# Euclidean algorithm

I found something interesting about solving this question:

If p is the number of pairs of values (x, y) that can make the equation 84x + 140y = 28 true, what is true about p?

I noticed while solving that I could simplify the equation:

84x - 140y = 28

42x – 70y = 14

21x – 35y = 7

3x – 5y = 1

Noting that 3x = 5y + 1 means that 3x must end in 5 + 1 = 6 or 0 + 1 = 1 to allow y to be divisible by 5.

This means 3x should end in either 1 or 6 to be equal to 5y + 1

Finding the following values for x and matching values for y:

3x = 6, 21, 36, 51, 66, 81, 96, 111, 126, 141, 156

5y = 5, 20, 35, 50, 65, 80, 95, 110, 125, 140, 155

Then finding the series of x and y:

X = 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52

Y = 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31

X seems to be increasing by 5, and Y by 3.

Which means that you can find pairs of x and y by the formulas:

x = 2+5a and y = 1+3a

Leading to a sequence that will satisfy 84x + 140y = 28.

Would this also be more generally true? I also thought it might be an interesting additional question to the quiz.

On another note, this text editor does not seem to like enters in paragraphs. Any way I can fix this?

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Yes, you just applied Euclidean algorithm.

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# Arc length question

##### Member
The length of the minor arc of a circle is 10cm, while the area of the sector AOB is 150cm2.

a) Form two equations involving r and θ, where θ is measured in radians.

b) Solve these equations simultaneously to find r and θ.

Help to solve? Cant understand the question very well.

I think the arc length formula was

length=$$\displaystyle \frac{n}{360}\cdot2\pi(r)$$

$$\displaystyle \therefore10=\frac{n}{360}\cdot2\pi(r)$$

The question states, Form two equations involving r and θ, where θ is measured in radians.

So would we have to arrange the formula to find the r and $$\displaystyle \theta$$

#### MarkFL

Staff member
The formulas you want are:

Circular arc-length:

$$\displaystyle s=r\theta\tag{1}$$

Area of circular sector

$$\displaystyle A=\frac{1}{2}r^2\theta\tag{2}$$

Now, if we solve (1) for $\theta$, we find:

$$\displaystyle \theta=\frac{s}{r}$$

Now, substituting this into (2), we obtain:

$$\displaystyle A=\frac{1}{2}r^2\left(\frac{s}{r}\right)=\frac{rs}{2}\implies r=\frac{2A}{s}\implies\theta=\frac{s^2}{2A}$$

Now, just plug in the given values for $A$ and $s$.

##### Member
The formulas you want are:

Circular arc-length:

$$\displaystyle s=r\theta\tag{1}$$

Area of circular sector

$$\displaystyle A=\frac{1}{2}r^2\theta\tag{2}$$

Now, if we solve (1) for $\theta$, we find:

$$\displaystyle \theta=\frac{s}{r}$$

Now, substituting this into (2), we obtain:

$$\displaystyle A=\frac{1}{2}r^2\left(\frac{s}{r}\right)=\frac{rs}{2}\implies r=\frac{2A}{s}\implies\theta=\frac{s^2}{2A}$$

Now, just plug in the given values for $A$ and $s$.
Im finding trouble understanding the question.
"Form two equations involving r and $$\displaystyle \theta$$" Basically means transpose the formula or make the equation equal to r and $$\displaystyle \theta$$?

Since the arc length = $$\displaystyle s=r\theta$$ (I thought arc length was length=$$\displaystyle \frac{n}{360}\cdot2\pi(r)$$?

r=$$\displaystyle s\theta$$?

For b) we just sub in the values? Where s is the length?

#### MarkFL

Staff member
You are given values for the arc-length $s$ and the area $A$, and so you want to use the formulas relating $r$ and $\theta$ to these given values (which I gave as (1) and (2)) to be able to express both $r$ and $\theta$ as functions of $s$ and $A$ (which I did using some algebra), so that you can use these given values to determine $r$ and $\theta$.

Once you have $r$ and $\theta$ as functions of $s$ and $A$, it is simply a matter of using the given values to evaluate $r$ and $\theta$.

#### SuperSonic4

##### Well-known member
MHB Math Helper
Im finding trouble understanding the question.
"Form two equations involving r and $$\displaystyle \theta$$" Basically means transpose the formula or make the equation equal to r and $$\displaystyle \theta$$?
It's asking to make two equations for the length of an arc and the area of a sector. As long as your equations have $$\displaystyle r$$ and $$\displaystyle /theta$$ in them then you're fine for this step.

MarkFL has given you the equations you need (his TeX is also better than mine! )

$$\displaystyle \displaystyle s=r\theta\tag{1}$$

$$\displaystyle \displaystyle A=\frac{1}{2}r^2\theta\tag{2}$$

As you see both of these equations have $$\displaystyle r$$ and $$\displaystyle \theta$$ in them so Part A of the question has been solved already

Since the arc length = $$\displaystyle s=r\theta$$ (I thought arc length was length=$$\displaystyle \frac{n}{360}\cdot2\pi(r)$$?

r=$$\displaystyle s\theta$$?
$$\displaystyle \frac{n}{360}\cdot 2\pi r$$ is the equation for arc length where $$\displaystyle n$$ is in degrees. Since the question asks for radians you want $$\displaystyle s = r\theta$$. See the spoiler below for why they are equivalent

By definition a full circle has 360 degrees and $$2\pi$$ radians so you can say $$360^{\circ} = 2\pi$$ (we omit the c symbol for radians)

Your formula for degrees is $$s = \frac{\theta}{360} \cdot 2\pi r$$ but since we know that $$360^{\circ} = 2\pi$$ it can be subbed in

$$s = \frac{\theta}{2\pi} \cdot 2\pi r = r \theta$$ - the radian expression

For b) we just sub in the values? Where s is the length?
Sub in the values given (10cm and 150cm²) for s and A respectively so you have two equations and two unknowns ($$\displaystyle r$$ and $$\displaystyle \theta$$).

You can now solve simultaneously using either the substitution method MarkFL showed in post 2. He found $$\displaystyle \theta$$ in terms of $$\displaystyle r$$ which means he was able to substitute $$\displaystyle \frac{s}{r}$$ wherever $$\displaystyle \theta$$ appeared.

He did in terms of $$\displaystyle s$$ in post 2 but if you find it easier you may use $$\displaystyle s=10$$ for your example

##### Member
It's asking to make two equations for the length of an arc and the area of a sector. As long as your equations have $$\displaystyle r$$ and $$\displaystyle /theta$$ in them then you're fine for this step.

MarkFL has given you the equations you need (his TeX is also better than mine! )

$$\displaystyle \displaystyle s=r\theta\tag{1}$$

$$\displaystyle \displaystyle A=\frac{1}{2}r^2\theta\tag{2}$$

As you see both of these equations have $$\displaystyle r$$ and $$\displaystyle \theta$$ in them so Part A of the question has been solved already

$$\displaystyle \frac{n}{360}\cdot 2\pi r$$ is the equation for arc length where $$\displaystyle n$$ is in degrees. Since the question asks for radians you want $$\displaystyle s = r\theta$$. See the spoiler below for why they are equivalent

By definition a full circle has 360 degrees and $$2\pi$$ radians so you can say $$360^{\circ} = 2\pi$$ (we omit the c symbol for radians)

Your formula for degrees is $$s = \frac{\theta}{360} \cdot 2\pi r$$ but since we know that $$360^{\circ} = 2\pi$$ it can be subbed in

$$s = \frac{\theta}{2\pi} \cdot 2\pi r = r \theta$$ - the radian expression

Sub in the values given (10cm and 150cm²) for s and A respectively so you have two equations and two unknowns ($$\displaystyle r$$ and $$\displaystyle \theta$$).

You can now solve simultaneously using either the substitution method MarkFL showed in post 2. He found $$\displaystyle \theta$$ in terms of $$\displaystyle r$$ which means he was able to substitute $$\displaystyle \frac{s}{r}$$ wherever $$\displaystyle \theta$$ appeared.

He did in terms of $$\displaystyle s$$ in post 2 but if you find it easier you may use $$\displaystyle s=10$$ for your example
Thanks this really did help

##### Member
The formulas you want are:

Circular arc-length:

$$\displaystyle s=r\theta\tag{1}$$

Area of circular sector

$$\displaystyle A=\frac{1}{2}r^2\theta\tag{2}$$

Now, if we solve (1) for $\theta$, we find:

$$\displaystyle \theta=\frac{s}{r}$$

Now, substituting this into (2), we obtain:

$$\displaystyle A=\frac{1}{2}r^2\left(\frac{s}{r}\right)=\frac{rs}{2}\implies r=\frac{2A}{s}\implies\theta=\frac{s^2}{2A}$$

Now, just plug in the given values for $A$ and $s$.
Beginning to understand this a lot easier.
Just a question on the last part.
How did you rearrange to get $$\displaystyle \theta$$?

It went from, $$\displaystyle A=\frac{1}{2}r^2\left(\frac{s}{r}\right)$$ which I understand to $$\displaystyle =\frac{rs}{2}$$?

#### MarkFL

Staff member
Beginning to understand this a lot easier.
Just a question on the last part.
How did you rearrange to get $$\displaystyle \theta$$?
$$\displaystyle s=r\theta$$

Divide through by $r$

$$\displaystyle \frac{s}{r}=\frac{\cancel{r}\theta}{\cancel{r}}$$

$$\displaystyle \theta=\frac{s}{r}$$

It went from, $$\displaystyle A=\frac{1}{2}r^2\left(\frac{s}{r}\right)$$ which I understand to $$\displaystyle =\frac{rs}{2}$$?
$$\displaystyle A=\frac{1}{2}r^2\left(\frac{s}{r}\right)=\frac{\cancel{r}\cdot r\cdot s}{2\cancel{r}}=\frac{rs}{2}$$


# Optimal route consisting of rowing then walking

Problem

You're in a boat on point A in the water, and you need to get to point B on land. Your rowing speed is 3km/h, and your walking speed 5km/h.

See figure:

Find the route that takes the least amount of time.

My idea

I started by marking an arbitrary route:

From here, I figure the total time is going to be $$T = \frac{R}{3\mathrm{km/h}} + \frac{W}{5\mathrm{km/h}}$$

Since this is a function of two variables, I'm stuck.

A general idea is to express $W$ in terms of $R$ to make it single-variable, and then apply the usual optimization tactics (with derivatives), but I'm having a hard time finding such an expression.

Any help appreciated!

EDIT - Alternative solution?

Since the distance from A to the right angle (RA) is traveled 3/5 times as fast as the distance between RA and B, could I just scale the former up?

That way, I get A-RA being a distance of $6\cdot\frac53 = 10\mathrm{km}$, which makes the hypotenuse $\sqrt{181}$ the shortest distance between A and B. And since we scaled it up, we can consider it traversable with walking speed rather than rowing speed!

Thoughts?

• but .. where is the river / land border ? – G Cab Oct 5 '17 at 21:56
• The horizontal leg of the triangle would simultaneously be the border and the walking route. It sounds strange, but the idea is that the distance between the shoreline and the walking path is "negligible", thus they can be seen as one-and-the-same. – Alec Oct 6 '17 at 8:27
• Thanks, now it is clear. Another question, do you know (accept) trigonometric approach? – G Cab Oct 6 '17 at 21:12
• @GCab - Yes. I've reached an answer of 3.4hours with that approach. However, $\frac{\sqrt{181}}{5}$ doesn't give the same answer, and I can't figure out why that wouldn't work, and if there's some adjustment that needs to be made. Or if it needs to be discarded completely. – Alec Oct 7 '17 at 11:01
• I think the answer is here, so wikipedia gets the bonus: en.wikipedia.org/wiki/Fermat%27s_principle – Ethan Bolker Oct 8 '17 at 14:23

• a) the solution

The formula has already been indicated by wgrenard and AdamBL $$T = {1 \over 3}\sqrt {36 + \left( {9 - W} \right)^{\,2} } + {1 \over 5}W$$

differentiating that $${{dT} \over {dW}} = {{5W + 3\sqrt {36 + \left( {9 - W} \right)^{\,2} } - 45} \over {15\,\sqrt {36 + \left( {9 - W} \right)^{\,2} } }}$$ and equating to $0$ gives \eqalign{ & {{dT} \over {dW}} = 0\quad \Rightarrow \quad 3\sqrt {36 + \left( {9 - W} \right)^{\,2} } = 45 - 5W\quad \Rightarrow \cr & \Rightarrow \quad 9\left( {36 + \left( {9 - W} \right)^{\,2} } \right) = 25\left( {9 - W} \right)^{\,2} \quad \Rightarrow \cr & \Rightarrow \quad 9 \cdot 36 = 16\left( {9 - W} \right)^{\,2} \quad \Rightarrow \quad W = 9 - \sqrt {{{9 \cdot 36} \over {16}}} = 9 - {9 \over 2} = {9 \over 2} \cr} which is a minimum, because the function is convex as already indicated.

Thus $$\left\{ \matrix{ W_m = 9/2 \hfill \cr T_m = 17/5 \hfill \cr R_m = 15/2 \hfill \cr} \right.$$

• b) Scaling

Your idea of scaling according to speed is quite entangling.
That means (if I understood properly) that you are transforming the triangle from space to time units.
But, by introducing different scaling factors for the two coordinates, you undermine the Euclidean norm, which does not " transfer" between the two systems (if assumed valid in one, shall be modified in the other).

Consider for example the transformation sketched below.

From the mathematical point of view it is a linear scale transformation $$\left( {\matrix{ {y_1 } \cr {y_2 } \cr } } \right) = \left( {\matrix{ {1/v_1 } & 0 \cr 0 & {1/v_2 } \cr } } \right) \left( {\matrix{ {x_1 } \cr {x_2 } \cr } } \right)$$

Now, with constant $v_1, \,v_2$, any path in $x$ will transform in the corresponding path in $y$ (going through corresponding points).
If the path is a curve parametrized through a common parameter $\lambda$, not influencing the $v_k$'s, then, at any given value of $\lambda$ the point on the $x$ plane will transform into the corresponding point in $y$ plane $$\left( {\matrix{ {y_{1}(\lambda) } \cr {y_{2}(\lambda) } \cr } } \right) = \left( {\matrix{ {1/v_1 } & 0 \cr 0 & {1/v_2 } \cr } } \right) \left( {\matrix{ {x{_1}(\lambda) } \cr {x_{2}(\lambda) } \cr } } \right)$$ and the minimal path in one plane will be the corresponding minimal path in the other.

But we shall also have that $$\frac{d}{{d\lambda }}\left( {\matrix{ {y_{1}(\lambda) } \cr {y_{2}(\lambda) } \cr } } \right) = \left( {\matrix{ {1/v_1 } & 0 \cr 0 & {1/v_2 } \cr } } \right) \frac{d}{{d\lambda }}\left( {\matrix{ {x{_1}(\lambda) } \cr {x_{2}(\lambda) } \cr } } \right)$$ that is that the "velocities" compose vectorially.

Therefore if $\lambda$ is the time, you shall go from $A$ to $C$ with a composition of a vertical rowing speed and a horizontal walking speed (a "$\infty$-thlon"), which takes the same time as rowing $AH$ and walking $HC$.

When, instead, you just row on $AC$, then you shall change the above matrix - for that segment only - according to the $\angle AC$, and of course you loose the correspondence minimal to minimal as based on the Euclidean norm (straight line $A'B'$).

• The question has recieved a nice handful of good answers. I'm still somewhat perplexed as to the alternative solution, which I was hoping to focus on with the bounty, but since you're the only one who even mentioned it, the bounty goes here. I would be interested to learn more about the entangling nature of that strategy though. Do you have any links, or simply a concept I should look into? – Alec Oct 12 '17 at 7:23
• Thanks for appreciation. I will try and add more elaboration to clear the "scaling" matter. – G Cab Oct 12 '17 at 8:48
• @Alec: I expanded on the 2nd part: wish I succeeded and make it more clear and .. convincing. – G Cab Oct 12 '17 at 13:31

The portion of the line on the top left is $9-w$. So by the Pythagorean theorem $R^2 = 36 + (9-w)^2$. I think this is the relationship you are looking for.

You do not need to worry about the hypotenuse in the larger right triangle.

Consider the right triangle that has R as the hypotenuse. Left side of this triangle will have length $6$ and the top side will have length $(9-W)$. By the Pythagorean theorem, $R^{2} = 36+(9-W)^2$.

Solving for R and inserting in your original equation yields $T= \frac{1}{3} \sqrt{36+(9-W)^{2}} + \frac{1}{5} W$. This function is strictly convex, so you can simply minimize it by solving $\frac{d\ T}{d\ W}=0$ (the 'usual optimization tactics').

Regarding your alternative solution, I am unsure what you are arguing. The shortest distance between the points A and B will always be the line connecting them. As you have shown in your drawing, $|AB|=3\sqrt{13} \ne\sqrt{181}$.

• No, I mean if we've scaled the left-most side to 10km (by multiplying 5/3), the hypotenuse will be $\sqrt{10^2 + 9^2} = \sqrt{181}$. – Alec Oct 4 '17 at 7:39
• Yes, that is correct. If I understand you correctly, though, you are arguing that it will always be optimal to row the entire way, since the hypotenuse will always be the shortest distance from A to B. That's obviously not the case. – AdB Oct 5 '17 at 15:37
• Nono, I'm not arguing that it's faster to row always. I'm saying that if we scale the rowing distance up by 5/3, then we can consider that distance to be traversable by walking speed instead of rowing speed. – Alec Oct 5 '17 at 17:09

Snell's Law is usually applied to optics, but it is based on the quickest path through two media in which the speed of light differs. Snell's Law says that $$n_1\sin(i_1)=n_2\sin(i_2)\tag1$$ where $i_k$ is the angle of incidence to the boundary of the path and $n_k$ is inversely proportional to the speed in the particular medium ($n_kv_k=c$).

We can adapt this to the current situation by noting that $(1)$ is equivalent to $$\frac{\sin(i_1)}{v_1}=\frac{\sin(i_2)}{v_2}\tag2$$ If we travel at all along the shore, $\sin(i_2)=1$ (the angle of incidence is $90^\circ$). Since $v_1=3$ and $v_2=5$, we must have $\sin(i_1)=\frac35$, which implies that $\tan(i_1)=\frac34$.

If $\tan(i_1)=\frac34$, and the width of the river is $6$ km, then the downriver distance must be $\frac34\cdot6=4.5$ km.

This path takes $\frac{7.5}3+\frac{4.5}5=3.4$ hours.

• Wait, so we must find an angle such that we only travel 4.5km by water? I must be interpreting that wrong, because the shortest possible water-distance must be 6km, right? – Alec Oct 8 '17 at 20:50
• No, the downriver distance is $4.5$ km, the river is still $6$ km wide, so that makes that water distance $\sqrt{4.5^2+6^2}=7.5$ km. – robjohn Oct 8 '17 at 22:55
• Ah, gotcha! Yeah, that was a nice take on this problem. I never would have connected this to Snell's Law on my own. – Alec Oct 9 '17 at 7:47


# Magnetic Dollars

Imgur

Suppose that I have two urns, with one magnetic dollar in each urn. I begin randomly throwing more magnetic dollars in the general vicinity of the urns, and the coins fall in the urns by a simple rule:

Let's say that Urn A has $$x$$ coins, and Urn B has $$y$$ coins. The probability that the next coin falls into Urn A is $$\dfrac{x}{x+y}$$, and the probability that the next coin falls into Urn B is $$\dfrac{y}{x+y}$$.

You keep throwing magnetic dollars until there is a total of $$1,000,000$$ magnetic dollars in total.

What would you bet would be the price on average, of the urn with the smaller amount of coins?

$$\ 10$$, perhaps? Maybe $$\ 100$$ or $$\ 500$$? Post your bet or write it down on a piece of paper before looking at the next section.

The less-priced urn, on average, is actually worth a grand total of a quarter of a million dollars! Don't worry if you guessed wrong; many professional mathematicians also guessed much lower than this. In fact, when a group of mathematicians were asked this question and were asked to bet, most people only bet $$\ 10$$ and only one person bet over $$\ 100$$.

But why does the lower-priced urn price so high? You may want to try the problem out yourself before I go over a very nice and elegant solution. See if you can find it!

Tried it out yet? In the case that you have, let's see how this problem can be so elegantly solved, as I claimed.

Suppose that you have a deck of cards; one red, and $$999,998$$ white. Currently, you just have a red card. Now every turn, you place a white card in any available slot. For example, in the first move, you have $$2$$ available slots: one above the red card, and one below. In the second move, you have $$3$$ available slots, and so on.

But wait! Let's say that the empty slots above the red card are the magnetic dollars in Urn A, and the empty slots below the red card are the magnetic dollars in Urn B. Notice that if you had $$x$$ empty slots above the red card and $$y$$ empty slots below the red card, then the probability that the next card will be above the red card is $$\dfrac{x}{x+y}$$, and the probability that the next card will be below the red card is $$\dfrac{y}{x+y}$$! We've found a one-to-one correspondence between the original magnetic dollar problem and this new card problem!

Finally, we know that in a random placements of white cards in this fashion will result in a uniform distribution of where the red card is, every single final position is of equal probability. This means that in the original problem, the probability of $$42$$ magnetic dollars being in Urn A is the same as the probability of $$314,159$$ magnetic dollars in Urn A. Therefore, the average price of the lower priced urn is clearly $$250,000$$. $$\Box$$

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The result is certain not intuitive, nor obvious.

You can look at this problem John's Red And Blue Balls, which is based off the same idea.

Staff - 4 years, 8 months ago

I know some of you have been waiting for me to post for a long time. Sorry for the inactivity, I had a bunch of homework and stuff.

Well, here is my next #CosinesGroup posts. The problem in my opinion is a very cool problem because of the awesome elegant solution.

This problem has also been called "Polya's Urns", but I looked that up and "Polya's Urns" actually covers a lot of random complicated stuff, so I kept it at magnetic dollars.

Hope you enjoy! Feedback is appreciated.

- 4 years, 8 months ago

This is the solution in Peter Winkler ' Mathematical Puzzles. An excellent book.

- 2 years, 2 months ago

How do you do this?

- 4 years, 7 months ago

I'm curious to know why only one mathematician guessed over $100. I personally guessed around$150,000 becuase while the probability might be pretty low at some point, in A MILLION tries, it would be bound to happen quite a bit.

- 4 years, 8 months ago

Well, their reasoning would probably be then assuming a sort of "reverse" normal distribution curve. The average of all the games would probably be pretty low.

They were asked to bet immediately, depriving them of the chance to think it over. Their first instinct told them that the more coins that get in a urn, the higher probability of getting more coins in, giving a feedback loop. This means that the smaller urn probably won't really have any coins at all.

- 4 years, 8 months ago


# Asymptotic behavior of $x_{n+1}=x_n+\frac{1}{x_n}, \space\space\space x_0=1$

I have defined a sequence $x_n$ as follows: $$x_{n+1}=x_n+\frac{1}{x_n}, \space\space\space x_0=1$$ After convincing myself that there is no nice closed-form formula for $x_n$, I decided to try and find an asymptotic formula for $x_n$ (unfortunately, I have very little experience with asymptotic formulae). I noticed that the recurrence is equivalent to $$\Delta x_{n}=\frac{1}{x_n}$$ and so a solution can be approximated by solving the corresponding differential equation for $y(t)$: $$y'=\frac{1}{y}$$ This differential equation (with initial value $y(0)=1$) yielded $$y=\sqrt{2t+1}$$ which led me to believe that $$x_n\approx \sqrt{2t+1}$$ When I plotted $x_n$ and $y(n)$ side by side, it did indeed appear that they were very close to each other. However, this isn't enough for me... I would like to prove that $y(n)$ is a good approximation for $x_n$, either by proving that $$\lim_{n\to\infty}(x_n-\sqrt{2n+1})=0$$ ...or, even better, by finding a zero-approaching function $f$ satisfying $$x_n=\sqrt{2n+1}+O(f(x))$$ This is where I got stuck. How do can I prove (or disprove?) the statement in the first equality, and find $f$ satisfying the second?

NOTE: There may be a closed-form that I wasn't able to find. In fact, the similar sequence $$y_{n+1}=\frac{y_n}{2}-\frac{1}{2y_n}$$ has closed form formula $$y_n=\frac{1}{\tan(2^n\arctan(y_0^{-1}))}$$ ...but even if you do find a closed form, I would still like to know how to prove the above statements, since the techniques would be useful to know for future problems.

• I think $\int_n^{n+1} f(x) dx = \frac 1 {\int_0^n f(x)}$ is more accurate. But its probably nastier to solve. – mathreadler Nov 29 '17 at 0:15
• See OEIS sequence A073833 and related sequences. – Robert Israel Nov 29 '17 at 0:21
• Probably the first step, based on this guess, would be to write a recurrence for $a_n := x_n^2$: $a_{n+1} = a_n + 2 + \frac{1}{a_n}$. So, to start a bootstrapping argument, it would be easy to prove $a_n \ge 2n - 1$ for each $n$... And then $a_{n+1} \le a_n + 2 + \frac{1}{2n-1}$ so also $a_n \le 2n + \frac{1}{2} \log(n) - C$, etc. ... – Daniel Schepler Nov 29 '17 at 0:28
• Here is a class of arguments that could work. You can try to prove inductively both an upper and a lower bound simultaneously, say $\sqrt{2n+a} \le x_n \le \sqrt{2n+b}$ for some $a, b$, because when bounding $x_{n+1}$ in terms of $x_n$ you need both a lower and an upper bound on $x_n$ to establish either an upper bound or a lower bound on $x_n + \frac{1}{x_n}$. – Qiaochu Yuan Nov 29 '17 at 0:55
• Possible duplicate of Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$ – Aryabhata Nov 29 '17 at 3:10

You can show it's a quite nice bound by doing the following. First let $$f(x)=x+\frac{1}{x}$$ and then define $b_n=\sqrt{2n+1}$, $\epsilon_n=x_n-b_n$, and $\xi_n=f(b_n)-b_{n+1}$. First notice that $$f(x+\epsilon)-f(x)\le \epsilon$$ when $x,\epsilon \ge 0$ (we will later show that $\epsilon_n\ge 0$, for now assume this). We then have that \begin{aligned} \epsilon_{n+1} &= x_{n+1}-b_{n+1} \\ &= f(x_n)-f(b_n)+\xi_n \\ &= f(b_n+\epsilon_n)-f(b_n)+\xi_n \\ &\le \epsilon_n+\xi_n \end{aligned} Combined with $\epsilon_0=0$, this tells us that $$0\le \epsilon_{n}\le \sum_{i=0}^{\infty}\xi_i\approx 0.56$$ The above being a convergent series (according to Wolfram by comparison). This of course gives $$\sqrt{2n+1}\le x_n\le \sqrt{2n+1}+0.56$$ for all $n$, making it a very good approximation.
To show that $\epsilon_n\ge 0$ we first see that $\epsilon_0=0$. Then assume $\epsilon_n\ge 0$, by the above we have $\epsilon_{n+1}=f(b_n+\epsilon_n)-f(b_n)+\xi_n$. It is easy to see that $f$ is increasing on $x\ge 1$, and that $b_n\ge 1$, thus since $\epsilon_n\ge 0$ we see that $f(b_n+\epsilon_n)-f(b_n)\ge 0$ finally yielding $\epsilon_{n+1}\ge \xi_n\ge 0$, where $\xi_n\ge 0$ is easy to verify since we have an explicit formula for $\xi_n$. This completes the induction.
The bound numerically looks like it can be improved and it might also be good to very find the series that Wolfram is comparing ours to. And this still doesn't resolve whether $\lim_{n\to \infty}(x_n-b_n)=0$.


# Thread: Help in combinatorics exercise

1. ## Help in combinatorics exercise

Hey everyone I'm stuck in the exercise, I will be happy to work out a solution
Find how many integers - n, have the attribute:
n is divisible by 7 and is not divisible by any natural number - k that sustains:
Thanks friends.

2. ## Re: Help in combinatorics exercise

You are looking for numbers that have 7 as a factor, but do not have 2,3, or 5, as a factor.

Let $A_k = \{n \in \mathbb{N}|1 \le n \le 7770\text{ and }k\text{ divides }n\}$

You are looking for:

\begin{align*}\left|A_7 \setminus (A_2 \cap A_3 \cap A_5)\right| = & |A_7| - |A_{7\cdot 2}| - |A_{7\cdot 3}| - |A_{7\cdot 5}| \\ & + |A_{7\cdot 2\cdot 3}| + |A_{7\cdot 2\cdot 5}| + |A_{7\cdot 3\cdot 5}| \\ & - |A_{7\cdot 2\cdot 3\cdot 5}|\end{align*}

This is because if $p,q$ are distinct primes, then $A_p\cap A_q = A_{pq}$, and also from the Inclusion/Exclusion principle. Additionally, for any prime $p$, you have:

$|A_p| = \left\lfloor \dfrac{7770}{p} \right\rfloor$

I hope you get the idea. So, the answer is:

$\dfrac{7770}{7}-\left(\dfrac{7770}{7\cdot 2} + \dfrac{7770}{7\cdot 3} + \dfrac{7770}{7\cdot 5}\right) + \left(\dfrac{7770}{7\cdot 2\cdot 3} + \dfrac{7770}{7\cdot 2\cdot 5} + \dfrac{7770}{7\cdot 3\cdot 5} \right) - \dfrac{7770}{7\cdot 2\cdot 3\cdot 5} = 296$

Edit: To verify, you can use Excel.
In cell A1, type the formula: =MOD(ROW(),2)>0
(this will be TRUE if the row number is NOT divisible by 2)
In cell B1, type: =MOD(ROW(),3)>0
C1: =MOD(ROW(),4)>0
D1: =MOD(ROW(),5)>0
E1: =MOD(ROW(),6)>0
F1: =MOD(ROW(),7)=0
(notice this one you want it to be equal to zero because that means the row number is divisible by 7)
G1: =MOD(ROW(),8)>0
H1: =MOD(ROW(),9)>0
I1: =MOD(ROW(),10)>0
J1: =AND(A1:I1)
Then select cells A1:J1 and copy the formulas down to row 7770. Then, in any open cell, add the formula: =COUNTIF(J1:J7770,TRUE) and it will tell you how many rows satisfied the conditions you set for divisibility.

3. ## Re: Help in combinatorics exercise

Thanks a friend for helping me very much

4. ## Re: Help in combinatorics exercise

Edit: To verify, you can use Excel.
In cell A1, type the formula: =MOD(ROW(),2)>0
(this will be TRUE if the row number is NOT divisible by 2)
In cell B1, type: =MOD(ROW(),3)>0
C1: =MOD(ROW(),4)>0
D1: =MOD(ROW(),5)>0
E1: =MOD(ROW(),6)>0
F1: =MOD(ROW(),7)=0
(notice this one you want it to be equal to zero because that means the row number is divisible by 7)
G1: =MOD(ROW(),8)>0
H1: =MOD(ROW(),9)>0
I1: =MOD(ROW(),10)>0
J1: =AND(A1:I1)
Then select cells A1:J1 and copy the formulas down to row 7770. Then, in any open cell, add the formula: =COUNTIF(J1:J7770,TRUE) and it will tell you how many rows satisfied the conditions you set for divisibility.[/QUOTE]

This site shows me something interesting that I did not know, can you please repeat the last part?

5. ## Re: Help in combinatorics exercise

Originally Posted by yossa
Edit: To verify, you can use Excel.
In cell A1, type the formula: =MOD(ROW(),2)>0
(this will be TRUE if the row number is NOT divisible by 2)
In cell B1, type: =MOD(ROW(),3)>0
C1: =MOD(ROW(),4)>0
D1: =MOD(ROW(),5)>0
E1: =MOD(ROW(),6)>0
F1: =MOD(ROW(),7)=0
(notice this one you want it to be equal to zero because that means the row number is divisible by 7)
G1: =MOD(ROW(),8)>0
H1: =MOD(ROW(),9)>0
I1: =MOD(ROW(),10)>0
J1: =AND(A1:I1)
Then select cells A1:J1 and copy the formulas down to row 7770. Then, in any open cell, add the formula: =COUNTIF(J1:J7770,TRUE) and it will tell you how many rows satisfied the conditions you set for divisibility.

This site shows me something interesting that I did not know, can you please repeat the last part?
I'm not sure what you are asking. If you do not have Microsoft Excel, then any OpenOffice spreadsheet will work.

6. ## Re: Help in combinatorics exercise

Originally Posted by SlipEternal
I'm not sure what you are asking. If you do not have Microsoft Excel, then any OpenOffice spreadsheet will work.
I'll explain myself, I have Microsoft Excel, I was able to place everything in a simple phase a - "Then select cells A1:J1 and copy the formulas down to row 7770. Then, in any open cell, add the formula: =COUNTIF(J1:J7770,TRUE) and it will tell you how many rows satisfied the conditions you set for divisibility." not so figured out

7. ## Re: Help in combinatorics exercise

Select cells A1:J1. Right click and choose Copy (or CTRL+C on the keyboard). On the keyboard, press CTRL+G (this will bring up a dialog that says GoTo in the title. In the Reference line, put A7770. Now, while holding the Shift key (do not let go), do the following:
1. Press and release the End key
2. Press and release the Up arrow key
3. Press and release the right arrow key nine times (so you will have every cell in the range A1:J7770 highlighted).

Release the Shift key, right click, and press Paste (or CTRL+V on the keyboard).

8. ## Re: Help in combinatorics exercise

Originally Posted by SlipEternal
Select cells A1:J1. Right click and choose Copy (or CTRL+C on the keyboard). On the keyboard, press CTRL+G (this will bring up a dialog that says GoTo in the title. In the Reference line, put A7770. Now, while holding the Shift key (do not let go), do the following:
1. Press and release the End key
2. Press and release the Up arrow key
3. Press and release the right arrow key nine times (so you will have every cell in the range A1:J7770 highlighted).

Release the Shift key, right click, and press Paste (or CTRL+V on the keyboard).

i get it thanks


• Predator Prey Model Simulation Matlab
• Design, simulation and analysis in Simulink. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. The link to this assignment on github is here. ) Wilensky, U. GIBSON1*, DAVID L. I have a program called Predator Prey that's in the collection of programs that comes with NCM, Numerical Computing with MATLAB. This dataset includes the source computer code and supporting data files for the predator-prey simulation model (parameterized for summer flounder, Paralichthys dentatus) developed to investigate bottom-up effects defined to be temporal pulses in prey abundance on predator growth, production, and fisheries management. What are synonyms for Predator and prey?. This model is common, e. The predator-prey population-change dynamics are modeled using linear and nonlinear time series models. As a mathematical consequence of the herd behavior, they considered competition models and predator-prey systems. This video will show you the basics and give you an idea of what working in MATLAB looks like. Lotka-Volterra predator prey model. It was developed independently by Alfred Lotka and Vito Volterra in. While creating a model for combined predator and prey strategy would inform an estimation of overall fitness throughout an animal’s lifetime, an overarching model of this sort would be extremely complex and is beyond the scope of our paper. In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. However, the organisms in Holland’s simulation are very simple and do not involve any behavioral model. Describing the dynamics of such models occasionally requires some techniques of model analysis. Simulation of CTMC model I Use CTMC model to simulate predator-prey dynamics I Initial conditions are X(0) = 50 preys and Y(0) = 100 predators 0 5 10 15 20 25 30 0 50 100 150 200 250 300 350 400 Time Population Size X (Prey) Y (Predator) I Prey reproduction rate c 1 = 1 reactions/second I Rate of predator consumption of prey c 2 = 0:005. Student Challenge: Set up a level of hunting that keeps the populations of predators and prey at healthy levels. Predator and Prey Interactions Level A: Up and Down in the Wild: Predator and Prey. Both a detailed large eddy simulation of the dynamics and microphysics of a precipitating shallow boundary layer cloud system and a simpler model built upon basic physical principles, reproduce predator-prey behavior with rain acting as the predator and cloud as the prey. In particular, it describes how to structure a model and how to define species - that are the key components of GAMA models. They will, however, also be modiﬁed during this exercise. Related Data and Programs: FD_PREDATOR_PREY, a MATLAB program which solves a pair of predator prey ODE's using a finite difference approximation. The model is derived and the behavior of its solutions is discussed. Here is how Volterra got to these equations: The number of predatory shes immediately after WWI was much larger than. The traditional mathematical model describing the predator-prey interactions consists of the following system of differential equations. C=Death rate of predators. Click on the link below to download a zipped archive of the original iThink (. Models of interacting populations. Yes, it is agent-based model. We now replace the difference equation model used there with a more sophisticated differential equation model. This example implements best practices with MATLAB and Robotics System Toolbox. I have a Predator-Prey Model: dR/dt = λR - aRF dF/dt = -μF + bRF Where λ and μ are growth rates of rabbits (R) and foxes (F) respectively, treated in isolation. Read "A Fractional Predator-Prey Model and its Solution, International Journal of Nonlinear Sciences and Numerical Simulation" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. PY - 2009/1. In this research article, we considered an ecological prey predator fishery model system with a generalized case where both the patches are accessible to both prey and predator. Let's use the example function lotka that uses $\alpha = 0. For more information about iThink or to download a free trial version, visit www. Open a diary file in Matlab in order to save your work. These are ordinary differential equations that are straightforward to solve. An if-then. One such pair of systems is the population of Foxes and Rabbits. The total prey population is divided into two subdivisions, namely susceptible prey population and infected prey population. We show that food availability and predators' densities influence patterns of prey distribution. Use model blocks to import, initialize, and simulate models from the MATLAB ® environment into a Simulink model. Lotka was born in Lemberg, Austria-Hungary, but his parents immigrated to the US. The WATOR simulation was one of the first of these. October 30, 2017 Post source code In this post, I'll explore using R to analyze dynamical systems. , model the interactions of two industrial sectors. EE 5323 Homework 2. Learn to use MATLAB and Simulink for Simulation and other science and engineering computations. Usage of Boids for a prey-predator simulation. For a systematic approach to some of this we turn to Dynamical systems theory. This is to be able to compare with the behaviour of a corresponding stochastic and dynamic model. 2Mathematics Department , Faculty of Science , Al-Azhar University, Assiut 71511, Egypt. GIBSON1*, DAVID L. Usage of Boids for a prey-predator simulation- + Dailymotion. As the population of the prey increases then the predator population will increase. Models of interacting populations. The model is used to study the ecological dynamics of the lion-buﬀalo-Uganda Kob prey-predator system of Queen Elizabeth National Park, Western Uganda. This will help us use the lotka model with different values of alpha and beta. Modelling Predator-Prey Interactions with ODE Modelling Predator-Prey Interactions with ODE Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Modelling Predator-Prey Interactions with ODE Outline Outline of Topics Predator-Prey Models The Lotka-Volterra (LV) model. Simulate Identified Model in Simulink. Simulate Identified Model in Simulink. Or copy & paste this link into an email or IM:. fd1d_predator_prey_test. Determine the equilibrium points and their nature for the system. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. Originally, the HANDY Model is derived from a predator-prey model as indicated in [1]. ODEs are frequently used in biology to model population dynamics. 1007/s11859-015-1054-4. Save the first part of this model. Introduction This chapter, originally intended for inclusion in [4], focuses on mod-eling issues by way of an example of a predator-prey model where the. Yes, it is agent-based model. Vector with the named parameters of the model: k1. DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM WITH CHOICE by Douglas Magomo August 2007 Studies of predator-prey systems vary from simple Lotka-Volterra type to nonlinear systems involving the Holling Type II or Holling Type III functional response functions. Matlab code for the examples discussed below is in this compressed folder. If no predators, prey population grows at natural rate: for some constant a > 0,. Since we are considering two species, the model will involve two equations, one which describes how the prey population changes and the second which describes how the predator population changes. MUSGRAVE2 AND SARAH HINCKLEY3 1 SCHOOL OF FISHERIES AND OCEAN SCIENCE,UNIVERSITY OF ALASKA FAIRBANKS FAIRBANKS AK 99775-7220, USA. A STAGE-STRUCTURED PREDATOR-PREY MODEL HAL SMITH 1. My book that's available on the MathWorks website. The x_t denote the number of snow hares (prey) and y_t be the. But these functions also arise in the other sciences. In recent years, many authors have explored the dynamic relationship between predators and their preys. Some predator-prey models use terms similar to those appearing in the Jacob-Monod model to describe the rate at which predators consume prey. Finally, you will see a demonstration of the concepts above through an autonomous object tracking example. There is a simulation speed slider on the bottom of the model page (I think it has. What is the carrying capacity for moose in the simulation model of Isle Royale, prior to any changes in the weather?. Predator Prey Models in Real Life. Di erential Equations (Aggregate) Models with MATLAB and Octave A Predator-Prey Example Di erential equations in biology are most commonly associated with aggregate models. Participants are assigned a role in the food chain, participate in the simulation, collect and analyze results, and assess factors affecting their survival. Back to Eduweb Portfolio. In the special case in which both predator and prey are from the same species, predation is called cannibalism. Part 1: create the model. We define a prey (mouse) and predator (cat) model. However, the organisms in Holland’s simulation are very simple and do not involve any behavioral model. Lotka was born in Lemberg, Austria-Hungary, but his parents immigrated to the US. Many of our resources are part of collections that are created by our various research projects. The Predator-Prey Model Simulation. The prey–predator algorithm was used to evaluate the best performance of the heat sinks. This model reflects the point in time where the predator species has evolved completely and no longer competes for the initial food source. Software Programming And Modelling For Scientific Researchers. Fall 2017 Math 636 Predator-Prey Models 1. Using the Lotka-Volterra predator prey model as a simple case-study, I use the R packages deSolve to solve a system of differential equations and FME to perform a sensitivity analysis. In the Lotka Volterra predator-prey model, the changes in the predator population y and the prey population x are described by the following equations: Δxt=xt+1−xt=axt−bxtyt Δyt=yt+1−yt=cxtyt−dyt. The prey population increases when there are no predators, and the predator population decreases when there are no prey. We will assume that the predators are Greater Californian Killer Foxes and the prey are Lesser Fluffy Rabbits. Predator-Prey Models in Excel. This paper describes the GA model using a new selection method inspired by predator-prey interactions. I The main hypothesis: The prey-predator interaction is the only factor. What are synonyms for Predator and prey?. in pursuit of her chosen prey. Predator-Prey Cycles. Predator-Prey Simulation Lab. Then, the model was further developed to include density dependent prey growth and a functional response of the form developed by C. ’ Note that this model can be considered as a simple predator-prey model. For the eBay data, we have >>polyﬁt(year,income,1) ans = 100. Simulink is also practical. 2Modeling Predator-Prey Dynamics and Climate Inﬂuence 2. PloS one, 7, e28924 [paper3, 1 predator - 2 prey model] Unfortunately, there are very few technical documents available on how to implement point process IBMs and corresponding moment equations. Antonyms for Predator and prey. A STAGE-STRUCTURED PREDATOR-PREY MODEL HAL SMITH 1. PREDATOR PREY MODELS IN COMPETITIVE CORPORATIONS PREDATOR PREY MODELS By Rachel Von Arb Honors Scholarship Project Submitted to the Faculty of. This represents our first multi-species model. Recently, a new type of mathematical model was introduced into biology to study the pattern formation, i. MATLAB Code: function yp = lotka(t,y). / Spatiotemporal dynamics of a diffusive Leslie-Gower predator-prey model with ratio-dependent functional response. Wilkinson and T. It also assumes no outside influences like disease, changing conditions, pollution, and so on. In addition, the amount of food needed to sustain a prey and the prey life span also affect the carrying capacity. Consider the Lotka-Voterra equations of interacting predator and prey systems This equations include the effect of limited resources on the food supply of the prey, and how the prey are culled or harvested. The predators depend on the populations of these prey organisms. To illustrate the use of WSMCreateModel, the component "hare" in the predator-prey model was created in Mathematica. If your students are unable to run the simulation at their own workstations then it may be played on an overhead projector. Initial populations sizes can be selected by the user and are randomly distributed in a square ‘environment’, (dimensions=km,. Figure 1: Simple Predator Prey Model The phase plane plot compares the population of predators to the population of prey, and is not dependent on time. The grid is enclosed, so a critter is not allowed to move off the edges of the world. Predator-Prey Models from Iterated Prisonerʼs Dilemma model. Shiflet and G. Use model blocks to import, initialize, and simulate models from the MATLAB ® environment into a Simulink model. Consider for example, the classic Lotka-Volterra predator prey. Classic population models including the logistic map, predator-prey systems, and epidemic models will be used to motivate dynamics concepts such as stability analysis, bifurcations, chaos, and Lyapunov exponents. Spring (3) Shaw. Universitas Negeri Surabaya. I - Ecological Interactions: Predator and Prey Dynamics on the Kaibab Plateau - Andrew Ford ©Encyclopedia of Life Support Systems (EOLSS) 1. We show the different types of system behaviors for various parameter values. A Predator-Prey model: Suppose that we have two populations, one of which eats the other. Round 1 Data Analysis: Produce a "ﬁnished product" graph of the data from the simulation. In this paper, we have considered a prey–predator model where both prey and predator live in herds. Course Goals: Expose students to the process of model building, the simulation and computation with mathematical models, and the interpretation and analysis of simulation results. Run the simulation again now with the controls below, paying attention to the animals in the enclosure at the top. Both prey and predator harvesting or combined harvesting and maximum sustainable yield have been discussed. Determine the equilibrium points and their nature for the system. The solution is also given in Taylor’s series. Models of interacting populations. Multi-Team Prey-Predator Model with Delay Shaban Aly1, 2 and M. Predation has been described as a clean. The second model (Daypr) is more realistic. wednesday, june 19, 2019. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. Go through the m-ﬁle step-by-step. The Canadian lynx is a type of wild felid, or cat, which is found in northern forests across almost all of Canada and Alaska. GIBSON1*, DAVID L. Predator Prey Models in MatLab James K. Many of our resources are part of collections that are created by our various research projects. Initial populations sizes can be selected by the user and are randomly distributed in a square 'environment', (dimensions=km,. As part of our. This pair of interacting populations is a classic example of the predator‐prey dynamical system. of Shiflet;: The Fox and the Rabbit Many dynamic systems are interdependent systems. My book that's available on the MathWorks website. Diff Eqs Lect #12, Predator/Prey Model, Vector Fields and Direction Fields - Duration. , 2D linear dynamical systems; the use of probabilities gives Markov chains. However, the organisms in Holland’s simulation are very simple and do not involve any behavioral model. 01$ and \$\beta = 0. It is a simple program originally described by A. ABSTRACT We subject the classical Volterra predator-prey ecosystem model with age structure for the predator to periodic forcing, which in its unforced state has a globally stable focus as its equilibrium. There has been growing interest in the study of Prey-Predator models. Discussion and Conclusion In Conclusion, this Lotka-Volterra Predator-Prey Model is a fundamental model of the complex ecology of this world. If x is the population of zebra, and y is the population of lions, description of the population dynamics with the help of coupled differential equations. albena, june 20-25, 2019. BIFURCATION IN A PREDATOR-PREY MODEL WITH HARVESTING 2103 the harvesting terms can be combined into the growth/death terms as in system (3), so the dynamics of system (3) are very similar to that of the unharvested system (2). It also highlights the modularity of MATLAB and ROS by showing the algorithm using real and simulated TurtleBot ® robotic platforms, as well as a webcam. If your students are unable to run the simulation at their own workstations then it may be played on an overhead projector. zeszyty naukowe politechniki ŚlĄskiej 2018 seria: organizacja i zarzĄdzanie z. It was developed independently by Alfred Lotka and Vito Volterra in. Canadian lynx feed predominantly on snowshoe hares. The most popular example is the population of the snowshoe hare and the lynx. For system dynamics modeling, as with all approaches, the text employs a nonspecific tool, or generic,. Prey population x(t); Predator population y(t) 2. Boids simulation on Matlab. The predator population starts to decrease and, let me do that same blue color. Course Readings. println("Will think about this more. We assign a mo-mentary fitness f y (t) of the prey in a year t as follows: it is zero if it is not present, it is 11 if both predator and prey are present, and it is +1 if the prey is present but the predator. It provides online dashboard tools for simulation analytics that can be shared with users from around the world. zip contains versions of some programs converted to work with SciLab. Now ode45 is used to perform simulation by showing the solution as it changes in time. The Puma-Prey Simulator demonstrates the natural balance of a healthy ecosystem, in contrast with the changes that occur as a result of human encroachment. (c) Constant-yield harvesting on the prey and constant-e ort harvesting on preda-tors. We describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator-prey model proposed by R. Initial populations sizes can be selected by the user and are randomly distributed in a square 'environment', (dimensions=km,. 8, in steps of 0. Learn to use MATLAB and Simulink for Simulation and other science and engineering computations. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty. 05 and Euler's method, to model the population numbers over the next 5 years. National Science Foundation. Predator Prey Multi Agent Simulation Model (JAVA & REPAST). Suppose in a closed eco-system (i. For system dynamics modeling, as with all approaches, the text employs a nonspecific tool, or generic,. Objective: Students will simulate predator prey interactions using cards. Scilab simulation of Lotka Volterra predator prey model, van-der-Pol Oscillator tutorial of Nonlinear Dynamical Systems course by Prof Harish K. Attentional strategies for dynamically focusing on multiple predators/prey, click here. Be sure to stay to the end to find out where to go next to learn MATLAB in depth. Denning, "Computing is a natural science" MatlaB Tutorial. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. Matlab code for the examples discussed below is in this compressed folder. Some predator-prey models use terms similar to those appearing in the Jacob-Monod model to describe the rate at which predators consume prey. Surabaya, Indonesia. These equations have given rise to a vast literature, some of which we will sample in this lecture. A FORMAL MODEL OF EMOTIONAL STATE. The results developed in this article reveal far richer dynamics compared to the model without harvesting. Pillai of IIT Bombay. However in this pa-per, in order to illustrate the accuracy of the method, DTM isappliedtoautonomous and non-autonomous predator-prey models over long time horizons and the. In the Lotka-Volterra model, there is one populations of animals (predator) that feeds on another population of animals (prey). The Lotka-Volterra model is the simplest model of predator-prey. 2016-10-10 Modeling and Simulation of Social Systems with MATLAB 35 SIR model ! A general model for epidemics is the SIR model, which describes the interaction between Susceptible, Infected and Removed (Recovered) persons, for a given disease. In the Lotka Volterra predator-prey model, the changes in the predator population y and the prey population x are described by the following equations: Δxt=xt+1−xt=axt−bxtyt Δyt=yt+1−yt=cxtyt−dyt. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. I have a program called Predator Prey that's in the collection of programs that comes with NCM, Numerical Computing with MATLAB. The Dynamical System. If x is the population of zebra, and y is the population of lions, description of the population dynamics with the help of coupled differential equations. In [9] the DTM was applied to a predator-prey model with constant coefﬁ-cients over a short time horizon. For You Explore. The second is a study of a dynamical system with a simple bifurcation, and the third problem deals with predator-prey models. Denning, "Computing is a natural science" MatlaB Tutorial. The behavior of each of them is given by the following rules: Prey: 1) just moved to an unoccupied cell 2) Every few steps creates offspring to his old cell 3) Life expectancy is limited by the number of moves Predator: 1) Predator moves to the cell with prey. Given the differences between these two types of models, why would it be difficult to determine accurate values for the four parameters in the Lotka-Volterra predator-prey model (a, rprey, m, b) for animals in the real world?. Department of Mathematics. 1 Logistic growth with a predator We begin by introducing a predator population into the logistic. lab 11 predator prey simulation lab answers. PY - 2009/1. Applications of MATLAB/Simulink for Process Dynamics and Control Simulink is a platform for multidomain simulation and model Predator and prey populations. The objective of this paper is to study systematically the dynamical properties of a predator-prey model with nonlinear predator harvesting. Using the Lotka-Volterra predator prey model as a simple case-study, I use the R packages deSolve to solve a system of differential equations and FME to perform a sensitivity analysis. Find Alien at affordable prices. AU - Ruan, Shigui. The prey are blue, and the predators are yellow. Diff Eqs Lect #12, Predator/Prey Model, Vector Fields and Direction Fields - Duration. Suppose in a closed eco-system (i. Suppose there are two species of animals, a prey and a predator. Software Programming And Modelling For Scientific Researchers. The Puma-Prey Simulator demonstrates the natural balance of a healthy ecosystem, in contrast with the changes that occur as a result of human encroachment. Forecasting performance of these models is compared. predator-prey simulations 1 Hopping Frogs an object oriented model of a frog animating frogs with threads 2 Frogs on Canvas a GUI for hopping frogs stopping and restarting threads 3 Flying Birds an object oriented model of a bird deﬁning a pond of frogs giving birds access to the swamp MCS 260 Lecture 36 Introduction to Computer Science. My book that's available on the MathWorks website. 2Mathematics Department , Faculty of Science , Al-Azhar University, Assiut 71511, Egypt. zip contains all Matlab program files listed here. Introduction This chapter, originally intended for inclusion in [4], focuses on mod-eling issues by way of an example of a predator-prey model where the. Model equations In this paper, we study the numerical solutions of 2-component reaction–diffusion. Both prey and predator harvesting or combined harvesting and maximum sustainable yield have been discussed. Read "A Fractional Predator-Prey Model and its Solution, International Journal of Nonlinear Sciences and Numerical Simulation" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Predator-prey model with delay. iseesystems. 03(2015), Article ID:56937,10 pages 10. " – Simulation as a basic tool. This is referred to. • Use Euler’s method and Runge-Kutta in MATLAB to obtain numerical approximations. Continuous time (ODE) version of predator prey dynamics: Equilibrium points (2) •~(20. A mathematical analysis shows that prey refuge plays a crucial role for the survival of the species and that the harvesting effort on the predator may be used as a control to prevent the cyclic behaviour of the system. We are trying to understand as the population grows in one of the species what the effect is on the other species which co inhabit that environment. The second model is an extension of the logistic model to species compe-tition. Some predator-prey models use terms similar to those appearing in the Jacob-Monod model to describe the rate at which predators consume prey. Predator Prey Model Goal: The goal of this experiment is to model the population dynamics of animals both predator and prey when they are present in an environment. Rabbits and Wolves: Experiment with a simple ecosystem consisting of grass, rabbits, and wolves, learning about probabilities, chaos, and simulation. Homework 5, Phase Portraits. Using no barriers and a random distribution of 100 beans, run 1 trial as done during the baseline data trials. Suppose in a closed eco-system (i. Date: 22nd August, 2007 Lab #1: Predator-Prey Simulation ==> OBJECTIVE: To simulate predator prey interactions and record the numbers of predator and prey in their "ecosystem" and prepare a graph. Zhao, "Mathematical and dynamic analysis of a prey-predator model in the presence of alternative prey with impulsive state feedback control," Discrete Dynamics in Nature and Society, vol. FD1D_WAVE, a MATLAB program which applies the finite difference method to solve the time-dependent wave equation in one spatial dimension. Stan is used to encode the statistical model and perform full Bayesian inference to solve the inverse problem of inferring parameters from noisy data. Here is how Volterra got to these equations: The number of predatory shes immediately after WWI was much larger than. [paper2, 2 predator - 1 prey model] Barraquand, F. The model is first applied to a system with two-dimensions, but is then extended to include more. Introduction (1 week) # - "What is a model?" A. The objective of this paper is to study systematically the dynamical properties of a predator-prey model with nonlinear predator harvesting. Dewdney in Scientific American magazine. Simulate Identified Model in Simulink. The Dynamical System. We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. INTRODUCTION global properties of the orbit structure. Here, the predators are the police and the prey the gang members. One of such models that simulates predator-prey interactions is the Lotka-Volterra Model. The grid is enclosed, so a critter is not allowed to move off the edges of the world. The model of Lotka and Volterra is not very realistic. A game in the everyday sense—“a competitive activity. Ballesteros, Intuition, functional responses and the formulation of predator–prey models when there is a large disparity in the spatial domains of the interacting species, Journal of Animal Ecology, 77, 5, (891-897), (2008). sciencedaily. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. Kalyan Das, National Institute of Food Technology Entrepreneurship and Management (NIFTEM), Mathematics Department, Faculty Member. The book is related to aircraft control, dynamics and simulation. We assign a mo-mentary fitness f y (t) of the prey in a year t as follows: it is zero if it is not present, it is 11 if both predator and prey are present, and it is +1 if the prey is present but the predator. GIBSON1*, DAVID L. Y1 - 2009/1. I - Ecological Interactions: Predator and Prey Dynamics on the Kaibab Plateau - Andrew Ford ©Encyclopedia of Life Support Systems (EOLSS) 1. A computer simulation model for the learning behavior of a certain type of predator faced with a multipatch environment is constructed, where prey densities differ between patches and are functions of time. Various computer models have been created to simulate the predator-prey relationship within an ecosystem. Mathematical Modeling: Models, Analysis and Applications covers modeling with all kinds of differential equations, namely ordinary, partial, delay, and stochastic. Lotka-Volterra predator prey model. Predator-Prey Model, University of Tuebingen, Germany. Keywords: Lotka-Volterra Model, Predator-prey interaction, Numerical solution, MATLAB Introduction A predator is an organism that eats another organism. We'll start with a simple Lotka-Volterra predator/prey two-body simulation. Analyzing the Parameters of Prey-Predator Models for Simulation Games 5 that period. Circles represent prey and predator initial conditions from x = y = 0. Model equations In this paper, we study the numerical solutions of 2-component reaction–diffusion. Run the simulation again now with the controls below, paying attention to the animals in the enclosure at the top. Finally, as we’ll see in Chapter xx, there is a deep mathematical connection between predator-prey models and the replicator dynamics of evolutionary game theory. These functions are for the numerical solution of ordinary differential equations using variable step size Runge-Kutta integration methods. The Lotka-Volterra model is the simplest model of predator-prey. I have a Predator-Prey Model: dR/dt = λR - aRF dF/dt = -μF + bRF Where λ and μ are growth rates of rabbits (R) and foxes (F) respectively, treated in isolation. Predators consume energy at a fixed rate over time; if their internal energy level becomes too low, they die. Go through the m-ﬁle step-by-step. Back to Eduweb Portfolio. They use a simplified version of the Lotka-Volterra equations and generate graphs showing population change. Lotka-Volterra predator-prey model. Section 5-4 : Systems of Differential Equations. Introduction This chapter, originally intended for inclusion in [4], focuses on mod-eling issues by way of an example of a predator-prey model where the. The model is intended to represent a warm—blooded vertebrate predator and its prey. MATLAB files for the discrete time model: predprey_discrete. Abstract—We describe and analyze emergent behavior and its effect for a class of prey-predators’ simulation models. Each collection has specific learning goals within the context of a larger subject area. Fussmann*† & Nelson G. Rapid evolution drives ecological dynamics in a predator–prey system Takehito Yoshida*, Laura E. Below are examples that show how to solve differential equations with (1) GEKKO Python, (2) Euler's method, (3) the ODEINT function from Scipy. I Also used in economic theory, e. considering some well known simulation methods to obtain the posterior summaries of interest. More generally, any of the data in the Lotka-Volterra model can be taken to depend on prey density as appropriate for the system being studied. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. For You Explore. One is the. In this model, prey, which represents the decision space vector, will be placed on the vertices of a two-dimensional lattice. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. The predator-prey population-change dynamics are modeled using linear and nonlinear time series models. It is a simple program originally described by A. This will help us use the lotka model with different values of alpha and beta. Open the first file for this module by typing on the Matlab command line: ppmodel1. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty. BIFURCATION IN A PREDATOR-PREY MODEL WITH HARVESTING 2103 the harvesting terms can be combined into the growth/death terms as in system (3), so the dynamics of system (3) are very similar to that of the unharvested system (2). One animal in the simulation is a predator. Lotka-Volterra Model The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. Stan is used to encode the statistical model and perform full Bayesian inference to solve the inverse problem of inferring parameters from noisy data. This model reflects the point in time where the predator species has evolved completely and no longer competes for the initial food source. Circles represent prey and predator initial conditions from x = y = 0. Using the Lotka-Volterra predator prey model as a simple case-study, I use the R packages deSolve to solve a system of differential equations and FME to perform a sensitivity analysis. Predation has been described as a clean. 1007/s11859-015-1054-4. The parameters preyPop and predPop are the initial sizes of the prey and predator populations (M(0) and W(0)), respectively, dt (Δt) is the time interval used in the simulation, and months is the number of months (maximum value of t) for which to run the simulation. You may wish to introduce disturbances in the cycle such as killing off the lynx or starving the rabbits. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 7, 2013 Outline Numerical Solutions Estimating T with MatLab Plotting x and y vs time Plotting Using a Function Automated Phase Plane Plots. 8, in steps of 0. Load the sample project containing the Lotka-Volterra model m1. Determine the equilibrium points and their nature for the system. The Modeling Commons contains more than 2,000 other NetLogo models, contributed by modelers around the world. In the Lotka Volterra predator-prey model, the changes in the predator population y and the prey population x are described by the following equations: Δxt=xt+1−xt=axt−bxtyt Δyt=yt+1−yt=cxtyt−dyt. The model is derived and the behavior of its solutions is discussed. to investigate the key dynamical properties of spatially extended predator–prey interactions. Prey Simulation Lab Introduction In this lab project the objective is to simulate the relationship over generations of prey vs. It is a simple program originally described by A. This is a spatial predator-prey model from population ecology. Represent and interpret data on a line graph. The prey should exhibit mild oscillations, and the predator should fluctuate little. The solution is also given in Taylor’s series. Trajectories are closed lines. Models of interacting populations. System of differential equations. Denning, "Computing is a natural science" MatlaB Tutorial. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the bivariate truncated generalized Cauchy model fits the data better than the other models. Nonlinear model predictive control (planning) for level control in a surge tank, click here. Make phase plane plot. Princeton University Press, Princeton, NJ (2006). , wolf spider) and prey species (e. They use a simplified version of the Lotka-Volterra equations and generate graphs showing population change. Article (PDF Available) Simulation of Predator-Prey in MATLAB. Study the eﬀects of e. And an improved model with Logistic blocking effect is proposed. Dynamics of the system. Here is some data that approximates the populations of lynx and snowshoe hares observed by the Hudson Bay Company beginning in 1852. Usage of Boids for a prey-predator simulation. See our discounts on Alien, check our site and compare prices. This model takes the form of a pair of ordinary differential equations, one representing a prey species, the other its predator. Graph the population number (hares. Prey populations. Continuous time (ODE) version of predator prey dynamics: Equilibrium points (2) •~(20. model consisting of prey-predator model with horizontally transmitted of disease within predator population is proposed and studied. As the students work on constructing a model (Circulate Constructing a Model - Rabbit, Constructing a Model - Fox) I rotate the room and offer support where needed. Predator prey offers this graphic user interface to demonstrate what we've been talking about the predator prey equations. http://simulations. Predators/prey respond to other predators/prey and move of their own volition. zeszyty naukowe politechniki ŚlĄskiej 2018 seria: organizacja i zarzĄdzanie z. Wilkinson and T. Predator, which deals with objective functions, will also be placed on the same lattice randomly. Modified Model with "Limits to Growth" for Prey (in Absence of Predators) In the original equation, the population of prey increases indefinitely in the absence of predators. % the purpose of this program is to model a predator prey relationship % I will be using. In the last section we make a review of the paper and share with the future plans. To ﬁnd the ratios of the errors, we will. We present an individual-based predator-prey model with, for the first time, each agent behavior being modeled by a fuzzy cognitive map (FCM), allowing the evolution of the agent behavior through the epochs of the simulation. The prey still relies on the food source, but the predator relies solely on the former competitor. This lesson allows students to explore the interactions of two animal populations (wolves and moose) within an ecosystem. Represent and interpret data on a line graph. Analysis of the main equation guides in the correct choice of parameter values. The model of Lotka and Volterra is not very realistic. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. Plot the prey versus predator data from the stochastically simulated lotka model by using a custom function (plotXY). The persistence of food chains is maximized when prey species are neither too big nor too small relative to their predator. PY - 2009/1. The model. One animal in the simulation is a predator. 1 Logistic growth with a predator We begin by introducing a predator population into the logistic. As part of our. The Lotka-Volterra equations can be written simply as a system of first-order non-linear ordinary differential equations (ODEs). The second project of the semester was the predator prey model. Gillespie, predator-prey simulation GillespieSSA test. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty. Set the solver type to SSA to perform stochastic simulations, and set the stop time to 3. A modified predator–prey model with transmissible disease in both the predator and prey species is proposed and analysed, with infected prey being more vulnerable to predation and infected predators hunting at a reduced rate. Software Programming And Modelling For Scientific Researchers. & Murrell, D. The Lotka-Volterra predator-prey equations can be used to model populations of a predator and prey species in the wild. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. Make Life easier. Deterministic Models can be classified dimensionally as 0D, 1D, 2D or 3D. Models of interacting populations. master program for. View at Publisher · View at Google Scholar. under the existence of the interior equilibrium point E 2 ∗ = (x 2 ∗, y 2 ∗). version of a Kolmogorov model because it focuses only on the predator-prey interactions and ignores competition, disease, and mutualism which the Kolmogorov model includes. The simulation uses rule-based agent behavior and follows a prey-predator structure modulated by a number of user-assigned parameters. Innovation Process Simulation on the Base Predator and Prey. Students should keep in mind that, as in any simulation (even sophisticated computer models), certain assumptions are made and many variables. in the literature [2-7,10,17]. Predator-prey model, design, simulation and analysis in Simulink. Predator prey offers this graphic user interface to demonstrate what we've been talking about the predator prey equations. The most popular example is the population of the snowshoe hare and the lynx. Lotka (1925) and Vito Volterra (1926). Let the initial values of prey and predator be [20 20]. Matt Miller, Department of Mathematics, University of South Carolina email: miller@math. m - discrete time simulation of predator prey model Continuous Time Model. In addition to discussing the well posedness of the model equations, the results of numerical experiments are presented and demonstrate the crucial role that habitat shape, initial data, and the boundary conditions play in determining the spatiotemporal dynamics of predator-prey interactions. Predator vs. Prey-predator model has received much attention during the last few decades due to its wide range of applications. The model for this simulation was created using iThink Systems Thinking software from isee systems. Participants are assigned a role in the food chain, participate in the simulation, collect and analyze results, and assess factors affecting their survival. Consider a population of foxes, the predator, and rabbits, the prey. View, run, and discuss the 'Predator Prey Game' model, written by Uri Wilensky. 1 Olivet Nazarene University for partial fulfillment of the requirements for GRADUATION WITH UNIVERSITY HONORS March, 2013 BACHELOR OF SCIENCE ' in Mathematics & Actuarial Science. The model is fit to Canadian lynx 1 1 Predator: Canadian lynx. We implemented this model in Matlab to. The prey still relies on the food source, but the predator relies solely on the former competitor. Date: 22nd August, 2007 Lab #1: Predator-Prey Simulation ==> OBJECTIVE: To simulate predator prey interactions and record the numbers of predator and prey in their "ecosystem" and prepare a graph. Modeling and Simulation Krister Wiklund, Joakim Lundin, Peter Olsson, Daniel V˚agberg 1 Predator-Prey,model A In this exercise you will solve an ODE-system describing the dynamics of rabbit and fox populations. So the prey population increases, and you see that the other way around. ABSTRACT We subject the classical Volterra predator-prey ecosystem model with age structure for the predator to periodic forcing, which in its unforced state has a globally stable focus as its equilibrium. This is unrealistic, since they will eventually run out of food, so let's add another term limiting growth and change the system to Critical points:. http://simulations. itmx) model file. Now what’s truly exciting is this, we made a lot of assumptions when deriving this model, and even still the information extrapolated from this model can be found in actual physical models. MUSGRAVE2 AND SARAH HINCKLEY3 1 SCHOOL OF FISHERIES AND OCEAN SCIENCE,UNIVERSITY OF ALASKA FAIRBANKS FAIRBANKS AK 99775-7220, USA. Initially at time t=0, the population of prey is some value say x0 and. Leslie-Gower Predator-Prey Model 202 (Pratiwi et al) Numerical Simulation of Leslie-Gower Predator-Prey Model with Stage-Structure on Predator Rima Anissa Pratiwi, Agus Suryanto*, Trisilowati Departemant of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia Abstract. Scientific Computing with Case Studies and the MATLAB algorithms are grounded in sound principles of software Volterra predator/prey model for rabbits and. Predator-Prey Cycles. Predator, which deals with objective functions, will also be placed on the same lattice randomly. A high-dimensional predator-prey reaction-diffusion system with Holling-type III functional response, where the usual second-order derivatives give place to a fractional derivative of order α with 1 < α ≤ 2. FD1D_WAVE, a MATLAB program which applies the finite difference method to solve the time-dependent wave equation in one spatial dimension. Retrieved June 21, 2019 from www. Synonyms for Predator and prey in Free Thesaurus. Dewdney in Scientific American magazine. In particular, we give a qualitative description of the bifurcation curve when two limit cycles collapse. They use a simplified version of the Lotka-Volterra equations and generate graphs showing population change. Models of interacting populations. The predator is represented by coyotes, the prey by rabbits, and the prey's food by grass, although the model can apply to any three species in an ecological food chain. At α=1 the model is purely predator-prey. DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM WITH CHOICE by Douglas Magomo August 2007 Studies of predator-prey systems vary from simple Lotka-Volterra type to nonlinear systems involving the Holling Type II or Holling Type III functional response functions. Make Life easier. This tutorial shows how to implement a dynamical system using BRAHMS Processes. The Lotka-Volterra equations can be written simply as a system of first-order non-linear ordinary differential equations (ODEs). In the model to be formulated, it is now assumed that instead of a (deterministic) rate of predator and prey births and deaths, there is a probability of a predator and prey birth or death. We extend it to explore the interaction between population and evolutionary dynamics in the context of predator–prey and morphology–behavior coevolution. The model is used to study the ecological dynamics of the lion-buﬀalo-Uganda Kob prey-predator system of Queen Elizabeth National Park, Western Uganda. Andrew, Nick, and I worked on this project. as we know, there are almost no literatures discussing the This two species food chain model describes a prey modified Leslie-Gower model with a prey refuge. The ﬁrst consists in scaling of a homogeneous and a nonhonogeneous differential equation. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. 6 CHAPTER 1. There are many kind of prey-predator models in mathematical ecology. Lotka-Volterra predator-prey model. The prey population increases when there are no predators, and the predator population decreases when there are no prey. GIBSON1*, DAVID L. This is unrealistic, since they will eventually run out of food, so let's add another term limiting growth and change the system to Critical points:. 22 contributions in the last year Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Sun Mon Tue Wed Thu Fri Sat. I give the analysis of dispersal relation of wave behavior in detail. This was done using the. The interactions of ecological models may occur among individuals of the same species or individuals of different species. Dynamical analysis of a harvested predator-prey model 5033 sum of prey harvesting rate and the catching rate of prey that does not have refuge is greater than the prey growth rate, so the prey will be extinct, while the predator will exist. Since we are considering two species, the model will involve two equations, one which describes how the prey population changes and the second which describes how the predator population changes. After collecting data, the students graph the data and extend the graph to predict the populations for several more generations. Predator and Prey I. Lotka-Volterra model, realized as a computer program. predators decline, and the prey recover, ad infinitum.


# How to prove the result of this definite integral?

During my work I came up with this integral:

$$\mathcal{J} = \int_0^{+\infty} \frac{\sqrt{x}\ln(x)}{e^{\sqrt{x}}}\ \text{d}x$$

Mathematica has a very elegant and simple numerical result for this, which is

$$\mathcal{J} = 12 - 8\gamma$$

where $\gamma$ is the Euler-Mascheroni constant.

I tried to make some substitutions, but I failed. Any hint to proceed?

Enforce the substitution $x\to x^2$. Then, we have

\begin{align} \mathcal{I}&=4\int_0^\infty x^2\log(x)e^{-x}\,dx\\\\ &=4\left.\left(\frac{d}{da}\int_0^\infty x^{2+a}e^{-x}\,dx\right)\right|_{a=0}\\\\ &=4\left.\left(\frac{d}{da}\Gamma(a+3)\right)\right|_{a=0}\\\\ &=4\Gamma(3)\psi(3)\\\\ &=4(2!)(3/2-\gamma)\\\\ &=12-8\gamma \end{align}

as was to be shown!

Note the we used (i) the integral representation of the Gamma function

$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt$$

(ii) the relationship between the digamma and Gamma functions

$$\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$$

and (iii) the recurrence relationship

$$\psi(x+1)=\psi(x)+\frac1x$$

In addition, we used the special values

$$\Gamma(3)=2!$$

and

$$\psi(1)=-\gamma$$

• Our answers are closed, but slightly different:) Tell me if you want me to remove mine. Thanks! – Olivier Oloa Mar 12 '16 at 20:02
• Awesome method! So fast and elegant haha, I should have thought about >.< Well.. this is more experience! Thanks!! – Von Neumann Mar 12 '16 at 20:03
• @OlivierOloa Olivier, we are friends here on MSE. I would not begin to ask that you remove your very solid answer. - Mark – Mark Viola Mar 12 '16 at 20:05
• @1over137 Thank you! And you're welcome. My pleasure. - Mark – Mark Viola Mar 12 '16 at 20:06
• @Dr.MV Good answer! (+1) – Olivier Oloa Mar 12 '16 at 20:07

Hint. One may recall that $$\int_0^\infty u^{s}e^{-u}\:du=\Gamma(s+1), \quad s>0,\tag1$$ giving, by differentiating under the integral sign, \begin{align} \int_0^\infty u^{2}\ln u \:e^{-u}\:du&=\left.\left(\Gamma(s+1)\right)'\right|_{s=2}\\\\ &=\left.\left(s(s-1)\Gamma(s-1)\right)'\right|_{s=2}\\\\ &=3+2\Gamma'(1)\\\\ &=3-2\gamma,\tag2 \end{align} where we have used $\Gamma'(1)=-\gamma$.

Then, one may rewrite the initial integral as $$2\int_0^\infty \sqrt{x}\:(\ln \sqrt{x}) \:e^{-\sqrt{x}}\:dx,$$ then perform the change of variable $x=u^2$, $dx=2udu$, obtaining \begin{align} \int_0^\infty \sqrt{x}\ln x \:e^{-\sqrt{x}}\:dx&=4\int_0^\infty u^{2}\ln u \:e^{-u}\:du.\tag3 \end{align} Considering $(2)$ and $(3)$ gives the announced result.

• Cool way to solve it! Don't remove please, it's useful. I like to have more than 1 perspective! – Von Neumann Mar 12 '16 at 20:02

Start from the well-known integral

$$-\gamma=\int_0^\infty\exp(-x)\log x\,\mathrm dx$$

A round of integration by parts yields

\begin{align*} -\gamma&=\int_0^\infty\exp(-x)\log x\,\mathrm dx\\ &=\int_0^\infty x\exp(-x)\log x\,\mathrm dx-\int_0^\infty x\exp(-x)\,\mathrm dx\\ 1-\gamma&=\int_0^\infty x\exp(-x)\log x\,\mathrm dx \end{align*}

A second round gives

\begin{align*} 1-\gamma&=\int_0^\infty x\exp(-x)\log x\,\mathrm dx\\ &=\int_0^\infty x\exp(-x)(x-1)(\log x-1)\,\mathrm dx\\ &=\int_0^\infty x\exp(-x)\,\mathrm dx-\int_0^\infty x^2\exp(-x)\,\mathrm dx-\int_0^\infty x\exp(-x)\log x\,\mathrm dx+\int_0^\infty x^2\exp(-x)\log x\,\mathrm dx\\ 3-2\gamma&=\int_0^\infty x^2\exp(-x)\log x\,\mathrm dx \end{align*}

where the last integral is the one obtained by Dr. MV after an appropriate substitution. Thus, the original integral is equal to $12-8\gamma$.

• I like this one since it does not require appeal to special functions. And thank you for the reference. -Mark – Mark Viola Apr 24 '17 at 19:19
• When I saw this question, I immediately thought of that integral for $\gamma$, and experimented a bit to see if things would work out. I'm glad it did. – J. M. is a poor mathematician Apr 25 '17 at 1:55


# Math Help - Probibality problem, Pls help

1. ## Probibality problem, Pls help

A market research study is being conducted to determine if a product modification will be well received by the public. A total of 910 consumers are questioned regarding this product. The table below provides information regarding this sample.

_______Positive Reaction.....Neutral Reaction.....Negative Reaction
Male............190....................70......... .................110
Female..........210...................200......... ................130

(a) What is the probability that a randomly selected male would find this change unfavorable (negative)?
(b) What is the probability that a randomly selected person would be a female who had a positive reaction?
(c) If it is known that a person had a negative reaction to the study, what is the probability that the person is male?

2. Originally Posted by stoorrey
(a) What is the probability that a randomly selected male would find this change unfavorable (negative)?
= number on males with negative reaction divided by the total number of males

Originally Posted by stoorrey
(b) What is the probability that a randomly selected person would be a female who had a positive reaction?
= number of females with a positive reaction divided by the total number of people

Originally Posted by stoorrey
(c) If it is known that a person had a negative reaction to the study, what is the probability that the person is male?

This is conditional probabilty, let A be the person with a negative reaction and B be a male

You require $P(B/A) = \frac{P(A\cap B)}{P(A)}$

3. ## Thanks a lot

Thanks a lot for your help pickslides

but pls can you elaborate part c a bit more

i dont know how to use the formula.

4. $P(B/A) = \frac{P(A\cap B)}{P(A)}$

= (number of males with a negative reaction divided by total number of people in the survey) divided by (number of people (males + females) with a negative reaction divided by total number of people in the survey)

5. Originally Posted by pickslides
$P(B/A) = \frac{P(A\cap B)}{P(A)}$

= (number of males with a negative reaction divided by total number of people in the survey) divided by (number of people (males + females) with a negative reaction divided by total number of people in the survey)
Doesn't a condition alter the sample space? So shouldn't the answer be:

(number of males with a negative reaction)/(total number of people with a negative reaction)

The math works out the same way as in your solution, however your logic intrigues me.

6. You are correct to say the arithmetic will give the same solution. My explanation is from the definition in the equation supplied. Yours is a simplification knowing the number of total people surveyed will cancel out.

7. Hello, stoorrey!

pickslides is absolutely correct.
Vitruvian's approach to part (c) is also correct and more direct.

A market research study is being conducted to determine if a product modification
will be well received by the public.
A total of 980 consumers are questioned regarding this product.
The table below provides information regarding this sample.

$\begin{array}{c||c|c|c||c|}
& \text{Positive} & \text{Neutral} & \text{Negative} & \text{Total} \\ \hline \hline
\text{Male} & 190 & 70 & 110 & 370 \\ \hline
\text{Female} & 210 & 200 & 130 & 610 \\ \hline\hline\
\text{Total} & 400 & 340 & 240 & 980 \\ \hline \end{array}$
(a) What is the probability that a randomly selected male would have a negative reaction?
$P(\text{neg}\,|\,\text{male}) \;=\;\frac{n(\text{neg} \wedge \text{male})}{n(\text{male})} \;=\;\frac{110}{370} \;=\;\frac{11}{37}$

(b) What is the probability that a randomly selected person would be a female who had a positive reaction?
$P(\text{female}\wedge\text{pos}) \;=\;\frac{n(\text{female}\wedge\text{pos})} {n(\text{Total})} \;=\;\frac{210}{980} \;=\;\frac{3}{14}$

(c) If it is known that a person had a negative reaction to the study,
what is the probability that the person is male?
$P(\text{male}\,|\,\text{neg}) \;=\;\frac{n(\text{neg}\wedge\text{male})} {n(\text{neg})} \;=\;\frac{110}{240} \;=\;\frac{11}{24}$