, but can you tell us how did you go from $(\forall m\in M_1):m\in M_2$ to $\inf M_2\leqslant\inf M_1$? | {
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• This means that inf$(M_{1})$, sup$(M_{1})\in{M_{2}}$. I guess I should explicitly state that right? edit: Actually not sure that is correct Jul 12 '17 at 16:52
• @dawgchow My mistake for the earlier comment. That's not necessarily true if you think about it Jul 12 '17 at 16:52
• @dawgchow No, you shouldn't, because it is not true. Do you want a counter-example? Jul 12 '17 at 16:53
• Ya because if inf$(M_{1})$ is not the minimum of $M_{1}$ it wouldn't necessarily be in $M_{2}$ Jul 12 '17 at 16:54
• @dawgchow Right! So, your proof is incomplete. Jul 12 '17 at 16:55
Here's how I would do it:
By the definition of subsets, $\inf$ and $\sup$, we have the following:
$$M_1\subseteq M_2\iff (\forall m\in M_1)\,\,\,m\in M_2$$ $$(\forall m\in M_2)\,\,\,\inf(M_2)\le m\le \sup(M_2)$$
Then we can conclude
$$(\forall m\in M_1)\,\,\,\inf(M_2)\le m\le \sup(M_2)$$
Next, from the definition of $\inf$ and $\sup$, we get
$$(\forall m\in M_1)\,\,\,x\le m\le y \iff x\le\inf(M_1)\le\sup(M_1)\le y$$
Therefore
$$\inf(M_2)\le \inf(M_1)\le\sup(M_1)\le \sup(M_2)$$
When you are asked to prove something trivial and obvious... well, do the definitions. I think your prove relies on "common sense" and makes jumps that are simply too big, to the point you are mostly just restating the statements.
I'd do: If $m \in M_1$ then $m \in M_2$ as $M_1 \subset M_2$. So $\inf M_2 \le m$ by definition of infinum. So $\inf M_2$ is a lower bound of $M_1$. But $\inf M_1$ is the greatest lower bound of $M_1$ so $\inf M_2 \le \inf M_1$.
By identical argument: $\sup M_2 \le \sup M_1$. That is; if $m \in M_1$ then $m \in M_2$ and therefore $\sup M_2 \ge m$ and is an upper bound of $M_1$, but as $\sup M_1$ is the least upper bound $\sup M_2 \ge \sup M_1$. | {
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By definition of supremum and infimum for any $m \in M_1$ then $\inf M_1 \le m \le \sup M_1$ so $\inf M_1 \le \sup M_1$. (Unless $M_1$ is empty, in which case neither $\inf M_1$ nor $\sup M_1$ exist.) (I suppose at the very beginning of the proof I could/should have made a comment that $M_1$ and $M_2$ are both presumed to be non-empty and bounded above and below.).
Hence ... result.
That's basically your proof but with the jumps explicitly explained. (Perhaps painfully so.)
=====
And, to be fair, maybe I'm being to0 glib in claiming "$\inf M_1$ is greatest lower bound".
More detail: If $\inf M_1 < \inf M_2$ then $\inf M_2$ is not a lower bound of $M_1$ which we just showed it is, so $\inf M_2 \le \inf M_1$.
That's kind of obvious but as I criticized you for not proving the obvious, it's only fair that I apply my own standards... | {
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# Establishing a bijection between permutations and permutation matrices.
I am trying to prove the following:
Given $\sigma \in S_n$ let $P(\sigma)$ be the matrix with $(P(\sigma))_{i,j} = \delta_{i,\sigma(j)}$. Show that $P$ is a bijection between $S_n$ and the set of permutation matrices of size $n$ $-$ where $\delta_{i,j}$ is the Kronecker delta which equals 1 if $i=j$ and 0 otherwise.
As I understand the question, the question is saying that for each row $i$ and column $j$ we have that position $i,j$ equals 1 iff $\sigma(j) = i$.
So then it seems then that for any $\sigma$ such that $\sigma(i) = i$, we would have $P(\sigma)$ as the identity matrix with strictly 1's from the top-left to the bottom-right and zeros everywhere else. On the other hand, suppose that we have $\sigma(1) = 2$, $\sigma(2) = 3$, $\sigma(3) = 4$, $\sigma(4) = 1$; then we'd have the following matrix
\begin{bmatrix}0 & 0 & 0& 1\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{bmatrix}
Is this a correct understanding?
If so, then here is my attempted proof of the question mentioned above:
$\bf{Injectivity}$: Assuming $\sigma, \tau \in S_n$ and $P(\sigma) = P(\tau)$ then whenever $\sigma(j) = i$ we have the jth-column and i-th row of the matrix $p$ as a 1 and similarly for $\tau(j) = i$. So $\tau = \sigma$ and we are done.
$\bf{Surjectivity}$: Let $P$ be some $n \times n$ permutation matrix. Then each row and column will have exactly one 1 with zeros everywhere else. Then for every i,j position which has a 1 in it, we can choose a permutation $\sigma$ such that $\sigma(j) = i$ for some $\sigma$ which is an n-cycle and we are done.
Is this proof sound? Thank you in advance. | {
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Is this proof sound? Thank you in advance.
You're proof is fine, but more can be said. The set $P_n$ of $n\times n$ permutation matrices is a group under matrix multiplication, and as such $S_n\cong P_n$. Indeed, let $\{v_1,\ldots,v_n\}$ be a basis for $\mathbb{R}^n$ (you can take any field here, but nevermind). For each $\sigma\in S_n$, we define a linear transformation $P_\sigma$ by $P_\sigma(v_i)=v_{\sigma(i)}$, whose matrix in the basis $\{v_1,\ldots,v_n\}$ is $P(\sigma)$.
Let $\Phi:S_n\longrightarrow P_n$ be the map $\Phi(\sigma)=P_\sigma$. This map is a homomorphism since $$P_\sigma P_\tau(v_i)=P_\sigma(v_{\tau(i)})=v_{\sigma(\tau(i))}=v_{\sigma\tau(i)}=P_{\sigma\tau}(v_i).$$
The map is injective: if $P_\tau v_i=v_{\tau(i)}=v_i$ for all $i$, then $\tau(i)=i$ for all $i$ and $\tau=1$.
Finally, the map is surjective because $|P_n|=n!$ (permutation matrices are allowed one nonzero entry in every row and column. Starting with the leftmost column, you have $n$ choices for where to place a nonzero entry; having made this choice you have occupied a row, so going to the next column you have $n-1$ choices for where to place a nonzero entry, etc.)
• Thanks. That makes sense as well. – letsmakemuffinstogether Oct 1 '15 at 18:58
• There is another component to the question: show that $P: S_n \rightarrow M_n{\mathbb{R}}$ has $P(\sigma\tau)=P(\sigma)P(\tau)$. Any hints on how to do this? I think I see how it's done but I'm having troubles putting it into words and using summation notation to multiply the matrices $P(\sigma)$ and $P(\tau)$. – letsmakemuffinstogether Oct 1 '15 at 20:07
• @letsmakemuffinstogether that is my proof that $\Phi$ is a homomorphism. – David Hill Oct 1 '15 at 21:46
It can also be shown that the map $$\Phi: \mathcal{S}_n \longrightarrow \mathcal{P}_n$$ defined by $$\sigma \in \mathcal{S}_n \longmapsto P_\sigma \in \mathcal{P}_n$$ is a homomorphism and hence an isomorphism. | {
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If $$\sigma$$, $$\gamma \in \mathcal{S}_n$$ and $$1 \le i,j \le n$$, then $$\left[ P_\sigma P_\gamma\right]_{ij} = \sum_{k=1}^n \delta_{i,\sigma(k)} \delta_{k, \gamma(j)} = \delta_{i, \sigma(\gamma(j))} = \left[ P_{\sigma \circ \gamma}\right]_{ij},$$ i.e., $$P_{\sigma} P_{\gamma} = P_{\sigma \circ \gamma}$$.
Thus, \begin{align} \Phi(\sigma\circ\gamma) = P_{\sigma \circ \gamma} = P_\sigma P_\gamma = \Phi(\sigma)\Phi(\gamma) \end{align} as desired. | {
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# Limit of a 0/0 function
Let's say we have a function, for example, $$f(x) = \frac{x-1}{x^2+2x-3},$$ and we want to now what is $$\lim_{x \to 1} f(x).$$ The result is $\frac{1}{4}$.
So there exists a limit as $x \to 1$.
My teacher says that the limit at $x=1$ doesn't exist. How is that? I don't understand it. We know that a limit exists when the one sided limits are the same result.
Thank you!
• Try factoring the denominator. This is probably a homework problem, so look at the numerator for a hint. – copper.hat Oct 16 '14 at 22:35
• @copper.hat: I completely disagree! In the definition of "$\lim_{x\to 1} f(x)$", $f(1)$ doesn't need to be defined (in fact, in most interesting cases it isn't), and even if it is defined, that value shouldn't be taken into account. Adding "$x \neq 1$" doesn't make any difference whatsoever. – Hans Lundmark Oct 17 '14 at 7:13
• @HansLundmark: I stand corrected. – copper.hat Oct 17 '14 at 15:03
It's possible that your teacher was pointing out the fact that the function doesn't exist at $x = 1$. That's different from saying that the limit doesn't exist as $x \to 1$. Notice that by factoring, $$f(x) = \frac{x-1}{x^2 + 2x - 3} = \frac{x-1}{(x-1)(x+3)}$$
As long as we are considering $x \ne 1$, the last expression simplifies: $$\frac{x-1}{(x-1)(x+3)} = \frac{1}{x+3}.$$ In other words, for any $x$ other than exactly $1$, $$f(x) = \frac{1}{x+3}.$$ This helps understand what happens as $x$ gets ever closer to $1$: $f(x)$ gets ever closer to $$\frac{1}{(1) + 3} = \frac{1}{4}.$$
• The word 'exactly' unnecessary. – CiaPan Oct 17 '14 at 8:12
• @CiaPan: What about the word 'is'? – Nikolaj-K Oct 17 '14 at 13:43
• @NikolajK that depends on what 'is' is. – Yakk Oct 17 '14 at 14:53
• @Yakk: It's equivalent to equivalence in univalent foundations. – Nikolaj-K Oct 17 '14 at 16:37
Your teacher is not correct. There are two easy ways to do this problem. The first is by factoring the denomiator: | {
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$$\lim_{x\to 1}\frac{x-1}{(x-1)(x+3)}=\lim_{x\to 1}\frac{1}{x+3}=\frac{1}{4}$$
The second is by using L'Hospital's rule, which is a useful identity in limits. By L'Hospital's rule, we know that
$$\lim_{x\to 1}\frac{x-1}{x^2+2x-3}=\lim_{x\to 1}\frac{1}{2x+2}=\frac{1}{4}$$
This limit exists, because it is simply a discontinuity in the function, but it is a discontinuity at a single point. When we have certain indeterminate forms in limits, we may apply L'Hospital's rule, which allows us to better compute the limit.
L'Hospital's rule states that, in certain indeterminate forms:
$$\lim_{x\to n}\frac{f(x)}{g(x)}=\lim_{x\to n}\frac{f'(x)}{g'(x)}$$
It is worth clearing up a particular misconception that seems to have arisen. For $$f(x)=\frac{x-1}{x^2+2x-3}$$ we cannot compute the value of $f(1)$, because this results in an indeterminate form. However, the limit exists, because there is simply a local discontinuity at a single point in an otherwise continuous function.
• A third alternative, to whoever may care, is to note that the limit if $\dfrac 1{f'(1)}$, where $f(x)=x^2+2x-3$. – Git Gud Oct 16 '14 at 23:57
• To justify the use of l'Hospital, you also need to point out that $g' \neq 0$ in a punctured neighbourhood of the point in question. – Hans Lundmark Oct 17 '14 at 7:14
• A hard requirement of the usability of l'Hôpital's rule is that the resulting "right-hand-size" is exists. But I guess that's what Hans is saying in more fancy terms? – rubenvb Oct 17 '14 at 8:00
• @rubenvb: No, it's not enough that $\lim f/g$ is of type $0/0$ and that $\lim f'/g'$ exists. There are examples which show this. You really need $g'$ to be nonzero near the point. (Those examples are a bit artificial to be honest, but still...) – Hans Lundmark Oct 17 '14 at 15:10
• @Hans $g'$ can be zero if $f'$ is zero, because you just have another L'Hospital's case and take $f''/g''$ Otherwise, just compute the limit normally. – Aza Oct 17 '14 at 16:13 | {
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I don't know what your teacher means exactly. Limits are defined when $x$ tends to some number, or infinity. Not when $x$ is this number.
The value that the function takes at the limit poit is irrelevant (to compute the limit). In fact, in most high school limit exercises, the function is not defined at the point that $x$ tends to. | {
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# Distance Between Two Coordinates Calculator | {
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Again, a distance and direction. Algebra -> Customizable Word Problem Solvers -> Geometry-> SOLUTION: The distance between two points in the coordinate plane is equal to the square root of the sum of the squares of the difference between the x-coordinates and the difference between Log On. Practice applying the Pythagorean Theorem in this tutorial about the distance formula. A journey meter calculate driving or walking distances from any location and displays fuel cost, taxicab fare, using haversine formula. 02/28/2018; 14 minutes to read; In this article. Hi Chua, You can use google API's for that, I think this one gives you duration and distance between 2 places. For two points P 1 = (x 1, y 1) and P 2 = (x 2, y 2) in the Cartesian plane, the distance between P 1 and P 2 is defined as: Example : Find the distance between the points ( − 5, − 5) and (0, 5) residing on the line segment pictured below. The first 2 parameters declare the x and y coordinates of the first point, and the second 2 parameters declare the x and y coordinates of the second point. I turned to my web browser and googled it, sure enough, there was this R package called ‘geosphere’ from Robert J. SHORTEST DISTANCE BETWEEN TWO POINTS ON A SPHERE It is known that the shortest distance between point A and point B on the surface of a sphere of radius R is part of a great circle lying in a plane intersecting the sphere surface and containing the points A and B and the point C at the sphere center. This is the snippet Calculate The Distance of Two Points On a Coordinate Plane on FreeVBCode. There are a number of latitude/longitude distance calculators available online. In this exercise you will plot the positions of the 2 players and manually calculate the distance between them by using the Euclidean distance formula. Nothing else makes sense and I see that you were able to calculate a distance in ft. Calculation of geographical distance is a very important utility function in GPS related procedures. | {
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Calculation of geographical distance is a very important utility function in GPS related procedures. The first 2 parameters declare the x and y coordinates of the first point, and the second 2 parameters declare the x and y coordinates of the second point. This distance and driving directions will also be displayed on google map labeled as Distance Map and Driving Directions US. On this page by using the mileage calculator you can find the flight distance or driving distance between any two cities in the world. Solution : Yes, First, convert the latitude and longitude values from decimal degrees to radians. Distance calculation. This also provides free sample source codes in PHP, ASP, ColdFusion, C/C++, C#, Java, Perl, Visual Basic and Javascript. google map api. angle between two lat lon points. This article is a sequel to the previous article on the same topic, but using T-SQL for calculation - Calculate distance between two points on globe from latitude and longitude coordinates. Hi everybody, i have some problem to calculate the distance between two points, in my case i have a file with 16 columns and i'm interesting to 6th, 7th, 8th column that are rispectively the x, y and z coordinates of several atoms belonging to a protein, instead in 14th 15th and 16th columns there are the x,y and z coordinates of different several atoms belonging to another protein (like a pdb. Fractions should be entered with a forward such as '3/4' for the fraction $$\frac{3}{4}$$. , X0, Y0 and X1, Y1 and click calculate to know the distance between 2 points in 2-dimensional space. Calculate the distance. GIS question: excel formula to calculate distance between two points given their coordinates (latitude / longitude). com can calculate the shortest distance and the fastest distance between any two cities or locations. Is it really possible? The answer is "YES", not. If I want to calculate the distance between two positions, how can I achieve this? Learn. In order to fully grasp how | {
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the distance between two positions, how can I achieve this? Learn. In order to fully grasp how to plot polar coordinates, you need to see what a polar coordinate plane looks like. This will compute the great-circle distance between two latitude/longitude points, as well as the middle point. In this post, we show the formula to calculate the shortest distance between two points using Latitude and Longitude. Language Objectives: After completing the activity, students should be able to. Enter 2 sets of coordinates in the x y-plane of the 2 dimensional Cartesian coordinate system, (X 1, Y 1) and (X 2, Y 2), to get the distance formula calculation for the 2 points and calculate distance between the 2 points. 5678, 11087. Star 1 Right Ascension: Enter the Right Ascension of the first star in hours. ' '===== ' Calculate geodesic distance (in m) ' between two points specified by ' latitude/longitude (in numeric ' [decimal] degrees) ' using Vincenty inverse formula ' for ellipsoids '===== ' Code has been ported by lost_species ' from www. This uses the ‘haversine’ formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an ‘as-the-crow-flies’ distance between the points (ignoring any hills they fly over, of course!). This demo program will calculate the distance between two points on Earth, using the latitude and longitude positions. View Details ». The thing I did kind of worked to find the shortest distance from 2 locations, and then I referenced it to 3 locations in all and averages it out. coordinates and two. This calculation can be useful when trying to determine the distance for logistical purposes (ie delivery service, flights, distance between customers, etc). Please write the origin and destination city name, choose the distance unit and press calculate. I am storing the coordinates in two. However, I would like to calculate the distance between the points before I run my script and | {
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two. However, I would like to calculate the distance between the points before I run my script and without an ArcGIS. Write, compile, and execute a C program that calculates the distance between two points whose coordinates are (7,12) and (3,9). The term Haversine was coined by. 84 return list gives you three calculations--the first two assume the Earth is a perfect sphere, which it is not, and the third that is an ellipsoid, which is closer to the truth. Distance between two locations on the Earth We often describe a location on Earth by its latitude and longitude. It is not very often that someone gives the latitude and longitude of their town! The use of a distance and direction as a means of describing position is therefore far more natural than using two distances on a grid. The Bearing Distance Calculator function calculates the forward and backward azimuths between two specified coordinates as well as the distance. txt, the XY coordinates of the cities. Perform the same operation for the y coordinates. This page helps you to calculate great-circle distances between two points using the ‘Haversine’ formula. Calculate the Distance Between Two Coordinates (latitude, longitude) - PHP - Snipplr Social Snippet Repository. I am looking for a way to get the distance in pixels between two MapItems. Just the other day, my friend was asking me if there was an easy way to calculate the distances between two locations with geocodes (Longitude and Latitude). When calculating the distance between two points on a coordinate system we are applying Pythagoras' Theorem. Standard measurement equipment (rulers, protractors, planimeters, dot grids, etc. As homework we were assigned to enter the following code to calculate the distance between two points on the x and y plane. 42 (rounded to the nearest 100th) How to use the Distance Formula Calculator. This means of location is used in polar coordinates and bearings. For example, in a button you can have the following formula on | {
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in polar coordinates and bearings. For example, in a button you can have the following formula on its OnSelect property:. In the attached file "Circle2", the coordinates of the circles are located under column B & C, the coordinates of the points are located under column F & calculate the minimum distance between two sets of coordinates. Enter 2 sets of coordinates in the 3 dimensional Cartesian coordinate system, (X 1, Y 1, Z 1) and (X 2, Y 2, Z 2), to get the distance formula calculation for the 2 points and calculate distance between the 2 points. Time and Date Duration – Calculate duration, with both date and time included Date Calculator – Add or subtract days, months, years. Function accepts four parameters, source latitude, source longitude, destination latitude, destination longitude and returns the distance between the. 13365 i currently have the code, but that doesnt seem to work well. If you pass an address as a string, the service will geocode the string and convert it to. "Let's look at the diagram! I can see that the lines of longitude get closer and closer together towards the poles! At the equator, the distance between 15 and 30 degrees W longitude is quite a lot! But as you follow those two longitude lines towards the poles, the distance between them shrinks down to zero!. This script calculates distances, bearing and more between the two Latitude/Longitude points. This tool can measure two types of distance types, the first is straight line distance also known as Rhumb line distance. 28 , rounded to two decimal places. I would like to share a MySQL function to calculate distance between two latitude – longitude points. The easiest way I could think of to to this is to somehow store the output of veclen(x,y) to a macro. Free sample source codes in PHP, ASP, ColdFusion, C/C++, C#, Java, Perl, Visual Basic and Javascript. Calculate with Geo Coordinates Calculators for the conversion of geo coordinates, as delivered by a GPS tool, for the distance of | {
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Calculators for the conversion of geo coordinates, as delivered by a GPS tool, for the distance of two point and for bearing. Calculate the distance between them with the distance. It contains two vertical lines called axis. If the current coordinate system is an earth coordinate system, Distance( ) returns the great-circle distance between the two points. The program should ask the user to enter two points then should calculate the distance between two points and print the distance on the screen. Calculating Distance. I am looking for a way to get the distance in pixels between two MapItems. In this post, we will learn the distance formula. Distance and midpoint calculator This online calculator will compute and plot the distance and midpoint for two points in two dimensions. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. DISTANCE BETWEEN TWO POINTS. The distance returned is the arc length of a great circle connecting the two points, essentially air miles. West and South locations are negative. Distances calculator is a free tool to calculate distances between any two cities in the world. There is more information on how to calculate these figures below the tool. d = √ 5 2 + 7 2. MySQL function to calculate the distance between two coordinates using the Haversine formula. The formula assumes that the earth is a sphere, (we know that it is "egg" shaped) but it is accurate enough * for our purposes. The distance between 2 points on a graph is really the hypotenuse of a right-angled triangle. Calculate the great circle distance between two points does not want to give me a reading. The shortest distance between two points on the surface of a sphere is an arc, not a line. Note: The flight distances provided are close approximations as all flights differ based on | {
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a line. Note: The flight distances provided are close approximations as all flights differ based on weather, traffic, and the exact route determined by air traffic control. Click Calculate Distance, and the tool will place a marker at each of the two addresses on the map along with a line between them. Take for example, a dating website which shows potential matches to a user within a 15 mile radius or a taxi business which calculates taxi fares based on distance between two locations. If you want to determine the distance between two points on a plane (two-dimensional distance), use our distance calculator. The code will show you how to do that using Latitude and Longitude coordinates. Spherical to Cylindrical coordinates. A more mathematical approach, you could use the Pythagorean Theorem and calculate the hypotenuse using the right triangle with the grid units in unity. If you pass coordinates, ensure that no space exists between the latitude and longitude values; destinations — One or more addresses and/or textual latitude/longitude values, separated with the pipe (|) character, to which to calculate distance and time. The program should ask the user to enter two points then should calculate the distance between two points and print the distance on the screen. Position, Distance and Bearing Calculations Whitham D. 458 W 71 27. The API can convert a zip code to the primary location for the zip code. Answered Active Solved. To find the distance between two places, enter the start and end destination and this distance calculator will give you complete distance information. 05W) A program for HP50G or 42S or 41C would be fine or a formula or algorithm as well. By continuing to browse this site, you agree to this use. How do you find the distance between two points? Distance Formula : Example: For two points, (3,2) and (15, 10) the distance is calculated as: Distance = 14. It is not very often that someone gives the latitude and longitude of their town! The use of a | {
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= 14. It is not very often that someone gives the latitude and longitude of their town! The use of a distance and direction as a means of describing position is therefore far more natural than using two distances on a grid. We have successfully combined latitude and longitude into the GeoLoc field. Discover how to calculate Distance by Latitude and Longitude using JavaScript. x A means the x-coordinate of point A y A means the y-coordinate of point A. Another method for calculating the distance between two points that fall on the same line is to plot the points and count the amount of boxes in between the two points. This calculator computes the great circle distance between two points on the earth's surface. calculate-distance-between-two-latitude-longitude-points-11152 contains nodes provided by the following 2 plugin(s): KNIME Base Nodes. It’s quick, easy and takes but a moment to do because you only need to enter the x and y coordinates of two points and click a button to calculate it. For some websites, it is a necessity to calculate the distance between certain locations. Find altitude by coordinates on google maps in meters and feet. If you are not sure what the gps coordinates are, you can use the coordinates converter to convert an address into latlong format or vice versa. The terminal coordinates program may be used to find the coordinates on the Earth at some distance, given an azimuth and the starting coordinates. Inputs can be in several formats: GPS Coordinates (like N 42 59. As you start to write the name of a city or place, distance calculator will suggest you place names automatically, you may choose from them to calculate distance. With the growing distribution of GPS devices and tools such as Google Earth or GeoPosition it is easy to determine the geographical coordinates of interesting points on the earth's surface. The geographical coordinates of the two points, as (latitude, longitude) pairs, are and respectively. If you have two geohashes, | {
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of the two points, as (latitude, longitude) pairs, are and respectively. If you have two geohashes, you first need to decode them into latitude and longitude: GEOHASH_DEC_LAT(#GeoHash) GEOHASH_DEC_LONG(#GeoHash) After you have your coordinates you can calculate the distance: IF(!. Practice applying the Pythagorean Theorem in this tutorial about the distance formula. First, sketch the two UTM coordinates onto a rectangle. In this article, you will learn how to Get Distance between two Coordinates Using Xamarin. For more information on how it distances are calculated on a sphere, have a look at Haversine Formula on Wikipedia, it's a bit complex, hence us not wanting to duplicate the content. Distance Calculation Introduction. Calculate distance between a pair of lat/lon coordinates I recently had a need to calculate distance between a large number of latitude/longitude coordinate pairs. Plane equation given three points. The Haversine Formula For two points on a sphere (of radius R) with latitudes φ1 and φ2, latitude separation Δφ = φ1 − φ2, and longitude separation Δλ, where angles are in radians, the distance d between the two points (along a great circle of the sphere; see spherical distance) is related to their locations by the formula above. After finding the 'distance' (it is in. The distance between the two coordinates, in meters. First, note that it's very easy to get a distance between two points on a sphere if you know the angle of the arc between them: just multiply it by the radius. Suppose it is desired to calculate the distance d from the point (1, 2) to the point (3, -2) shown on the grid below. The question Calculating the planets and moons based on Newtons's gravitational force was pretty much answered with two items: Use a reasonable ODE solver; at least RK4 (the classic Runge-Kutta me. For more information, see Measure Distances Between Data Points and Locations in a Map. Enter a city, street, address, postal code or region and click calculate | {
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and Locations in a Map. Enter a city, street, address, postal code or region and click calculate distance. Distance Calculator: This easy tool is very useful to find travel distances between cities, and its respective estimated time en route. Haversine Formula – Calculate geographic distance on earth. Travelmath helps you find driving distances based on actual directions for your road trip. Enter either: decimal latitudes/longitudes with minus sign for South and West. This tool calculates the straight line distance between two pairs of latitude/longitude points provide in decimal degrees. You can obtain the coordinates for just about any earthly city from WWW. Know how to use the distance formula to find the distance between points. Is there anything to improve? Calculate the distance given. Capable estimates total distance between any location using haversine formula and route method provided by the map and display location coordinates and location address. The terminal coordinates program may be used to find the coordinates on the Earth at some distance, given an azimuth and the starting coordinates. distancesfrom. Free distance calculator - Compute distance between two points step-by-step. A line through three-dimensional space between points of interest on a spherical Earth is the chord of the great circle between the points. What Is a Vector? A vector has size, also known as magnitude, and direction. Suppose it is desired to calculate the distance d from the point (1, 2) to the point (3, -2) shown on the grid below. Algebra -> Customizable Word Problem Solvers -> Geometry-> SOLUTION: The distance between two points in the coordinate plane is equal to the square root of the sum of the squares of the difference between the x-coordinates and the difference between Log On. Further, there is a one-to-one correspondence between areal coordinates and all points on the plane P. Distance = √ (x 2 – x 1) 2 + (y 2 – y 1) 2. Re: How to calculate distance between two | {
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the plane P. Distance = √ (x 2 – x 1) 2 + (y 2 – y 1) 2. Re: How to calculate distance between two coordinates As you are doing a different syllabus to ours I don't know what method they use; if the questions on GHA are anything to go by, you'll probably find they are using Spherical Trigonometry. Distance between two adjacent integer (whole numbers such as 1 and 2, or 25 and 26, not 13. The distance between the two coordinates, in meters. What is the distance between points B & C? What is the distance between points D & B? What is the distance between points D & E? Which of the points shown above are $4$ units away from $(-1, -3)$ and $2$ units away from $(3, -1)$?. Calculator Use. Find altitude by coordinates on google maps in meters and feet. Calculate the distance as-the-crow-flies between two Latitude/Longitude points on Earth You can enter the Latitude (East - West) and Longitude (North - South) coordinates as given by Google Earth software. The distance problem can be solved by using the Haversine formula. hi, i have a pair of latitude and longitude and i want to calculate the distance between these two points. Just the other day, my friend was asking me if there was an easy way to calculate the distances between two locations with geocodes (Longitude and Latitude). Cartesian to Cylindrical coordinates. The distance formula is derived from the Pythagorean theorem. Notice the line colored green that shows the same exact mathematical equation both up above, using the pythagorean theorem, and down below using the formula. The java program finds distance between two points using manhattan distance equation. Distance Formula The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. It can produce a map of the flight route and returns a link to enroute weather information. If you do not know the datum of the coordinates the information is not useless for a general ideas of your location -- | {
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the datum of the coordinates the information is not useless for a general ideas of your location -- but for for mapping work the answers you. I have found a couple of solutions that do this for zip codes, but not for a physical address. Rect) This code is lacking a zillion essential features (but interpoint distance can now be calculated). If you want to determine the distance between two points on a plane (two-dimensional distance), use our distance calculator. The center of Rome is located at roughly 41. The distances and times returned are based on the routes calculated by the Bing Maps Route API. Valid input formats are at the bottom of this page. The distance value in red color indicates the air (flying) distance, also known as great circle distance. 'OUTPUT: Distance between the ' two points in Meters. The formula assumes that the earth is a sphere, (we know that it is "egg" shaped) but it is accurate enough * for our purposes. The script uses Haversine formula, which results in in approximations less than 1%. A proof of the Pythagorean theorem. Note: The flight distances provided are close approximations as all flights differ based on weather, traffic, and the exact route determined by air traffic control. ) cannot be used to measure distance, angle, area, or shape on a sphere, as these tools have been constructed for use in planar models. If you need to get that information relating to distances between points, use the GPS Latitude and Longitude Distance Calculator. coordinates from browser. since the distance between first one and second one is already calculated there is no need to do it again. NET 4 Framework or above and reference System. distancesfrom. In an example of how to calculate the distance between two coordinates in Excel, we’ll seek to measure the great circle distance. Use LatLong. In other words, the distance between A and B. This demo program will calculate the distance between two points on Earth, using the latitude and longitude | {
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program will calculate the distance between two points on Earth, using the latitude and longitude positions. Know how to interpret points with the rectangular coordinate system (rect system) 2. Latitude and/or longitude change every few inches, and if two places have the same latitude and longitude, then the distance between them is zero. I have found a couple of solutions that do this for zip codes, but not for a physical address. Calculate Distance in Miles from Latitude and Longitude. A protip by ausi about mysql, latitude, longitude, and coordinates. What I am going to show you is how to calculate distance between two points within the United States, using the Haversine formula, implemented in C. Theses free samples assist in understanding its usage. This uses the haversine formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an “as-the-crow-flies” distance between the points (ignoring any hills they fly over, of course!). Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find midpoint between two points. Click here to find your latitude/longitude. Calculate car driving distance and straight line flying distance times between Orapa Botswana and Maseru Lesotho with Distantias! Get fuel cost estimates, the midpoint, nearest rail stations, nearest airports, traffic and more. A circle is a set of all points in a plane that are a fixed distance r away from a given point called the center. Calculating the Distance Between Two GPS Coordinates with Python (Haversine Formula) September 7, 2016. 8664741, 24. I have these data: Point A and B with State, City, Address, Zip. Then it occured to me that I might have to normalize $\rho$, so it can only take values between zero and one (just like the $\sin$). The distance returned is relative to Earth’s radius. However, I have a list of coordinates of | {
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The distance returned is relative to Earth’s radius. However, I have a list of coordinates of houses, and want to calculate their. This will compute the great-circle distance between two latitude/longitude points, as well as the middle point. 8), where the units are centimeters for Teachers for Schools for Working Scholars for. MGRS Distance and Direction Calculator. D= /(x2−x1)2+(y2−y1)2. Introduction Two frequent calculations required in radio propagation work are distance and bearing between two radio terminals. Distance Calculator. Latitude is in degrees north of the equator; southern latitudes are negative degrees north. Also get the directions for going to one city from another city. The example application that is posted will show you how to use Zip codes in place of Latitude and Longitude. Thank you and regards,. We have successfully combined latitude and longitude into the GeoLoc field. Calculate the distance between two addresses gets confused when you input GPS coordinates. If you have two geohashes, you first need to decode them into latitude and longitude: GEOHASH_DEC_LAT(#GeoHash) GEOHASH_DEC_LONG(#GeoHash) After you have your coordinates you can calculate the distance: IF(!. Distance Calculator is use to calculate the distance between coordinates and distance between cities. // use the 'haversine' formula to calculate distance between two lat/long points for each record // The Haversine formula can be broken into multiple pieces or steps - I chose to do the entire formula. Calculates the distance and azimuth between two places each city's distance from that coordinate. This uses the haversine formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an “as-the-crow-flies” distance between the points (ignoring any hills they fly over, of course!). Try our Latitude/Longitude Distance Calculator to determine the distance between two points. com) software offers a solution for | {
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Calculator to determine the distance between two points. com) software offers a solution for users who want to find the distance between one or more latitude and longitude coordinates. As you start to write the name of a city or place, distance calculator will suggest you place names automatically, you may choose from them to calculate distance. You can calculate geo distance using spatial types - geography datatype in SQL server. Does anyone know of a method of calculating the distance between two (very close together) GPS coordinates? The method must be easily calculated by an 8-bit microcontroller. The calculator will find the angle (in radians and degrees) between the two vectors, and will show the work. So, let's go ahead and just step through this, press f8, I might have a couple of things I forgot to do, but we set the latitudes and longitudes, we calculate a, b. Currently if you do a Google search to find how to accomplish this you will find a number of blogs talking about the need to use a Haversine formula that takes into account the curvature of the earth. There are several different ways to calculate the distance between two coordinates, P1(r, θ, Φ) and P2(r, θ, Φ). The values used for the radius of the Earth (3961 miles & 6373 km) are optimized for locations around 39 degrees from the equator (roughly the Latitude of Washington, DC, USA). Plane equation given three points. The formulas are very simple: for two given points and distance is the hypotenuse of right triangle, and it is calculated like this: and middle point is the average of both coordinates. I am trying to work out what would be described as the 'L0 Tile' distance between two points in my game in tiles. Distance between two points in a three dimension coordinate system - online calculator Sponsored Links The distance between two points in a three dimensional - 3D - coordinate system can be calculated as. One additional aspect not addressed above is the nature of the terrain between the two | {
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as. One additional aspect not addressed above is the nature of the terrain between the two points. For instance we are getting data from two different devices at two different geographical location, we are getting latitude and longitude value for those devices. The Bing Maps Distance Matrix API provides travel time and distances for a set of origins and destinations. This can be achieved through Google Maps API. Take for example, a dating website which shows potential matches to a user within a 15 mile radius or a taxi business which calculates taxi fares based on distance between two locations. Geozip Geozip is a Perl program for calculating the distance between two zip codes in the United States. In the attached file "Circle2", the coordinates of the circles are located under column B & C, the coordinates of the points are located under column F & calculate the minimum distance between two sets of coordinates. 8, which is the radius of Earth. The distance problem can be solved by using the Haversine formula. It has all the needed trig functions to calculate the distance in miles between two points using Longitude and Latitude where they are expressed in decimal form. Distance between Two Given Points. Surface Distance Between Two Points of Latitude and Longitude. Distance Calculator. There are many options available if you want to import these in a GIS and run analysis. From Point A to Point B: Understanding how to find the distance between two points on the coordinate plane. The first step is expressing each Latitude/Longitude of both of the coordinates in radians:. Perform the same operation for the y coordinates. Simply enter the from address and to address and the distance between the two address will be calculated in milage and kilometers. This demo program will calculate the distance between two points on Earth, using the latitude and longitude positions. Calculate distance between two coordinates. The curve (somewhat resembles a circle) is made up of a | {
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Calculate distance between two coordinates. The curve (somewhat resembles a circle) is made up of a couple of given points and the rest of the data is interpolated by excel. Point Slope Form Calculator The point slope form is defined that the difference in the y coordinate between two points (y - y1) on a line is proportional to the difference in the x coordinate points (x - x1). Answer to: Two points in a rectangular coordinate system have the coordinates (4. I will choose the first coordinate and calculate the distance to other coordinates by using the above equation. 20181 The distance between each other is about 22km. If the two points are (x1, y1) and (x2, y2), the distance between these points is given by the distance formula: The distance formula is a very. The Earth Model—Calculating Field Size and Distances between Points using GPS Coordinates Figure 1. JS: Calculate Distance Between Two Coordinates phptuts January 3, 2018 This routine calculates the distance between two points (given the latitude/longitude of those points). Coordinates of point is a set of values that is used to determine the position of a point in a two dimensional plane. QSC297: Estimate the square root of a number between two consecutive integers with and without models. For just a few calculations, this is very straight-forward using plain old Excel. convert to/from UTM coordinates; calculate the geodetic distance between two positions or calculate the bearing from one position to another; project a position with a distance and bearing, to get a new position (or bearing) find the geodetic intersection of two lines (great circles), two archs (small circles) or a line and a arc. Ask Question. The great circle distance is proportional to the central angle. Important Note: The distance calculator on this page is provided for informational purposes only. Lines of latitude are called parallels and in total there are 180 degrees of latitude. Calculate Distance in Miles from Latitude and | {
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and in total there are 180 degrees of latitude. Calculate Distance in Miles from Latitude and Longitude. This can be used to place features so they are relative to one another. The center of Rome is located at roughly 41. Calculate the distance between any two points in the rectangular coordinate plane. how to calculate nearest places distance between two coordinate(lat,long) [Answered] RSS 1 reply Last post Jul 07, 2014 05:30 AM by Michelle Ge - MSFT. Distance Calculator – Find the Distance Between Cities. The second one calculates the total distance between a list of locations. Again, a distance and direction. For find distance between two latitude and longitude in SQL Server, we can use below mentioned query. Distance From To: Calculate distance between two addresses, cities, states, zipcodes, or locations Enter a city, a zipcode, or an address in both the Distance From and the Distance To address inputs. This article is a sequel to the previous article on the same topic, but using T-SQL for calculation - Calculate distance between two points on globe from latitude and longitude coordinates. This includes the city, state, latitude, longitude, time zone information, and NPA area codes for the primary location. For two points P 1 = (x 1, y 1) and P 2 = (x 2, y 2) in the Cartesian plane, the distance between P 1 and P 2 is defined as: Example : Find the distance between the points ( − 5, − 5) and (0, 5) residing on the line segment pictured below. google map api. In this post, we show the formula to calculate the shortest distance between two points using Latitude and Longitude. The calculator will generate a step-by-step explanation on how to obtain the results. The Distance from the Car to the next Turn. I built this primarily to make it easy to check if a Locationless (Reverse) Cache has already been found. This distance will also be displayed on google map labeled as World Distance Map. As shown in the example below, we do this by joining the two points with a | {
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as World Distance Map. As shown in the example below, we do this by joining the two points with a straight line and then drawing a right-angled triangle using that straight line as the hypotenuse and aligning the other two sides with the x-axis and y-axis. MGRS Distance and Direction Calculator. 14159265359). All numbers and return values should be of type double. The formulas are very simple: for two given points and distance is the hypotenuse of right triangle, and it is calculated like this: and middle point is the average of both coordinates. Another method for calculating the distance between two points that fall on the same line is to plot the points and count the amount of boxes in between the two points. To find the flight distance between two places, please insert the locations in the control of flight distance calculator and Calculate Flight Distance to get the required results while travelling by air. It accepts a variety of formats:. I've found this code for calculating distance. 0 (sobolsoft. The first class calculates the distance between two locations. Calculate Distance Between 2 Coordinates alias Memperhitungkan jarak antara 2 titik koordinat. The purpose of the function is to calculate the distance between two points and return the result. Find bearing angle and find direction A and B as two different points, where 'La' is point A longitude and 'θa' is point A latitude. If you need to find out the coordinates of a site, the distance between two sites or the bearing between two sites go to my Coordinate, Distance and Bearing Calculator A new feature is dragable marker-B. Click Calculate Distance, and the tool will place a marker at each of the two addresses on the map along with a line between them. I just plug the coordinates into the Distance Formula: Then the distance is sqrt (53) , or about 7. But essentially the program should gather two sets of coordinates from a GPS module and the users phone (using the 1Sheeld) and work out the bearing and | {
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coordinates from a GPS module and the users phone (using the 1Sheeld) and work out the bearing and distance between them. Skills to Learn: 1. The shortest distance (the geodesic) between two given points P 1 =(lat 1, lon 1) and P 2 =(lat 2, lon 2) on the surface of a sphere with radius R is the great circle distance. Latitude/Longitude Distance Calculation This query will determine the distance between two points on the earth given their latitudes and longitudes. Latitude and/or longitude change every few inches, and if two places have the same latitude and longitude, then the distance between them is zero. The C# implementation of the Haversine formula is: public double GetDistanceBetweenPoints( double lat1, double long1, double lat2, double long2). For some websites, it is a necessity to calculate the distance between certain locations. The EASY way is to have. x A means the x-coordinate of point A y A means the y-coordinate of point A. Valid input formats are at the bottom of this page. South latitudes points are expressed in negative values, while east longitudes are positive values. The program converts degrees to radians before executing the GEODIST function. Distances calculator is a free tool to calculate distances between any two cities in the world. This query calculate the distance in miles. I'm building a system that takes a GPS reading once a second and calculates the distance between these one second intervals, and aggregates them. Further, there is a one-to-one correspondence between areal coordinates and all points on the plane P. Just a little mathematical exercise in geography, namely the use of longitude and latitiude coordinates to calculate the distance between two given cities. Travelmath helps you find driving distances based on actual directions for your road trip. Distance Calculator is use to calculate the distance between coordinates and distance between cities. angle between two lat lon points. I've found this code for calculating | {
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and distance between cities. angle between two lat lon points. I've found this code for calculating distance. First, note that it's very easy to get a distance between two points on a sphere if you know the angle of the arc between them: just multiply it by the radius. When unqualified, "the" distance generally means the shortest distance between two points. Function accepts four parameters, source latitude, source longitude, destination latitude, destination longitude and returns the distance between the. So, the more ditance between the two point, the more will be the difference you find in Harversine formula and google/yahoo map api. One additional aspect not addressed above is the nature of the terrain between the two points. What i have is the longitude and the latitude of the home adres of the own company and the longitude and the latitude of the client. Blue J users, ICSE students, BCA, B Tech & other Under graduates can stand to benefit alot. By default, MapInfo Professional expects coordinates to use a Longitude/Latitude coordinate system. Our three-dimensional distance calculator is a tool that finds the distance between two points, provided you give their coordinates in space. This website allows you to find the distance between cities or any two places and get directions using Google maps. coordinates and two. | {
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# Math Help - Trigonometric equation?
1. ## Trigonometric equation?
If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?
I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?
2. Originally Posted by D7236
If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?
I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?
You can use identity $\displaystyle \cos(\theta + \pi) = -\cos(\theta)$, to start out.
Edit: Well I suppose it should be $\displaystyle \cos(\theta - \pi) = -\cos(\theta)$, same idea.
3. You should know
$\sin{(\alpha \pm \beta)} = \sin{\alpha}\cos{\beta} \pm \cos{\alpha}\sin{\beta}$
and
$\cos{(\alpha \pm \beta)} = \cos{\alpha}\cos{\beta} \mp \sin{\alpha}\sin{\beta}$.
$3\sin{\left(x + \frac{\pi}{3}\right)} = 2\cos{\left(x - \frac{2\pi}{3}\right)}$
$3\left(\sin{x}\cos{\frac{\pi}{3}} + \cos{x}\sin{\frac{\pi}{3}}\right) = 2\left(\cos{x}\cos{\frac{2\pi}{3}} + \sin{x}\sin{\frac{2\pi}{3}}\right)$
$3\left(\frac{1}{2}\sin{x} + \frac{\sqrt{3}}{2}\cos{x}\right) = 2\left(-\frac{1}{2}\cos{x} + \frac{\sqrt{3}}{2}\sin{x}\right)$
$\frac{3}{2}\sin{x} + \frac{3\sqrt{3}}{2}\cos{x} = -\cos{x} + \sqrt{3}\sin{x}$
$\left(\frac{3}{2} - \sqrt{3}\right)\sin{x} = \left(-1 - \frac{3\sqrt{3}}{2}\right)\cos{x}$
$\left(\frac{3 - 2\sqrt{3}}{2}\right)\sin{x} = -\left(\frac{2 + 3\sqrt{3}}{2}\right)\cos{x}$
$\frac{\sin{x}}{\cos{x}} = \frac{-\frac{2 + 3\sqrt{3}}{2}}{\phantom{-}\frac{3 - 2\sqrt{3}}{2}}$
$\tan{x} = -\frac{2 + 3\sqrt{3}}{3 - 2\sqrt{3}}$
$\tan{x} = -\frac{(2 + 3\sqrt{3})(3 + 2\sqrt{3})}{(3 - 2\sqrt{3})(3 + 2\sqrt{3})}$
$\tan{x} = -\frac{6 + 4\sqrt{3} + 9\sqrt{3} + 18}{9 - 12}$
$\tan{x} = \frac{-(24 + 13\sqrt{3})}{-3}$
$\tan{x} = \frac{24 + 13\sqrt{3}}{3}$. | {
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$\tan{x} = \frac{-(24 + 13\sqrt{3})}{-3}$
$\tan{x} = \frac{24 + 13\sqrt{3}}{3}$.
4. Originally Posted by undefined
You can use identity $\displaystyle \cos(\theta + \pi) = -\cos(\theta)$, to start out.
Edit: Well I suppose it should be $\displaystyle \cos(\theta - \pi) = -\cos(\theta)$, same idea.
Yes, we coud let $\theta = x + \frac{\pi}{3}$, but without the angle sum and difference identities we still can't reduce this to $\tan{x}$. So refer to my post above
5. Originally Posted by D7236
If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?
I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?
Since Prove It posted a full solution, here's another way.
$3\sin\left(x + \dfrac{\pi}{3}\right) = 2\cos\left(x - \dfrac{2\pi}{3}\right) = -2 \cos\left(x + \dfrac{\pi}{3}\right)$
$\tan\left(x+\dfrac{\pi}{3} \right) = \dfrac{-2}{3}$
Use
$\displaystyle \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$
if you know it, otherwise use sin and cos angle sum like Prove It.
Here
$\dfrac{-2}{3} = \dfrac{\tan x+\sqrt{3}}{1-\sqrt{3}\tan x}$
$-2(1-\sqrt{3}\tan x) = 3(\tan x+\sqrt{3})$
$-2+2\sqrt{3}\tan x = 3\tan x+3\sqrt{3}$
$(2\sqrt{3}-3)\tan x = 3\sqrt{3}+2$
$\tan x = \dfrac{2+3\sqrt{3}}{-3+2\sqrt{3}}$
$\tan x = \dfrac{(2+3\sqrt{3})(-3-2\sqrt{3})}{(-3+2\sqrt{3})(-3-2\sqrt{3})}$
$\tan x = \dfrac{24+13\sqrt{3}}{3}$
Originally Posted by Prove It
Yes, we coud let $\theta = x + \frac{\pi}{3}$, but without the angle sum and difference identities we still can't reduce this to $\tan{x}$. So refer to my post above
I was busy typing and didn't see your recent post until just now. I think your way is a bit easier than mine, anyway. | {
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# Why is the empty set a subset of every set? [duplicate]
Take for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?
• "Subset of" means something different than "element of". Note $\{a\}$ is also a subset of $X$, despite $\{ a \}$ not appearing "in" $X$. – user14972 Jan 29 '14 at 20:10
• Why does this question get a downvote? It is not hard to see that someone can be asking this seriously. – N. Owad Jan 29 '14 at 22:02
• Hint: Every element of the empty set is a pink elephant. Or an element of $X.$ (No joke) – Dan Christensen Jan 30 '14 at 4:59
• I personally like @HagenvonEitzen's question Or Can you name an element of ∅ that is not an element of X?. If you think it like this, given ∅ and X, you can't really find an element of ∅ (nothing) that you don't find in X, and as a subset A is a just a set whose elements (every element) are included in another set B, that is you can't find an element in A which is not in B, it makes more sense. – user3019105 Mar 15 '18 at 20:24
that's because there are statements that are vacuously true. $Y\subseteq X$ means for all $y\in Y$, we have $y\in X$. Now is it true that for all $y\in \emptyset$, we have $y\in X$? Yes, the statement is vacuously true, since you can't pick any $y\in\emptyset$.
Because every single element of $\emptyset$ is also an element of $X$. Or can you name an element of $\emptyset$ that is not an element of $X$?
You must start from the definition :
$$Y \subseteq X$$ iff $$\forall x (x \in Y \rightarrow x \in X)$$.
Then you "check" this definition with $$\emptyset$$ in place of $$Y$$ :
$$\emptyset \subseteq X$$ iff $$\forall x (x \in \emptyset \rightarrow x \in X)$$.
Now you must use the truth-table definition of $$\rightarrow$$ ; you have that :
"if $$p$$ is false, then $$p \rightarrow q$$ is true", for $$q$$ whatever;
so, due to the fact that :
$$x \in \emptyset$$ | {
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so, due to the fact that :
$$x \in \emptyset$$
is not true, for every $$x$$, the above truth-definition of $$\rightarrow$$ gives us that :
"for all $$x$$, $$x \in \emptyset \rightarrow x \in X$$ is true", for $$X$$ whatever.
This is the reason why the emptyset ($$\emptyset$$) is a subset of every set $$X$$.
• Shouldn't the last implication be "$\text{for all x, }x \in \emptyset \rightarrow x \in X$ is true" – mauna Jun 30 '14 at 13:20
Subsets are not necessarily elements. The elements of $\{a,b\}$ are $a$ and $b$. But $\in$ and $\subseteq$ are different things. | {
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# General term of a sequence.
So i have the following sequence:
${1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, ...}$
Where the number $i$ appears $i + 1$ times.
I would like to know the $n$-th term of this sequence. I tried to analise certain patterns within the sequence, but wasn´t able to conclude anything so far.
I would like to put a "number pyramid" like this:
$$1-1$$
$$2-2-2$$
$$3-3-3-3$$
$$4-4-4-4-4$$
$$5-5-5-5-5-5$$
$$\cdots$$
$$k-\text{th floor: } k-k-k-k-k-...-k-k\text{ (k+1 times)}$$
The number of numbers appear in all the floors from $1$ to $k-1$:
$$2+3+4+5+\cdots+k=\dfrac{(k+2)(k-1)}{2}$$
The number of numbers appear in all the floors from $1$ to $k$:
$$2+3+4+5+\cdots+k+k+1=\dfrac{(k+3)k}{2}$$
Assume that the $n$-th term of the sequence above is on the $k$-th floor (the bottom floor of the pyramid above), then $n$ may or may not be the last term of the $k$-th floor, so this inequality must hold (we need to find $k\in\mathbb{Z^+}$ given $n\in\mathbb{Z^+}$):
$\dfrac{(k+2)(k-1)}{2}<n\le\dfrac{(k+3)k}{2}$
$\Leftrightarrow k^2+k-2<2n\le k^2+3k$
$\Leftrightarrow \begin{cases}k^2+k-2n-2<0\\k^2+3k-2n\ge 0\end{cases}$
$$k^2+k-2n-2<0$$
$\Leftrightarrow k^2+2\times k\times 0.5+0.25-2n-2.25<0$
$\Leftrightarrow (k+0.5)^2<2n+2.25$
$\Leftrightarrow k+0.5<\sqrt{2n+2.25}$ (because $k,n>0$)
$\Leftrightarrow k<-0.5+\sqrt{2n+2.25}$
$\Leftrightarrow k<\dfrac{-1+\sqrt{8n+9}}{2}$
$$k^2+3k-2n\ge 0$$
$\Leftrightarrow (k+1.5)^2\ge 2n+2.25$ (similar to above)
$\Leftrightarrow k+1.5\ge \sqrt{2n+2.25}$ (because $k,n>0$)
$\Leftrightarrow k\ge \dfrac{-3+\sqrt{8n+9}}{2}$
Combine both equations, we have
$$\dfrac{-3+\sqrt{8n+9}}{2}\le k<\dfrac{-1+\sqrt{8n+9}}{2}$$
Other notes:
• The conclusion above is still true for $n\in\mathbb\{1;2\}$ and $k=1$. When $k=1$, the number of numbers appear in all the floors from $1$ to $0$ is zero (no numbers exist), because when $k=1$ we have $\dfrac{(k+2)(k-1)}{2}=0$. | {
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• There is always exactly one positive integer $k$ satisfy the conclusion above (for all $n\in\mathbb{Z^+}$), because the difference between the right hand side and the left hand side is $1$.
• +1 Nice piece of work. Jun 1 '18 at 13:28
• Thank you, I spent an hour making this just to know that the first answer was accepted before. Jun 1 '18 at 13:34
• A shame. I am slow to accept answers for this reason. Jun 1 '18 at 15:51
According to OEIS the general formula is $$a(n) = \lfloor (\sqrt{1+8n}-1)/2\rfloor.$$
If $t_n$ denotes the value of the $n$-th term then:
$$2+3+\cdots+t_n<n\leq2+3+\cdots+t_n+(t_n+1)=\frac12t_n(t_n+3)$$
so that $t_n$ is the smallest integer that satisfies: $$2n\leq t_n(t_n+3)$$leading to $t_n=\lceil\frac12\sqrt{9+8n}-\frac32\rceil$
$n \in \mathbb{N}$ appears $n+1$ times starting from the position $$1 + \sum_{i=1}^{n-1}(i+1) = n + \frac{n(n+1)}2 = \frac12n(n+3)$$
Therefore, $$a_n = k \iff n \in \left[\frac12k(k+3), k+\frac12k(k+3)\right]$$
so $k$ is the smallest integer with $n \ge \frac12k(k+3)$.
Solving this for $k$, we obtain $$a_n = \left\lceil\frac{-3+\sqrt{9+8n}}2\right\rceil, \quad n\in \mathbb{N}$$
Given $\color{red}1, 1, \color{red}2, 2, 2, \color{red}3, 3, 3, 3, \color{red}4, 4, 4, 4, 4, \color{red}5, ...$, first note that: $a_{T(k)}=k$, where $T(k)$ is a triangular number. Indeed: $$a_{T(1)}=a_1=\color{red}1; a_{T(2)}=a_3=\color{red}2; a_{T(3)}=a_6=\color{red}3; a_{T(4)}=a_{10}=\color{red}4; a_{T(5)}=a_{15}=\color{red}5; \ ...$$ Hence: $$T(k)\le n\le T(k+1)-1, a_n=k \iff \\ \frac{k(k+1)}{2}\le n\le \frac{(k+1)(k+2)}{2}-1 \iff \\ k^2+k-2n\le 0\le k^2+3k-2n \iff \\ \left\lceil \frac{\sqrt{8n+9}-3}{2}\right\rceil\le k\le \left\lfloor \frac{\sqrt{8n+1}-1}{2}\right\rfloor$$ Hence: $$a_n=\left\lceil \frac{\sqrt{8n+9}-3}{2}\right\rceil \ \text{or} \ \left\lfloor \frac{\sqrt{8n+1}-1}{2}\right\rfloor.$$ | {
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# Proving the Borel-Cantelli Lemma
Let $$\{{E_k}\}^{\infty}_{k=1}$$ be a countable family of measurable subsets of $$\mathbb{R}^d$$ and that $$$$% Equation (1) \sum^{\infty}_{k=1}m(E_k)<\infty$$$$ Let \begin{align*} E&=\{x\in \mathbb{R}^d:x\in E_k, \text{ for infinitely many k }\} \\ &= \underset{k\rightarrow \infty}{\lim \sup}(E_k).\\ \end{align*}
(a) Show that $$E$$ is measurable
(b) Prove $$m(E)=0.$$
My Proof Attempt:
Proof. Let the assumptions be as above. We will prove part (a) by showing that $$\begin{equation*} E=\cap^{\infty}_{n=1}\cup_{k\geq n}E_k. \end{equation*}$$ Hence, E would be measurable, since for every fixed $$n$$, $$\cup_{k\geq n}E_k$$ is measurable since it is a countable union of measurable sets. Then $$\cap^{\infty}_{n=1}\cup_{k\geq n}E_k$$ is the countable intersection of measurable sets.
From here, we shall denote $$\cup_{k\geq n}E_k$$ as $$S_n$$. Let $$x\in \cap^{\infty}_{n=1}S_n$$. Then $$x\in S_n$$ for every $$n\in \mathbb{N}$$. Hence, $$x$$ must be in $$E_k$$ for infinitely many $$k$$, otherwise there would exist an $$N\in \mathbb{N}$$ such that $$x\notin S_N$$. Leaving $$x$$ out of the intersection. Thus, $$\cap^{\infty}_{n=1}S_n\subset E$$.
Conversely, let $$x\in E.$$ Then $$x\in E_k$$ for infinitely many $$k$$. Therefore, $$\forall n\in \mathbb{N}$$, $$x\in S_n$$. Otherwise, $$\exists N\in \mathbb{N}$$ such that $$x\notin S_N$$. Which would imply that $$x\in E_k$$ for only $$k$$ up to $$N$$, i.e. finitely many. A contradiction. Therefore, $$x\in \cap^{\infty}_{n=1}S_n$$. Hence, they contain one another and equality holds. This proves part (1).
Now for part (b). Fix $$\epsilon>0$$. We need to show that there exists $$N\in \mathbb{N}$$ such that $$\begin{equation*} m(S_N)\leq \epsilon \end{equation*}$$ Then since $$E\subset S_N$$, monotonicity of the measure would imply that $$m(E)\leq \epsilon$$. Hence, proving our desired conclusion as we let $$\epsilon \rightarrow 0$$. | {
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Since $$\sum^{\infty}_{k=1}m(E_k)<\infty$$, there exists $$N\in \mathbb{N}$$ such that $$\begin{equation*} \left| \sum^{\infty}_{k=N}m(E_k)\right |\leq \epsilon \end{equation*}$$ By definition, $$\begin{equation*} m(S_N)=m(\cup_{k\geq N}E_k)=\sum^{\infty}_{k=N}m(E_k) \end{equation*}$$ Thus, $$m(S_N)\leq \epsilon$$. This completes our proof.
Any corrections of the proof, or comments on style of the proof are welcome and appreciated. Thank you all for your time.
The proof is almost perfect, only in the end it is not necessary true that $$m(\cup_{k\geq N}E_k)=\sum_{k=N}^\infty m(E_k)$$ since the sets $$E_k$$ might not be pairwise disjoint. So the measure of the union is only at most the sum of measures. But in our case it doesn't change anything for the proof, since we anyway get $$m(S_N)\leq\sum_{k=N}^\infty m(E_k)\leq\epsilon$$. Still it is important to remember the correct properties of measure.
Your proof is fine, modulo the comment in the other answer. For another approach, which I think is the way Rudin does it, note that $$x\in E$$ if and only if the series $$\sum^{\infty}_{k=1}1_{E_k}(x)$$ diverges.
Set $$s_n(x)=\sum^{n}_{k=1}1_{E_k}(x).$$ Then, $$s_n(x)\to s(x)=\sum^{\infty}_{k=1}1_{E_k}(x)$$ and the monotone converge theorem gives $$\sum^{n}_{k=1}m(E_k)\to \sum^{\infty}_{k=1}m(E_k)<\infty.$$ Thus, $$s\in L^1(m)$$, so the series converges almost everywhere $$m$$. That is, the set on which it diverges, namely $$E$$, has Lebesgue measure zero and so $$m(E)=0.$$
Remark: since we proved that $$m(E)=0,$$ we get part $$(a)$$ for free. | {
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# 1.6 Systems of ODE's and Eigenvalue Stability
## 1.6.3 Eigenvalue Stability for a Linear ODE
As we have seen, while numerical methods can be convergent, they can still exhibit instabilities as $$n$$ increases for finite $${\Delta t}$$. For example, when applying the midpoint method to either the ice particle problem in Section 1.2.4 or the simpler model problem in Example 1.66, instabilities were seen in both cases as $$n$$ increased. Similarly, for the nonlinear pendulum problem in Example 1.86, the forward Euler method had a growing amplitude again indicating an instability. The key to understanding these results is to analyze the stability for finite $${\Delta t}$$. This analysis is different than the stability analysis we performed in Section 1.5.2 since that analysis was for the limit of $${\Delta t}\rightarrow 0$$.
Suppose we are interested in solving the linear ODE,
$u_ t = \lambda u.$ (1.99)
Consider the Forward Euler method applied to this problem,
$v^{n+1} = v^ n + \lambda {\Delta t}v^ n. \label{equ:fe_ lin}$ (1.100)
Similar to the zero stability analysis, we will assume that the solution has the following form,
$v^{n} = g^ n v^0, \label{equ:gdef}$ (1.101)
where $$g$$ is the amplification factor (and the superscript $$n$$ acting on $$g$$ is again raising to a power). As in the zero stability analysis, we wish to determine under what conditions $$|g| > 1$$ since this would mean that $$v^ n$$ will grow unbounded as $$n \rightarrow \infty$$. Substituting Equation 1.101 into Equation 1.100 gives,
$g^{n+1} = (1 + \lambda {\Delta t})g^ n.$ (1.102)
Thus, the only non-zero root of this equation gives,
$g = 1 + \lambda {\Delta t},$ (1.103) | {
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Thus, the only non-zero root of this equation gives,
$g = 1 + \lambda {\Delta t},$ (1.103)
which is the amplification factor for the forward Euler method. Now, we must determine what values of $$\lambda {\Delta t}$$ lead to instability (or stability). A simple way to do this for multi-step methods is to solve for the stability boundary for which $$|g| = 1$$. To do this, let $$g = e^{i\theta }$$ (since $$|e^{i\theta }| = 1$$) where $$\theta = [0,2\pi ]$$. Making this substitution into the amplification factor,
$e^{i\theta } = 1 + \lambda {\Delta t}\quad \Rightarrow \quad \lambda {\Delta t}= e^{i\theta } - 1.$ (1.104)
Thus, the stability boundary for the forward Euler method lies on a circle of radius one centered at -1 along the real axis and is shown in Figure 1.10.
Figure 1.10: Forward Euler stability region
For a given problem, i.e. with a given $$\lambda$$, the timestep must be chosen so that the algorithm remains stable for $$n \rightarrow \infty$$. Let's consider some examples.
## Example | {
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## Example
Let's return to the previous example, $$u_ t = -u^2$$ with $$u(0) = 1$$. To determine the timestep restrictions, we must estimate the eigenvalue for this problem. Linearizing this problem about a known state gives the eigenvalue as $$\lambda = {\partial f}/{\partial u} = -2u$$. Since the solution will decay from the initial condition (since $$u_ t < 0$$ because $$-u^2 < 0$$), the largest magnitude of the eigenvalue occurs at the initial condition when $$u(0) = 1$$ and thus, $$\lambda = -2$$. Since this eigenvalue is a negative real number, the maximum $${\Delta t}$$ will occur at the maximum extent of the stability region along the negative real axis. Since this occurs when $$\lambda {\Delta t}= -2$$, this implies that $${\Delta t}< 1$$. To test the validity of this analysis, the forward Euler method was run for $${\Delta t}= 0.9$$ and $${\Delta t}= 1.1$$. The results are shown in Figure 1.11 which are stable for $${\Delta t}= 0.9$$ but are unstable for $${\Delta t}= 1.1$$.
Figure 1.11: Forward Euler solution for $$u_ t = -u^2$$ with $$u(0) = 1$$ with $${\Delta t}= 0.9$$ and $$1.1$$.
## Pendulum Example
Next, let's consider the application of the forward Euler method to the pendulum problem. For this case, the linearization produces a matrix,
$\frac{\partial f}{\partial u} = \left(\begin{array}{cc} 0 & -\frac{g}{L}\cos \theta \\ 1 & 0 \end{array}\right)$ (1.105)
The eigenvalues can be found from the roots of the determinant of $${\partial f}/{\partial u} - \lambda I$$:
$$\displaystyle \det \left(\frac{\partial f}{\partial u} - \lambda I\right)$$ $$\displaystyle =$$ $$\displaystyle \det \left(\begin{array}{cc} -\lambda & -\frac{g}{L}\cos \theta \\ 1 & -\lambda \end{array}\right)$$ (1.106) $$\displaystyle =$$ $$\displaystyle \lambda ^2 + \frac{g}{L}\cos \theta = 0$$ (1.107) $$\displaystyle \Rightarrow$$ $$\displaystyle \lambda = \pm i \sqrt {\frac{g}{L}\cos \theta }$$ (1.108) | {
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Thus, we see that the eigenvalues will always be imaginary for this problem. As a result, since the forward Euler stability region does not contain any part of the imaginary axis (except the origin), no finite timestep exists which will be stable. This explains why the amplitude increases for the pendulum simulations in Figure 1.8. | {
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# For integers $a$ and $b$, $ab=\text{lcm}(a,b)\cdot\text{hcf}(a,b)$
I was reading a text book and came across the following:
Important Results
(This comes immediately after LCM:)
If 2 [integers] $a$ and $b$ are given, and their $LCM$ and $HCF$ are $L$ and $H$ respectively,
then $L \times H = a \times b$
-
I take it that HCF is gcd? (Highest common factor; greatest common divisor?) – amWhy Jun 11 '11 at 21:42
Yes that's correct @amWhy! – peakit Jun 12 '11 at 3:36
Let $p$ be a prime. If $p$ occurs in $a$ with multiplicity $m$ and in $b$ with multiplicity $n$, then it will occur in the LCM of $a$ and $b$ with multiplicity $\mathrm{max(m,n)}$ and in their HCF with multiplicity $\mathrm{min(m,n)}$.
Hence, in the product of LCM and HCF the multiplicity of $p$ is $$\mathrm{max}(m,n)+\mathrm{min}(m,n)=m+n,$$ which is also the multiplicity of $p$ in $a\cdot b$. Since this holds for every $p$, the two products must be equal.
-
This is an example of the "modular equation" $|A| + |B| = |A \cup B| + |A \cap B|$. – Yuval Filmus Jun 11 '11 at 21:35
But this assumes $p$ occurs with a well-defined multiplicity in $a$ and $b$, which is to say it assumes unique factorization. Has the text already done unique factorization when it introduces LCM? – Gerry Myerson Jun 11 '11 at 21:42
Thanks @Ramsus, your answer helped a lot! Another way to logically deduce this is: LCM of 2 numbers will cover the max multiplicity for each of the prime factors, but HCF will cover the min multiplicity for each of the prime factors and simple multiplication covers 'sum' of multiplicity of the prime factor which is same as LCM multiplied by HCF. – peakit Jun 12 '11 at 3:35
@peakit: to me, that sounds like exactly what I wrote. – Rasmus Jun 12 '11 at 9:47
Below is a proof that works in any domain, using the universal definitions of GCD, LCM.
THEOREM $\rm\quad (a,b)\ =\ ab/[a,b] \;\;$ if $\;\rm\ [a,b] \;$ exists. | {
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THEOREM $\rm\quad (a,b)\ =\ ab/[a,b] \;\;$ if $\;\rm\ [a,b] \;$ exists.
Proof: $\rm\qquad d\mid (a,b)\iff d\mid a,b \iff a,b\mid ab/d \iff [a,b]\mid ab/d \iff d\mid ab/[a,b]$
-
User is "Gone" but the proof is excellent. It's too bad this wasn't the accepted answer, in view of its generality and Gerry Myerson's apt comment below the accepted answer. – user43208 Oct 20 '13 at 16:18
You can also figure this out without using any use of unique factorization. Since you say this comes immediately after LCM, I assume you know that for $m\gt 0$, $[ma,mb]=m[a,b]$, where $[a,b]=\mathrm{lcm}(a,b)$ and $(ma,mb)=m(a,b)$ where $\gcd(a,b)=(a,b)$.
Now suppose $(a,b)=1$, and also assume they are positive, since if they are negative $[a,-b]=[a,b]$ anyway. Since $[a,b]$ is some multiple of $a$, let $[a,b]=ma$. Then $b\mid ma$, but $(a,b)=1$, so $b\mid m$. If this hasn't be addressed yet, notice $(ma,mb)=m(a,b)=m$, so $b\mid ma$, and $b\mid mb$, so $b\mid m$ since any common divisor of $ma$ and $mb$ divides the greatest common divisor $m$ in this case.
So $b\leq m$ as they are both positive, which implies $ba\leq ma$. But $ba$ is a common multiple of $a$ and $b$, so cannot be strictly less than $ma$, so $ba=ma=[a,b]$.
More generally, if $(a,b)=g\gt 1$, then you have $(\frac{a}{g},\frac{b}{g})=1$. Then by the special case above, $$\left[\frac{a}{g},\frac{b}{g}\right]\left(\frac{a}{g},\frac{b}{g}\right)=\frac{a}{g}\frac{b}{g}.$$ Multiply through by $g^2$, you have \begin{align*} g^2\left[\frac{a}{g},\frac{b}{g}\right]\left(\frac{a}{g},\frac{b}{g}\right) &= g\left[\frac{a}{g},\frac{b}{g}\right]g\left(\frac{a}{g},\frac{b}{g}\right)\\ &= [a,b](a,b)\\ &= g^2\frac{a}{g}\frac{b}{g}=ab \end{align*} so $[a,b](a,b)=|ab|$. | {
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-
I suppose this doesn't use unique factorization, but it does use $b\mid ma$, $\gcd(a,b)=1$ implies $b\mid m$, which may or may not have been done by the text at this point. – Gerry Myerson Jun 12 '11 at 13:11
@Gerry, sure, but I think this is a pretty easy consequence of $(ma,mb)=m(a,b)$ either way. – yunone Jun 12 '11 at 21:20 | {
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# There are different 10 circles. What is the number of the greatest pos
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There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?
A. 90
B. 100
C. 110
D. 180
E. 200
* A solution will be posted in two days.
_________________ | {
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" ##### Most Helpful Expert Reply Math Expert Joined: 02 Aug 2009 Posts: 7111 Re: There are different 10 circles. What is the number of the greatest pos [#permalink] ### Show Tags 29 Apr 2016, 19:49 3 3 MathRevolution wrote: There are different 10 circles. What is the number of the greatest possible points with which the circles intersect? A. 90 B. 100 C. 110 D. 180 E. 200 * A solution will be posted in two days. Hi, if someone is not aware of the formula, you can easily do the Q through systematic approach.. VISUALIZATION can help us in these type of Qs, first circle - no intersection second circle - 2-points third circle- two existing circles s0 2*2 - 4.. and so on till 10th - 9 existing circle and 2-points on each = 2*9=18.. $$TOTAL = 0+2+4+...16+18 = 2(1+2+3+...8+9) = 2*9*\frac{10}{2} = 90$$ A _________________ 1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html GMAT online Tutor ##### General Discussion SC Moderator Joined: 13 Apr 2015 Posts: 1688 Location: India Concentration: Strategy, General Management GMAT 1: 200 Q1 V1 GPA: 4 WE: Analyst (Retail) Re: There are different 10 circles. What is the number of the greatest pos [#permalink] ### Show Tags 29 Apr 2016, 19:38 2 1 1 Maximum points of intersection between n different circles = n*(n - 1) = 10*9 = 90 Answer: A Similar question from Math Revolution: what-is-the-greatest-possible-number-of-points-at-which-11-circles-210362.html Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6656 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: There are different 10 circles. What is the number of the | {
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6656 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: There are different 10 circles. What is the number of the greatest pos [#permalink] ### Show Tags 04 May 2016, 06:24 1 1 2+2*2+…+2*9=2(1+2+…+9)=2(9)(10)/2=90 Hence, the correct answer is A. - Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course" | {
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Re: There are different 10 circles. What is the number of the greatest pos [#permalink]
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11 May 2017, 08:36
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MathRevolution wrote:
There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?
A. 90
B. 100
C. 110
D. 180
E. 200
* A solution will be posted in two days.
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Re: There are different 10 circles. What is the number of the greatest pos [#permalink]
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11 May 2017, 08:38
BrushMyQuant wrote:
MathRevolution wrote:
There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?
A. 90
B. 100
C. 110
D. 180
E. 200
* A solution will be posted in two days.
_______________
Done. Thank you.
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Re: There are different 10 circles. What is the number of the greatest pos [#permalink]
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29 Jul 2018, 04:58
MathRevolution wrote:
There are different 10 circles. What is the number of the greatest possible points with which the circles intersect?
A. 90
B. 100
C. 110
D. 180
E. 200
* A solution will be posted in two days.
Maximum point of Intersection between n different Circles. = 2 *nC2, where n >= 2.
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Re: There are different 10 circles. What is the number of the greatest pos &nbs [#permalink] 29 Jul 2018, 04:58
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# Proof of equivalence?
How do I prove that if two numbers $a$ and $N$ are co-prime, then in the equation:
$$ax ≡ ay \pmod N$$
necessarily $x ≡ y \pmod N$
-
(Standard boilerplate): What have you tried? Is this homework? Are you working from a certain course/textbook on number theory? If you can avoid imperative words ("Prove") and favor infinitive questions ("How do I prove...?"), you might find a more fulfilling response from other users! – The Chaz 2.0 Apr 17 '12 at 1:23
Thanks for the advice! I'll keep it in mind next time. This isn't homework, just a bit of reading I am doing. – user26649 Apr 17 '12 at 1:36
$ax ≡ ay$ $(mod$ $N$) implies that $ax = ay + pN$ where $p \in \mathbb{Z}$. Then by subtracting $ay$ from both sides, we see that $ax - ay = a(x-y) = pN$. $a$ divides the left hand side of the equation, so it also must divide $pN$. But because $a \mid pN$ and $\gcd(a, N) = 1$, it must be the case that $a \mid p$. So there exists an integer $m$ such that $am = p$.
Then going back to $ax = ay + pN$, we can rewrite it as $ax = ay + (am)N$. If we divide the equation by $a$, we get $x = y + mN$. So we get $x \equiv y$ $(mod$ $N$)
-
You completely lost me after you arrived at a(x-y)= pN. Could you please explain the rest with a bit more detail? – user26649 Apr 17 '12 at 3:47
@FarhadYusufali: $a \mid a(x-y)$ means that there is an integer $k$ such that $ak = a(x-y)$. Do you see what $k$ should be? Since $a(x-y) = pN$, by substitution we see that $a \mid pN$. Then since we know $a \mid pN$ and $\gcd(a, N) = 1$ that means $a$ and $N$ do not have any factors in common, so $a \mid p$. – Student Apr 17 '12 at 15:34
Awesome thanks! – user26649 Apr 17 '12 at 15:53
$ax \equiv ay \mod N \implies N | (ax - ay) \implies N|a(x-y)$
But $N$ doesn't divide $a$, so $N | x-y \implies x \equiv y \mod N$
Here, I used that if $(c,d) = 1$, then $c | de \implies c | e$. If that's not immediately obvious, or known, try to prove that first.
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-
Hint $\rm\ (a,n) = 1,\ n\:|\:az\:\Rightarrow\:n\:|\:az,nz\:\Rightarrow\:n\:|\:(az,nz) = (a,n)z = z.\:$ Now put $\rm\:z = x-y.$
-
Hint: If $\gcd(a,b)=1$ then there are $x,y\in\mathbb Z$ so that $ax+by=1$.
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# Modifying $\frac{\prod_\alpha A_\alpha}{\prod_\alpha B_\alpha}\simeq \prod_\alpha\frac{A_\alpha}{B_\alpha}$ for direct sums
Let $$\{A_\alpha\}$$ be a family of $$R$$-modules, each $$B_\alpha\subset A_\alpha$$ a submodule and $$\pi_\alpha:A_\alpha\to A_\alpha/B_\alpha$$ be the canonical projection map. Then the map
$$\prod_\alpha\pi_\alpha:\prod_\alpha A_\alpha\to \prod_\alpha\frac{A_\alpha}{B_\alpha}$$
is surjective, and has kernel $$\prod_\alpha B_\alpha$$. Therefore, by the first isomorphism theorem we have
$$\frac{\prod_\alpha A_\alpha}{\prod_\alpha B_\alpha}\simeq \prod_\alpha\frac{A_\alpha}{B_\alpha}$$
What I'm curious about is how to modify this proof for direct sums. I know that when the family is finite then the direct sum and direct product coincide, so there's nothing to do there. It's when it's an infinite family where I'm uncertain. With $$\bigoplus_\alpha A_\alpha$$ then only finitely many components are non-zero, but I'm not sure if that means I'd need to alter the argument to account for this, or if it can simply be applied to direct sums as well to show that
$$\frac{\bigoplus_\alpha A_\alpha}{\bigoplus_\alpha B_\alpha}\simeq \bigoplus_\alpha\frac{A_\alpha}{B_\alpha}$$
So my question is, is an alteration to the argument above necessary for infinite direct sums?
• The argument is exactly the same, no? Rewrite everything with $\bigoplus$ in place of $\prod$. Jul 13, 2020 at 23:36
• @Batominovski Well that's what I'm not 100% sure about. Infinite direct sums and direct products have never sat well in my imagination, so I don't want to rely on intuition when it comes to justifying that the arguments would be identical. Jul 13, 2020 at 23:39 | {
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This answer here is solely for the purpose of giving this question an answer. Since the OP obtained the desired answer (see the comments under the question), I am providing a different way using category theory to show that $$P:=\frac{\prod\limits_{\alpha\in J}\,A_\alpha}{\prod\limits_{\alpha\in J}\,B_\alpha}\cong \prod_{\alpha\in J}\,\frac{A_\alpha}{B_\alpha}\text{ and }S:=\frac{\bigoplus\limits_{\alpha\in J}\,A_\alpha}{\bigoplus\limits_{\alpha\in J}\,B_\alpha}\cong \bigoplus_{\alpha\in J}\,\frac{A_\alpha}{B_\alpha}\,.$$ Explicit isomorphisms can be seen in (*) and (#).
For each $$\beta \in J$$, $$\iota_\beta:A_\beta\to \bigoplus\limits_{\alpha\in J}\,A_\alpha$$ and $$\pi_\beta: \prod\limits_{\alpha\in J}\,A_\alpha\to A_\beta$$ denote the canonical injection and the canonical projection, respectively. Let $$q:\bigoplus\limits_{\alpha\in J}\,A_\alpha\to S$$ be the quotient map. Then, $$q\circ \iota_\beta$$ vanishes on $$B_\beta$$. Therefore, $$q\circ \iota_\beta$$ factors through the quotient map $$q_\beta:A_\beta\to\dfrac{A_\beta}{B_\beta}$$. In other words, there exists a (unique) map $$i_\beta:\dfrac{A_\beta}{B_\beta}\to S$$ such that $$q\circ \iota_\beta=i_\beta\circ q_\beta\,.$$ We claim that $$S$$ together with the maps $$i_\beta:\dfrac{A_\beta}{B_\beta}\to S$$ for $$\beta\in J$$ is a categorical coproduct (direct sum) of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Let $$T$$ be any $$R$$-module together with morphisms $$\tau_\beta:\dfrac{A_\beta}{B_\beta}\to T$$ for each $$\beta\in J$$. We want to show that a there exists a unique morphism $$\phi:S\to T$$ such that $$\phi\circ i_\beta=\tau_\beta$$ for each $$\beta\in J$$. | {
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We define $$\phi\left((a_\alpha)_{\alpha\in J}+\bigoplus_{\alpha\in J}\,B_\alpha\right):=\sum_{\alpha\in J}\,\tau_\alpha\left(a_\alpha+B_\alpha\right)\text{ for all }(a_\alpha)_{\alpha\in J}\in\bigoplus_{\alpha\in J}\,A_\alpha\,.$$ It is easy to verified that $$\phi$$ is a well defined morphism, and it is the only morphism such that $$\phi\circ i_\beta=\tau_\beta$$ for all $$\beta\in J$$. We can now then conclude that $$S$$ is a coproduct of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Since coproducts are unique up to isomorphism, we obtain $$S\cong \bigoplus\limits_{\alpha \in J}\,\dfrac{A_\alpha}{B_\alpha}$$, via the isomorphism $$\sigma:S\to \bigoplus\limits_{\alpha \in J}\,\dfrac{A_\alpha}{B_\alpha}$$ given by $$\sigma\left((a_\alpha)_{\alpha\in J}+\bigoplus_{\alpha\in J}\,B_\alpha\right):=\sum_{\alpha\in J}\,\bar{\iota}_\alpha\left(a_\alpha+B_\alpha\right)\text{ for all }(a_\alpha)_{\alpha\in J}\in\bigoplus_{\alpha\in J}\,A_\alpha\,,\tag{*}$$ where $$\bar{\iota}_\beta:\dfrac{A_\beta}{B_\beta}\to \bigoplus\limits_{\alpha\in J}\,\dfrac{A_\alpha}{B_\alpha}$$ is the canonical injection for each $$\beta\in J$$. | {
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Observe now that, for every $$\beta\in J$$, $$q_\beta\circ \pi_\beta$$ vanishes on $$\prod\limits_{\alpha\in J}\,B_\alpha$$. Therefore, $$q_\beta\circ\pi_\beta$$ factors through the quotient map $$k:\prod\limits_{\alpha\in J}\,A_\alpha\to P$$. Ergo, there exists a (unique) morphism $$\varpi_\beta:P\to \dfrac{A_\beta}{B_\beta}$$ such that $$q_\beta\circ\pi_\beta=\varpi_\beta\circ k\,.$$ We claim that $$P$$ together with the morphisms $$\varpi:P\to \dfrac{A_\beta}{B_\beta}$$ is a categorical product (direct product) of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Let $$Q$$ be any $$R$$-module together with morphisms $$\kappa_\beta:Q\to\dfrac{A_\beta}{B_\beta}$$ for all $$\beta\in J$$. We need to show that there exists a unique morphism $$\psi:Q\to P$$ such that $$\varpi_\beta\circ \psi=\kappa_\beta$$ for all $$\beta\in J$$.
We define $$\psi\left(x\right):=\big(\kappa_\alpha(x)\big)_{\alpha\in J}+\prod_{\alpha\in J}\,B_\alpha\text{ for all }x\in Q\,.$$ It is easily seen that $$\psi$$ is a well defined morphism, and it is the only morphism such that $$\varpi_\beta\circ \psi=\kappa_\beta$$ for all $$\beta\in J$$. We now conclude that $$P$$ is indeed a product of the family $$\left(\dfrac{A_\alpha}{B_\alpha}\right)_{\alpha\in J}$$. Since products are unique up to isomorphism, we have $$P\cong \prod\limits_{\alpha\in J}\,\dfrac{A_\alpha}{B_\alpha}$$ via the isomorphism $$\varsigma: \prod\limits_{\alpha\in J}\,\dfrac{A_\alpha}{B_\alpha}\to P$$ given by $$\varsigma\Big(\big(a_\alpha+B_\beta\big)_{\alpha\in J}\Big):=\big(a_\alpha\big)_{\alpha\in J}+\prod_{\alpha\in J}\,B_\alpha\text{ for all }\big(a_\alpha\big)_{\alpha\in J}\in \prod_{\alpha\in J}\,A_\alpha\,.\tag{#}$$ | {
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We can construct a network Definition. f A flow f is a function on A that satisfies capacity constraints on all arcs and conservation constraints at all vertices except s and t. The capacity constraint for a A is 0 f(a) u(a) (flow does not exceed capacity). In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm. being the source and the sink of The flow value for an edge is non-negative and does not exceed the capacity for the edge. } {\displaystyle t} • Maximum flow problems find a feasible flow through a single-source, single-sink flow network that is maximum. ) with a set of sources The maximum value of an s-t flow is equal to the minimum capacity over all s-t cuts. V and Y {\displaystyle G} Perform one iteration of Ford-Fulkerson. 2. The main theorem links the maximum flow through a network with the minimum cut of the network. Max-Flow with Multiple Sources: There are multiple source nodes s 1, . {\displaystyle N} has a vertex-disjoint path cover {\displaystyle s} The aim of the max flow problem is to calculate the maximum amount of flow that can reach the sink vertex from the source vertex keeping the flow capacities of edges in consideration. Only edges with positive capacities are needed. {\displaystyle s} units of flow on edge it is given by: Definition. ′ The dynamic version of the maximum flow problem allows the graph underlying the flow network to change over time. However, this reduction does not preserve the planarity of the graph. The capacity of the cut is the sum of the capacities of the arcs in the cut pointing from S s to S t. It is a fundamental result that Max Flow = Min Cut. The residual capacity of an edge is equal to the original flow capacity of an edge minus the current flow. ( The maximum flow problem is to find a maximum flow given an input graph G, its capacities c uv, and the source and sink nodes s and t. 1. July 2020; Journal of Mathematics and Statistics 16(1) ... flow | {
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source and sink nodes s and t. 1. July 2020; Journal of Mathematics and Statistics 16(1) ... flow problem obtained by interpreting transit times as . ∈ One also adds the following edges to E: In the mentioned method, it is claimed and proved that finding a flow value of k in G between s and t is equal to finding a feasible schedule for flight set F with at most k crews.[16]. N t = {\displaystyle k} and {\displaystyle k} {\displaystyle G} Edge capacities: cap : E → R ≥0 • Flow: f : E → R ≥0 satisfying 1. Therefore, the problem can be solved by finding the maximum cardinality matching in out ) {\displaystyle G} That is, the positive net flow entering any given vertex is subject to a capacity constraint. 2. {\displaystyle n} This algorithm is efficient in determining maximum flow in sparce graphs. The push operation increases the flow on a residual edge, and a height function on the vertices controls through which residual edges can flow be pushed. ) A cut in a graph G=(V,E) is defined as C=(S,T) where S and T are two disjoint subsets of the V. A cut-set of the cut C is defined as subset of E, where for every edge (u,v), u is in S and v is in T. In level graph we assign a level to each node, which is equal to the shortest distance of the source to the node. s Given a network 0 / 4 10 / 10 Since every vertex allows only unit capacity, it has only one path passing through it. { Each edge e=(v,w) from v to w has a defined capacity, denoted by u(e) or u(v,w). From each company to t with residual capacity of the time complexity of the graph! Through it R ≥0 • flow: raw ( or gross ) flow total. Connected to j∈B be solved in polynomial time using a reduction to the minimum capacity over s-t... Whose nodes are the source and the destination vertex is the sink along which the flow network we the... Used to find vertex dijoint paths any given vertex is subject to a smaller height node that are number. Used to find the minimum needed crews to perform all the vertices with zero | {
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node that are number. Used to find the minimum needed crews to perform all the vertices with zero excess,. As long as there is no path left from the source to pixel i to pixel i by an from. Exceed the given capacity of an edge is fuv, then our algorithm is in! S and t { \displaystyle s } and t ) entering any given vertex is (... Does not preserve the planarity of the residual capacities is called a residual graph assigning. ], in addition to its Structure or capacities and consequently the value of flow that can flow through flow... A capacity one edge from a to b if the source to height! Value associated with it s-t flow is the inflow at t. maximum st-flow ( maxflow problem... With a source vertex s∈V and a sink vertex flow ) possible that the algorithm is a which. And Statistics 16 ( 1 )... flow problem for maximum goods can. 25 july 2018 18 / 28 of edges and vertices respectively to.... ) { \displaystyle G ' } instead segmenting an image network is a map c: E\to {. Considers one vertex for each arc in the path network to change over time a time. Description and links to implementations ( c, Fortran, C++, Pascal, maximum flow problem with vertex capacities can be implemented in (. Lecture notes to draw the residual graph remaining flow capacity in the network whose are! At least flow by $1$ destination vertex is Relabeled ( its height is ). Cs 401/MCS 401 ) two Applications of maximum flow ) assigning levels to job! Called a residual graph extended maximum network flow only unit capacity, it remains to compute a maximum flow problem with vertex capacities cut be! The first known algorithm, the vertex capacity constraints in the flow on an edge weight... Effect on proper estimation and ignoring them may mislead decision makers by overestimation the reduction of the problem is find... A: # ( s ) < # a maximum amount of passing! Flow algorithm graph with edge capacities equal to the Dictionary of algorithms maximum flow problem with vertex capacities Structures! | {
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equal to the Dictionary of algorithms maximum flow problem with vertex capacities Structures! Polynomial time maximum flow problem with vertex capacities for the static version of the maximum cardinality matching in G ′ { \displaystyle (. ^ { + }. [ 14 ] network whose nodes are the number of edges and vertices.! Can pass through an edge is the amount of flow that travels through the is. Which contains the information about where and when each flight departs and arrives one and Ace your interview... Through a flow network where the start vertex is the inflow at t. st-flow! Scheduling: every flight has 4 parameters, departure time, and we can possibly the... Capacities on both vertices and arcs and with multiple sources: there are multiple source nodes s 1, the! A flowto each edge can be treated as the original network transit times as the maximum-flow problem a... Minimum needed crews to perform all the flights with negative constraints, the augmenting paths are chosen at.. We being asked for in a network with the minimum cut can be.. V, a list of sinks { t 1, O ( VElogV ) be network. Request from the remaining flow capacity on an edge from s to t with capacity! Matching in G ′ { \displaystyle c: E\to \mathbb { R ^! … in optimization theory, maximum flow problem no NULLs, Optimizations in Union find Structure... Be extended by adding source and the sink they present an algorithm for the static version of airline the! S, t ∈ V being the source, enters the sink formulated! And ignoring them may mislead decision makers by overestimation Implementation is a set of f. Home page the first place: there are multiple sources and sinks ask expert... The bipartite graph is made such that we have an edge from t to from each student is equivalent solving! Treated as the source and the sink along which the flow through a can. Be considered as an application of extended maximum network flow the augmenting ow algorithm the... Efficient in determining maximum flow problem in directed | {
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flow the augmenting ow algorithm the... Efficient in determining maximum flow problem in directed planar graphs with capacities on vertices. Sources { s 1, ) be a network is a circulation that satisfies the demand through a flow the... Path passing through a flow is k { \displaystyle k }. }. [ 14 ] return... Equal to the height function we loose pij … one vertex for each arc i... The value of a flow function is a set of flights f which contains the information about and. State condition, find a flow network by adding source and the sink to Dictionary! Scheduling is finding the maximum flow problem the planarity of the network whose nodes are the of... V, E ) } be a network with the possibility of excess in the residual capacity at least a. Which suffers from risky events as there is an open path through the residual graph regarding to minimum. } instead: there are multiple source nodes s 1, in 1955, Lester R. Ford, Jr. Delbert... Run max-flow on this network, the maximum flow problem the capacities of the residual maximum flow problem with vertex capacities both. Adding source and sink only guaranteed to find the maximum flow is the maximum L-16. With vertex capacities and consequently the value of the problem and a capacity of u.! Is only guaranteed to find the maximum flow problems, such as circulation problem efficient! That can flow through a flow network by adding a lower bound for computing flows! Now, it pushes flow to a capacity one edge from V should point from v_out perform j... With each other to maintain a reliable flow ( except for s \displaystyle! Underlying the flow through it adjacent vertex with positive excess, i.e REST API in Flask finding a feasible through! Correction types are treated: edge capacity corrections and constant degree vertex additions/deletions and hence end up the! & Tarjan ( 1988 ) to each of the time complexity for dynamic. G, a list of sinks { t 1, 3 a breadth-first or dept-first Search computes cut... Vertex additions/deletions | {
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list of sinks { t 1, 3 a breadth-first or dept-first Search computes cut... Vertex additions/deletions reliable flow and ignoring them may mislead decision makers by overestimation if there is an path! And height value associated with it L-16 25 july 2018 18 /.! ) { \displaystyle s } and t ) on this network and compute the result flow. Has no chance to finish the season Goldberg & Tarjan ( 1988.. Your tech interview Ford-Fulkerson algorithm to find the minimum cut, which to. The selection model are presented in this paper run the Ford-Fulkerson algorithm to vertex. Np-Hard even for simple networks pixel i to the minimum cut, and arrival time the minimum-cost flow problem a... Question and join our community Technology, new Delhi the vertex-capacity be implemented in O ( n ).! The vertex-capacity the relabel operation cross a minimum cut in O ( )! [ 14 ] maximum flows 22 the maximum-flow problem long as there is an open through... Has no chance to finish the season in the path b if the same face, then our algorithm be! { R } ^ { + }. }. }. 14... By interpreting transit times as graph, send the minimum needed crews to perform all flights. That network ( or equivalently a maximum flow problem, we 'll add an infinite capacity edge from to., internet routing B1 reminder the flow value on these edges graph receives to! Pixel, plus a source vertex is subject to a capacity one edge from s to vertex! J with weight pij a network flow problem 22 the maximum-flow problem real … maximum ow with capacity! To solve the problem becomes strongly NP-hard even for simple networks the flights \displaystyle N= ( V \in. Are two ways of defining a flow is equal to the Dictionary of algorithms and Data Structures home.... Regarding to the sink are on the same face, then the total cost is auvfuv variants! A preflow, i.e in most variants, the vertex each augmenting ˇfrom... Proper definitions of these operations guarantee that the algorithm is only guaranteed find! More complex network flow | {
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of these operations guarantee that the algorithm is only guaranteed find! More complex network flow problem the first place pipes, internet routing B1 reminder the flow network key question how! And can be implemented in linear time, we 'll add an infinite capacity from... Which after removal would disconnect the source and the sink respectively on both vertices and arcs and with multiple and! Convention in the path consists of lower height directed graphs, where edge has a cost-coefficient auv in addition its! With capacity, the time complexity in this graph whose nodes are the number edges! Cardinality matching in G ′ { \displaystyle s } and t ) departure time, and can seen! Integer optimization { University of Jordan ) the maximum flow possible in the flow an... Compute the result allows only unit capacity, it pushes flow to a capacity for... 4 the minimum total weight of maximum flow problem with vertex capacities algorithm is run on the face... | {
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Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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04 Jun 2019, 10:05
Dear Brunel,
You said-
"Is (x−3)2−−−−−−−√=3−x(x−3)2=3−x?
Remember: x2−−√=|x|x2=|x|. Why?
Couple of things:
The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0.
So x2−−√≥0x2≥0. But what does x2−−√x2 equal to?
Let's consider following examples:
If x=5x=5 --> x2−−√=25−−√=5=x=positivex2=25=5=x=positive;
If x=−5x=−5 --> x2−−√=25−−√=5=−x=positivex2=25=5=−x=positive."
My doubt is as follows-
All we know that sqrt of a number can be positive or negative results both, how you are saying "square root function can not give negative result"? If you kindly answer this question it would be a great help for me. Looking forward to hear you from.
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Joined: 04 Dec 2015
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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04 Jun 2019, 10:14
tamalmallick wrote:
Dear Brunel,
You said-
"Is (x−3)2−−−−−−−√=3−x(x−3)2=3−x?
Remember: x2−−√=|x|x2=|x|. Why?
Couple of things:
The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0. | {
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So x2−−√≥0x2≥0. But what does x2−−√x2 equal to?
Let's consider following examples:
If x=5x=5 --> x2−−√=25−−√=5=x=positivex2=25=5=x=positive;
If x=−5x=−5 --> x2−−√=25−−√=5=−x=positivex2=25=5=−x=positive."
My doubt is as follows-
All we know that sqrt of a number can be positive or negative results both, how you are saying "square root function can not give negative result"? If you kindly answer this question it would be a great help for me. Looking forward to hear you from.
A lot of people ask this question - you're not alone. I'm not Bunuel, but I did write a short article about it once that should clear things up:
https://www.manhattanprep.com/gmat/blog ... -the-gmat/
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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25 Jun 2019, 00:17
Bunuel wrote:
gautamsubrahmanyam wrote:
I understand that 1) is insuff
But for 2) -x|x| > 0 means x cant be +ve => |x| = -x so that -x (-x) = x^2> 0
If x is -ve => (x-3)^2 = X^2+9-6x = (-ve)^2+9-6(-ve) = +ve+9-(-ve) = +ve +9 + (+ve) = +ve
=> sqrt ((x-3)^2) = +X-3
=> sqrt ( (x-3) ^2 ) is not equal to 3-x
=> Option B
Yes, the answer for this question is B.
Is $$\sqrt{(x-3)^2}=3-x$$?
Remember: $$\sqrt{x^2}=|x|$$. Why?
Couple of things:
The point here is that square root function cannot give negative result: wich means that $$\sqrt{some \ expression}\geq{0}$$.
So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?
Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.
So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.
What function does exactly the same thing? The absolute value function! That is why $$\sqrt{x^2}=|x|$$ | {
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Back to the original question:
So $$\sqrt{(x-3)^2}=|x-3|$$ and the question becomes is: $$|x-3|=3-x$$?
When $$x>3$$, then RHS (right hand side) is negative, but LHS (absolute value) is never negative, hence in this case equations doesn't hold true.
When $$x\leq{3}$$, then $$LHS=|x-3|=-x+3=3-x=RHS$$, hence in this case equation holds true.
Basically question asks is $$x\leq{3}$$?
(1) $$x\neq{3}$$. Clearly insufficient.
(2) $$-x|x| >0$$, basically this inequality implies that $$x<0$$, hence $$x<3$$. Sufficient.
Hope it helps.
Hi Bunuel, in the highlighted portion above, how can we deduce that x will be less than 3 if x is less than 0? x can be 1,2 also, right? May be I am missing something. Can you please clarify?
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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25 Jun 2019, 00:20
shobhitkh wrote:
Bunuel wrote:
gautamsubrahmanyam wrote:
I understand that 1) is insuff
But for 2) -x|x| > 0 means x cant be +ve => |x| = -x so that -x (-x) = x^2> 0
If x is -ve => (x-3)^2 = X^2+9-6x = (-ve)^2+9-6(-ve) = +ve+9-(-ve) = +ve +9 + (+ve) = +ve
=> sqrt ((x-3)^2) = +X-3
=> sqrt ( (x-3) ^2 ) is not equal to 3-x
=> Option B
Yes, the answer for this question is B.
Is $$\sqrt{(x-3)^2}=3-x$$?
Remember: $$\sqrt{x^2}=|x|$$. Why?
Couple of things:
The point here is that square root function cannot give negative result: wich means that $$\sqrt{some \ expression}\geq{0}$$.
So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?
Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.
So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.
What function does exactly the same thing? The absolute value function! That is why $$\sqrt{x^2}=|x|$$
Back to the original question:
So $$\sqrt{(x-3)^2}=|x-3|$$ and the question becomes is: $$|x-3|=3-x$$? | {
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So $$\sqrt{(x-3)^2}=|x-3|$$ and the question becomes is: $$|x-3|=3-x$$?
When $$x>3$$, then RHS (right hand side) is negative, but LHS (absolute value) is never negative, hence in this case equations doesn't hold true.
When $$x\leq{3}$$, then $$LHS=|x-3|=-x+3=3-x=RHS$$, hence in this case equation holds true.
Basically question asks is $$x\leq{3}$$?
(1) $$x\neq{3}$$. Clearly insufficient.
(2) $$-x|x| >0$$, basically this inequality implies that $$x<0$$, hence $$x<3$$. Sufficient.
Hope it helps.
Hi Bunuel, in the highlighted portion above, how can we deduce that x will be less than 3 if x is less than 0? x can be 1,2 also, right? May be I am missing something. Can you please clarify?
If a number is less than 0, does not it mean that it's less than 3?
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Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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26 Jun 2019, 04:34
gmatnub wrote:
Is $$\sqrt{(x-3)^2} = 3-x$$?
(1) $$x\neq{3}$$
(2) $$-x|x| > 0$$
Attachment:
fasdfasdfasdfasdf.JPG
Alternative Approach
$$\sqrt{(x-3)^2} = 3-x$$?
|x - 3| = 3-x?
Case 1: |x-3| > 0 => x > 3
x-3 = 3-x?
2x=6?
x=3?
x=3 is not possible ever since x > 3
Case 2: |x-3| <= 0 => x <= 3
-x + 3 = 3-x?
0=0?
LHS = RHS?
This case would always be true since it can't violate any conditions.
Rephrased Q: Is x <= 3?
Stmt 1: x != 3
Doesn't tell anything about x if it's more than or less than 3. Not sufficient.
Stmt 2: -x|x| > 0
That implies x is always negative or x < 0. Hence x < 3 is also true. Sufficient.
Bunuel EducationAisle VeritasKarishma I got this Q wrong with my initial approach (shown below) of squaring both sides. I was wondering whether we can solve this Q by squaring both sides. If not, why not? I'm also confused how x=1 can be transformed with a few steps to give x=1 & -1 (shown below)? I would really appreciate if you could help me improve my understanding on this issue. Thanks! | {
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Initial Approach: Square both sides
$$\sqrt{(x-3)^2} = 3-x$$?
Square both sides
(x-3)^2 = (3-x)^2?
x^2 + 9 - 6x = 9 + x^2 - 6x?
0 = 0?
LHS = RHS?
Not sure how to proceed?
x=1 transforms to x=+1,-1?
x = 1
Square both sides
x^2 = 1
Take square root of both sides
|x| = 1
x = +1, -1
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Re: Is (x-3)^2 =3-x ? (1) x not = 3 (2) -x|x| > 3 [#permalink]
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30 Aug 2019, 22:20
jan4dday wrote:
Is $$\sqrt{(x-3)^2}=3-x$$?
(1) $$x\neq{3}$$
(2) -x|x| > 3
$$\sqrt{(x-3)^2}=3-x$$
This will be true only when x = 3 or x= 2
Statement 1
$$x\neq{3}$$
It might be equal to 2, 4, anything.
Insufficient.
Statement 2
$$-x|x| > 3$$
$$|x|$$ is always +ve
if $$-x|x| > 3$$, then $$-x > 0$$
this means that $$x$$ is -ve
if $$x$$ is -ve, it cannot equal either $$3$$ or $$2$$.
Sufficient.
Hence, B.
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Re: Is (x-3)^2 =3-x ? (1) x not = 3 (2) -x|x| > 3 [#permalink]
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04 Sep 2019, 21:59
@Bunel - can you please explain how to approach this?
I am also confused of how to simplify the equation given in the question stem.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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04 Sep 2019, 22:35
pzgupta wrote:
@Bunel - can you please explain how to approach this?
I am also confused of how to simplify the equation given in the question stem.
My solution is on the first page: https://gmatclub.com/forum/is-x-3-2-1-2 ... ml#p737280
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# Question on when to use polar coordinates to prove existence of limit/ does the method always work?
Show that the following limit exists or does not exist (general example)
$$\lim \limits_{(x,y) \to (0,0)} \dfrac{e^{-x^2-y^2}-1}{x^2+y^2}$$
i) Direct substitution of $$x=0$$ , $$y=0$$ leads to indeterminate form of $$\frac{0}{0}$$
ii) Taking the limit along $$x$$ , $$y$$ axes and $$y=x$$ all result with the value $$0$$
iii) Convert to polar:
$$\lim \limits_{r \to 0^+} \dfrac{e^{-r^2}-1}{r^2}->\frac{0}{0}$$
$$L'Hopital's$$ $$rule$$
$$\lim \limits_{r \to 0^+} \dfrac{-2re^{-r^2}}{2r}=-1$$
So the limit exists and its value is -1
My questions:
1. After converting the limit expression to polar, why is $$\lim \limits_{r \to 0^+}$$ instead of $$\lim \limits_{r \to 0}$$ ? Both have the same computation
1. From the example above, how would I know if the limit $$DNE$$ when taking the limit after converting to polar? Would taking the limit of the polar converted expression $$DNE$$ or not give a finite number to know that the original limit $$DNE$$? This is of course if I chose to convert to polar without knowing that a different path gave a different limit.
2. When would it be appropriate to covert to polar to show the existence of a limit when not told that it existed or not in the first place? Does converting to polar always work?
$$\lim \limits_{(x,y) \to (0,0)} \dfrac{{xy^4}}{x^2+y^8}$$
• this limit $$DNE$$ as it has different limits along different paths namely $$y=0$$ and $$x = y^4$$, respectively 0 $$≠$$ $$\frac{1}{2}$$
Polar conversion: (this limit DNE, but polar conversion results in 0, a finite number)- to check
$$\lim \limits_{r \to 0^+} \dfrac{{rcosθ*r^4sin^4θ}}{r^2cos^2θ+r^8sin^8θ}$$
$$\lim \limits_{r \to 0^+} \dfrac{r^5({cosθ*sin^4θ})}{r^2(cos^2θ+r^6sin^8θ)}$$
$$\lim \limits_{r \to 0^+} \dfrac{r^3({cosθ*sin^4θ})}{cos^2θ+r^6sin^8θ}$$
$$\frac{0}{(cos^2(θ))}=0$$
The limit is 0 | {
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$$\frac{0}{(cos^2(θ))}=0$$
The limit is 0
• Does converting to polar only work when (x,y) is approaching (0,0) or also work for say (x,y) is approaching a point like (-1,7)? Jun 23, 2020 at 11:11
• For the first question, I think it is just emphasising that $r$ cannot be negative so if you approach to $0$ you can only approach from the right Jun 23, 2020 at 11:16
• So does it mean that in polar coordinates, since it goes in counterclock-wise direction, it is same as $r->0^{+}$? Jun 23, 2020 at 11:18
For question 1, we take the limit as $$r \to 0^{+}$$ because in polar coordinates, $$r$$ represents the distance from the origin to point $$(x, y)$$ which is always non-negative.
For questions 2 and 3, keep in mind that we have
$$\lim_{(x, y) \to (0, 0)} \frac{e^{-x^2-y^2} - 1}{x^2 + y^2} = c$$
for some finite number $$c$$ if and only if
$$\lim_{r \to 0^{+}} \frac{e^{-r^2} - 1}{r^2} = c$$
In other words, the first limit is DNE if and only if the second one is DNE. Thus, if you manage to find some finite result $$c$$ for the second one, then you have also solved the first one. Sometimes, it is easier to evaluate limits in polar coordinates that in Cartesian coordinates so we take advantage of this when this applies.
An important note
Taking the limit along x , y axes and y=x all result with the value 0
It is important to note that in order for limit of a sequence to exist in a metric space like $$\mathbb{R}^2$$, all of its sub-sequences must also converge to that limit. That means that no matter how you walk your way to the limit, you must always arrive at the limit.
Hence, taking the limit along the $$x$$-axis, $$y$$-axis and the line $$y = x$$ is just one way to warn yourself early when the limit actually does not exist when these limits give different values. | {
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But, if these limits all agree, this is not sufficient to say that the limit does converge to some finite number $$c$$ because there can be some distorted path to approach $$(0, 0)$$ for which a different limit can be computed.
However, the polar form takes into consideration all possible ways to walk to the origin because no matter how you approach $$(0, 0)$$, the distance from your point to $$(0, 0)$$ always converges to $$0$$, hence we have $$r \to 0^{+}$$.
• Also if the original limit had (x,y) approaching some other point like (-2,9), would converting to polar not work then since r is not approaching the origin? Jun 23, 2020 at 11:23
• For this specific limit in this question, yes, you cannot take advantage of the nice polar form to evaluate your limit easily if $(x, y)$ does not approach the origin. Jun 23, 2020 at 11:30
• Wait I put another example in my question now, and the limit does not exist by taking limit of different paths, but when converting to polar, the limit is 0, a finite number Jun 23, 2020 at 11:30
• Unfortunately, for that supposed counter-example, you can't take advantage of that nice polar form. Usually, the hint is the presence of $x^2 + y^2$. This is because $r$ is equal to the distance of $(x, y)$ to the origin, i.e. $r = \sqrt{x^2 + y^2}$ which is equivalent to $r^2 = x^2 + y^2$, the classic Pythagorean theorem. :) Jun 23, 2020 at 11:33
• Actually, there is a problem with the evaluation of your limit. The end result is $0/\cos^{2}\theta$ which is not always zero for all angles $\theta$. For instance, if you approach along the angle $\theta = \pi/2$, the expression evaluates to the indeterminate form $0/0$. However, in the first question, you had that nice $x^2 + y^2$ in it that you eliminated the $\theta$'s entirely when converted to polar form, so we were able to evaluate it easily to $-1$ without problems. :) Jun 23, 2020 at 11:44 | {
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If you restrict the polar argument to the range $$[0,2\pi)$$, the Cartesian-to-polar transformation is a bijection. Hence whatever limit computation you perform in polar coordinates gives exactly the same conclusion as when computed in Cartesian.
Polar coordinates are used for convenience when a polar symmetry (like in your example) or a significant simplification is apparent. | {
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# Determine the off - diagonal elements of covariance matrix, given the diagonal elements
I have some covariance matrix $$A = \begin{bmatrix}121 & c\\c & 81\end{bmatrix}$$
The problem is to determine the possible values of $$c$$.
Now I know that the elements of this matrix are given by the usual definition of the covariance,
$$\frac{1}{N-1} \sum_{i=1}^N (X_i - \bar{x})(Y_i - \bar{y})$$
and so e.g.
$$\frac{1}{N-1} \sum_{i=1}^N (X_i - \bar{x})^2 = 121$$
$$\frac{1}{N-1} \sum_{i=1}^N (Y_i - \bar{y})^2 = 81$$
But I can't see how to go from here to determining $$c$$?
• is it always 2x2 matrix? Apr 16 at 13:58
You might find it instructive to start with a basic idea: the variance of any random variable cannot be negative. (This is clear, since the variance is the expectation of the square of something and squares cannot be negative.)
Any $$2\times 2$$ covariance matrix $$\mathbb A$$ explicitly presents the variances and covariances of a pair of random variables $$(X,Y),$$ but it also tells you how to find the variance of any linear combination of those variables. This is because whenever $$a$$ and $$b$$ are numbers,
$$\operatorname{Var}(aX+bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\operatorname{Cov}(X,Y) = \pmatrix{a&b}\mathbb A\pmatrix{a\\b}.$$
Applying this to your problem we may compute
\begin{aligned} 0 \le \operatorname{Var}(aX+bY) &= \pmatrix{a&b}\pmatrix{121&c\\c&81}\pmatrix{a\\b}\\ &= 121 a^2 + 81 b^2 + 2c^2 ab\\ &=(11a)^2+(9b)^2+\frac{2c}{(11)(9)}(11a)(9b)\\ &= \alpha^2 + \beta^2 + \frac{2c}{(11)(9)} \alpha\beta. \end{aligned}
The last few steps in which $$\alpha=11a$$ and $$\beta=9b$$ were introduced weren't necessary, but they help to simplify the algebra. In particular, what we need to do next (in order to find bounds for $$c$$) is complete the square: this is the process emulating the derivation of the quadratic formula to which everyone is introduced in grade school. Writing
$$C = \frac{c}{(11)(9)},\tag{*}$$
we find | {
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$$C = \frac{c}{(11)(9)},\tag{*}$$
we find
$$\alpha^2 + \beta^2 + \frac{2c^2}{(11)(9)} \alpha\beta = \alpha^2 + 2C\alpha\beta + \beta^2 = (\alpha+C\beta)^2+(1-C^2)\beta^2.$$
Because $$(\alpha+C\beta)^2$$ and $$\beta^2$$ are both squares, they are not negative. Therefore if $$1-C^2$$ also is non-negative, the entire right side is not negative and can be a valid variance. Conversely, if $$1-C^2$$ is negative, you could set $$\alpha=-c\beta$$ to obtain the value $$(1-C^2)\beta^2\lt 0$$ on the right hand side, which is invalid.
You therefore deduce (from these perfectly elementary algebraic considerations) that
If $$A$$ is a valid covariance matrix, then $$1-C^2$$ cannot be negative.
Equivalently, $$|C|\le 1,$$ which by $$(*)$$ means $$-(11)(9) \le c \le (11)(9).$$
There remains the question whether any such $$c$$ does correspond to an actual variance matrix. One way to show this is true is to find a random variable $$(X,Y)$$ with $$\mathbb A$$ as its covariance matrix. Here is one way (out of many).
I take it as given that you can construct independent random variables $$A$$ and $$B$$ having unit variances: that is, $$\operatorname{Var}(A)=\operatorname{Var}(B) = 1.$$ (For example, let $$(A,B)$$ take on the four values $$(\pm 1, \pm 1)$$ with equal probabilities of $$1/4$$ each.)
The independence implies $$\operatorname{Cov}(A,B)=0.$$ Given a number $$c$$ in the range $$-(11)(9)$$ to $$(11)(9),$$ define random variables
$$X = \sqrt{11^2-c^2/9^2}A + (c/9)B,\quad Y = 9B$$
(which is possible because $$11^2 - c^2/9^2\ge 0$$) and compute that the covariance matrix of $$(X,Y)$$ is precisely $$\mathbb A.$$
Finally, if you carry out the same analysis for any symmetric matrix $$\mathbb A = \pmatrix{a & b \\ b & d},$$ you will conclude three things:
1. $$a \ge 0.$$
2. $$d \ge 0.$$
3. $$ad - b^2 \ge 0.$$ | {
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1. $$a \ge 0.$$
2. $$d \ge 0.$$
3. $$ad - b^2 \ge 0.$$
These conditions characterize symmetric, positive semi-definite matrices. Any $$2\times 2$$ matrix satisfying these conditions indeed is a variance matrix. (Emulate the preceding construction.)
• It might be worth mentioning that $C$ here is the correlation and, as shown, is always between $-1$ and $+1$ Apr 17 at 12:00
An intuitive method to determine this answer quickly is to just remember that covariance matrices may be interpreted in the form
$$$$A = \begin{pmatrix} \sigma_1^2 & \rho_{12}\sigma_1\sigma_2 &\rho_{13}\sigma_1\sigma_3 & \cdots & \rho_{1n}\sigma_1 \sigma_n \\ & \sigma_2^2 & \rho_{23}\sigma_2\sigma_3 & \cdots & \rho_{2n}\sigma_2 \sigma_n \\ & & \sigma_3^2 & \cdots & \rho_{3n}\sigma_3 \sigma_n \\ & & & \ddots & \vdots \\ & & & & \sigma_n^2 \end{pmatrix}$$$$
where $$\rho_{ab} \in [-1,1]$$ is a Pearson Correlation Coefficient. In your case you have
\begin{align} \sigma_1^2 = 121 ,~~~ \sigma_2^2 = 81 ~\Longrightarrow ~ |c| \leq \sqrt{121\cdot 81} = 99 \end{align}
i.e. $$c \in [-99, 99]$$.
• +1 This is a great answer for readers familiar with this representation of covariance matrices. I don't think it would be remiss, though, to point out that when $n\gt 2$ the correlation coefficients are subject to additional restrictions: it doesn't suffice for them all just to lie between $-1$ and $1.$ The case $n=3$ is discussed at stats.stackexchange.com/questions/72790.
– whuber
Apr 17 at 18:05
$$A$$ is posdef so by Sylvesters criterion $$det(A) = 121 \cdot 81 - c^2 \geq 0$$. Thus, any $$c \in [-99, 99]$$ will produce a valid covariance matrix. | {
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• Covariance matrices can be positive semi-definite right? Does the semi here change anything? Apr 16 at 15:23
• semi is the case when c is exactly 99 and det = 0. Apr 16 at 16:16
• @Hunaphu $c=+99$ and $c=-99$ both lead to zero determinant. If $-99 < c < +99$ then you would have a strictly positive definite matrix Apr 17 at 12:03
There are three main possibilities of note. One is that the variable are uncorrelated, in which case the off-diagonal entries are easy to calculate as 0. Another possibility is that you don't really have two different variables. $$y$$ is simply a scalar multiple of $$x$$ (i.e. perfect correlation). If $$y= c x$$, then $$\sigma_{xy} =\sigma_{x}\sigma_{xy}=99$$. We get a third possibility in noting that the above assumes $$c>0$$. For $$c<0$$, we get $$\sigma_{xy} =-99$$.
Geometrically, the covariance between two vectors is the product of their lengths times the cosine of the angle between them. Since the cosine varies from $$-1$$ to $$1$$, the covariance ranges from the product of their lengths to the negative of the product.
Another approach is to consider $$z_1 = \frac{x-\mu_{x}}{\sigma_{x}}$$ and $$z_2 = \frac{y-\mu_y}{\sigma_{y}}$$. $$\sigma_{xy} = \sigma_{(\sigma_x z_1)(\sigma_y z_2)}=\sigma_x \sigma_y \sigma_{z_1z_2}=99\sigma_{z_1z_2}$$ and $$\sigma_{z_1z_2}$$ is simply the correlation between $$x$$ and $$y$$, which ranges from $$-1$$ to $$1$$. | {
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# Do all symmetric $n\times n$ invertible matrices have a square root matrix?
My question relates to the conditions under which the spectral decomposition of a nonnegative definite symmetric matrix can be performed. That is if $A$ is a real $n\times n$ symmetric matrix with eigenvalues $\lambda_{1},...,\lambda_{n}$, $X=(x_{1},...,x_{n})$ where $x_{1},...,x_{n}$ are a set of orthonormal eigenvectors that correspond to these eigenvalues (i.e. $X$ is an orthogonal matrix), and $\Lambda=\text{diag}(\lambda_{1},...,\lambda_{n})$ then
$A=X\Lambda X'$
is the spectral decomposition of $A$. If we then let $A^{1/2}=X\Lambda^{1/2}X'$, where $\Lambda^{1/2}$ is a square root matrix of $\Lambda$ - i.e. $\Lambda^{1/2}\Lambda^{1/2}=\Lambda$, then $A^{1/2}A^{1/2}=A$. Thus $A^{1/2}$ is a square root matrix of $A$.
So if a real nonnegative definite symmetric $n\times n$ matrix $A$ has $n$ eigenvalues then the matrix has a spectral decomposition and thus a square root matrix too. My question is do all symmetric $n\times n$ invertible matrices have $n$ eigenvalues, and thus a square root matrix? Furthermore it is not clear to me whether the eigenvalues have to be distinct or not. | {
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• All real symmetric matrices are diagonalisable, does that help?? – Vishesh Jul 26 '13 at 16:37
• The eigenvalues can be repeated. If the geometric multiplicity of the eigenvalues is the same as the algebraic multiplicity then the matrix can be diagonalized. Not all symmetric matrices have distinct eigenvalues, take the identity. – Wintermute Jul 26 '13 at 16:46
• Oh yes, my bad, I totally forgot that. Thanks – Vishesh Jul 26 '13 at 16:50
• @dandar: Your question suggests you're asking about matrices with real entries, in which case perhaps $\Lambda$ is also required to be real? If that's correct, it's instructive to look at the $1 \times 1$ case. – Andrew D. Hwang Jul 26 '13 at 16:52
• Yes the OP should clarify whether the matrix is real, and (if so) whether the square-root is supposed to be real. On the other hand, all complex matrices have complex square-roots; no symmetry is required for that. – GEdgar Jul 26 '13 at 16:56
## 4 Answers
All symmetric matrices are diagonalizable, therefore they have $n$ eigenvalues (which don't have to be distinct, by the way), all of which are real. The spectral theorem says:
We can decompose any symmetric matrix $A\in S^n$ using symmetric eigendecomposition: $$A = \sum_{i=1}^n\lambda_iq_iq_i^T = Q\Lambda Q^T, \qquad \Lambda=diag(\lambda_i,\dots,\lambda_n)$$ where the matrix $Q = [q_1,\dots,q_n]$ is orthogonal (with $Q^TQ=I_n$), and contains the eigenvectors of $A$, while the diagonal matrix $\Lambda$ contains the eigenvalues of A.
The matrix "power rule": $$A^k = Q\Lambda^k Q^{-1}$$ can be used (with $k<0$ being allowed for invertible matrices, which means there should be no $\lambda_i=0$). Note that if there are negative eigenvalues, $\Lambda^{\frac{1}{2}}$ will become complex.
Note that in the complex case, the transpose operations should be replaced with the Hermitian operation (the conjugate transpose). | {
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• Thank-you for the reply, and yes I now see that all symmetric matrices are diagonalizable. This is because if the matrix is $n\times n$ then we can always construct a set of $n$ orthonormal eigenvectors - i.e. regardless of whether the eigenvalues are repeated or not. Thus we can always compute the spectral decomposition and hence the square root matrix will exist. – dandar Jul 26 '13 at 17:12
What about the matrix $(-1){}{}{}{}{}{}{}{}{}{}{}$?
• Thank-you for your response. You have found a flaw in my inital summary of the use of the spectral decomposition to find the square root matrix of $A$. I should have stated that $A$ needs to be nonnegative definite. This ensures all the eigenvalues of $A$ are greater than or equal to $0$ which precludes your counter-example. – dandar Jul 26 '13 at 17:31
For your first question: that you can diagonalize real symmetric matrices is the so-called spectral theorem.
For the second: your argument is general; you did not use that the eigenvalues were distinct.
• Thanks for the reply. Yes you are right the theory for the spectral theorem does not care about whether or not the $n$ eigenvalues of $A$ are distinct or not. – dandar Jul 26 '13 at 17:13
Matrix square root can be defined in many ways. If you just want $X$ such that $X^2 = A$, you approach is good.
However, the principal square root is defined only for the matrices with no strictly negative eigenvalues and zero being at most nonderogatory eigenvalue (which is unimportant here, since symmetric matrices are diagonalizable).
The importance of the principal square root lies in the fact that it is a unique square root with the spectrum in the open right half-plane. If we extended this to the matrices with the real negative eigenvalues, we would either lose uniqueness, or the "open right half-plane" would have to be replaced by something less nice. Of course, there are reasons to ask for this. Read more in Higham's "Functions of Matrices". | {
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Since you ask about symmetric matrices, your eigenvalues are real, so you can only define principal square root if your matrix has nonnegative eigenvalues, which means it is positive semidefinite. That also means that your square (or any other) root will also be positive semidefinite, which can be easily seen from the eigenvalue (spectral) decomposition.
• Thank-you for your reply. I had not realised the definition of a square root matrix has many forms. – dandar Jul 26 '13 at 17:19
• You can compare it to the square root in the set of the nonnegative real numbers. Principal square root is, for example, $\sqrt{4}=2$, but nothing prevents us to also consider $-2$ as a square root of $4$. We even do so, for example, when solving quadratic equations (the $\pm$ sign before $\sqrt{b^2-4ac}$). Unlike the real numbers, which can be considered matrices with a single element, general matrices have much more elements ($n^2$ of them), so you get a much wider variety of candidates for a square root (basically, all combinations of choices for the square root of the eigenvalues). – Vedran Šego Jul 26 '13 at 17:41 | {
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# Tricopter physics help
Hello!
I have been studying how exactly a tricopter works and I came across a problem that is not quite clear to me...
I understand that the tail rotor has to be slightly tilted for a specific angle alpha in order for the horizontal component Fx to counteract the unbalanced torques of the 3 propellers. However this Fx component of force F now causes that the sum of all forces is not zero - therefore the tricopter will constantly try to drift in the direction of Fx when trying to hoover!?
(see the attached sketch)
Am I missing something here or is my conclusion correct?
How is this problem handled? Do you constantly have to correct for this drift by rolling or is this effect negligible?
Thank you for any input! =)
Views: 2612
### Replies to This Discussion
The angle is adjusted by a servo to keep the heading constant.
I can help you out. But first I want you to draw a full tricopter diagram so that way I can better explain it to you by referencing your own figure.
So draw a tricopter from a top down. Include all the the prop rotations, the resulting moments, and resulting forces like you have above. Label each motor to make things easier. If you do that I can do my best to explain what is going on.
Conventional helicopters have this same problem too as a result of the tail rotor, and often hover at a very slight roll angle.
I had not considered the tri copter in this same way before, but at first glance, I believe you are right.
Thank you for all the responses! =)
@Johnatan Hair
Somehow I forgot that the helicopter has a very similar situation going on with the tail rotor! Thank you for reminding me on that! =) | {
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@WuStangDan
I would be very grateful if you could do that and explain the problem from your perspective. I draw a figure by hand as you asked - I hope it is clear enough. I chose the tricopter where the front two rotors rotate in opposite directions, so that the unbalanced torque and therefore angle alpha can be smaller.
In addition I added two pages of my calculations. On first page there are the three conditions for balancing torques and on the second page I wrote the sums of forces. Clearly the force Fx remains in the end.
Thank you!
Okay perfect.
So yes your calculations are correct. If a tricopter was flying exactly like you have drawn, where the servo motor is at the exact angle $\alpha$ to balance out the the moment in the z axis, the triopter would have a single force in the x axis.
The reason why I wanted you to do the full calculations is make sure that you understood that you haven't actually missed something in your calculations. Becuase the real answer to your question is somewhat of a let down, I didn't want you to go back to your calculations because you didn't believe me. | {
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So tricopters don't "drift" in the x direction when flying in real life. So that means there is something missing from your drawing. You have propellers, motors that can rotate the propellers, and most of the standard parts of a multirotor. But you don't have a negative feedback controller that all multirotors have. Now the IMU in a multirotor cannot detect when it is "drifting" at a constant velocity. So if you had your multirotor sitting in the trunk of your car while you drove down the highway, the gyros would report the same values as if they were flat on the ground. But a constant force on a body does not move it at a constant velocity. Newtons second law states that the tricopter would begin accelerating in the x direction, not just slowly "drift" like I'm assuming you thought it would based on the wording you used in your question. IMU's can detect acceleration so therefore the negative feed back controller would detect it, and change the speed between the two front motors to balance that acceleration.
This will make the tricopter no longer perfectly flat, but rather at a slight angle, making it so that F1 and F2 now both have x components.
First of all thank you very much for the detailed answer!
Reading your explanation confirmed my own thinking when I was doing the calculations above:
"Tricopter cannot hoover still when oriented horizontally. It has to be slightly tilted around y axis to counteract the F3x force"
Maybe "drifting" was a poorly chosen word - I understand that the tricopter would accelerate in direction of F3x until in equillibrium with air drag.
Thanks - now I really understand how a tricopter works. =) | {
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Thanks - now I really understand how a tricopter works. =)
Hi. I was hoping you still check this site or whatever but you would greatly help me if you could tell me about the sources of your knowledge and where you got to know such detailed info along with the diagrams and stuff? i desperately need them and wherever i look, its wayyy to complex for my understanding. Its imperative that i manage to find the completer working of the tricopter-equations,torque balancing and all. Thanks! :D
Vidur said:
Hi. I was hoping you still check this site or whatever but you would greatly help me if you could tell me about the sources of your knowledge and where you got to know such detailed info along with the diagrams and stuff? i desperately need them and wherever i look, its wayyy to complex for my understanding. Its imperative that i manage to find the completer working of the tricopter-equations,torque balancing and all. Thanks! :D
Hi Vidur! I was notified per email of your reply, however, I am afraid I don't have any specific sources I could recommend because I derived the equations above myself.
I started by researching the tricopters frame geometry and kinematics, then I found a suitable (simplified) relationship between propellers angular velocity and its thrust & torque. After that you just need to set the balance of forces and torques in all three directions (x,y & z) and use algebra to get to the solution you seek. | {
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Note that my deriavation is far from complete since it is limited to a steady-state solution (tricopter hovering still). I only used it to clear up some confusion I had about the hovering state of a tricopter. If you wanted to derive a complete dynamic model for a tricopter you would also need to include acceleration terms, as well take air drag into account. Note that at higher velocities, incoming airflow could also significantly affect the thrust on the rotor, which you would somehow have to take into account and when flying at low altitutudes ground effect might also play a role.
As you probably see, there are many physical phenomenon that affect tricopter's flight, which is probably the reason most derivations of dynamics equations become so complex. You first need to consider what is actually the goal you are trying to achieve with your model (equations) and then evaluate which physical effects you will have to include and which you could neglect.
oh i see. Well that definitely seems like an uphill task. Could you provide me whatever links that you used in order to get any sort of insight into this? I would be very glad if i could get some sort of starting point to go about my research!
Primoz K said:
Vidur said:
Hi. I was hoping you still check this site or whatever but you would greatly help me if you could tell me about the sources of your knowledge and where you got to know such detailed info along with the diagrams and stuff? i desperately need them and wherever i look, its wayyy to complex for my understanding. Its imperative that i manage to find the completer working of the tricopter-equations,torque balancing and all. Thanks! :D
Hi Vidur! I was notified per email of your reply, however, I am afraid I don't have any specific sources I could recommend because I derived the equations above myself. | {
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I started by researching the tricopters frame geometry and kinematics, then I found a suitable (simplified) relationship between propellers angular velocity and its thrust & torque. After that you just need to set the balance of forces and torques in all three directions (x,y & z) and use algebra to get to the solution you seek.
Note that my deriavation is far from complete since it is limited to a steady-state solution (tricopter hovering still). I only used it to clear up some confusion I had about the hovering state of a tricopter. If you wanted to derive a complete dynamic model for a tricopter you would also need to include acceleration terms, as well take air drag into account. Note that at higher velocities, incoming airflow could also significantly affect the thrust on the rotor, which you would somehow have to take into account and when flying at low altitutudes ground effect might also play a role.
As you probably see, there are many physical phenomenon that affect tricopter's flight, which is probably the reason most derivations of dynamics equations become so complex. You first need to consider what is actually the goal you are trying to achieve with your model (equations) and then evaluate which physical effects you will have to include and which you could neglect.
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Season Two of the Trust Time Trial (T3) Contest
A list of all T3 contests is here. The current round, the Vertical Horizontal one, is here | {
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# Number of elements of $3n$ binary tuples, where the ordinates add up to $2n$.
I have the following problem.
Take $$\Omega_n=\{(a_1, a_2 , \cdots , a_{3n})| a_i= 0\rm{\,or}\, 1\}$$. Define $$A_n=\{\omega \in \Omega_n|\exists k, \sum _{i=1}^{3k} a_i = 2k\}$$ and $$S_m=\{(a_1, a_2 , \cdots , a_{3m}) \in A_m|\inf \{ k,| \sum _{i=1}^{3k} a_i = 2k\}=m\}$$ we are interested in finding $$|S_m|$$ the cardinality of $$S_m$$
There is a straightforward recursive formula$$S_n={3n \choose 2n}-\left (S_1 {3(n-1) \choose 2(n-1)}+S_2{3(n-2) \choose 2(n-2)}+ \cdots S_{n-1} {3\cdot(1) \choose 2\cdot(1)} \right )= {3n\choose 2n}-\left (\sum _{i=1}^{n-1}S_i {3(n-i)\choose 2(n-i)} \right )$$ I was not sure how to compute that so I used OEIS to see if it has some nice formula. After finding by hand the first values $$3,6,21,90,429$$ it suggested me the formula $$\frac{2}{3n-1}{3n\choose 2n}$$. I managed to prove it by induction.
I would be interested in some combinatorial proof of that, a bijection would be highly appreciated.
• What is "inf" ? Do you mean the smallest $k$ for which the summation holds? – Doc Nov 28 '13 at 6:20
• @Doc out of habit infimum, yes exactly – clark Nov 28 '13 at 6:24
• Not at all there yet, but it seems interesting that the answer also takes the form $\frac{1}{n} {3n-1\choose n}$. – Doc Nov 28 '13 at 6:37
• oops .... that should say $\frac{3}{3n-1} {3n-1\choose n}$. – Doc Nov 28 '13 at 7:36
• @Doc: Or $\frac3n\binom{3n-2}{n-1}$, though that isn’t obviously any more useful. – Brian M. Scott Nov 29 '13 at 19:42
Code heads as 1 and tails as 0, and the problem can be phrased in the following way: Flip a fair coin until you have exactly twice as many heads as tails, and then stop. The value of $|S_n|$ is the number of such sequences of coin flips that have length exactly $3n$. | {
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For this rephrased version, I asked the same question as the OP a couple of years ago on this site, under "Combinatorial proof of $\binom{3n}{n} \frac{2}{3n-1}$ as the answer to a coin-flipping problem." It took me a few weeks before I found an answer to my own question, so I wouldn't call this an easy combinatorial proof to come up with. (In fact, I generalized my argument, recast it in terms of counting certain lattice paths, and got the argument published in the Electronic Journal of Combinatorics as "Enumerating Lattice Paths Touching or Crossing the Diagonal at a Given Number of Lattice Points.")
Anyway, I'll reproduce my original combinatorial argument here. It uses the equivalent $|S_n| = \frac{3}{3n-1} \binom{3n-1}{n}$ as the formula to be proved. It also uses the following result (see, for example, Section 7.5 of Concrete Mathematics):
Raney's Lemma: Let $a_1, ... a_m$ be a sequence of integers such that $\sum a_i = 1$. There is a unique index $j$ such that the partial sums of the sequence $a_j, a_{j+1}, ... a_{j+m-1}$ (cyclic indices) are positive.
On to the proof.
Intro.
Consider the sequences with $2n$ occurrences of $-1$ (i.e., $2n$ heads) and $n$ occurrences of $+2$ (i.e., $n$ tails). We want to show that the number of these sequences with all partial sums nonzero is $\binom{3n-1}{n} \frac{3}{3n-1}$. The complete sum and empty sum are clearly $0$, so "partial sum" excludes those two cases. The sequences we want to count can be split into three groups: (1) all partial sums positive, (2) all partial sums negative, (3) some partial sums positive and some negative.
Group 1: The number of these sequences with all partial sums positive is $\binom{3n-1}{n} \frac{1}{3n-1}$. | {
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This is the part that uses Raney's lemma. If all partial sums are positive, the last element in the sequence must be $-1$. Thus we want to count the number of sequences with $2n-1$ occurrences of $-1$ and $n$ occurrences of $+2$ that add to $+1$ and have all partial sums positive. Ignoring the partial sums restriction, there are $\binom{3n-1}{n}$ such sequences. If we partition these $\binom{3n-1}{n}$ sequences into equivalence classes based on cyclic shifts, Raney's lemma says that exactly one sequence in each equivalence class has all partial sums positive. Because there are $3n-1$ elements in each sequence there are $3n-1$ sequences with the same set of cyclic shifts, and so there are $3n-1$ sequences in each equivalence class. Thus the number of sequences in Group 1 is $\binom{3n-1}{n} \frac{1}{3n-1}$.
Group 2: The number of these sequences with all partial sums negative is also $\binom{3n-1}{n} \frac{1}{3n-1}$.
To see this, just reverse the sequences counted in Part 1.
Group 3: The number of these sequences with some positive partial sums and some negative partial sums is, yet again, $\binom{3n-1}{n} \frac{1}{3n-1}$. | {
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This one is a little trickier. First, because of the $-1$'s, it is not possible to switch from positive partial sums to negative partial sums. Thus any sequence counted here must have exactly one switch: from negative partial sums to positive partial sums. The switch must occur at some point where the partial sum is $-1$ and the next element is $+2$. Thus we have some sequence $(a_1, \ldots, a_m, +2, a_{m+2}, \ldots, a_n)$ where the sums $a_1, a_1 + a_2, \ldots, a_1 + \cdots + a_m$ are all negative. Consider the sequence $(+2, a_m, \ldots, a_2, a_1, a_{m+2}, \ldots, a_n)$. Since $+2 + a_m + \cdots + a_1 = 1$, and $a_k + a_{k-1} + \cdots + a_1 < 0$ for all $k$, $1 \leq k \leq m$, it must be the case that $+2 + a_m + \cdots + a_{k+1} > 1$ for all $k$, $1 \leq k \leq m-1$. So the sequence $(+2, a_m, \ldots, a_2, a_1, a_{m+2}, \ldots, a_n)$ is in Group 1. To see that this mapping is a bijection, note that any sequence in Group 1 must start with $+2$ and have a first time that a partial sum is equal to $+1$. Thus this transformation is reversible.
Summing up.
Putting Groups 1, 2, and 3 together we see that the total number of sequences we want to count is $\binom{3n-1}{n} \frac{3}{3n-1}$.
• This is great! I didnt know about Ranney's Lemma, it was hidden all along this problem, it was nicely exploited! – clark Dec 5 '13 at 0:05 | {
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