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# Complement Rule for Probability

## If P(A) is is the probability that event A happens then the probability that event A doesn't happen is 1-P(A).

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Identify Overlapping, Disjoint, and Complementary Events

Source: https://www.flickr.com/photos/artform/3191659950/in/photolist-aqRPD2-dmAbcS-5S36iQ-4QrZxN-856iH9-wSiwW-6yrGH-ctWHA5-7MQkE-7KsGxA

Instead of staying home to watch cartoons on Saturday morning, Stella decided to go with her dad to his squash club. However, when she got there she found there wasn’t much to do but look at all the different squash players. So Stella decided to make a game of it. Stella has black hair and blue eyes, so she decided to count how many of the people she saw also had those characteristics. By the end of her dad’s game, Stella had counted 27 members in total. Of those 27, 19 had black hair, 14 had blue eyes, and 11 had both black hair and blue eyes. How can Stella tell if these events are disjoint or overlapping?

In this concept, you will learn to identify and distinguish between overlapping, disjoint and complementary events.

### Guidance

When you spin a spinner or roll a die to calculate a probability, some probabilities have events in common and some don’t. This is where you can begin to talk about identifying disjoint events. Disjoint events are events that don’t have any outcomes in common.

Take a look at this situation.

Consider spinning this spinner:

• Event \begin{align*}A\end{align*}: {yellow}
• Event \begin{align*}B\end{align*}: {blue}

Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are disjoint events because they have no outcomes in common - the arrow either lands on blue or yellow.

You can use a Venn diagram to show when events overlap and when they don’t overlap. A Venn diagram shows a way of representing whether events have any outcomes in common. It has round shapes that overlap or don’t overlap. The Venn diagram for disjoint events shows no overlap between the two events.

Not all events are disjoint. There are many events that are connected to each other.

Let’s look at another example.

Consider this spinner and the events \begin{align*}R\end{align*} (red) and \begin{align*}T\end{align*} (top).

• Event \begin{align*}R\end{align*}: {red-top, red-bottom }
• Event \begin{align*}T\end{align*}: {red-top, blue-top }

Clearly, both events share an outcome  red-top  so the two are called overlapping events. The Venn diagram for overlapping events shows that the two events overlap, or share 1 or more outcomes.

Complementary events are events whose probability sum adds up to 1.0 or 100 percent.

Events \begin{align*}Y\end{align*} and \begin{align*}G\end{align*} in this spinner above are complementary.

Complementary events are either-or events. Either the spinner above lands on green or it lands on yellow. There are no other outcomes.

### Guided Practice

Are event red and event blue complementary or disjoint events? Why?

First, remember the definitions of overlapping and complementary events.

Overlapping events are events that have outcomes in common.

Complementary events are events whose probability sum adds up to 1 or 100%.

Next, look at the spinner. 50% of the spinner is red and 50% is blue. There is no section on the spinner that has both red and blue.

The answer is complementary. Therefore, the two events are complementary.

### Examples

#### Example 1

For the spinner below, if event \begin{align*}A\end{align*} is bottom and event \begin{align*}B\end{align*} is blue, are the events disjoint or overlapping. Are the events complementary?

First, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Complementary events are events whose probability sum adds up to 1 or 100%.

Next, look at the spinner to determine what type of event is occurring.

Event \begin{align*}A\end{align*} is the sections with the word bottom.

Event \begin{align*}B\end{align*} includes the blue sections.

Then determine whether these events are complementary, disjoint or overlapping.

They are not complementary because the red - top is not included in the two events.

They are not disjoint as there is a blue - bottom.

Therefore the events are overlapping.

#### Example 2

If event \begin{align*}A\end{align*} includes the numbers 1, 6, 7, and 9 and event \begin{align*}B\end{align*} includes the numbers 2, 3, 4, 5, and 8. Is this an example of disjoint or overlapping events?

First, let’s list all of the numbers in Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

Event \begin{align*}A\end{align*}: 1, 6, 7, 9.

Event \begin{align*}B\end{align*}: 2, 3, 4, 5, 8.

Next, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.

The answer is that these events are disjoint as there are no numbers common to both events.

#### Example 3

If Event \begin{align*}A\end{align*} is the set of even numbers from 0 to 20 and Event \begin{align*}B\end{align*} is the set of numbers that are multiples of 4 and less than 30, Is this an example of disjoint or overlapping events?

First, let’s list all of the numbers in Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

Event \begin{align*}A\end{align*}: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Event \begin{align*}B\end{align*}: 4, 8, 12, 16, 20, 24, 28.

Next, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.

The answer is that these events are overlapping as there are numbers common to both events.

Credit: Michael Pedersen
Source: https://www.flickr.com/photos/46289172@N04/4429959240/in/photolist-aqRPD2-dmAbcS-5S36iQ-4QrZxN-856iH9-wSiwW-6yrGH-ctWHA5-7MQkE-7KsGxA

Remember bored Stella at the squash club?

Stella needs to determine if the events of black hair and blue eyes are disjoint or overlapping.

First, let’s list all of the numbers in Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

Event \begin{align*}A\end{align*}: black hair

Event \begin{align*}B\end{align*}: 4, blue eyes

Next, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.

Remember that there are 19 people with black hair and 11 also have blue eyes, therefore 8 have eyes of a different color.

Also, 14 people have blue eyes and 11 of these also have black hair, therefore 3 have hair of a different color.

Looking at the Venn diagram above, it is clear that these events are overlapping as they have outcomes in common.

### Explore More

Solve the problems. For overlapping events, tell which events overlap.

1. For a flip of a coin, are events \begin{align*}H\end{align*} (heads) and \begin{align*}T\end{align*} (tails) disjoint or overlapping?

2. For a flip of a coin, are events \begin{align*}H\end{align*} (heads) and \begin{align*}T\end{align*} (tails) complementary or non-complementary?

3. Why?

4. For a single toss of a number cube, are \begin{align*}E\end{align*} (even) and \begin{align*}T\end{align*} (3) disjoint events or overlapping events?

5. Why?

6. For a single toss of a number cube, are \begin{align*}E\end{align*} (even) and \begin{align*}S\end{align*} (6) disjoint events or overlapping events?

7. For a single toss of a number cube, are \begin{align*}G3\end{align*} (greater than 3) and \begin{align*}O\end{align*} (odd) disjoint events or overlapping events?

8. For a single toss of a number cube, are \begin{align*}E\end{align*} (even) and \begin{align*}O\end{align*} (odd) complementary events or non-complementary events?

9. For a single spin, are \begin{align*}B\end{align*} (blue) and \begin{align*}G\end{align*} (green) disjoint events or overlapping events?

10. For a single spin, are \begin{align*}G\end{align*} (green) and \begin{align*}L\end{align*} (left) disjoint events or overlapping events?

11. For a single spin, are \begin{align*}Y\end{align*} (yellow) and \begin{align*}R\end{align*} (red) complementary or non-complementary events?

12. For a single spin, are \begin{align*}R\end{align*} (right) and \begin{align*}L\end{align*} (left) complementary or non-complementary events?

13. For a light switch, are ON and OFF disjoint or overlapping events?

14. For a light switch, are ON and OFF complementary or non-complementary events?

15. For an oven, are ON and OFF disjoint or overlapping events?

### Vocabulary Language: English

complement

complement

A mutually exclusive pair of events are complements to each other. For example: If the desired outcome is heads on a flipped coin, the complement is tails.
Complementary Events

Complementary Events

Complementary events can occur in a single trial of a given experiment.
Complement rule

Complement rule

The Complement Rule states that the sum of the probabilities of an event and its complement must equal 1, or for the event A, P(A) + P(A') = 1.
Disjoint Events

Disjoint Events

Disjoint or mutually exclusive events cannot both occur in a single trial of a given experiment.
enumerate

enumerate

Enumerate means to catalogue or list members independently.
Venn diagrams

Venn diagrams

A diagram of overlapping circles that shows the relationship among members of different sets.

### Explore More

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# Five true or false questions on abstract algebra

Which of the following statements are true:
1. every countable group G has only countably many distinct subgroups.
2. any automorphism of the group $\mathbb{Q}$ under addition is of the form $x→qx$ for some $q\in\mathbb{Q}$
3. all non-trivial proper subgroups of $(\mathbb{R},+)$ are cyclic.
4. every infinite abelian group has at least one element of infinite order.
5. there is an element of order $51$ in the multiplicative group $(\mathbb{Z}/103\mathbb{Z})^*$

My thoughts:
1. true as union of uncountable number of countable set is uncountable
2. true as any homomorphism must be one of those form
3. false as $(\mathbb{Q},+)$ is not cyclic.
4. false example circle group.
5. true by Fermat's little theorem.

Are my guesses correct?

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5 is true, but I don't think Fermat is involved. 4 is false, but I don't know what the circle group is --- if its the group of rotations of the circle, it certainly has elements of infinite order. – Gerry Myerson Dec 10 '12 at 3:56
Your justification of (1) is weird, to say the least: what has that to do with the question asked?. (2) This is no mathematical justification at all. (3) Good. (4) Not good, as the circle group has lots of elements of infinite order. You can take the Prufer group, though. (5) Well, yes...but you can't apply Cauchy's Theorem directly as $\,51\,$ is not a prime – DonAntonio Dec 10 '12 at 4:22

(1) Let $G$ be the direct sum of copies of $\Bbb Z/2\Bbb Z$ indexed by $\Bbb N$. $G$ is countably infinite, but each subset $A\subseteq\Bbb N$ generates a distinct subgroup $G_A=\{x\in G:x_n=0\text{ for all }n\in\Bbb N\setminus A\}$ of $G$, and $\Bbb N$ has uncountably many subsets.

(4) The statement is indeed false; the group $G$ above is an infinite Abelian group in which every non-zero element has order $2$. However, the circle group is not a counterexample: $e^{i\theta}$ has infinite order iff $\frac{\theta}{2\pi}$ is irrational, so the circle group has uncountably many elements of infinite order and only countably many of finite order.

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does Q/Z provide a counterexample for (4)? – jim Dec 10 '12 at 10:58
@jim: Yes: $n\left(\frac{m}n+\Bbb Z\right)=0+\Bbb Z$. – Brian M. Scott Dec 10 '12 at 11:00

For part 5, note that $$|(\mathbb{Z}/103\mathbb{Z})^*| = \phi(103) = 102$$ and since $(\mathbb{Z}/103\mathbb{Z})^*$ is an abelian group we know that there exists a subgroup for every divisor of $102$. Hence we know that there exists some subgroup, $H$, such that $|H| = 51$. Now if you know that $(\mathbb{Z}/103\mathbb{Z})^*$ is actually cyclic, and since each subgroup of a cyclic group is also cyclic, there exists some $a \in (\mathbb{Z}/103\mathbb{Z})^*$ such that $\langle a \rangle = H$. Thus $|H| = |\langle a \rangle | = |a| = 51$.

-

## protected by user26857Dec 2 '15 at 22:35

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Start » Exercises » Engineering Mechanics I » Forces on a crank drive

# Forces on a crank drive

This exercise is about a statics investigation of the forces on a crank drive with a crosshead guidance for a specified rotation angle. The kinematics of a crank drive is considered here.

The force F acts on the piston of a steam engine. The following parameters have to be calculated:

a) the force in the driving rod,

c) the torque in the crankshaft.

## Solution

The following video is in german language.

### Sketch for the solution

$\require{cancel}$ $\newcommand{\myvec}[1]{{\begin{pmatrix}#1\end{pmatrix}}}$

The angle α between the force F and the driving rod results from

$\tag{1} \alpha = arctan \left( \frac{r}{l} \right)$

Between the force F and the force FS in the driving rod exists the following relationship

$\tag{2} F = F_S \cdot \cos \alpha$

$\tag{3} F_S = \frac{F}{\cos \alpha}$

The cosine of the arctangent α leads to

$\tag{4} \cos \left( arctan \left( \frac{r}{l} \right) \right) = \frac{1}{\sqrt{\left( \frac{r}{l}\right)^2 + 1}}$

So this is the force in the driving rod:

$\tag{5} F_S = F \cdot \sqrt{\left( \frac{r}{l}\right)^2 + 1}$

The transversal force in the crosshead guiding results from

$\tag{6} F_Q = F_S \cdot \sin \alpha$

The sine of the arctangent α leads to

$\tag{7} \sin \left( arctan \left( \frac{r}{l} \right) \right) = \frac{\frac{r}{l}}{\sqrt{\left( \frac{r}{l}\right)^2 + 1}}$

So the transversal force is

$\tag{8} F_Q = F \cdot \bcancel{\sqrt{\left( \frac{r}{l}\right)^2 + 1}} \cdot \frac{\frac{r}{l}}{\bcancel{\sqrt{\left( \frac{r}{l}\right)^2 + 1}}}$

$\tag{9} F_Q = F \cdot \frac{r}{l}$

The torque at the crankshaft is

$\tag{10} M_K = F_S \cdot r$

$\tag{11} M_K = F \cdot r \cdot \sqrt{\left( \frac{r}{l}\right)^2 + 1}$

This exercise is part of the collection Engineering Mechanics 1 -- Statics.


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# Proper time of accelerated clocks

1. Feb 18, 2008

### yuiop

In other threads the topic of how the proper time of a clock under acceleration has come up a number of times so I have decided to analyse this subject in a little more detail.

For this analysis we have these conditions:

a)Two clocks are initially at rest separated by a distance $L_o$ along the x axis.
b) Both clocks are simultaneously born-rigid accelerated along the x axis (maintaining constant proper separation).
c) The experiment happens in flat space.
d) When the rear clock reaches velocity v relative to the initial frame it stops accelerating and maintains constant velocity v.
e) When the front clock reaches velocity v relative to the initial frame it stops accelerating and maintains constant velocity.
f) In the final state both clocks have constant and equal velocity and are separated by the proper distance $L_o$

Referring to the attached diagram:

$a_F$ is the acceleration of the front clock proportional to $c^2/x_F$

$a_R$ is the acceleration of the rear clock proportional to $c^2/x_R$

$t_F$ is the coordinate time that the front clock clock takes to get to velocity v.

$t_R$ is the coordinate time that the rear clock clock takes to get to velocity v.

The dashed green line from the origin going through $T_{R1}$ and $T_F$ is a line of simultaneity in the accelerated clock frame.

The red lines are constant velocity phase after the acceleration phase.

When the front clock stops accelerating the proper time that has elapsed ($t_f$) is given by the formula:

(Eq 1) $$T_F = { c \over a_F} \sinh^{-1} \left({a_F t_F \over c} \right)$$

When the rear clock stops accelerating the proper time that has elapsed ($t_R$) is given by the formula:

(Eq 2) $$T_{R1} = { c \over a_R} \sinh^{-1} \left({a_R t_R \over c} \right)$$

The rear clock accelerates faster and gets to velocity v quicker than the front clock. The proper time that elapses from the coordinate time the rear clock stops accelerating to coordinate time that the front clock stops accelerating is simply the coordinate time difference adjusted by the Lorentz factor:

(Eq 3) $$T_{R2} - T_{R1} = (t_f - t_r)\sqrt{1-v^2/c^2}$$

The difference in proper time between the front clock and the rear clock after accelerating is given by

(Eq 4) = (Eq1) - (Eq 2) - (Eq 3)

The following equations are generally known:

(Eq 5) $$a=\sqrt{\left({c^4 \over x^2-c^2 t^2} \right)}$$

(Eq 6) $$v={a t \over \sqrt{\left(1+(a t /c)^2\right)}}$$

and the acceleration of the front clock in terms of acceleration of the rear clock can easily be derived as :

(Eq 7) $$a_F = {a_R c^2 \over (L_o a_R +c^2)$$

By substituting (Eq 5), (Eq 6) and (Eq 7) into (Eq 4) and solving the time difference between the front and rear clocks as seen in the initial frame is given by:

(Eq 8) $$T_F - T_{R2} = {-L_o v \over c^2} + {L_o \over c} \sinh^{-1} \left({ v \over c \sqrt{1-v^2/c^2}} \right)$$

When the clocks are synchronised again it is well known that:

(Eq 9) $$T_F - T_{R2} = {-L_o v \over c^2}$$

It is interesting to note that the difference in the proper times of the clocks after born-rigid acceleration and before synchronisation (Eq 8) is completely determined by the final velocity and the proper spatial separation and is independent of the rate of acceleration or time taken to transfer from one frame to another.

It is obvious from (Eq 8) that if constant proper separation is maintained the clocks will not remain naturally synchronised.

It is further conjectured that there is no acceleration scheme that can keep two spatially separated clocks synchronised when transferring the clocks from one inertial reference frame to another.*

I have left quite a few gaps in the derivation to try and keep it brief, but I will fill the gaps in, if anyone is really interested.

[EDIT] *I found a counter proof to this conjecture in post #9. (This does not change the validity of (Eq 8) in the above context of born-rigid transportation.

#### Attached Files:

• ###### acceleration.JPG
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Last edited: Feb 19, 2008
2. Feb 18, 2008

### meemoe_uk

Interesting. The result is counter-intuitive because of the asymmetry.
What about a simular situation where the clocks are alined in a perpendicular direction instead of in the direction of acel?
Any asymmetry in the final time frame there?

Has this equation been experimentally verified? Reference?

3. Feb 18, 2008

### 1effect

Very interesting, nice work.
How do you reconcile (eq3) with the fact that, as seen from your Minkowski diagram:

$$T_{R2} = { c \over a_R} \sinh^{-1} \left({a_R t_r \over c} \right)$$ ?

4. Feb 18, 2008

### yuiop

No, clocks exactly perpendicular to a clock on the x axis and syncronised with that clock, will remain syncronised with that clock when the acceleration occurs purely along the x axis.

I used equations from the following sources:

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)

http://www.mathpages.com/home/kmath422/kmath422.htm

The final result obtained in (Eq 8) is my own work. I am hoping the experts on this forum can confirm that the maths and logic is correct.

P.S. The diagram was created using a free geometry software program (called CaR) that can plot functions and evaluate expressions. That software at least confirms that (Eq 8) = (Eq 1) - (Eq 2) - (Eq 3)

For software that does not handle hyperbolic functions (Eq 8) can be expressed in the alternative form:

$$T_F - T_{R2} = {-L_o v \over c^2} + L_o c \log \left( \left({ v \over c \sqrt{1-v^2/c^2}} \right) + \sqrt{\left({ v \over c \sqrt{1-v^2/c^2}} \right)^2 +1 \right)$$

using the general relation

$$\sinh^{-1}(x) = \log\left(x+ \sqrt{x^2+1}\right)$$

where log is natural logarithm.

Last edited: Feb 18, 2008
5. Feb 18, 2008

### yuiop

The rear clock maintains constant velocity from $T_{R1}$ to $T_{R2}$ (the straight diagonal red line) so the hyperbolic motion equations are not required for that segment and the straight forward Lorentz transformation of the time inteval is sufficient to obtain (Eq 3).

The total proper time experienced by the front clock is given by (Eq 1)

The total proper time experienced by the rear clock is (Eq 2) + (Eq 3)

The rear clock has a combination of hyperbolic motion and constant velocity.

6. Feb 18, 2008

### 1effect

OK, I understand, you have some typos in the text and in the diagram that threw me off.
So, you are trying to calculate $$Ttotal_F-Ttotal_R$$, right?
But, in eq(8) you are calculating:

$$Ttotal_F-T_R_2$$, unless it is another typo?

7. Feb 18, 2008

### yuiop

Yes

$$Ttotal_F-Ttotal_R = (Eq 1) - (Eq2) - (Eq 3) = T_F - T_{R1} - (T_{R2}-T_{R1}) = T_F - T_{R1} - T_{R2} + T_{R1} = T_F - T_{R2}$$

So

$$Ttotal_F-Ttotal_R = Ttotal_F-T_R_2$$

8. Feb 18, 2008

### 1effect

kev,

Thank you for the previous clarifications. I hope that I am helping you cleaning up the post.

Now, this is somewhat misleading. The accelerations $$a_F$$ and $$a_R$$ do not appear in the final result because of the extra conditions that you set on them (measure the elapsed proper time difference when the clocks have achieved the same speed $$v$$ while maintaining the same proper separation $$L_0$$). This is why $$v$$ replaces $$a_F$$ and $$a_R$$ in the final formula of eq(8)

This is well known, indeed.

This is an artifact of the fact that the two clocks will describe different trajectories in space-time, so, when you try to calculate the total elapsed proper time via integration, you may get different results (I don't think proper time is a conservative field :-) )

Last edited: Feb 18, 2008
9. Feb 18, 2008

### yuiop

As always, constructive criticism is always welcome ;)

Accelerating while maintaining constant proper separation is the definition of born-rigid acceleration which is probably the most used acceleration method in discussions of accelerating rockets. It is admittedly an ideal situation, that is probably not often found in nature except as an approximation. The result that clocks go out of synchronisation under ideal born-rigid acceleration is not obvious to everyone.

As far as I can tell the equation I derived in post #1 is correct but unfortunately (for me) I have found a counter example to my own conjecture that I added to the end of the post. “It is further conjectured that there is no acceleration scheme that can keep two spatially separated clocks synchronised when transferring the clocks from one inertial reference frame to another. ( peh! :P) I have found that if we allow for the propagation delay of the acceleration impulse along the length of a rear engined rocket and allow the rocket to compress well beyond its natural length contracted length then it IS possible to find an acceleration pattern that keeps the clocks synchronised. This acceleration pattern allows the clock at the nose of the rocket to follow a longer path through space-time and time dilate to a lesser extent than the rear clock. In the attached diagram the front of the rocket does not start accelerating until coordinate time t2 which is later than the coordinate time t1 when the rear of rocket has finished accelerating. This is because of the minimum time for a signal travelling at light speed to propagate from the rear to the front. In practice this delay would probably be even longer if the impulse wave propagates through the material at the speed of sound.

#### Attached Files:

• ###### acceleration2.JPG
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Last edited: Feb 18, 2008
10. Feb 18, 2008

### 1effect

Thank you, I know very well what Born-rigid motion means. I was commenting about why $$a_R$$ and $$a_F$$ do not show in eq(8). Speaking of eq(8), I did not redo your calculations but the fraction $$\frac{L_0 v}{c^2}$$ looks very suspect since the result is made up of irrational expressions (square roots and sinh). The only way you may arrive to the rational term is if you did some approximations.

You may want to double-check eq (8), it may not be correct.

11. Feb 19, 2008

### chjoaygame

please clarify a few points for me

Hi Kev, Some readers are clued up, e.g. "

12. Feb 19, 2008

### yuiop

13. Feb 20, 2008

### yuiop

Irrational numbers are simply numbers that cannot be expressed as a ratio of whole natural numbers. I assume you actually meant transcendental expressions which sinh is but square is not. As far as the expression contained within the sinh function is concerned I only did straight substitutions of equivalent values into the function but otherwise performed no mathematical operations on it.

Square roots are not transcendental functions but they can present problems due to having real or imaginary roots. As long as c is greater than zero and v is less than c there are not too many problems in handling them here.

The expression $$\frac{L_0 v}{c^2}$$ comes directly from substitutions into

(Eq 3) $$T_{R2} - T_{R1} = (t_f - t_r)\sqrt{1-v^2/c^2}$$

which does not contain the transcendental sinh function.

By re-writing the rhs of (Eq 3) as

(Eq 11) $$\left(t_f\sqrt{1-v^2/c^2} - t_r\sqrt{1-v^2/c^2}\right)$$

and substituting $$t={v \over a\sqrt{1-v^2/c^2}$$ into (Eq 11) we get

(Eq 12) $${v \over a_F} - {v \over a_R}$$

then substituting $$a_F = {c^2a_R \over (L_0 a_R +c^2)}$$ into (Eq 12) we get

(Eq 13) $${v(L_0a_R+c^2) \over c^2a_r} - {v \over a_R} = {vL_0a_r+vc^2-vc^2 \over c^2a_R} = {L_0 v \over c^2}$$

Nothing too dodgy there I think.

Last edited: Feb 20, 2008
14. Feb 20, 2008

### 1effect

Thank you, this explains very well my point about the solution being independent of $$a_F$$ and $$a_R$$ due to imposing that:
-proper separation $$L_0$$ is maintained
-both clocks arrive to the same terminal speed $$v$$

Last edited: Feb 20, 2008
15. Feb 20, 2008

### chjoaygame

reply kev on Proper time of accelerated clocks

Hi Kev.

Thank you for the link to the article on Born-rigidity. It might even perhaps be said at one point to support Aristotle's much ridiculed notion about inertia! I will have to think about it.

You write (with a slight editorial tidy-up by me): #If the higher clock sent signals at one second intervals to the lower clock they would appear to be arriving more than once per second to an observer at the bottom referring to the lower clock.# This is the most concisely commonsensical and perspicuously meaningful plain English statement I have read about this. Thank you. Though this comment of yours has a clear meaning, the phenomenon it describes is still baffling. Would it concomitantly be the case that #If the lower clock sent signals at one second intervals to the higher clock they would appear to be arriving less than once per second to an observer at the top referring to the higher clock.#? I think so.

Presumably we are talking about light pulses. A light pulse can vary in its frequency, its wavelength, its speed, its duration, and its energy, its polarisation, and its coherence. Whence come the extra pulses, and whither go the missing ones? To or from the past, the future, a storage place part way up the tower, the very matter at the top or bottom of the tower? How do the pulses change on their way? If this went on for long enough, what would happen: would it mean that the light pulses on the way up were supplied by actually eroding the earth at the bottom and sending it up to the top where their energy was stored, till there was no more earth left at the bottom, and the tower got very very top heavy? Or that the top of the tower would be eroded by the need to supply energy for the downgoing pulses until there was no more top of the tower? Does conservation of mass-energy apply here? Does this fit with the notion that photons are particles that can be counted as they travel? Would the height of the tower gradually change because of all this? In answers to these questions, what is a priori or ex cathedra postulate, and what is observed or observable fact?

How about sending not light pulses, but rather atoms of ponderable matter, at speeds less than light speed? Same questions, except for different degrees of freedom for atoms.

Let's leave the clock design and construction problem to another day.

Sincerely,

Christopher

16. Feb 21, 2008

### yuiop

Hi Christopher.

Yes

Observable facts:

(a) The Pound Rebka Harvard Tower experiment. The frequency of waves sent up a tower was observer to be reduced by a detector at the top of the tower. Conversely signals sent down the tower were seen to increase in frequency.

(b) Atmospheric muon observations. The half life of muons is known to be too short for muons created at the top of the Earth's atmosphere to live long enough to make to it to sea level. However far too many muons are detected at sea level. This is accounted for by time dilation extending the effective life of the muons long enough for them to make the complete journey.

(c) Many particle laboratories routinely measure the half life of transient particles as extended by the time dilation factor when the particles are moving at relativistic speeds or when stored in cyclotrons that keep the particles accelerating in circles.

Time dilation is routinely observed in laboratories every day all over the world. I would be surprised if there was a single employed physicist alive today that doubted the validity of time dilation.

However, some would argue that experiment (a) does not directly show that the lower clock is running slower than the top clock. They say the wavelength of the photon is continually stretching as the photon climbs up the gravity well (and the frequency of the photon is slowing down and therefore the photon is losing energy). Where has this energy gone that the photon has lost? In classical physics an object thrown upwards loses kinetic energy and gains potential energy, and this potential energy is stored in the gravitational field. In general relativity there isn't a gravitational field that can store potential energy. A distant observer watching the photon climbing up the gravitational well would see the coordinate speed of the photon as apparently increasing so he could argue that the loss of the photon's energy as calculated by Energy=h(frequency) is compensated by the apparent increase in the photon's velocity. Energy=h(frequency) can expressed as Energy = hc/(wavelength). In this view the loss of energy due to increased wavelength of the photon, is compensating by the apparent increase in c as measured by the distant observer. However this is a very controversial view and it should always be stressed that c is always measured as 299792458 m/s by any local observer.

Personally, I have no doubt that atoms sent down the tower at a frequency of once per second (as measured by a clock at the top) will be seen to be arriving at intervals of less than one second, as measured by a clock at the base (and vice versa going the other way).

Others may differ.

P.S. A red light shone from the top of a sufficiently high tower would appear red to an observer at the base and a blue light shone from the base would appear red to an observer at the top of the tower. I think most people would agree with that.

Last edited: Feb 21, 2008
17. Feb 21, 2008

### 1effect

You mean the other way around, right? :-)

18. Feb 21, 2008

### yuiop

Quote from that link: "The effect is very small but measurable on Earth using the Mossbauer effect and was first observed in the Pound-Rebka experiment.[32] However, it is significant near a black hole, and as an object approaches the event horizon the red shift becomes infinite." Increasing red shift is synonomous with increasing wavelength which is the same thing as decreasing frequency.

Quote from this link http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html#c2

"the gain in energy for a photon which falls distance h is ..." gain in energy for a falling photon is the same as saying the frequency of the falling photon is increasing.

From this wikipedia article on gravitational redshift: http://en.wikipedia.org/wiki/Gravitational_redshift

" In physics, light or other forms of electromagnetic radiation of a certain wavelength originating from a source placed in a region of stronger gravitational field (and which could be said to have climbed "uphill" out of a gravity well) will be found to be of longer wavelength when received by an observer in a region of weaker gravitational field. "

Longer wavelength = reduced frequency.

So no, not the other way round.

19. Feb 21, 2008

### 1effect

"Likewise, gravitational blue shifts are associated with light emitted from a source residing within a weaker gravitational field (i.e. , top of the tower) observed within a stronger gravitational field (i.e. bottom of the tower), while gravitational redshifting implies the opposite conditions."

You are right, we are saying the same thing, just in a reverse order. My bad.

Last edited: Feb 21, 2008


text:
# parallelMap

0th

Percentile

##### Maps a function over lists or vectors in parallel.

Uses the parallelization mode and the other options specified in parallelStart.

Libraries and source file can be initialized on slaves with parallelLibrary and parallelSource.

Large objects can be separately exported via parallelExport, they can be simply used under their exported name in slave body code.

Regarding error handling, see the argument impute.error.

##### Usage
parallelMap(fun, ..., more.args = list(), simplify = FALSE,
use.names = FALSE, impute.error = NULL, level = NA_character_,
show.info = NA)
##### Arguments
fun

[function] Function to map over ....

...

[any] Arguments to vectorize over (list or vector).

more.args

[list] A list of other arguments passed to fun. Default is empty list.

simplify

[logical(1)] Should the result be simplified? See sapply. Default is FALSE.

use.names

[logical(1)] Should result be named by first vector if that is of class character? Default is FALSE.

impute.error

[NULL | function(x)] This argument can be used for improved error handling. NULL means that, if an exception is generated on one of the slaves, it is also thrown on the master. Usually all slave jobs will have to terminate until this exception on the master can be thrown. If you pass a constant value or a function, all jobs are guaranteed to return a result object, without generating an exception on the master for slave errors. In case of an error, this is a simpleError object containing the error message. If you passed a constant object, the error-objects will be substituted with this object. If you passed a function, it will be used to operate on these error-objects (it will ONLY be applied to the error results). For example, using identity would keep and return the simpleError-object, or function(x) 99 would impute a constant value (which could be achieved more easily by simply passing 99). Default is NULL.

level

[character(1)] If a (non-missing) level is specified in parallelStart, this call is only parallelized if the level specified here matches. Useful if this function is used in a package. Default is NA.

show.info

[logical(1)] Verbose output on console? Can be used to override setting from options / parallelStart. Default is NA which means no overriding.

Result.

• parallelMap
##### Examples
# NOT RUN {
parallelStart()
parallelMap(identity, 1:2)
parallelStop()
# }

Documentation reproduced from package parallelMap, version 1.4, License: BSD_2_clause + file LICENSE

### Community examples

Looks like there are no examples yet.


text:
User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:22:15Z http://mathoverflow.net/feeds/user/23964 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126850/gram-series-for-more-general-integrals Gram series for more general integrals unknown (yahoo) 2013-04-08T13:06:27Z 2013-04-08T13:06:27Z
given the Gram series

$$\pi (x) \sim \sum_{n=1}^{\infty} \frac{1}{n}\frac{log^{n}(x)}{n!\zeta (n+1)}$$

can this result be generalized to more integral transforms ?

$$g(x)= \int_{0}^{\infty}dyK(xy)f(y)dy$$

for exapmle in Gramm series for the prime counting function the integral equation is

$$log\zeta(s)= s\int_{0}^{\infty}\frac{1}{e^{xy}-1}\pi(e^{y})dy$$
 http://mathoverflow.net/questions/126839/numerical-evaluation-of-a-triple-integral-can-be-made Numerical evaluation of a triple integral can be made ? unknown (yahoo) 2013-04-08T11:14:56Z 2013-04-08T11:14:56Z
let be the integral

$$\int_{0}^{\infty}dx\int_{0}^{\infty}dy\int_{0}^{\infty}dz \frac{f(x,y,z)}{(1+x^{2}+y^{2}+z^{2})^{s}}$$

here $s$ is a parameter so the integral converges

now let us make the change to polar coordinates , however the function $f(x,y,z)$ will not be invariant under rotations so there will be an extra integration over the angular variables

$$\int_{\Omega} d \Omega \int_{0}^{\infty}r^{2}f(r, \Omega )(1+r^{2})^{-s}$$

my question is can we ALWAYS integrate with Numerical (MOnte carlo method) inside the angular variables $\int_{\Omega}d \Omega$ so we are left only with a sum of one dimensional integrals ??

$$\sum_{i} \int_{0}^{\infty}dr f(r, \Omega _{i})r^{2}(1+r^{2})^{-s}.$$
 http://mathoverflow.net/questions/116690/bessel-function-with-complex-argument-and-index Bessel function with complex argument and index unknown (yahoo) 2012-12-18T10:03:32Z 2012-12-18T10:03:32Z
let be a Bessel function $J_{u} (x)$

if both the index 'u' and the argument 'x' are complex

$$J_{ia}(ib)$$

for real 'a' and 'b' what is then the name for this function ??

if only the index is complex $J_{ia}(b)$ or the argumentis complex $J_{a}(ib)$

what is the name of this function ??
 http://mathoverflow.net/questions/103926/how-can-i-solve-a-boundary-value-numerically-on-an-infinite-interval how can i solve a boundary value numerically on an infinite interval ?? unknown (yahoo) 2012-08-04T09:36:31Z 2012-08-05T21:08:08Z
let be the differential equation $-y''(x)+x^{4}y(x)-E_{n}y(x)=0$ with the boundary conditions $y(0)=0=y(\infty)$

how could i use the shooting method or other numerical method to solve this equation ? , my only idea is to set $R=10000$ for example and solve $y(0)=0=y(R)$

of course i also could made the substitution $u= \frac{x}{x-1}$ so the new boundary conditions could become $y(0)=0=y(1)$ but know the differential equation would be singular at the point $u=1$
 http://mathoverflow.net/questions/127122/taylor-series-in-lgox-for-the-square-root Comment by 2013-04-11T11:37:13Z 2013-04-11T11:37:13Z oh sorry tehn Ben if possible erase teh questio please :) http://mathoverflow.net/questions/126839/numerical-evaluation-of-a-triple-integral-can-be-made Comment by 2013-04-10T11:00:13Z 2013-04-10T11:00:13Z my idea is to change to polar coordinates :) and then to apply numerical integration ONLY to the angular variables so we are left with a set of ONE dimensional integral in the form $\int_{0}^{\infty}drr^{n-1}f(r, \Omega _{i} )$ http://mathoverflow.net/questions/126839/numerical-evaluation-of-a-triple-integral-can-be-made Comment by 2013-04-08T19:01:55Z 2013-04-08T19:01:55Z my question was about if my numerical method is correct a) change to polar coordinates b) numerical integration ONLY over the angular variables. http://mathoverflow.net/questions/120017/why-mathematicians-do-not-accept-the-wu-sprung-semiclassical-model-as-a-solution Comment by 2013-01-27T13:47:03Z 2013-01-27T13:47:03Z mathdl.maa.org/images/upload_library/22/Ford/… inverse problems in physics.. see form equation (6.1) we can get the inverse of the potentia $f^{-1}(x)$ from the eigenvalue staircase in this case the staricase is the Riemann zeta function http://mathoverflow.net/questions/120017/why-mathematicians-do-not-accept-the-wu-sprung-semiclassical-model-as-a-solution Comment by 2013-01-27T13:41:02Z 2013-01-27T13:41:02Z Hua Wu and D. W. L. Sprung, "Riemann zeta and a fractal potential", Physical Review E 48 (1993) 2595. Fractal fits to Riemann zeros P B Slater, Canadian Journal of Physics, 2007, 85(4): 345–357, 10.1139/p07-050 http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/112070#112070 Comment by 2012-11-12T13:06:15Z 2012-11-12T13:06:15Z as far as i know an operator of the form $-D^{2}+f(x)$ has only Real Eigenvalues doesn't it ? , of course as a phsyicist i could be wrong :D , look on the web WU-SPRUNG potential PHYSICS OF THE RIEMANN HYPOTHESIS and similar questions or simply use a computer to get the eigenvalues to see i am right http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/112070#112070 Comment by 2012-11-11T14:48:38Z 2012-11-11T14:48:38Z the idea is that using WKB method , or semiclassical theory , look at this en.wikipedia.org/wiki/Old_quantum_theory one could get a Hamiltonian with a density of states given by the Riemann Weil formula and whose energies are the Riemann Zeros the potential is given implicitly of course http://mathoverflow.net/questions/111809/how-to-deal-with-divergent-fourier-series Comment by 2012-11-08T16:07:33Z 2012-11-08T16:07:33Z given any 'x' positive i would like to find a method to sum the divergent series with $a=1/2$ or with a a real number on the interval $(0,1)$


text:
Let the set $$\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$$ Susan makes a list as follows: for each two-element subset of $$\mathcal{S},$$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

(第二十一届AIME1 2003 第3题)


text:
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## G = C2×S3≀C2order 144 = 24·32

### Direct product of C2 and S3≀C2

Aliases: C2×S3≀C2, C3⋊S3⋊D4, (C3×C6)⋊D4, S32⋊C22, C32⋊(C2×D4), C32⋊C4⋊C22, C3⋊S3.1C23, (C2×S32)⋊5C2, (C2×C32⋊C4)⋊3C2, (C2×C3⋊S3).6C22, SmallGroup(144,186)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C2×S3≀C2
 Chief series C1 — C32 — C3⋊S3 — S32 — S3≀C2 — C2×S3≀C2
 Lower central C32 — C3⋊S3 — C2×S3≀C2
 Upper central C1 — C2

Generators and relations for C2×S3≀C2
G = < a,b,c,d,e | a2=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 414 in 86 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C2 [×6], C3 [×2], C4 [×2], C22 [×9], S3 [×8], C6 [×6], C2×C4, D4 [×4], C23 [×2], C32, D6 [×12], C2×C6 [×2], C2×D4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C22×S3 [×2], C32⋊C4 [×2], S32 [×4], S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3≀C2 [×4], C2×C32⋊C4, C2×S32 [×2], C2×S3≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, S3≀C2, C2×S3≀C2

Character table of C2×S3≀C2

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F size 1 1 6 6 6 6 9 9 4 4 18 18 4 4 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 linear of order 2 ρ3 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 1 linear of order 2 ρ4 1 -1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 linear of order 2 ρ8 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ9 2 2 0 0 0 0 -2 -2 2 2 0 0 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 0 0 0 0 -2 2 2 2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 0 2 0 2 0 0 -2 1 0 0 1 -2 -1 0 0 -1 orthogonal lifted from S3≀C2 ρ12 4 4 -2 0 -2 0 0 0 1 -2 0 0 -2 1 0 1 1 0 orthogonal lifted from S3≀C2 ρ13 4 -4 0 2 0 -2 0 0 -2 1 0 0 -1 2 -1 0 0 1 orthogonal faithful ρ14 4 -4 -2 0 2 0 0 0 1 -2 0 0 2 -1 0 1 -1 0 orthogonal faithful ρ15 4 -4 0 -2 0 2 0 0 -2 1 0 0 -1 2 1 0 0 -1 orthogonal faithful ρ16 4 -4 2 0 -2 0 0 0 1 -2 0 0 2 -1 0 -1 1 0 orthogonal faithful ρ17 4 4 2 0 2 0 0 0 1 -2 0 0 -2 1 0 -1 -1 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 -2 0 -2 0 0 -2 1 0 0 1 -2 1 0 0 1 orthogonal lifted from S3≀C2

Permutation representations of C2×S3≀C2
On 12 points - transitive group 12T77
Generators in S12
(1 4)(2 3)(5 9)(6 10)(7 11)(8 12)
(1 6 8)(2 7 5)(3 11 9)(4 10 12)
(1 6 8)(2 5 7)(3 9 11)(4 10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 10)(6 9)(7 12)(8 11)

G:=sub<Sym(12)| (1,4)(2,3)(5,9)(6,10)(7,11)(8,12), (1,6,8)(2,7,5)(3,11,9)(4,10,12), (1,6,8)(2,5,7)(3,9,11)(4,10,12), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,10)(6,9)(7,12)(8,11)>;

G:=Group( (1,4)(2,3)(5,9)(6,10)(7,11)(8,12), (1,6,8)(2,7,5)(3,11,9)(4,10,12), (1,6,8)(2,5,7)(3,9,11)(4,10,12), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,10)(6,9)(7,12)(8,11) );

G=PermutationGroup([(1,4),(2,3),(5,9),(6,10),(7,11),(8,12)], [(1,6,8),(2,7,5),(3,11,9),(4,10,12)], [(1,6,8),(2,5,7),(3,9,11),(4,10,12)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(5,10),(6,9),(7,12),(8,11)])

G:=TransitiveGroup(12,77);

On 12 points - transitive group 12T78
Generators in S12
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)
(1 9 8)(3 6 11)
(2 5 10)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 12)(6 11)(7 10)(8 9)

G:=sub<Sym(12)| (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (1,9,8)(3,6,11), (2,5,10)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,12)(6,11)(7,10)(8,9)>;

G:=Group( (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (1,9,8)(3,6,11), (2,5,10)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,12)(6,11)(7,10)(8,9) );

G=PermutationGroup([(1,3),(2,4),(5,12),(6,9),(7,10),(8,11)], [(1,9,8),(3,6,11)], [(2,5,10),(4,12,7)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,12),(6,11),(7,10),(8,9)])

G:=TransitiveGroup(12,78);

On 18 points - transitive group 18T63
Generators in S18
(1 2)(3 16)(4 17)(5 18)(6 15)(7 14)(8 11)(9 12)(10 13)
(1 4 6)(2 17 15)(3 14 13)(5 11 12)(7 10 16)(8 9 18)
(1 3 5)(2 16 18)(4 14 11)(6 13 12)(7 8 17)(9 15 10)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(4 6)(7 10)(8 9)(11 12)(13 14)(15 17)

G:=sub<Sym(18)| (1,2)(3,16)(4,17)(5,18)(6,15)(7,14)(8,11)(9,12)(10,13), (1,4,6)(2,17,15)(3,14,13)(5,11,12)(7,10,16)(8,9,18), (1,3,5)(2,16,18)(4,14,11)(6,13,12)(7,8,17)(9,15,10), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (4,6)(7,10)(8,9)(11,12)(13,14)(15,17)>;

G:=Group( (1,2)(3,16)(4,17)(5,18)(6,15)(7,14)(8,11)(9,12)(10,13), (1,4,6)(2,17,15)(3,14,13)(5,11,12)(7,10,16)(8,9,18), (1,3,5)(2,16,18)(4,14,11)(6,13,12)(7,8,17)(9,15,10), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (4,6)(7,10)(8,9)(11,12)(13,14)(15,17) );

G=PermutationGroup([(1,2),(3,16),(4,17),(5,18),(6,15),(7,14),(8,11),(9,12),(10,13)], [(1,4,6),(2,17,15),(3,14,13),(5,11,12),(7,10,16),(8,9,18)], [(1,3,5),(2,16,18),(4,14,11),(6,13,12),(7,8,17),(9,15,10)], [(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(4,6),(7,10),(8,9),(11,12),(13,14),(15,17)])

G:=TransitiveGroup(18,63);

On 24 points - transitive group 24T261
Generators in S24
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 17)(14 18)(15 19)(16 20)
(1 19 6)(2 20 7)(3 8 17)(4 5 18)(9 23 13)(10 24 14)(11 15 21)(12 16 22)
(1 19 6)(2 7 20)(3 8 17)(4 18 5)(9 23 13)(10 14 24)(11 15 21)(12 22 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)

G:=sub<Sym(24)| (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,17)(14,18)(15,19)(16,20), (1,19,6)(2,20,7)(3,8,17)(4,5,18)(9,23,13)(10,24,14)(11,15,21)(12,16,22), (1,19,6)(2,7,20)(3,8,17)(4,18,5)(9,23,13)(10,14,24)(11,15,21)(12,22,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)>;

G:=Group( (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,17)(14,18)(15,19)(16,20), (1,19,6)(2,20,7)(3,8,17)(4,5,18)(9,23,13)(10,24,14)(11,15,21)(12,16,22), (1,19,6)(2,7,20)(3,8,17)(4,18,5)(9,23,13)(10,14,24)(11,15,21)(12,22,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23) );

G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,24),(6,21),(7,22),(8,23),(13,17),(14,18),(15,19),(16,20)], [(1,19,6),(2,20,7),(3,8,17),(4,5,18),(9,23,13),(10,24,14),(11,15,21),(12,16,22)], [(1,19,6),(2,7,20),(3,8,17),(4,18,5),(9,23,13),(10,14,24),(11,15,21),(12,22,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23)])

G:=TransitiveGroup(24,261);

On 24 points - transitive group 24T262
Generators in S24
(1 18)(2 19)(3 20)(4 17)(5 13)(6 14)(7 15)(8 16)(9 22)(10 23)(11 24)(12 21)
(1 7 12)(2 9 8)(3 10 5)(4 6 11)(13 20 23)(14 24 17)(15 21 18)(16 19 22)
(1 12 7)(2 9 8)(3 5 10)(4 6 11)(13 23 20)(14 24 17)(15 18 21)(16 19 22)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 22)(14 21)(15 24)(16 23)(17 18)(19 20)

G:=sub<Sym(24)| (1,18)(2,19)(3,20)(4,17)(5,13)(6,14)(7,15)(8,16)(9,22)(10,23)(11,24)(12,21), (1,7,12)(2,9,8)(3,10,5)(4,6,11)(13,20,23)(14,24,17)(15,21,18)(16,19,22), (1,12,7)(2,9,8)(3,5,10)(4,6,11)(13,23,20)(14,24,17)(15,18,21)(16,19,22), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,22)(14,21)(15,24)(16,23)(17,18)(19,20)>;

G:=Group( (1,18)(2,19)(3,20)(4,17)(5,13)(6,14)(7,15)(8,16)(9,22)(10,23)(11,24)(12,21), (1,7,12)(2,9,8)(3,10,5)(4,6,11)(13,20,23)(14,24,17)(15,21,18)(16,19,22), (1,12,7)(2,9,8)(3,5,10)(4,6,11)(13,23,20)(14,24,17)(15,18,21)(16,19,22), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,22)(14,21)(15,24)(16,23)(17,18)(19,20) );

G=PermutationGroup([(1,18),(2,19),(3,20),(4,17),(5,13),(6,14),(7,15),(8,16),(9,22),(10,23),(11,24),(12,21)], [(1,7,12),(2,9,8),(3,10,5),(4,6,11),(13,20,23),(14,24,17),(15,21,18),(16,19,22)], [(1,12,7),(2,9,8),(3,5,10),(4,6,11),(13,23,20),(14,24,17),(15,18,21),(16,19,22)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,9),(6,12),(7,11),(8,10),(13,22),(14,21),(15,24),(16,23),(17,18),(19,20)])

G:=TransitiveGroup(24,262);

On 24 points - transitive group 24T263
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 16)(10 13)(11 14)(12 15)(17 23)(18 24)(19 21)(20 22)
(1 10 12)(2 11 9)(3 18 20)(4 17 19)(5 13 15)(6 14 16)(7 24 22)(8 23 21)
(1 10 12)(2 9 11)(3 20 18)(4 17 19)(5 13 15)(6 16 14)(7 22 24)(8 23 21)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3)(2 4)(5 7)(6 8)(9 17)(10 20)(11 19)(12 18)(13 22)(14 21)(15 24)(16 23)

G:=sub<Sym(24)| (1,5)(2,6)(3,7)(4,8)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (1,10,12)(2,11,9)(3,18,20)(4,17,19)(5,13,15)(6,14,16)(7,24,22)(8,23,21), (1,10,12)(2,9,11)(3,20,18)(4,17,19)(5,13,15)(6,16,14)(7,22,24)(8,23,21), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(2,4)(5,7)(6,8)(9,17)(10,20)(11,19)(12,18)(13,22)(14,21)(15,24)(16,23)>;

G:=Group( (1,5)(2,6)(3,7)(4,8)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (1,10,12)(2,11,9)(3,18,20)(4,17,19)(5,13,15)(6,14,16)(7,24,22)(8,23,21), (1,10,12)(2,9,11)(3,20,18)(4,17,19)(5,13,15)(6,16,14)(7,22,24)(8,23,21), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(2,4)(5,7)(6,8)(9,17)(10,20)(11,19)(12,18)(13,22)(14,21)(15,24)(16,23) );

G=PermutationGroup([(1,5),(2,6),(3,7),(4,8),(9,16),(10,13),(11,14),(12,15),(17,23),(18,24),(19,21),(20,22)], [(1,10,12),(2,11,9),(3,18,20),(4,17,19),(5,13,15),(6,14,16),(7,24,22),(8,23,21)], [(1,10,12),(2,9,11),(3,20,18),(4,17,19),(5,13,15),(6,16,14),(7,22,24),(8,23,21)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3),(2,4),(5,7),(6,8),(9,17),(10,20),(11,19),(12,18),(13,22),(14,21),(15,24),(16,23)])

G:=TransitiveGroup(24,263);

On 24 points - transitive group 24T264
Generators in S24
(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)
(2 22 17)(4 19 24)(5 14 10)(7 12 16)
(1 21 20)(3 18 23)(6 11 15)(8 13 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

G:=sub<Sym(24)| (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,24)(18,21)(19,22)(20,23), (2,22,17)(4,19,24)(5,14,10)(7,12,16), (1,21,20)(3,18,23)(6,11,15)(8,13,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,24)(18,21)(19,22)(20,23), (2,22,17)(4,19,24)(5,14,10)(7,12,16), (1,21,20)(3,18,23)(6,11,15)(8,13,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );

G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,15),(10,16),(11,13),(12,14),(17,24),(18,21),(19,22),(20,23)], [(2,22,17),(4,19,24),(5,14,10),(7,12,16)], [(1,21,20),(3,18,23),(6,11,15),(8,13,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)])

G:=TransitiveGroup(24,264);

C2×S3≀C2 is a maximal subgroup of
C2.AΓL1(𝔽9)  S32⋊D4  C4⋊S3≀C2  D6≀C2  C62⋊D4
C2×S3≀C2 is a maximal quotient of
S32⋊Q8  C4.4S3≀C2  C32⋊C4⋊Q8  C32⋊D85C2  C32⋊D8⋊C2  C3⋊S3⋊D8  C32⋊Q16⋊C2  C3⋊S32SD16  C3⋊S3⋊Q16  S32⋊D4  C4⋊S3≀C2  C62.9D4  C62.12D4  C62.13D4  C62.15D4  D6≀C2  C62⋊D4

Polynomial with Galois group C2×S3≀C2 over ℚ
actionf(x)Disc(f)
12T77x12-2x10-x8-2x6+5x4-6x2+4226·56·114·5092
12T78x12-12x10+54x8-104x6+57x4+36x2-16-256·312·56

Matrix representation of C2×S3≀C2 in GL4(ℤ) generated by

 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1
,
 0 1 0 0 -1 -1 0 0 0 0 -1 -1 0 0 1 0
,
 -1 -1 0 0 1 0 0 0 0 0 -1 -1 0 0 1 0
,
 0 0 1 0 0 0 -1 -1 -1 0 0 0 0 -1 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,-1,0,0,1,-1,0,0,0,0,-1,1,0,0,-1,0],[-1,1,0,0,-1,0,0,0,0,0,-1,1,0,0,-1,0],[0,0,-1,0,0,0,0,-1,1,-1,0,0,0,-1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C2×S3≀C2 in GAP, Magma, Sage, TeX

C_2\times S_3\wr C_2
% in TeX

G:=Group("C2xS3wrC2");
// GroupNames label

G:=SmallGroup(144,186);
// by ID

G=gap.SmallGroup(144,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,121,964,730,142,299,455]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

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text:
# zbMATH — the first resource for mathematics

## Proceedings of the Royal Society of Edinburgh

 Short Title: Proc. R. Soc. Edinburgh Publisher: Oliver & Boyd, Edinburgh ISSN: 0370-1646 Successor: Proceedings of the Royal Society of Edinburgh. Section B. BiologyProceedings of the Royal Society of Edinburgh. Section A. Mathematics Comments: No longer indexed
 Documents Indexed: 651 Publications (1870–1940)
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#### Latest Issues

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#### Authors

 102 Muir, Thomas 89 Tait, Peter Guthrie 28 Thomson, William (Lord Kelvin) 27 Aitken, Alexander Craig 19 Turnbull, Herbert Westren 17 Sang, Edward 16 Ince, Edward Lindsay 14 MacRobert, Thomas Murray 14 Peddie, W. 11 Kermack, William Ogilvie 10 Whittaker, Edmund Taylor 9 Anglin, A. H. H. 9 Fuchs, Klaus 9 Sandeman, Ian 8 Knott, Cargill Gilston 8 McKendrick, Anderson Gray 8 Metzler, William Henry 8 Ruse, Harold Stanley 7 Born, Max 7 Brownlee, J. 7 McLaren, John 6 Cayley, Arthur 6 Erdélyi, Arthur 6 McCrea, William Hunter 6 Sommerville, Duncan M’Laren Young 5 Chrystal, George 5 Copson, Edward Thomas 5 Etherington, I. M. H. 5 Green, George 5 Plarr, Gustav 4 Ferrar, W. L. 4 Hitchcock, Frank Lauren 4 Hogben, Lancelot Thomas 4 Jackson, Frank Hilton 4 Kirkman, Thomas Penyngton 4 Ledermann, Walter 4 MacFarlane, Alexander 4 Meksyn, David 4 Milne-Thomson, Louis Melville 4 Milne, J. R. 4 Thomson, Joseph John 4 Wedderburn, Joseph 4 Young, William Henry 3 Gray, Andrew 3 Humbert, Pierre 3 Lawley, Derrick Norman 3 Levy, Hyman 3 Nanson, Edward John 3 Ritchie, J. B. 3 Whipple, Francis John Welsh 3 Williamson, John 2 Bartlett, Maurice Stevenson 2 Blackwell, A. 2 Brown, Crum 2 Collar, A. R. 2 Forsyth, Andrew Russell 2 Geronimus, Yakov Lazarevich 2 Gonin, Henri Theodore 2 Gray, James Gordon 2 Houstoun, R. A. 2 Innes, Robert Thorburn Ayton 2 Lamb, Horace 2 Lefort, J. 2 Meyer, Wilhelm Franz 2 Oppenheim, Alexander 2 Saddler, W. 2 Smith, W. R. 2 Spain, Barry 2 Tweedie, Charles 2 Walker, Arthur Geoffrey 1 Aitken, C. A. 1 Allan, F. E. 1 Allardice, Robert Edgar 1 Anderson, E. M. 1 B., T. H. 1 Barrett, J. A. S. 1 Bateman, Harry 1 Bauschinger, T. 1 Becker, Ludwig 1 Bell, H. 1 Black, J. S. 1 Black, Th. P. 1 Briggs, Henry 1 Burgess, J. 1 Burnside, William 1 Carothers, S. D. 1 Colson, C. G. 1 Conrady, A. E. 1 Crawford, W. J. 1 Cremona, Luigi 1 Crum Brown, Alexander 1 Dalton, John P. 1 Dirac, Paul Adrien Maurice 1 E. M. H. 1 Eldridge, J. A. 1 Elliott, A. C. 1 Erskine Murray, J. 1 Everett, Joseph David 1 Ewing, A. 1 Filon, Louis Napoleon George ...and 57 more Authors
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#### Fields

 13 Manifolds and cell complexes (57-XX) 8 Number theory (11-XX) 7 Geometry (51-XX) 6 Combinatorics (05-XX) 6 Linear and multilinear algebra; matrix theory (15-XX) 6 Real functions (26-XX) 5 Differential geometry (53-XX) 4 Algebraic geometry (14-XX) 4 Nonassociative rings and algebras (17-XX) 3 Mathematical logic and foundations (03-XX) 3 Ordinary differential equations (34-XX) 3 Probability theory and stochastic processes (60-XX) 2 Field theory and polynomials (12-XX) 2 Special functions (33-XX) 2 Sequences, series, summability (40-XX) 2 Fluid mechanics (76-XX) 2 Biology and other natural sciences (92-XX) 1 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Associative rings and algebras (16-XX) 1 Functions of a complex variable (30-XX) 1 Difference and functional equations (39-XX) 1 Integral transforms, operational calculus (44-XX) 1 Integral equations (45-XX) 1 Convex and discrete geometry (52-XX) 1 Algebraic topology (55-XX) 1 Numerical analysis (65-XX) 1 Mechanics of particles and systems (70-XX) 1 Mechanics of deformable solids (74-XX)

#### Citations contained in zbMATH Open

164 Publications have been cited 1,250 times in 1,205 Documents Cited by Year
On a quadrature formula for trigonometric integrals. JFM 55.0946.02
Filon, L. N. G.
1929
On Bernoulli’s numerical solution of algebraic equations. JFM 52.0098.05
Aitken, A. C.
1926
On least squares and linear combination of observations. Zbl 0011.26603
Aitken, A. C.
1935
The gravitational field of a distribution of particles rotating about an axis of symmetry. Zbl 0016.28302
van Stockum, W. J.
1937
The estimation of factor loadings by the method of maximum likelihood. Zbl 0027.23503
Lawley, D. N.
1940
Genetic algebras. Zbl 0027.29402
Etherington, I. M. H.
1939
On the functions which are represented by the expansions of the interpolation-theory. JFM 45.1275.02
Whittaker, E. T.
1914
Further investigations into the periodic Lame functions. Zbl 0027.21201
Ince, E. L.
1940
Random paths in two and three dimensions. Zbl 0027.33903
McCrea, W. H.; Whipple, F. J. W.
1940
On the colouring of maps. JFM 12.0408.02
Tait, P. G.
1880
Remarks on the preceeding communication. JFM 12.0409.02
Tait, P. G.
1880
Genetic algebras. JFM 66.1209.01
Etherington, I. M. H.
1939
The distribution of gene ratios for rare mutations. JFM 56.1107.09
Fisher, R. A.
1930
The periodic Lame functions. Zbl 0027.21103
Ince, E. L.
1940
On the theory of statistical regression. Zbl 0008.02402
Bartlett, M. S.
1933
On fourfold sampling with and without replacement. Zbl 0013.21405
Aitken, A. C.; Gonin, H. T.
1935
On non-associative combinations. Zbl 0021.29404
Etherington, I. M. H.
1939
Studies in practical mathematics. II. The evaluation of the latent roots and latent vectors of a matrix. Zbl 0017.14701
Aitken, A. C.
1937
The confluent hypergeometric functions of two variables. JFM 47.0930.03
Humbert, P.
1919
On least squares and linear combinations of observations. JFM 61.1300.05
Aitken, A. C.
1935
The rate of multiplication of microorganisms: A mathematical study. JFM 43.0304.02
McKendrick, A. G.; Pai, M. K.
1912
On the geometry of Dirac’s equations and their expression in tensor form. Zbl 0016.18104
Ruse, H. S.
1937
Relative co-ordinates. Zbl 0006.03002
Walker, A. G.
1932
Genetic algebras. JFM 65.1138.05
Etherington, I. M. H.
1939
A series formula for the roots of algebraic and transcendental equations. JFM 51.0096.03
Aitken, C. A.
1925
On the adelphic integral of the differential equations of dynamics. JFM 46.1173.02
Whittaker, E. T.
1916
Tables of the elliptic-cylinder functions. Zbl 0005.29503
Ince, E. L.
1932
The relation between mathematics and physics. Zbl 0021.18504
Dirac, P. A. M.
1939
Harmonic Riemannian spaces. Zbl 0027.26001
Copson, E. T.; Ruse, H. S.
1940
The theory of recurrent determinants in the historical order of development up to 1860. JFM 42.0169.01
Muir, Th.
1911
Studies in practical mathematics. I. The evaluation, with applications, of a certain triple product matrix. Zbl 0016.24102
Aitken, A. C.
1937
Division of space by congruent triangles and tetrahedra. JFM 49.0411.03
Sommerville, D. M. Y.
1923
Further numerical studies in algebraic equations and matrices. Zbl 0002.00703
Aitken, A. C.
1931
On the reciprocation of certain matrices. Zbl 0022.25801
Collar, A. R.
1939
On a problem concerning matrices with variable diagonal elements. JFM 66.0042.01
Ledermann, W.
1940
The periodic Lamé functions. JFM 66.1242.03
Ince, E. L.
1940
On Hamilton’s principle and the modified function in analytical dynamics. JFM 46.1172.02
Livens, G. H.
1919
Plane strain in a wedge, with applications to masonry dams. JFM 44.0933.02
Carothers, S. D.
1913
Tests for randomness in a series of numerical observations. Zbl 0016.41301
Kermack, W. O.; McKendrick, A. G.
1937
Reciprocity and the number 137. I. Zbl 0023.08902
Born, Max
1939
On the independence of linear and quadratic forms in samples of normally distributed variates. Zbl 0027.23302
Aitken, A. C.
1940
Studies in practical mathematics. II. The evaluation of the latent roots and the latent vectors of a matrix. JFM 63.1132.02
Aitken, A. C.
1937
Researches into the characteristic numbers of the Mathieu equation. III. JFM 53.0340.01
Ince, E. L.
1927
On the theory of graduation. JFM 49.0374.01
Whittaker, E. T.
1924
Some results in the mathematical theory of seiches. JFM 36.0807.01
Chrystal, G.
1905
On gravitational oscillations of rotating water. JFM 11.0689.01
Thomson, W.
1879
Further investigations into the periodic Lamé functions. JFM 66.1242.04
Ince, E. L.
1940
On the consistency of cardinal function interpolation. I, II. JFM 53.0221.03
Ferrar, W. L.
1927
The mathematical theory of random migration and epidemic distribution. JFM 42.0257.02
Brownlee, J.
1911
Deep water ship-waves. JFM 36.0805.02
Lord Kelvin
1905
On the eliminant of a set of quadrics, ternary or quaternary. JFM 30.0161.09
Nanson, E. J.
1899
On a class of permanent symmetric functions. JFM 15.0128.01
Muir, Th.
1883
On knots. JFM 09.0392.03
Tait, P. G.
1877
Vortex statics. JFM 08.0613.01
Thomson, W.
1876
On the equilibrium of vapour of a curves surface of liquid. JFM 07.0631.01
Thomson, W.
1870
Some confluent hypergeometric functions of two variables. Zbl 0063.01258
Erdélyi, Arthur
1940
Relative co-ordinates. JFM 58.1253.04
Walker, A. G.
1932
On singular pencils of Zehfuss, compound, and Schläflian matrices. Zbl 0014.39004
Ledermann, W.
1936
Some distributions associated with a randomly arranged set of numbers. Zbl 0017.27205
Kermack, W. O.; McKendrick, A. G.
1937
Reciprocity. 2. Scalar wave functions. Zbl 0027.28601
Born, Max; Fuchs, Klaus
1940
The periodic Lamé functions. JFM 66.0324.05
Ince, E. L.
1940
Further investigations into the periodic Lamé functions. JFM 66.0324.06
Ince, E. L.
1940
Some confluent hypergeometric functions of two variables. JFM 66.0325.04
Erdélyi, A.
1940
The estimation of factor loadings by the method of maximum likelihood. JFM 66.1304.06
Lawley, D. N.
1940
Integration of a certain system of linear partial differential equations of hypergeometric type. JFM 65.1268.02
Erdélyi, A.
1939
The gravitational field of a distribution of particles rotating about an axis of symmetry. JFM 63.1259.02
van Stokum, W. J.
1937
On the theory of statistical regression. JFM 59.0513.04
Bartlett, M. S.
1933
Researches into the characteristic numbers of the Mathieu equation. I, II. JFM 52.0462.02
Ince, E. L.
1926
On the theory of graduation. JFM 52.0525.03
Aitken, A. C.
1926
On the connexion between linear differential systems and integral equations. JFM 47.0945.02
Ince, E. L.
1919
On an integral equation whose solutions are the functions of Lamé. JFM 45.1305.02
Whittaker, E. T.
1914
On the conditions for the reversibility of the order of partial differentiation. JFM 40.0332.03
Young, W. H.
1909
On the front and rear of a free precession of waves in deep water. JFM 19.0996.03
Sir Thomson, W.
1887
Note on Mr. Muir’s solution of a problem of arrangement. JFM 09.0149.01
Cayley, A.
1877
Tait, P. G.
1877
On the motion of rigid solids in a liquid circulating irrotationally through perforations in them or in any fixed solid. JFM 07.0586.01
Thomson, W.
1872
On the forces experienced by solids immersed in a moving liquid. JFM 07.0586.05
Thomson, W.
1870
Fourier integrals. Zbl 0003.30403
MacRobert, Thomas M.
1931
On the orthogonal polynomials in frequencies of type B. Zbl 0005.25605
Aitken, A. C.
1932
Zeros and turning points of the elliptic-cylinder functions. Zbl 0005.29601
Ince, E. L.
1932
Integration of a certain system of linear partial differential equations of hypergeometric type. Zbl 0027.39903
Erdélyi, Arthur
1939
Random paths in two and three dimensions. JFM 66.1300.01
McCrea, W. H.; Whipple, F. J. W.
1940
On the independence of linear and quadratic forms in samples of normally distributed variates. JFM 66.1305.05
Aitken, A. C.
1940
On some persymmetric determinants. JFM 56.0857.02
Gueronimus, J.
1930
The general form of the orthogonal polynomials for simple series, with proofs of their simple properties. JFM 56.0405.02
Allan, F. E.
1930
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text:
My Math Forum Evaluating a limit

 Pre-Calculus Pre-Calculus Math Forum

 August 15th, 2018, 07:32 PM #1 Senior Member     Joined: Jan 2012 Posts: 123 Thanks: 2 Evaluating a limit Hi, Please evaluate the following limit: $\displaystyle \lim_{x->0}\left ( \tfrac{1}{x^{2}}-\cot^{2}x \right )$ Thanks. Last edited by skipjack; August 15th, 2018 at 08:41 PM.
 August 15th, 2018, 08:03 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 520 Thanks: 293 Math Focus: Dynamical systems, analytic function theory, numerics The limit is 2/3.
August 15th, 2018, 08:27 PM   #3
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Quote:
 Originally Posted by SDK The limit is 2/3.

Last edited by skipjack; August 15th, 2018 at 09:29 PM.

 August 15th, 2018, 08:44 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,509 Thanks: 2514 Math Focus: Mainly analysis and algebra Assuming that the limit exists and that therefore $$\lim_{x \to 0} \left(\frac1{x^2}-\cot^2{x}\right) = L$$ setting $2t=x$ we have that \begin{align}L &= \lim_{t \to 0} \left(\frac1{(2t)^2}-\cot^2{2t}\right) \\ &= \lim_{t \to 0} \left(\frac1{4t^2}-\frac{\cos^2{2t}}{\sin^2{2t}}\right) \\ &= \lim_{t \to 0} \left(\frac1{4t^2}-\frac{\cos^2{2t}}{(2\sin{t}\cos{t})^2}\right) \\ &= \lim_{t \to 0} \left(\frac1{4t^2}-\frac{(\cos^2{t}-\sin^2{t})^2}{4\sin^2{t}\cos^2{t}}\right) \\ 4L &= \lim_{t \to 0} \left(\frac1{t^2}-\frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) \\ 3L &= \lim_{t \to 0} \left(\frac1{t^2}-\frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) - \lim_{t \to 0} \left(\frac1{t^2}-\cot^2{t}\right) \\ &=\lim_{t \to 0} \left(\frac1{t^2}-\frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} - \frac1{t^2}+\cot^2{t}\right) \\ &=\lim_{t \to 0} \left(\frac{\cos^2{t}}{\sin^2{t}} - \frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) \\ &= \lim_{t \to 0} \left(\frac{\cos^4{t}}{\sin^2{t}\cos^2{t}} - \frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) \\ &= \lim_{t \to 0} \left(\frac{2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{ t}\cos^2{t}} \right) \\ &= \lim_{t \to 0} \left(2+\tan^2{t} \right) \\ &=2 \\ L&=\frac23 \end{align} Thanks to this for inspiration. Thanks from happy21 and topsquark
August 15th, 2018, 09:21 PM   #5
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Quote:
 Originally Posted by v8archie Assuming that the limit exists and that therefore $$\lim_{x \to 0} \left(\frac1{x^2}-\cot^2{x}\right) = L$$ setting $2t=x$ we have that \begin{align}L &= \lim_{t \to 0} \left(\frac1{(2t)^2}-\cot^2{2t}\right) \\ &= \lim_{t \to 0} \left(\frac1{4t^2}-\frac{\cos^2{2t}}{\sin^2{2t}}\right) \\ &= \lim_{t \to 0} \left(\frac1{4t^2}-\frac{\cos^2{2t}}{(2\sin{t}\cos{t})^2}\right) \\ &= \lim_{t \to 0} \left(\frac1{4t^2}-\frac{(\cos^2{t}-\sin^2{t})^2}{4\sin^2{t}\cos^2{t}}\right) \\ 4L &= \lim_{t \to 0} \left(\frac1{t^2}-\frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) \\ 3L &= \lim_{t \to 0} \left(\frac1{t^2}-\frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) - \lim_{t \to 0} \left(\frac1{t^2}-\cot^2{t}\right) \\ &=\lim_{t \to 0} \left(\frac1{t^2}-\frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} - \frac1{t^2}+\cot^2{t}\right) \\ &=\lim_{t \to 0} \left(\frac{\cos^2{t}}{\sin^2{t}} - \frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) \\ &= \lim_{t \to 0} \left(\frac{\cos^4{t}}{\sin^2{t}\cos^2{t}} - \frac{\cos^4{t}-2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{t}\cos^2{t}} \right) \\ &= \lim_{t \to 0} \left(\frac{2\sin^2{t}\cos^2{t}+\sin^4{t}}{\sin^2{ t}\cos^2{t}} \right) \\ &= \lim_{t \to 0} \left(2+\tan^2{t} \right) \\ &=2 \\ L&=\frac23 \end{align} Thanks to this for inspiration.
Brilliant! This method is very nice though uncommon, I think!

Thx.

 August 15th, 2018, 10:06 PM #6 Global Moderator   Joined: Dec 2006 Posts: 19,957 Thanks: 1844 \displaystyle \begin{align*}\cot^2\!x &= 1/(\sin^2\!x) - 1 \\ &= 1/(x^2(1 - x^2/3 \,+\; ...)) - 1 \\ &= (1/x^2)(1 + x^2/3 \,+\; ...) - 1 \\ &= 1/x^2 - 2/3 \,+\; ...\end{align*} so the required limit is 2/3. Thanks from happy21
August 16th, 2018, 06:06 AM   #7
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Quote:
 Originally Posted by happy21 Can you please explain?
If you show me some work I'll show you some work.

August 16th, 2018, 10:59 AM   #8
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Quote:
 Originally Posted by SDK If you show me some work I'll show you some work.
Yeah I did from the methods depicted above.

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text:
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is it (B) 5,7 ??
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B) 5,7 ?

there may be more answer such as 1,5

and 4,7  etc

but 5,7 matching to the option
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explain pls?
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what is the meaning of not pushed in?
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@himgta
when we want explore any node first we push them into the stack
and if the connected node of that node is already visited than there is no need to push that node into the stack
ex if the connected node of node A is already visited then there is no need to push A into the stack
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but if we push the every node exactly once in order to traverse it...isin't it?

can u post the detailed solution of the question?
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The answer is 5,7.

The question is little bit wrong or may be ambiguous. What i did is :-

First create a graph out of it and when you'll do the DFS of it you'll find in most of the case the vertex 5 or 7 will be at top of stack at will be then popped for backtracking they never go within the stack. So intuitively i answered like that.

The solution doesn't contain anything they simply give the answer. That is as far i know.

Please point out some mistake here :)
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that's correct...i also got 5,7 by same procedure.
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I am getting C as answer , there is no braching at V4, V5 and V7.
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DFS traversal will be like:- $V_6$->$V_3$->$V_1$->$V_2$->$V_4$->$V_8$
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I am just getting vertex 4
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will you please! check it out,

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c is the right answer acording to ace test series.
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I'm getting everything into the stack, this is my order 6-> 8-> 7-> 3-> 1->2->5->4
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The solution is ambiguous, If you follow the algorithm given on geeksforgeeks,

Each vertex is pushed at least once into the stack,Then how could you find the answer
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@Gurdeep Saini Do you have any reference for your statement above on not pushing an unbranched node into the stack ?

0

@Shivam Kasat it is he way of implementing the DFS algo , i.e.

when we want explore any node first we push them into the stack
and if the connected node of that node is already visited than there is no need to push that node into the stack
ex if the connected node of node A is already visited then there is no need to push A into the stack

0
implement it on note book then three case arise

1) 5,4 not pushed

2) 4,7 not pushed

3) 5,7 not pushed

and the given answer is the combination of all these

+1 vote


text:
PREPRINT

# Neutrino Origin of LHAASO's 18 TeV GRB221009A Photon

Vedran Brdar and Ying-Ying Li

Submitted on 3 November 2022

## Abstract

LHAASO collaboration detected photons with energy above 10 TeV from the most recent gamma-ray burst (GRB), GRB221009A. Given the redshift of this event, $z\sim 0.15$, photons of such energy are expected to interact with the diffuse extragalactic background light (EBL) well before reaching Earth. In this paper we provide the first neutrino-related explanation of the most energetic 18 TeV event reported by LHAASO. We find that the minimal viable scenario involves both mixing and transition magnetic moment portal between light and sterile neutrinos. The production of sterile neutrinos occurs efficiently via mixing while the transition magnetic moment portal governs the decay rate in the parameter space where tree-level decays via mixing to non-photon final states are suppressed. Our explanation of this event, while being consistent with the terrestrial constraints, points to the non-standard cosmology.

## Preprint

Comment: 5 pages, 2 figures

Subjects: High Energy Physics - Phenomenology; Astrophysics - Cosmology and Nongalactic Astrophysics; Astrophysics - High Energy Astrophysical Phenomena