text:
3:22 AM
@sammygerbil
Can anyone tell me how to solve the first part
Here's what I did
What to do next

3:45 AM
@JohnRennie morning.

@Nobodyrecognizeable morning :-)

@JohnRennie do you recall noethers theorem ?

Yes ... ?

@JohnRennie if you have time ill ask you a question and youll try to solve that with noethers theorem. Can i go ahead?

3:52 AM
9 hours ago, by sammy gerbil
@JohnRennie i know we can not know kinetic energy and potential energy is conserved or not using noethers theorem . So i want to ask you to check about the energy, momentum , angular momentum is conserved or not using noethers theorem.

I don't think Noether's theorem is useful here. Noether's theorem relates a symmetry of the action to a conserved quantity.

Now before proceeding i want to ask what do we mean by translational symmetry?
@JohnRennie can you demonstrate the noethers theorem by working out an example (just at classical level) if you go qm then ill have to study for that.

Didn't watched it yet

@LoopBack thats done already by sammy gerbil. Im just here to learn noethers theorem from johnrennie.
And yep thats a.

Nice

4:04 AM
@JohnRennie are you still here?

@Nobodyrecognizeable I don't think Noether's theorem is applicable to this sort of problem. The problem gives you $\mathbf r(t)$ so you can work out the velocity, and therefore the momentum, angular momentum and kinetic energy by differentiating.

@JohnRennie yep. Thats ok. But my request is can you demonstrate noethers theorem with an example

@Nobodyrecognizeable have you solved this question which you posted

@LoopBack which one ?

4:06 AM
53
@Nobodyrecognizeable question no. 53

I have just starred the message from where the solution starts you can have a look at that. @LoopBack

@Nobodyrecognizeable Were you able to prove it

@Nobodyrecognizeable well consider a free particle. The action is:
$$S[\mathbf{r}(t)] = \int_{t_1}^{t_2} \tfrac{1}{2}m\dot{\mathbf{r}}^2$$

@JohnRennie kinetic energy integrated over a time integral?

Suppose we displace the while system in space by some amount $r_0$ so that $\mathbf r \to \mathbf r + \mathbf r_0$. This clearly doesn't affect the action because when we differentiate the constant term disappears.

4:10 AM
@LoopBack its done by @sammygerbil.

@Nobodyrecognizeable I very close to proving the equation, but I m unable to resolve initial velocity of the ball in terms of $h$.

That means a displacement in space does not affect the action, and Noether's theorem tells us that must mean momentum is conserved. Which of course it is.

@JohnRennie how do you get the action?

In physics, action is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has the dimensions of [energy]⋅[time] or [momentum]⋅[length], and its SI unit is joule-second. == Introduction == Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an...

@JohnRennie the rest is fine..
@LoopBack its vertical velocity is $\sqrt 2gh$

4:15 AM
@Nobodyrecognizeable No it's not

@LoopBack why ?

@Nobodyrecognizeable I want to know the velocity just before first collision
@Nobodyrecognizeable velocity just before collision cannot be $\sqrt{2gh}$. Because the ball has travelled a lesser distance than h

@LoopBack see the vertical distance through which the ball has fallen is h. So the velocity should be $\sqrt 2gh$ in the vertical direction

@Nobodyrecognizeable but h is distance from the horizontal ground and not from the incline plane.

I should mention that was the wrong thing in the question. They wanted to say this that the point of impact is at vertically h distance away from the inclined plane. Cause that perfectly matches the answer. And except that the question can not be solved as its height isn't given.

4:21 AM
@Nobodyrecognizeable Btw if I substitute √2gh in my equation I will get the correct answer.

@LoopBack if the exact vertical distance isn't given no one will be able to solve this as youll not get a perfect initial velocity. Making the problem unsolvable or we shall be getting a general solution.
@LoopBack anyway goodbye.
@JohnRennie thanks for helping. Goodbye, professor have a nice day.

@Nobodyrecognizeable bye

@JohnRennie @sammygerbil good morning

@harambe morning :-)
You're unlikely to find Sammy here at 05:30 a.m. :-)

Oh. I think he is more active during night time

4:42 AM
@JohnRennie imagine two concentric spheres with potential difference between them. Does earthing any of the sphere change the potential difference between them

I'm not sure that's a well defined question because earthing either of the spheres will change the charge on it.

@JohnRennielet me post the question
In this question, the final potential difference remains same between the spheres

Suppose A weren't present, what would the field inside B be?

The field inside a charged spherical shell is ... ?

4:48 AM
Zero

So the field inside B due to the charge on B is zero. Then the field inside B is only due to the charge on A.
Yes?

Yes

And the potential is the integral of the field. That means the potential difference between A and B is determined only by the charge on A.

Doesn't this violate gauss law

@harambe can you expand on that. Why do you think this violates Gauss's law?

4:52 AM
Not gauss law. I meant isn't electric field zero inside a conductor zero at any situation
Property of conductor

Didn't we discuss this before? The field inside the metal of a conductor is always zero. The field in a space enclosed by a conductor is not necessarily zero.

Okay. I didn't think of it like that

@harambe Suppose you have an electron floating around in a conductor like copper.

Okay

This electron is mobile, because electrons in a conductor are mobile, so if any field is present the electron will feel a force from that field and it will move.

4:59 AM
Then they would collide bc of this?

But as the electrons move this causes a charge separation and that in turn causes a field inside the conductor. Eventually the field caused by the fact the electrons have moved will exactly balance out the external field. At that point the electrons will stop moving and the total field inside the conductor will be zero.
The point is that if there is any field inside the conductor the electrons will move, and they will only stop moving when the field is zero.

Okay

But this only applies in the metal of the conductor where there are mobile electrons.
In the air/vacuum/whatever inside a conducting shell there aren't any mobile electrons.

Yea. That makes sense

As it happens the field inside a spherical shell is zero due to Gauss's law, but this applies to non-conducting as well as conducting shells. It's unrelated to the reason why the field inside a conductor is zero.

5:46 AM
Hello :-)
You there John?
Isn't moment of inertia just Mk^2?

@JohnRennie are you free

@harambe yes

@user64829 Yeah, I guess he is writing it from another non-inertial frame and giving pseudo force too

@user64829 I think that's the parallel axis theorem ...

Q19

6:01 AM
But the axis is through the centre of the ball, why is he using parallel axis theorom?

I find all the options incorrect... Funny
@user64829 the axis is at the point of contact of the sphete with the ground

Although it's not clear from the question I would guess $Mk^2$ is the moment of inertia about the centre of the object. Then the moment of inertia about the contact point with the slope is given by the parallel axis theorem.

No, would You mind if I answer, di electric constant of water is some 80 and dielectric breakdown (you can look it up in the web) and answer the question

No
Let me explain my view

@user64829 the instantaneous axis of rotation is the point of contact with the slope.

6:04 AM
I think capacitance of the sphere is directly proportional to dielectric constant and the potential of the sphere is inversely proportional to the dielectric constant so won't these cancel each other @EshaManideep
Q=CV
So that's why I am confused

@JohnRennie Oh yeah, now I get it
Thanks

@harambe I'm not sure I understand answers (a) and (b). I wonder if they mean hold more charge for a given voltage
Otherwise I can't see what hold more charge means.

John Rennie, that's what he meant

@JohnRennie I think so too
If for a given voltage then I guess this answers it

A fireman wants to slide down a rope. The breaking load for the rope is (3/4)th of the weight of the man. Find the minimum acceleration of the fireman to slide down the rope.

6:18 AM
T= m(g-a)
Maximum tension must be 3T/4

@blue_eyed_... the net force on the man is the gravitational force minus the tension in the rope. Yes?

@JohnRennie, yeah
@JohnRennie, are these forces acting in opposite directions?

The force on the man is $mg$, and we are told that the maximum tension in the rope is $0.75mg$, so the net force on the man is $F = mg - 0.75mg = 0.25mg$
@blue_eyed_... yes, gravity is pulling the man down while the tension in the rope is pulling him up.

@JohnRennie, thanks. I got it

@blue_eyed_... Cool :-)

6:31 AM
John, you there?

6:48 AM
@user64829 hi

4 hours later…
10:36 AM
@JohnRennie are you free

@harambe hi

Q10
I calculated energy density which came to be $Q^2$/16πEo$R^2$

What is $C$?

I messed up. I think
Capacitance

OK. The field inside the smaller sphere and outside the larger sphere is zero. So the field only exists in the shell between the two spheres. Yes?

10:43 AM
Yes

So the volume is just $4/3 \pi R^3 - 4/3 \pi r^3$ where $R$ and $r$ are the radii of the two cylinders.

Yea

So given the capacitance you need to relate this to the radii of the two spheres and hence work out the volume.

C=4πEo r1r2/r2-r1
I have calculated energy from here
@JohnRennie how do I proceed from here

We're told that $r_2 = 2r_1$ so the capacitance is $C = 8 \pi \epsilon r_1$. Yes?

10:53 AM
Yes
Calculate d energy from Q^2/2C

@harambe why are you calculating the energy? The question asks for the volume not the energy?

I just thought it had something to do with it

If we call the radius of the small sphere $r$ then the capacitance is $C = 8\pi\epsilon r$.
So $r = C/(8\pi\epsilon)$
And the volume of the small sphere is:
$$V = \tfrac{4}{3} \pi \frac{C^3}{8^3 \pi^3 \epsilon^3}$$
OK so far?

@JohnRennie I get what you are doing.... I can't believe I missed this
I just thought of going to energy density and energy but it was actually simple...

@harambe the question looked harder than it was. The reference to energy was just a complicated way of asking what the voulem between the spheres is.

11:01 AM
Yes
I can do this now
Got the hint
@JohnRennie Got D
It looks right too

Yes, that's what I got :-)

11:36 AM
@JohnRennie can you tell me why the CGS unit of thermal conductivity is $\dfrac{kcal}{m. hrs .K}$ instead of $\dfrac{ergs}{sec. cm .K}$

I have absolutely no idea. Using any system of units other than SI strikes me as a road to madness.
5

@JohnRennie okay

Anonymous
11:56 AM
Hello, I have few questions. I tried them, but couldn't solve them entirely.

Anonymous

@JohnRennie Sometimes I haven't even gone to sleep by 5:30am! But yes usually I am asleep at that time.

Anonymous
In this question, the problem that I face is that it given that the spring is stretched between points A and B but it is not mentioned how much.

Anonymous
So, I just assumed that the distance b/w A and B to be its natural length.

Anonymous
For solving that, I assumed the part of spring to the left of point P to be one spring and the part to the right to another spring.

Anonymous
12:00 PM
(Assuming the one to the left as spring 1 and to the right as spring 2)

Anonymous
I will have $$k_1 = \dfrac{k × 0.5}{0.4}$$ and $$k_2 = \dfrac{k × 0.5}{0.1}$$

Anonymous
And I assumed that the displacement of the point P when the force is 5 N be x.

Anonymous
So, that will give me $$k_1x + k_2x = 5$$

Anonymous

@IceInkberry It is not necessary to know how much the spring is stretched between A and B because force is proportional to extension no matter how much extension there is already. So your method is correct.

Anonymous
12:04 PM

Anonymous
There isn't even an option for my answer.

Anonymous
$$k_1 = \dfrac{500}{4}$$ $$k_2 = \dfrac{100}{4}$$

Anonymous
$$\dfrac{500x}{4} + \dfrac{100x}{4} = 5$$

Anonymous
$$x = 0.033 m$$

Anonymous
@sammygerbil Do you see anything wrong in my method?

12:08 PM
@IceInkberry I don't see how you got $k_1=k\times 0.5/0.4$.

But k1x and k2x are in opposite directions.. Would it make sense to add them

Shouldn't that be $k_1=k\times 0.4/0.5$ and similar for $k_2$?

Anonymous
@sammygerbil Because $$k × l = constant$$

Anonymous
So, $$k × l = k_1 × l_1$$

Anonymous
For a spring cut, the spring constant varies inversely to its length.

12:11 PM
@DavidZ heyyy. I worked hard on those comments. Hrumph

Anonymous
11

To supplement the answer by Luboš Motl, I will come to this problem from a Material Science point of view. What you mean by the inherent property of the string is not the spring constant, in fact, it is Young's modulus $E$, which only depends on the properties of a material of a body but not it'...

@IceInkberry Ah yes you are right. You need a bigger force to produce the same extension.

Anonymous
@sammygerbil Yes.

I think harambe is right : $x$ should be +ve for the spring on the left and -ve for that on the right.
So you will have $k_1x-k_2x=5N$.

Anonymous
@sammygerbil I don't think that, because if you stretch the the spring, it will try to pull in the opposite direction and if you compress it, it will try to push it.

Anonymous
12:18 PM
Basically, the spring on right will push it to left and the spring on left will push it to right.

Anonymous
Also a reminder, it's one spring. I assumed it to be two.

@IceInkberry Yes I think you are right again. I'm not thinking well today!

Anonymous
@sammygerbil Haha, maybe you just need some sleep.

@IceInkberry Maybe. I have being playing football this morning, and walking. Often I have a nap afterwards!

Anonymous
@sammygerbil Oh well, nevermind. Have some sleep! I will ask it on main.

12:24 PM
@IceInkberry I get $8mm$ answer A.

Anonymous
@sammygerbil How?

Anonymous
I am surprised.

Anonymous

@IceInkberry $k_1=k\times \frac{0.5}{0.4}=1N/cm\times \frac54=\frac{5}{4} N/cm$.

Anonymous
@sammygerbil But isn't $$k = 1.00 N/cm ?$$

12:28 PM
Yes sorry corrected. I continue ...

Anonymous
Listening..

$k_2=1N/cm \times \frac{0.5}{0.1}=5 N/cm$.

Anonymous
@sammygerbil Understood.

Anonymous
I calculated $k_2$ incorrectly!

$(\frac54+5)x=\frac{25}{4}x=5$
$x=\frac45 cm=8mm$

Anonymous
12:30 PM
@sammygerbil Got that!

Anonymous
Pity on me.

Anonymous
Thank you! @sammygerbil

@IceInkberry It took us both to sort it out.

Anonymous
Yes, haha.

Anonymous
I have one more question, could you help me with it?

Anonymous
12:33 PM
(If you have time)

Anonymous

Anonymous
I don't know how to do it by intuition, so I tried the mathematical method.

Anonymous
(I'll have to make a diagram)

Anonymous
(A minute)

12:40 PM
np

Anonymous

Anonymous
I wasn't able to upload the image. Finally.

Anonymous
I wrote these equations.

Anonymous
Now, I tried to apply some sort of logic to the $2$nd equation.

Anonymous
When the ball is at the bottom most position, $$\theta = 90°$$ and $\alpha$ will be some angle.

Anonymous
12:43 PM
Basically $F = mg\sin{\theta}/\sin{\alpha}$

Anonymous
So, when $\theta = 90°$, $\sin{\theta} = 1$ and that divided by $\sin{\alpha}$ (which is less than $1$) means $sin\theta / sin\alpha ≥ 1$

Anonymous
But, when the ball is at topmost point, $\theta , \alpha = 0$ so $sin\theta / sin\alpha$ = 1$Anonymous So, that means$F$is decreasing as we pull up. Anonymous But, I don't understand how I should do it for$R$from the first equation since$F$is the variable there. Anonymous I also don't think that this is a valid method. It works only when the options are given. 12:50 PM @IceInkberry The options are a key part of the question. It is valid to use them - eg eliminate options you know must be wrong, whatever is left must be correct - assuming that one option is in fact correct. Anonymous @sammygerbil Oh, I see. So, I've eliminated **$A$** and **$D$** @IceInkberry Can you decide between B and C? Anonymous @sammygerbil No :/ @IceInkberry I approach such questions by looking at extremes. If the hemisphere was a sphere then at the lowest point the string would bear all of the weight of the ball and the normal reaction would be zero. When the ball reaches the top the situation is reversed : the string bears no weight and the normal reaction bears all of the weight. Anonymous @sammygerbil I did think about it, I even tried to manipulate the first equation that way, but the force$F$was again bothering me. Anonymous 12:59 PM The answer even surprised me, it is$C$. Anonymous I cannot intuitively think how the normal reaction wouldn't vary. Anonymous The tension's component ($T = F$here) is playing some role. @IceInkberry The answer surprises me also. I thought it should be B. Another "extreme" case is that the pulley is very much higher than the radius of the hemisphere. Then when the ball is at the side of the hemisphere the string hangs almost vertically down, so the normal reaction is very small. In this case R increases as the ball goes up. I don't see how it can remain unchanged. Anonymous That was incorrect. Anonymous Incorrect. Anonymous 1:09 PM I don't know. Anonymous @sammygerbil Yeah! Anonymous Maybe, I should move on. The question took a lot of time. I shouldn't waste any more. Anonymous Thank you for your help @sammygerbil @IceInkberry you are welcome Why is the answer to this question Y1+Y2/2 and not Y1+Y2? Considering the two wires as two springs with spring constants AY1/L and AY2/L since the two wires are in parallel why isn't effective spring constant AY/L taken as sum of both spring constants? 1:21 PM @Hema I think it is because the cross-sectional area is doubled. @sammygerbil thank you so much! Hello :-) 2:05 PM @IceInkberry @sammygerbil i think as the ball is moving in a circular path the sum of all the forces in radial direction should have been$\dfrac{mv²}{r}$2:26 PM @LoopBack The question says the small ball is pulled gradually. I think this means that$v=0$- ie the ball is in static equilibrium at each position around the rim of the hemisphere. 2:44 PM @LoopBack You left a message for me to contact you yesterday. Did you want to discuss further your solution to the problem of the two balls connected by a string? 3:39 PM A simple harmonic wave train of amplitude 3cm and frequency 200 Hz travels in the positive X axis with a velocity of 20 m/s. Calculate the displacement, velocity and acceleration of a particle situated at 50 cm from the origin at t=2 sec. @JohnRennie, @sammygerbil, please see... The answer in my book is 0 for displacement and acceleration. How is that? @blue_eyed_... If the displacement is zero then the acceleration will also be zero because the restoring force is proportional to displacement. @sammygerbil, could you please explain mathematically..? I mean by using equations The wavelength is 10cm so the particle is situated a whole number of wavelengths from the origin. ... @sammygerbil v=0 does it make sense. Absolutely not the body won't move @sammygerbil I wanted to ask you a doubt but later I solved it myself @sammygerbil, y=a sin(wt-kx) gives the displacement, is it? 3:49 PM @blue_eyed_... The particle moves vertically. Its motion is SHM. So the restoring force on it is proportional to displacement. This is one way of recognising SHM. @blue_eyed_... Yes that is correct. So you can differentiate to get acceleration. @LoopBack It is possible to move it very slowly, which is what gradually means. Then$v \to 0$. The question is comparing the static forces in various positions of rest. It is not asking about the dynamic forces while motion is occurring. @sammygerbil, first order derivative gives aw cos (wt-kx). Differentiating again seems quite difficult? @blue_eyed_... Surely not!$a$and$\omega$are constants. @sammygerbil, does second order derivative gives -aw^2 sin(wt-kx)? @blue_eyed_... Yes. And this is equal to$-\omega^2 y$. Acceleration is proportional to displacement. 3:58 PM @sammygerbil, what about the sine function @blue_eyed_... What about it? @sammygerbil, you told acceleration is equal to -w^2y. Why did you not include the sine? @blue_eyed_... The sine is included in the definition of$y=a\sin(\omega t-kx)$. @sammygerbil, oh.. I got it. Thank you @blue_eyed_... It need not be wt-kx right? it can be kx-wt also Since we do not know how the wave train started 4:08 PM @blue_eyed_... The next step is to find the value of$y$. The phase of the sine wave is$\phi=\omega t-kx)=2\pi (ft-x/\lambda)$. We are given$f=200Hz$and$C=20ms/s$so$\lambda=10cm$. Therefore$\phi=2\pi (200*2-50/10)$. ... @user64829$\sin(kx-wt)=-\sin(wt-kx)=\sin(wt-kx+\pi/2)$. That is correct we do not know the phase with which the wave started. However, with the values given in the question it does not matter. So displacement at the specified time and distance isn't zero unless the initial phase is N(pi) (where N is some integer) right? the instant and distance specified in the question I mean @user64829 Sorry that should be$sin(kx−wt)=−sin(wt−kx)=sin(wt−kx+\pi)$. @user64829 Try putting values into the equation and see what result you get! @user64829, yeah.. But the answer given was zero. So I was confused with that I think they were referring to most general case, where phase is 0 @user64829 Yes sorry you are correct we have to assume that the phase is zero at the origin$x=0$at$t=0$. Good point. If we had used a cosine wave instead of a sine we would have gotten a different answer. 4:20 PM Yeah, in that case phase would be pi/2 +/- @user64829 That's right. So the particle would be at maximum displacement = 3cm and maximum acceleration. Yeah 4:34 PM @sammygerbil I have solved this question and all the options are correct. But the solution says (b) is incorrect. But why ? Heat flowing through slab E= heat flowing through slab A=Maximum heat flowing Mind if I give my opinion? @LoopBack I think this is a question of what you mean by maximum. Maximum usually means it is bigger than any other value, rather than equal biggest. @user64829 Go ahead. @sammygerbil Exactly what I was going to tell haha 4:57 PM Transverse waves on a string have wave speed 8m/s, amplitude 0.07m and wave length 0.320 m. The waves travel in the negative X direction and at t=0s, the x=0 end of the string has its maximum upward displacement. (a) find the frequency, period and wave number of these waves. (b) write a wave function describing the wave. (c). Find the transverse displacement of a particle at x=0.36 m at time t=0.15 sec @sammygerbil, @JohnRennie, @user64829, please see this X=0 has max upward displacemnet, so phase = 0 @blue_eyed_... a wave equation is$A\sin(\omega t - kx)$or$A\cos(\omega t - kx)$. Since it's in negative x direction, squation will be Asin(wt + kx) @JohnRennie Shouldn't it be wt + Kx since it's travelling in negative x direction? In this case we are told that at$x=0, t=0$the amplitude is a maximum, i.e. equal to$A$, so it's going to be the cosine not the sine. Oh yeah sorry, cosine function What about symbol between wt and kx? 5:02 PM$\omega = 2\pi f$and you're given the frequency so you can work out$\omega$@JohnRennie, how do we know when we need to use cosine function and when we need to use sine function? @blue_eyed_... the most general form of the wave equation is: $$\psi(x,t) = A\sin(\omega t - kx) + B\cos(\omega t - kx)$$ And you have to determine the constants$A$and$B$from the initial conditions. @JohnRennie, where does that come from? @blue_eyed_... the wave equation is a linear equation, so any sum of solutions is also a solution. The sine is a solution and so is the cosine, so the sum of the two is also a solution. Consider what happens at$x=0, t=0$. If we substitute this into the above equation we get: $$\psi(0,0) = A\sin(0) + B\cos(0) = B$$ And we are told that the amplitude is a maximum when$x=0, t=0$so that must mean$B$is equal to the maximum amplitude. Yes? @JohnRennie, yeah that's true 5:08 PM And that in turn has to mean$A=0$So the behaviour at the origin means we end up with the solution: $$\psi(x,t) = B\cos(\omega t - kx)$$ Suppose we were told that$\psi(0,0) = 0$In that case we would have$B=0$and the solution would be: $$\psi(x,t) = B\sin(\omega t - kx)$$ @JohnRennie, Where did we use the information that the wave is travelling in negative x direction? @blue_eyed_... we haven't ... yet @JohnRennie, where do we need that? Is that used while adjusting the sign? Strictly speaking$k$is a vector called the wave vector so it has a direction. In practice for 1D motion we generally don't worry about this and write it as a scalar. In that case for a wave moving in the positive$x$direction we use$\omega t - kx$and for a wave moving in the negative$x$direction we swap the sign of$k$to get$\omega t + kx$. So the full equation for the wave in this question is going to be: $$\psi(x,t) = B\cos(\omega t + kx)$$ @JohnRennie, what does x actually represent in these wave equations? 5:14 PM It is the position on the$x$axis @JohnRennie, position of wave or particle? The wave is travelling along the$x$axis, so if this is a wave on a rope the particles that make up the rope are oscillating normal to the$x$axis. i.e. any particular bit of rope stays at constant$x$and just oscillates sideways. @JohnRennie, thank you. Got it 5:36 PM @sammygerbil, you there? @blue_eyed_... hello @sammygerbil, hi. I had a confusion. When do we use wt - kx and kx-wt in wave equation? If you are using the equation$y=A\sin(wt-kx)$this is a wave travelling in the +x direction starting with zero phase. That is, the slope is increasing as we move in the +x direction from the origin - ie slope is +ve at origin. Whereas if you use$y=A\sin(kx-wt)$this is also a wave travelling in the +x direction but the phase at the origin is now$\pi$. That is, the slope is decreasing as we move in +x direction - ie slope is -ve at the origin. @sammygerbil, that means in the second case the wave starts below x axis? The question we looked at with amplitude 3cm and frequency 200Hz did not say what the phase was at$x=0$and$t=0$so it was ambiguous. It was an ill-defined question. @blue_eyed_... Yes in the 2nd case$y=A\sin(kx-wt)$the wave moves down below the x axis as we move right. 5:51 PM @sammygerbil, how is the phase 0 and pi? @blue_eyed_... Good question. That is making me think ... I think I might have it the wrong way round. @sammygerbil, huh? I didn't get! If we have$y=A\sin(kx-wt)$and we take a "snap-shot" of the wave at$t=0$then we have$y=A\sin(kx)$. Starting from the origin and looking in the +x direction this goes up above the x axis - ie the slope is +ve. Whereas if we have$y=A\sin(wt-kx)$then the snap-shot at$t=0$is$y=A\sin(-kx)=-A\sin(kx)$. This goes down below the x axis as we look in the +x direction - ie the slope at the origin is -ve. @sammygerbil, how is the phase difference between them pi? @blue_eyed_... Because$\sin(x+\pi)=\sin(x)\cos(\pi)+\cos(x)\sin(\pi)=-\sin(x)+0$because$\sin(\pi)=0$.$\sin(x+\pi)$is the same as$\sin(x)\$ reflected about the x axis.
@blue_eyed_... It is (potentially) quite confusing because we have 3 variables (y, x and t) but it is difficult to plot a 3D graph. It is easier to plot y vs x for a particular value of t (eg t=0) or y vs t for a particular value of x (eg x=0).

6:56 PM
@blue_eyed_... Ping me tomorrow morning and I'll explain an easy way to understand what direction the wave moved in.

2 hours later…
8:49 PM
@IceInkberry Small metal ball pulled on fixed hemisphere : I have looked again at this question, using your equations in a spreadsheet to calculate the values of N and F. To my surprise F is constant! But I do not understand why this is.

2 hours later…
10:40 PM
Correction : normal reaction N is constant.

1 hour later…
11:41 PM
@sammygerbil I m 100% sure that the answer to this question is B
@sammygerbil Btw N is not constant
@sammygerbil I can show you why

@LoopBack go on

@sammygerbil Wait I'll upload an image to demonstrate it
@sammygerbil if the radius of the ball is negligible then according to geometry, the string will not be straight but will lie on the surface of the hemisphere. But that's not the case, so diagram B represents the correct situation.
Diagram B suggests that although string is not in contact with the hemisphere but it is parallel to a tangent at some point. I have marked that point in this diagram.
@sammygerbil this means that there will be another point in the hemisphere where the string will again make same angle with the vertical as it was making in the bottommost position.
@sammygerbil I have shown that by a dotted sphere
@sammygerbil I hope I m clear to you


text:
RAII and smart pointers

This topic is 2259 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

Recommended Posts

Can someone give me a few simple to understand examples of RAII and smart pointers? I read somewhere that RAII and smart pointers should be used for everything but other places say that pointers are useless and you should use references instead. For my part I don't know of any uses for pointers aside from passing values between functions economically or iterating through a linked list. Any major uses beside those?

Also in the examples of RAII I'm looking at all it is seems to be a class file with an empty constructor and empty destructor. Is it really that simple? Edited by Jarwulf

Share on other sites
Uhm... try to use polymorphism without pointers. Ok, references, but how often do you see code like
Object& obj = *(new DerivedObject);

Not to mention the cleanup:
delete &obj;

Besides, there is no such thing as "smart references" to help you avoid memory leaks and accessing invalid objects. Apart from that, pointers and references have only a few differences under the hood. It's not like they are doing completely different things. References are more convenient in syntax and unless someone deliberately does something stupid you shouldn't have to worry about them being valid.

RAII examples? Sure. std::fstream, std::vector and pretty much anything that keeps track of its resources and releases them when going out of scope. Like scoped_lock or unique_ptr.

If the constructor and destructor is really empty, it has little to do with RAII. Apparently the object doesn't "acquire" any resources during initialization and has nothing to release on destruction. That would be like calling an empty chassis an example of a light weight car.

An example for RAII and smart pointers.

 //without char* tmpBuffer = new char[chunkSize]; mutex.lock(); loadData(tmpBuffer); mutex.unlock(); delete[] tmpBuffer;

Oh no, loadData just threw an exception. I must ensure proper cleanup in all cases.

 //without char* tmpBuffer = new char[chunkSize]; mutex.lock(); try { loadData(tmpBuffer); } catch (...) //and I don't really know what kind of exception to expect { mutex.unlock(); delete[] tmpBuffer; } mutex.unlock(); delete[] tmpBuffer;

How about doing it the smart way (without a mess)?

 vector<char> tmpBuffer(chunkSize); scoped_lock lock(mutex); loadData(&tmpBuffer[0]);

See? No cleanup, no mess to deal with exceptions, no chance of someone screwing it up by spamming early returns all over your function.

Share on other sites
I generally don't use "pure" smart pointers (reference counted pointers that delete themselves) because in my opinion they muddle the concept of what owns the object and manages its lifetime. For most objects that have a complex lifetime that would require smart pointers, I find it makes sense to have something manage the lifetime of that object. That can involve reference counting pointers, but not ones that delete themselves automatically.

1. 1
2. 2
Rutin
24
3. 3
4. 4
JoeJ
18
5. 5

• 14
• 23
• 11
• 11
• 9
• Forum Statistics

• Total Topics
631768
• Total Posts
3002242
×

format: plain


text:
Next: Waveguide Network for Timoshenko's Up: Timoshenko's Beam Equations Previous: Dispersion

## MDKC and MDWDF for Timoshenko's System

Nitsche [131] showed how to write a MDKC and MDWD network corresponding to Timoshenko's system. In order to deal with the asymmetric coupling of the system of equations, he constructed a network using both MD capacitors and inductors, but here we will take the more conventional approach, and use a gyrator (For computational purposes, there is no essential difference between this representation and his.)

Consider again the Timoshenko system of (5.17) and (5.18). We can scale the variables just as for the transmission line (see §3.7). That is, we can write

 (5.22)

where the constants and , are strictly positive. We introduce, as before, the scaled time variable where is the space step/time step ratio. Then the Timoshenko system can be rewritten as

 (5.23a)

 (5.24a)

The constant proportional terms on the right-hand side appear anti-symmetrically, and can be interpreted as a lossless gyrator coupling. We can now write down a MDKC for the scaled system of equations; it is shown, along with element values in Figure 5.5. Its MDWD counterpart is pictured in Figure 5.6. Here, we have used the coordinate transformation defined in (3.18) with step sizes . We have used for the one-port time inductors.

An MDWD network can obviously also be designed to operate on alternating grids, just as in the case of the (1+1)D transmission line.

A comment is necessary regarding the gyrator in Figure 5.6. In order to deal with the delay-free loop which arises from the placement of a gyrator between two series junctions, we have set the corresponding ports of the series junctions on either side of the gyrator to be reflection-free. This, however, means that the two port resistances of the gyrator are not, in general, equal to the gyrator constant, which, in this case, will be . In terms of wave variables, the signal flow diagram of the gyrator will not be of the simple form of (2.26), but takes the more general form of (2.25) mentioned in §2.3.4. It is of course also possible to set only one of the ports connected to the gyrator to be reflection-free, (say ), and then the other port resistance to be , in which case the general gyrator form degenerates to a pair of scalings.

As for the parameters and , an optimal choice can be shown to be

which yields the bound

 (5.25)

which is the same as that which is derived in [131]. We will show how to improve upon this bound in §5.2.6.

Next: Waveguide Network for Timoshenko's Up: Timoshenko's Beam Equations Previous: Dispersion
Stefan Bilbao 2002-01-22


text:
Mysterious tufte-book error

I'm getting a flurry of errors in at line 175 and 243 of the following document:

\documentclass[a4paper,twoside,nobib]{tufte-book}
\usepackage[utf8]{inputenc}
\usepackage[normalem]{ulem}
\usepackage{soul}
\usepackage{amssymb}
%\usepackage{nopageno}
\usepackage{graphicx}
\graphicspath{ {images/} }
\usepackage[authordate,backend=biber,doi=true,url=true,isbn=false]{biblatex-chicago}
\AtEveryBibitem{
\ifentrytype{online}
{}
{\clearfield{urlyear}\clearfield{urlmonth}\clearfield{urlday}}}

% Define a \subtitle command
\newcommand{\subtitle}[1]{%
\gdef\@subtitle{#1}%
\gdef\plainsubtitle{#1}%
\ifthenelse{\isundefined{\hypersetup}}%
{}% hyperref is not loaded; do nothing
{\hypersetup{pdftitle={\plaintitle: \plainsubtitle}}}% set the PDF metadata title
}

% Redesign the title page
\renewcommand{\maketitlepage}{%
\cleardoublepage%
{%
\begin{fullwidth}%
\setlength{\parindent}{0pt}%
\fontsize{20}{24}\selectfont\textit{\plainauthor}\par
\vspace{1.75in}\fontsize{48}{54}\selectfont{\plaintitle}\par
\vspace{0.25in}\fontsize{20}{24}\selectfont{\plainsubtitle}\par
\vfill\fontsize{14}{14}\selectfont\textit{\plainpublisher}\par
\end{fullwidth}%
}%
\thispagestyle{empty}%
\clearpage%
}

\title{Tactile Paths}
\subtitle{on and through Notation for Improvisers}
\author{Christopher Williams}
\publisher{Academy of Creative and Performing Arts, University of Leiden}

\begin{document}

\frontmatter

\tableofcontents

\mainmatter

\chapter[Entextualization and Preparation in Patterson's \textit{Variations~for~Double-Bass}]{\texorpdfstring{Entextualization and Preparation in Patterson's \\ \textit{Variations~for~Double-Bass}}{Entextualization and Preparation in Patterson's \textit{Variations~for~Double-Bass}}} \marginnote[-2cm]{Media for this chapter may be found at \url{http://www.tactilepaths.net/patterson}}

\section{Introduction}\label{introduction}

Most composers of notation for improvisers are improvisers themselves.
This is no coincidence. As composer-pianist and improvisation scholar
Vijay Iyer has noted,

\begin{quote}
The most savvy composers writing for improvisors are those with personal
experience as improvisors -- those who possess an intimate understanding
of its parameters of expression, its interactive possibilities, and the
stakes involved in the commitment to process.\footnote{Throughout
the dissertation I use the -er'' spelling of the word
improviser''. However I respect the alternate -or'' spelling in
citations by others who choose this variant. Likewise I respect
British spelling norms in citations, while adopting American English
norms for myself.}
\end{quote} (Iyer 2004)

How, why, and what these artists notate can vary substantially. Some
transmit aspects of their own practice as improvisers (Malcolm
Goldstein); some develop it privately (Derek Bailey -- see Lash 2011);
and others agitate their ensembles (Misha Mengelberg -- see Schuiling
2016 and Whitehead 1998). Some luxuriate in the gray area between the
written and the improvised (Bob Ostertag), and others inscribe a gap
(see Richard Barrett's \emph{Blattwerk} (2002) or my \emph{Apples are Basic}(2008).
But in all these cases, the matter of what gets notated is nearly always
intertwined with ongoing improvisational practice.

The present chapter aims to articulate the dynamics of this intertwining
-- the process of inscription. It centers on the following question: how
do composer-improvisers use notation to share, challenge, or transform
their own ways of improvising?

By tracing my study, preparation, and multiple performances of Ben
Patterson's \emph{Variations for Double-Bass} (1962), I will flesh out a
deceptively straightforward answer: notation for improvisers
\emph{entextualizes} the ongoing improvisatory practices of its
composers and/or performers. Borrowing from anthropologist Karin Barber,
who, after Michael Silverstein and Greg Urban (1996), defines
entextualization as the process of rendering a given instance of
discourse as text, detachable from its local context'\,'' (Barber 2007,
30), I will pursue two related claims: (1) notation in this piece
emerges from and feeds back on improvisation, rather than simply
generating or freezing it, and (2) improvisation in this piece is a
continuous thread throughout processes of score-making, preparing and
rehearsing, and revision after performance -- above and beyond its most
obvious manifestation in concert performance. \emph{Variations} offers a
unique opportunity to address these issues, as it foregrounds aspects of
inscription, preparation, and revision that are often too private or
ephemeral to trace in other scores for improvisers.

\section{Variations for Double-Bass: Background}\label{variations-for-double-bass-background}

I first learned about Ben Patterson's \emph{Variations for Double-Bass}
in Fluxus\footnote{Fluxus is (or, according to some, was) a
heterogeneous network of artists including Patterson, George Brecht,
Geoff Hendricks, Dick Higgins, Alison Knowles, Yoko Ono, and many
others, founded'' in the early 1960s by George Maciunas.
Fluxdaughter and historian Hannah Higgins states, Since Fluxus
artists never seem to agree on anything, Fluxus has become a pain in
art's ass,' in the words of Fluxus artist Ben Vautier. Neither the
style nor the substance or significance of what they do produces
consensus among the artists. Production ranges from minimal
performances, called Events, to full-scale operas, and from graphics
and boxed multiples called Fluxkits to paintings on canvas. The
artists come from almost every industrialized nation, they span
several generations, and many even dislike each other. Accurately
portraying Fluxus therefore requires thinking about art in a way that
forgoes the normally definitive terms of style, medium, and political
sensibility.'' (2002, xiii)} catalogs and histories -- the piece has
something of a legendary aura around it. Composer and experimental music
scholar George E. Lewis describes it as going well beyond any previous
notion of extended technique then in force in the world of contemporary
classical music'' (2014, 95); elsewhere it is referred to, along with
other early pieces by Patterson such as \emph{Paper Piece} (Stegmayer
2012, 59-61), as a historical contribution to black performance
art.\footnote{See
\url{http://13.performa-arts.org/artists/benjamin-patterson}.} It was
premiered at Mary Bauermeister's Cologne salon in 1961, subsequently
performed at the famous Wiesbaden Fluxus exhibition of 1962\footnote{See
\emph{Fluxus Festival neuester Musik}
\url{http://www.hundertmark-gallery.com/videos.0.html}.}, and is now
considered by most scholars and enthusiasts to be a Fluxus classic.

So it was with some surprise that my request to Patterson for a score in
2001 was answered with an unceremonious thicket of typewritten text,
handwritten comments, unexplained Xs and arrows, cut and pasted
fragments from Verdi's \emph{Rigoletto}, editorial scribbles, and
photographs from an early performance. Its provenance was Patterson's
\emph{Black and White File} (1999), a working file'' (Patterson 1999,
Overview'') of the composer's scores from 1960-1999. Unpaginated and
bound in a simple two-ring binder of the type used by Germans for the
most banal of record-keeping, the seventeen \emph{Variations} are
unassumingly sandwiched between \emph{Duo} (1961), for voice and string
instrument, and \emph{Paper Piece} (1960), for an unspecified number of
performers playing (with) paper.

The humble presentation of this Do-It-Yourself collection stands in
marked contrast to scores by many of Patterson's Fluxus contemporaries
such as George Brecht's \emph{Water Yam} (1963) or the La Monte
Young-edited volume \emph{An Anthology of Chance Operations} (1963).
Compared with these Fluxus archetypes, both meticulously designed and
packaged by the movement's self-appointed spokesman George Maciunas,
\emph{Variations} seems more like a leaky sketch than a polished,
autonomous art object. Its informality is emphasized by its distribution
history: from 1999 to 2012, Patterson produced copies of the \emph{Black
and White File} to order and often gave them away for free, up until his
scores were published together in an anthology edited by Benedikt
Stegmayer (2012).

In my opinion Patterson's direct, unfussy approach to writing,
publishing, and distributing his scores does not reflect a lack of care;
nor is it merely cosmetic. In choosing to work this way Patterson
underlines practice's primacy, and notation's embeddedness within it:

\begin{quote}
My pieces, as they appear on paper, have neither material nor abstract value. They achieve value in performance, and then only the personal value that the participant himself perceives about his own behavior and / or that of the society during and / or after the experience. (In fact, any piece is just this: a person, who, consciously, does this or that. Everybody can do it.) (Patterson, as quoted in Lewis 2012, 988)
\end{quote}

Having performed \emph{Variations} on several occasions, I can attest
that the principle value of Patterson's notation is indeed personal and
reflective; one discovers this both in and \emph{en route} to concert
performance. However whereas Patterson implies that \emph{Variations'}
notation on paper'' is simply a means to an end -- prescribing and
preserving the piece for performance -- my own experience playing the
piece has revealed that the score is something more. In addition to
providing the performer a set of instructions or generative conditions
for performance, it is also itself performative, and the nature of that
performance -- always shifting, contingent, and reflective -- might be
best termed improvisation.

\section{Entextualization}\label{entextualization}

Explaining the performative, improvisational nature of
\emph{Variations}' notation -- as well as that of any other example of
notation for improvisers -- can benefit from the notion of
entextualization. A brief historical contextualization of the term
suggests why.

The concept emerged in the field of anthropology in the 1980s and 1990s
as a response to the rise of performance theory. According to Karin
Barber, performance theory sought to provide an alternative to the then
dominant view of culture as text:

\begin{quote}
Text implied a view of society as prescriptive, fixed and adhering to
rigid structures; performance implied a focus on what was improvised,
ephemeral, fluid, of the moment only -- but in that moment, vital and
responsive to the contingencies of context. {[}\ldots{}{]} Dwight
Conquergood elegantly sums up the opposition as a war of vocabulary,
where the benign forces of improvisation'', flow'', process'',
participation'', embodiment'', and dialogue'' are ranged against
the enemy lexicon -- fixity'', structure'', objectification'',
reification'', system'', distance'', and detachment''
(Conquergood 1989). {[}\ldots{}{]}
\end{quote}

\begin{quote}
But while performance theory provided wonderful conceptual tools for
thinking about emergence, it had a tendency to dismiss the whole idea of
the aspiration to fixity as a scriptocentric imposition. It thus offered
inadequate resources for understanding how the fluid is consolidated,
and why stunningly creative oral performers so often claim that their
texts have never changed by so much as a syllable. But out of
performance theory came its own inverse and complement: the concept of
entextualizuation''. {[}\ldots{}{]}
\end{quote}

\begin{quote}
The concept of entextualization {[}\ldots{}{]} opens the way to an
integrated vision of the generation of cultural forms from the bottom
up, in which misleadingly sharp binary oppositions can be allowed to
\end{quote}

This performative turn'' (Conquergood 1989) in anthropology (as well
as in theater studies and other disciplines in the humanities) has
resonated in music scholarship over the last twenty-five or so years.
New Musicology'' (also currently known as Critical Musicology''),
music performance studies, and in particular the field of improvisation
studies, of which I consider this dissertation to be a part, have all
turned away from hard textuality and embraced the role of agency,
contingency, collectivity, corporeality, and intersubjectivity in
musical discourse. Musicologist Floris Schuiling summarizes the shift
thus:

\begin{quote}
During the 1990s, musicologists increasingly started to address music's
entanglement with social and political issues, as well as the
ideological baggage that had prevented this earlier, and ethnomusicology
was swept up in the more general reflexive turn in anthropology that
problematized the notion of culture'' and the nature of ethnography
and fieldwork. In both fields, one outcome of these developments was a
shift in emphasis towards the concept of performance, to avoid either a
work-based approach or a totalising concept of culture, and to
foreground the forms of social and creative interaction that were now
increasingly seen to be essential to music's existence. (Schuiling 2016,
42-43)
\end{quote}

\end{document}

I can't understand what they refer to, but I believe it may be related to a chapter heading. The PDF looks right, but the sections after this point in the code stop showing up in the document viewer index, and subsequent chapters don't show up in the TOC.

Can anyone help? Many thanks.

• I get some warnings about underful boxes, but not 'errors' -- The line numbers given by you are quite ambigious -- it is not clear which lines are meant. What is the error? – user31729 Sep 8 '16 at 21:23
• here are the errors: pastebin.com/RB5pgRS2 – christopher Sep 8 '16 at 21:33
• I don't get this errors. I've compiled on TL 2016, on Linux with pdflatex. – user31729 Sep 8 '16 at 21:38
• On pdflatex I don't get the errors either... they came running latex from within texmaker on linux. Was I compiling incorrectly? – christopher Sep 8 '16 at 21:44
• As far as I can see you are not using any features that restricts you to latex. Use pdflatex rather in this case! – user31729 Sep 8 '16 at 21:49


text:
Exponential Discounting

Most of us seem to agree that the promise of a dollar in the future is worth less to us than a dollar today, even if the promise is certain to be fulfilled. Economists often assume ‘exponential discounting’, which says that a dollar promised at some time $s$ is worth

$\exp(-\alpha(s - t))$

dollars in hand at time $t.$ The constant $\alpha$ is connected to the ‘interest rate’.

Why are economists so wedded to exponential discounting? The main reason is probably that it’s mathematically simple. But one argument for it goes roughly like this: if your decisions today are to look rational at any future time, you need to use exponential discounting.

In practice, humans, pigeons and rats do not use exponential discounting. So, economists say they are ‘dynamically inconsistent’:

• Wikipedia, Dynamic inconsistency.

In economics, dynamic inconsistency or time inconsistency is a situation in which a decision-maker’s preferences change over time in such a way that a preference can become inconsistent at another point in time. This can be thought of as there being many different “selves” within decision makers, with each “self” representing the decision-maker at a different point in time; the inconsistency occurs when not all preferences are aligned.

I this ‘inconsistent’ could be a misleading term for what’s going on here. It suggests that something bad is happening. That may not be true.

Anyway, some of the early research on this was done by George Ainslie, and here is what he found:

Ainslie’s research showed that a substantial number of subjects reported that they would prefer $50 immediately rather than$100 in six months, but would NOT prefer $50 in 3 months rather than$100 in nine months, even though this was the same choice seen at 3 months’ greater distance. More significantly, those subjects who said they preferred $50 in 3 months to$100 in 9 months said they would NOT prefer $50 in 12 months to$100 in 18 months—again, the same pair of options at a different distance—showing that the preference-reversal effect did not depend on the excitement of getting an immediate reward. Nor does it depend on human culture; the first preference reversal findings were in rats and pigeons.

Let me give a mathematical argument for exponential discounting. Of course it will rely on some assumptions. I’m not claiming these assumptions are true! Far from it. I’m just claiming that if we don’t use exponential discounting, we are violating one or more of these assumptions… or breaking out of the whole framework of my argument. The widespread prevalence of ‘dynamic inconsistency’ suggests that the argument doesn’t apply to real life.

Here’s the argument:

Suppose the value to us at any time $t$ of a dollar given to us at some other time $s$ is $V(t,s).$

Let us assume:

1) The ratio

$\displaystyle{ \frac{V(t,s_2)}{V(t,s_1)} }$

is independent of $t.$ E.g., the ratio of value of a “dollar on Friday” to “a dollar on Thursday” is the same if you’re computing it on Monday, or on Tuesday, or on Wednesday.

2) The quantity $V(t,s)$ depends only on the difference $s - t.$

3) The quantity $V(t,s)$ is a continuous function of $s$ and $t.$

Then we can show

$V(t,s) = k \exp(-\alpha(s-t))$

for some constants $\alpha$ and $k.$ Typically we assume $k = 1$ since the value of a dollar given to us right now is 1. But let’s just see how we get this formula for $V(t,s)$ out of assumptions 1), 2) and 3).

The proof goes like this. By 2) we know

$V(t,s) = F(s-t)$

for some function $F$. By 1) it follows that

$\displaystyle{ \frac{F(s_2 - t)}{F(s_1 - t)} }$

is independent of $t,$ so

$\displaystyle{ \frac{F(s_2 - t)}{F(s_1 - t)} = \frac{F(s_2)}{F(s_1)} }$

or in other words

$F(s_2 - t) F(s_1) = F(s_2) F(s_1 - t)$

Ugh! What next? Well, if we take $s_1 = t,$ we get a simpler equation that’s probably still good enough to get the job done:

$F(s_2 - t) F(t) = F(s_2) F(0)$

Now let’s make up a variable $t' = s_2 - t,$ so that $s_2 = t + t'.$ Then we can rewrite our equation as

$F(t') F(t) = F(t+t') F(0)$

or

$F(t) F(t') = F(t+t') F(0)$

This is beautiful except for the constant $F(0).$ Let’s call that $k$ and factor it out by writing

$F(t) = k G(t)$

Then we get

$G(t) G(t') = G(t+t')$

A theorem of Cauchy implies that any continuous solution of this equation is of the form

$G(t) = \exp(-\alpha t)$

So, we get

$F(t) = k \exp(-\alpha t)$

or

$V(t,s) = k \exp(-\alpha(s-t))$

as desired!

By the way, we don’t need to assume $G$ is continuous: it’s enough to assume $G$ is measurable. You can get bizarre nonmeasurable solutions of $G(t) G(t') = G(t+t')$ using the axiom of choice, but they are not of practical interest.

So, assumption 3) is not the assumption I’d want to attack in trying to argue against exponential discounting. In fact both assumptions 1) and 2) are open to quite a few objections. Can you name some? Here’s one: in real life the interest rate changes with time.

By the way, nothing in the argument I gave shows that $\alpha \ge 0.$ So there could be people who obey assumptions 1)–3) yet believe the promise of a dollar in the future is worth more than a dollar in hand today.

Also, nothing in my argument for the form of $V(t,s)$ assumes that $s \ge t.$ That is, my assumptions as stated also concern the value of a dollar that was promised in the past. So, you might have fun seeing what changes, or does not change, if you restrict the assumptions to say they only apply when $s \ge t.$ The arrow of time seems to be built into economics, after all.

Also, you may enjoy finding the place in my derivation where I might have divided by zero, and figure out to do about that.

If you don’t like exponential discounting—for example, because people use it to argue against spending money now to fight climate change—you might prefer hyperbolic discounting:

• Wikipedia, Hyperbolic discounting.


text:
Lemma 34.6.7. Let $\mathit{Sch}_{syntomic}$ be a big syntomic site as in Definition 34.6.6. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{syntomic})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary syntomic covering of $T$.

1. There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{syntomic}$ which refines $\{ T_ i \to T\} _{i \in I}$.

2. If $\{ T_ i \to T\} _{i \in I}$ is a standard syntomic covering, then it is tautologically equivalent to a covering in $\mathit{Sch}_{syntomic}$.

3. If $\{ T_ i \to T\} _{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering in $\mathit{Sch}_{syntomic}$.

Proof. For each $i$ choose an affine open covering $T_ i = \bigcup _{j \in J_ i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemma 34.6.3 the refinement $\{ T_{ij} \to T\} _{i \in I, j \in J_ i}$ is a syntomic covering of $T$ as well. Hence we may assume each $T_ i$ is affine, and maps into an affine open $W_ i$ of $T$. Applying Sets, Lemma 3.9.9 we see that $W_ i$ is isomorphic to an object of $\mathit{Sch}_{syntomic}$. But then $T_ i$ as a finite type scheme over $W_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{syntomic}$ by a second application of Sets, Lemma 3.9.9. The covering $\{ V_ i \to T\} _{i \in I}$ refines $\{ T_ i \to T\} _{i \in I}$ (because they are isomorphic). Moreover, $\{ V_ i \to T\} _{i \in I}$ is combinatorially equivalent to a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{syntomic}$ by Sets, Lemma 3.9.9. The covering $\{ U_ j \to T\} _{j \in J}$ is a covering as in (1). In the situation of (2), (3) each of the schemes $T_ i$ is isomorphic to an object of $\mathit{Sch}_{syntomic}$ by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
# Classification of abelianness-forcing numbers

## Name

This result is attributed to Dickson, and is hence also called Dickson's theorem, though there are many results with that name.

## Statement

The following are equivalent for a natural number $n$:

1. Any group of order $n$ is an abelian group.
2. $n$ has prime factorization of the form $n = p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}$ with $k_i \le 2$ for all $i$ AND $p_i$ does not divide $p_j^{k_j} - 1$ for any $1 \le i,j \le r$.

## Proof

### (1) implies (2)

It suffices to prove the contrapositive, namely, that violating the conditions of (2) allows one to construct a non-abelian group of that order. There are two subcases.

#### Case of a prime-cube dividing the number

Given: A natural number $n$ such that there exists a prime number $p$ such that $p^3$ divides $n$

To prove: There exists a non-abelian group of order $n$

Proof: First, by classification of groups of prime-cube order, there exists a non-abelian group of order $p^3$. Specifically, we can take the unitriangular matrix group:UT(3,p). Call this group $H$.

Now define:

$G = H \times \mathbb{Z}/(n/p^3)\mathbb{Z}$

In other words, $G$ is the external direct product of this non-abelian group with a cyclic group of order $n/p^3$. We obtain that $G$ is non-abelian of order $n$.

#### Case of the divisibility condition being violated

Given: A natural number $n$, such that there exist distinct primes $p,q$ and a natural number $j \in \{1 , 2 \}$ such that $p$ divides $n$, and $p$ divides $q^j - 1$.

To prove: There is a non-abelian group of order $n$

Proof: Note that the product $pq^j$ divides $n$.

We can construct a non-abelian group $H$ of order $pq^j$ as follows: consider the additive group of the field of size $q^j$. The multiplicative group of this field is a cyclic group of size $q^j - 1$. Since $p$ divides $q^j - 1$, it has a subgroup of order $p$. Construct a semidirect product of the additive group of order $q^j$ with this subgroup of order $p$.

We can now construct a non-abelian group $G$ of order $n$ as the external direct product:

$G = H \times \mathbb{Z}/(n/pq^j)\mathbb{Z}$

### (2) implies (1)

We prove the claim by induction on the number. Note that any divisor of a number that satisfies the condition also satisfies the condition.

Base case for induction ($n = 1$): Obvious

Inductive step: The inductive hypothesis is that the result holds for all smaller numbers.

Given: A natural number $n$ has prime factorization of the form $n = p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}$ with $k_i \le 2$ for all $i$ AND $p_i$ does not divide $p_j^{k_j} - 1$ for any $1 \le i,j \le r$. $G$ is a group of order $n$.

To prove: $G$ is abelian

Proof:

Step no. Assertion/construction Facts used Give data used Previous steps used Explanation
1 Every proper subgroup of $G$ is abelian Fact (1) inductive hypothesis
arithmetic condition on $n$
Any divisor of $n$ satisfies the conditions necessary for the inductive hypothesis to apply. Applying the inductive hypothesis, we obtain that every subgroup is abelian.
2 $G$ is metabelian; specifically, either $G$ is abelian or it contains an abelian maximal normal subgroup $N$ such that the quotient group $G/N$ is cyclic of prime order $p$ for some $p$ dividing $n$ Fact (2) Step (1) Step-fact combination direct
3 If $G$ is non-abelian, the maximal normal subgroup $N$ has an automorphism $\sigma$ of order $p$ Step (2) [SHOW MORE]
4 $G$ must be abelian condition on no $p_i$ dividing $p_j^{k_j} - 1$ and all $k_i \le 2$ Step (3) [SHOW MORE]


text:
# How do you graph f(x)= 2+csc(2x+2)?

Dec 10, 2017

As shown below:

#### Explanation:

The first thing we must consider is the transformations that $\csc x$ must undergo to reach $2 + \csc \left(2 x + 2\right)$

So first it must be translated by $\left(0 , 2\right)$ or in other words, shifted upward by 2 units: yielding $2 + \csc x$

Then stretched by a scale factor of $\frac{1}{2}$ in the $x$ direction:
yielding $2 + \csc \left(2 x\right)$

Then translate by the vector $\left(- 1 , 0\right)$ to yield: $2 + \csc \left(2 \left(x + 1\right)\right)$
$= 2 + \csc \left(2 x + 2\right)$

$\implies$

$\csc x :$ graph{cscx [-10, 10, -5, 5]}

$2 + \csc x$: graph{cscx + 2 [-9.71, 10.29, -2.32, 7.68]}

$2 + \csc 2 x$ : graph{2+ csc(2x) [-9.71, 10.29, -2.32, 7.68]}

$2 + \csc \left(2 x + 2\right)$: graph{2+csc(2x+2) [-9.71, 10.29, -2.32, 7.68]}


text:
# Modality (philosophy)

In philosophy, modality describes the way something is, happens or is thought.

## Concept and types of modality

### Modalities in the strict sense

In the narrower, classic sense, the term modality denotes the alethic (truth-related) modality: in ontological terms, it is the way in which a state of affairs exists or, more logically, the truth of statements .

In addition to the simple (factual) truth, a distinction is usually made between necessity , possibility and impossibility and contingency .

The concept of the (alethic) modality is already used by the commentators on Aristotle .

According to Kant , categories of modality are possibility, reality and necessity, to which the modality of judgments (problematic (possible), assertoric (real), apodictic (necessary)) should correspond.

Leibniz differentiates between four basic categories of types of statements or facts : necessary, possible, impossible and contingently true (actual).

Classical theorists as well as modern semantics , philosophy of language , logic (especially modal logic ) and ontology have proposed different theories about the nature of these statements and their truth makers .

### Modalities in the broader sense

In addition to the alethic modalities mentioned, one also speaks of the modalities in the broader sense and means (among other things) the ways in which a statement can be true in relation to the knowledge of the telling, the time or the ought. This covers the doxastic (epistemic), temporal or deontic modalities.

### Overview

The different modalities can be traced back to each other, are in an analogous relationship and justify different modal logics :

 formula Modal logic i. e. S. Deontic logic Temporal logic Doxastic logic ${\ displaystyle \ Diamond}$ p It is possible that p It is permissible that p p applies sometime in the future (past) I think it is possible that p ${\ displaystyle \ Box}$ p It is necessary that p It is imperative that p p is always valid in the future (past) I consider it certain that p

## Modality and Possible Worlds Theory

Following Leibniz, Rudolf Carnap ( Meaning and Necessity 1947), Saul Kripke and David Kellogg Lewis suggested that modal statements should be understood as statements about or in possible worlds . A necessary truth is true in all of these worlds, a possible truth in at least one. These worlds, according to Lewis, are real and even concrete objects, not just concepts in the mind or abstract universals. This ontologically demanding option has theoretical advantages because it enormously simplifies the semantic evaluation of modal statements . In addition, there are ontologically irreducible relationships between these possible worlds, which makes the evaluation of conditionalities (if ... then ...) and v. a. of counterfactual conditionalities (if it weren't… then…) are very elegantly simplified. Nevertheless, many ontologists are not ready to add so many additional irreducible objects to their ontological inventory. Some therefore try to maintain the theoretical elegance of many-worlds semantics, but propose anti- realistic substitute theories regarding the ontological dignity of these worlds.

## Theories about actuality

Theories about actuality are a separate subject area in these debates : What is constitutive for the fact that the actual world is the actual and no other possible world? A much-discussed proposal understands "aktual" as indexical expression when: I say that a situation is aktual, then I say that he listened to the world that I (the speaker) inhabit (and I now their inventory with your finger show can ). (According to Lewis, who takes this position, there is no identity of objects across different worlds (so-called transworld identity), only counterparts; there is therefore only one speaker.) Alternatively, for example, that world can be determined as actual, in which we do the evaluation of an utterance .

## literature

• Nicolai Hartmann: Possibility and Reality. de Gruyter, Berlin 1938.
• Robert Merrihew Adams: Theories of Actuality. In: Noûs , 8/3 (1974), pp. 211-231, also in: Loux 1979.
• John Divers: Possible Worlds. Routledge, 2002.
• David Kellogg Lewis : On the Plurality of Worlds. Blackwell, 1986.
• Michael J. Loux (Ed.): The Possible and the Actual. Cornell University Press, Ithaca NY 1979.
• Uwe Meixner: Modality. Possibility, necessity, essentialism . Klostermann, Frankfurt am Main 2008, ISBN 978-3-465-04050-7 .
• Alvin Plantinga : Actualism and Possible Worlds. In: Theoria 42 / 1-3 (1976), pp. 139-160, also in: ders .: Essays in the Metaphysics of Modality. Matthew Davidson (Ed.): Oxford University Press, New York NY 2003, pp. 103–121 and also in Loux 1979.
• Alvin Plantinga: Two Concepts of Modality: Modal Realism and Modal Reductionism. In: James E. Tomberlin (ed.): Philosophical Perspectives , 1 (1987), Atascadero, CA: Ridgeview, pp. 189-231, also in: Plantinga 2003, 192-228.
• Alexander R. Pruss: The Actual and the Possible . In: Richard M. Gale (Ed.): The Blackwell Guide to Metaphysics . Blackwell 2002.
• Ted Sider : Reductive Theories of Modality (PDF; 310 kB). In: MJ Loux, DW Zimmerman (Eds.): The Oxford Handbook of Metaphysics. OUP 2003.
• Robert Stalnaker : Possible Worlds. In: Noûs , 10/1 (1976), pp. 65-75 also in Loux 1979.
• Peter van Inwagen : Two Concepts of Possible Worlds. In: Peter A. French, Theodore E. Uehling, Jr., Howard K. Wettstein (Eds.): Midwest Studies in Philosophy 11 (Studies in Essentialism), University of Minnesota Press, Minneapolis MN 1986, pp. 185-213; also in van Inwagen 2001, pp. 206–242.
• Peter van Inwagen: Ontology, Identity, and Modality : Essays in Metaphysics. Cambridge Studies in Philosophy. Cambridge University Press, Cambridge 2001.


text:
# zbMATH — the first resource for mathematics

Compute Distance To:
 Documents Indexed: 40 Publications since 1984 Biographic References: 1 Publication
all top 5

#### Co-Authors

 5 single-authored 12 Davis, Michael W. 7 Świątkowski, Jacek 5 Dymara, Jan 4 Leary, Ian James 4 Okun, Boris L. 3 Derdzinski, Andrzej 2 Behrstock, Jason A. 2 Gal, Światosław R. 2 Neumann, Walter David 2 Scott, Richard Allan 1 Arzhantseva, Goulnara N. 1 Bozejko, Marek 1 Bridson, Martin R. 1 Buczyński, Jarosław 1 Dobrowolski, Edward 1 Dranishnikov, Alexander Nikolaevich 1 Elsner, Tomasz 1 Hajduk, Bogusław 1 Jelisiejew, Joachim 1 Kropholler, Peter H. 1 Lafont, Jean-François 1 Meier, John 1 Michałek, Mateusz 1 Minasyan, Ashot 1 Spatzier, Ralf J. 1 Weinberger, Shmuel
all top 5

#### Serials

 3 Proceedings of the American Mathematical Society 3 Geometry & Topology 2 Bulletin of the London Mathematical Society 2 Commentarii Mathematici Helvetici 2 Duke Mathematical Journal 2 Journal of Differential Geometry 2 Topology 2 Annals of Global Analysis and Geometry 2 Algebraic & Geometric Topology 2 Groups, Geometry, and Dynamics 1 Advances in Mathematics 1 Colloquium Mathematicum 1 Fundamenta Mathematicae 1 Geometriae Dedicata 1 Publications Mathématiques 1 Inventiones Mathematicae 1 Journal of Pure and Applied Algebra 1 Transactions of the American Mathematical Society 1 Journal of Operator Theory 1 Discrete & Computational Geometry 1 IMRN. International Mathematics Research Notices 1 Topology Proceedings 1 Journal of Lie Theory 1 Selecta Mathematica. New Series 1 Journal of the European Mathematical Society (JEMS) 1 Journal of Topology
all top 5

#### Fields

 25 Manifolds and cell complexes (57-XX) 21 Group theory and generalizations (20-XX) 11 Differential geometry (53-XX) 6 Convex and discrete geometry (52-XX) 3 Algebraic geometry (14-XX) 3 Topological groups, Lie groups (22-XX) 3 Abstract harmonic analysis (43-XX) 3 Algebraic topology (55-XX) 2 Commutative algebra (13-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Functional analysis (46-XX) 2 Geometry (51-XX) 2 Global analysis, analysis on manifolds (58-XX) 1 Combinatorics (05-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Number theory (11-XX) 1 Dynamical systems and ergodic theory (37-XX)

#### Citations contained in zbMATH Open

34 Publications have been cited 625 times in 534 Documents Cited by Year
Convex polytopes, Coxeter orbifolds and torus actions. Zbl 0733.52006
1991
Cohomology of buildings and of their automorphism groups. Zbl 1140.20308
2002
Hyperbolization of polyhedra. Zbl 0723.57017
1991
Simplicial nonpositive curvature. Zbl 1143.53039
2006
Right-angled Artin groups are commensurable with right-angled Coxeter groups. Zbl 0982.20022
2000
Hyperbolic Coxeter groups of large dimension. Zbl 1068.20043
2003
Every Coxeter group acts amenably on a compact space. Zbl 0973.20029
Dranishnikov, A.; Januszkiewicz, T.
1999
Nonpositive curvature of blow-ups. Zbl 0924.53033
Davis, M.; Januszkiewicz, T.; Scott, R.
1998
Filling invariants of systolic complexes and groups. Zbl 1188.20043
2007
Fundamental groups of blow-ups. Zbl 1080.52512
Davis, M.; Januszkiewicz, T.; Scott, R.
2003
Weighted $$L^2$$-cohomology of Coxeter groups. Zbl 1173.20029
Davis, Michael W.; Dymara, Jan; Januszkiewicz, Tadeusz; Okun, Boris
2007
Infinite Coxeter groups do not have Kazhdan’s property. Zbl 0662.20040
Bożejko, M; Januszkiewicz, T.; Spatzier, R. J.
1988
For Coxeter groups $$z^{| g |}$$ is a coefficient of a uniformly bounded representation. Zbl 1038.20025
2002
Infinite groups with fixed point properties. Zbl 1197.20034
Arzhantseva, Goulnara; Bridson, Martin R.; Januszkiewicz, Tadeusz; Leary, Ian J.; Minasyan, Ashot; Świątkowski, Jacek
2009
Commensurability of graph products. Zbl 0998.20029
2001
Characteristic invariants of noncompact Riemannian manifolds. Zbl 0594.57011
1984
Quasi-isometric classification of some high dimensional right-angled Artin groups. Zbl 1226.20033
Behrstock, Jason A.; Januszkiewicz, Tadeusz; Neumann, Walter D.
2010
New Kazhdan groups. Zbl 0984.22002
2000
Compactly supported cohomology of buildings. Zbl 1281.20046
Davis, Michael W.; Dymara, Jan; Januszkiewicz, Tadeusz; Meier, John; Okun, Boris
2010
The $$\ell^2$$-cohomology of hyperplane complements. Zbl 1148.52012
Davis, M. W.; Januszkiewicz, T.; Leary, I. J.
2007
Cohomology of Coxeter groups with group ring coefficients. II. Zbl 1153.20038
Davis, Michael W.; Dymara, Jan; Januszkiewicz, Tadeusz; Okun, Boris
2006
Constructions of $$k$$-regular maps using finite local schemes. Zbl 1418.53005
Buczyński, Jarosław; Januszkiewicz, Tadeusz; Jelisiejew, Joachim; Michałek, Mateusz
2019
4-dimensional locally $$\mathrm{CAT}(0)$$-manifolds with no Riemannian smoothings. Zbl 1237.57015
Davis, M.; Januszkiewicz, T.; Lafont, J.-F.
2012
Cohomology of hyperplane complements with group ring coefficients. Zbl 1222.52024
Davis, Michael W.; Januszkiewicz, Tadeusz; Leary, Ian J.; Okun, Boris
2011
Relative hyperbolization and aspherical bordisms: An addendum to: “Hyperbolization of polyhedra”. Zbl 1028.57020
Davis, Michael W.; Januszkiewicz, Tadeusz; Weinberger, Shmuel
2001
Non-positively curved developments of billiards. Zbl 1244.37024
2010
For right-angled Coxeter groups $$z^{| g|}$$ is a coefficient of a uniformly bounded representation. Zbl 0824.20037
1993
Totally real immersions of surfaces. Zbl 1185.53065
2010
Commensurability and QI classification of free products of finitely generated Abelian groups. Zbl 1183.20025
Behrstock, Jason A.; Januszkiewicz, Tadeusz; Neumann, Walter D.
2009
New a-T-menable HNN-extensions. Zbl 1041.20028
2003
Even- vs. odd-dimensional Charney-Davis conjecture. Zbl 1208.52013
2010
Immersions of surfaces in spin$$^c$$-manifolds with a generic positive spinor. Zbl 1067.53037
2004
Arithmetic of an elliptic curve and circle actions on four-manifolds. Zbl 0851.57020
1993
Cobordism of polarized t-structures. Zbl 0770.57018
1992
Constructions of $$k$$-regular maps using finite local schemes. Zbl 1418.53005
Buczyński, Jarosław; Januszkiewicz, Tadeusz; Jelisiejew, Joachim; Michałek, Mateusz
2019
4-dimensional locally $$\mathrm{CAT}(0)$$-manifolds with no Riemannian smoothings. Zbl 1237.57015
Davis, M.; Januszkiewicz, T.; Lafont, J.-F.
2012
Cohomology of hyperplane complements with group ring coefficients. Zbl 1222.52024
Davis, Michael W.; Januszkiewicz, Tadeusz; Leary, Ian J.; Okun, Boris
2011
Quasi-isometric classification of some high dimensional right-angled Artin groups. Zbl 1226.20033
Behrstock, Jason A.; Januszkiewicz, Tadeusz; Neumann, Walter D.
2010
Compactly supported cohomology of buildings. Zbl 1281.20046
Davis, Michael W.; Dymara, Jan; Januszkiewicz, Tadeusz; Meier, John; Okun, Boris
2010
Non-positively curved developments of billiards. Zbl 1244.37024
2010
Totally real immersions of surfaces. Zbl 1185.53065
2010
Even- vs. odd-dimensional Charney-Davis conjecture. Zbl 1208.52013
2010
Infinite groups with fixed point properties. Zbl 1197.20034
Arzhantseva, Goulnara; Bridson, Martin R.; Januszkiewicz, Tadeusz; Leary, Ian J.; Minasyan, Ashot; Świątkowski, Jacek
2009
Commensurability and QI classification of free products of finitely generated Abelian groups. Zbl 1183.20025
Behrstock, Jason A.; Januszkiewicz, Tadeusz; Neumann, Walter D.
2009
Filling invariants of systolic complexes and groups. Zbl 1188.20043
2007
Weighted $$L^2$$-cohomology of Coxeter groups. Zbl 1173.20029
Davis, Michael W.; Dymara, Jan; Januszkiewicz, Tadeusz; Okun, Boris
2007
The $$\ell^2$$-cohomology of hyperplane complements. Zbl 1148.52012
Davis, M. W.; Januszkiewicz, T.; Leary, I. J.
2007
Simplicial nonpositive curvature. Zbl 1143.53039
2006
Cohomology of Coxeter groups with group ring coefficients. II. Zbl 1153.20038
Davis, Michael W.; Dymara, Jan; Januszkiewicz, Tadeusz; Okun, Boris
2006
Immersions of surfaces in spin$$^c$$-manifolds with a generic positive spinor. Zbl 1067.53037
2004
Hyperbolic Coxeter groups of large dimension. Zbl 1068.20043
2003
Fundamental groups of blow-ups. Zbl 1080.52512
Davis, M.; Januszkiewicz, T.; Scott, R.
2003
New a-T-menable HNN-extensions. Zbl 1041.20028
2003
Cohomology of buildings and of their automorphism groups. Zbl 1140.20308
2002
For Coxeter groups $$z^{| g |}$$ is a coefficient of a uniformly bounded representation. Zbl 1038.20025
2002
Commensurability of graph products. Zbl 0998.20029
2001
Relative hyperbolization and aspherical bordisms: An addendum to: “Hyperbolization of polyhedra”. Zbl 1028.57020
Davis, Michael W.; Januszkiewicz, Tadeusz; Weinberger, Shmuel
2001
Right-angled Artin groups are commensurable with right-angled Coxeter groups. Zbl 0982.20022
2000
New Kazhdan groups. Zbl 0984.22002
2000
Every Coxeter group acts amenably on a compact space. Zbl 0973.20029
Dranishnikov, A.; Januszkiewicz, T.
1999
Nonpositive curvature of blow-ups. Zbl 0924.53033
Davis, M.; Januszkiewicz, T.; Scott, R.
1998
For right-angled Coxeter groups $$z^{| g|}$$ is a coefficient of a uniformly bounded representation. Zbl 0824.20037
1993
Arithmetic of an elliptic curve and circle actions on four-manifolds. Zbl 0851.57020
1993
Cobordism of polarized t-structures. Zbl 0770.57018
1992
Convex polytopes, Coxeter orbifolds and torus actions. Zbl 0733.52006
1991
Hyperbolization of polyhedra. Zbl 0723.57017
1991
Infinite Coxeter groups do not have Kazhdan’s property. Zbl 0662.20040
Bożejko, M; Januszkiewicz, T.; Spatzier, R. J.
1988
Characteristic invariants of noncompact Riemannian manifolds. Zbl 0594.57011
1984
all top 5

#### Cited by 536 Authors

 15 Choi, Suyoung 14 Davis, Michael W. 13 Masuda, Mikiya 13 Sarkar, Soumen 12 Dranishnikov, Alexander Nikolaevich 11 Bahri, Anthony P. 10 Januszkiewicz, Tadeusz 10 Lü, Zhi 9 Wiemeler, Michael 8 Okun, Boris L. 8 Osajda, Damian 8 Świątkowski, Jacek 7 Aĭzenberg, Anton Andreevich 7 Bendersky, Martin 7 Cohen, Frederick Ronald 7 Ershov, Mikhail V. 7 Gitler Hammer, Samuel Carlos 7 Kuroki, Shintarô 7 Panov, Taras E. 7 Przytycki, Piotr 7 Ray, Nigel 6 Iriye, Kouyemon 6 Kishimoto, Daisuke 6 Park, Hanchul 6 Scott, Richard Allan 5 Bukhshtaber, Viktor Matveevich 5 Caprace, Pierre-Emmanuel 5 Devadoss, Satyan Linus 5 Dymara, Jan 5 Franz, Matthias 5 Haglund, Frédéric 5 Huang, Jingyin 5 Lafont, Jean-François 5 López de Medrano, Santiago 5 Monod, Nicolas 5 Nowak, Piotr W. 5 Suh, Dong Youp 5 Thomas, Anne 4 Belegradek, Igor 4 Bell, Gregory C. 4 Carbone, Lisa J. 4 Chepoi, Victor D. 4 Elsner, Tomasz 4 Kim, Jin Hong 4 Kim, Sang-Hyun 4 La Luz, José 4 Notbohm, Dietrich 4 Oppenheim, Izhar 4 Theriault, Stephen D. 4 Uma, Vikraman 4 Valette, Alain 3 Adiprasito, Karim Alexander 3 Bader, Uri 3 Bestvina, Mladen 3 Caspers, Martijn 3 Civan, Yusuf 3 Denham, Graham 3 Gaiffi, Giovanni 3 Gaifullin, Alexander A. 3 Gal, Światosław R. 3 Grbić, Jelena 3 Hausmann, Jean-Claude 3 Holm, Tara S. 3 Jaikin-Zapirain, Andrei 3 Kaji, Shizuo 3 Kamnitzer, Joel 3 Kassabov, Martin 3 Kleiner, Bruce 3 Limonchenko, Ivan Yur’evich 3 Liu, Dengpin 3 Lück, Wolfgang 3 Maxim, Laurentiu G. 3 Minasyan, Ashot 3 Olbermann, Martin 3 Osin, Denis V. 3 Petersen, Henrik Densing 3 Poddar, Mainak 3 Rémy, Bertrand 3 Sapir, Mark Valentinovich 3 Song, Jongbaek 3 Stafa, Mentor 3 Suciu, Alexander I. 3 Wise, Daniel T. 3 Wu, Jie 3 Yu, Li 3 Živaljević, Rade T. 2 Allen, David H. 2 Baralić, Đorđe 2 Barnard, Emily 2 Bayeh, Marzieh 2 Belolipetsky, Mikhail V. 2 Berestovskiĭ, Valeriĭ Nikolaevich 2 Billera, Louis J. 2 Bloch, Anthony M. 2 Bozejko, Marek 2 Brešar, Boštjan 2 Buyalo, Sergei V. 2 Cao, Xiangyu 2 Capdeboscq, Inna 2 Carlsson, Gunnar E. ...and 436 more Authors
all top 5

#### Cited in 132 Serials

 45 Algebraic & Geometric Topology 24 Geometry & Topology 23 Geometriae Dedicata 23 Topology and its Applications 22 Transactions of the American Mathematical Society 20 Advances in Mathematics 18 Proceedings of the Steklov Institute of Mathematics 17 Proceedings of the American Mathematical Society 16 Osaka Journal of Mathematics 15 Groups, Geometry, and Dynamics 12 Journal of Pure and Applied Algebra 11 Journal of Algebra 10 Journal of Functional Analysis 9 Duke Mathematical Journal 9 Mathematische Annalen 9 Chinese Annals of Mathematics. Series B 9 International Journal of Algebra and Computation 8 Forum Mathematicum 8 Journal of Topology and Analysis 7 Mathematical Proceedings of the Cambridge Philosophical Society 6 Mathematische Zeitschrift 5 Inventiones Mathematicae 5 Journal of Combinatorial Theory. Series A 5 Michigan Mathematical Journal 5 Boletín de la Sociedad Matemática Mexicana. Third Series 5 Journal of Topology 4 Israel Journal of Mathematics 4 Russian Mathematical Surveys 4 Bulletin of the London Mathematical Society 4 Journal of the Mathematical Society of Japan 4 Journal of the American Mathematical Society 4 Selecta Mathematica. New Series 4 Transformation Groups 4 Comptes Rendus. Mathématique. Académie des Sciences, Paris 3 Manuscripta Mathematica 3 Memoirs of the American Mathematical Society 3 Proceedings of the London Mathematical Society. Third Series 3 Tohoku Mathematical Journal. Second Series 3 Advances in Applied Mathematics 3 Publicacions Matemàtiques 3 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 3 Topology Proceedings 3 Annals of Mathematics. Second Series 3 Acta Mathematica Sinica. English Series 3 Journal of Algebra and its Applications 3 Journal of Homotopy and Related Structures 2 Communications in Mathematical Physics 2 Mathematical Notes 2 Journal of Geometry and Physics 2 Annales de l’Institut Fourier 2 Archiv der Mathematik 2 Compositio Mathematica 2 Publications Mathématiques 2 Journal für die Reine und Angewandte Mathematik 2 Proceedings of the Japan Academy. Series A 2 Siberian Mathematical Journal 2 Transactions of the Moscow Mathematical Society 2 European Journal of Combinatorics 2 Bulletin of the Korean Mathematical Society 2 Discrete & Computational Geometry 2 International Journal of Mathematics 2 Geometric and Functional Analysis. GAFA 2 Proceedings of the National Academy of Sciences of the United States of America 2 Bulletin of the American Mathematical Society. New Series 2 Turkish Journal of Mathematics 2 Sbornik: Mathematics 2 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 2 Kyoto Journal of Mathematics 2 Forum of Mathematics, Sigma 2 European Journal of Mathematics 2 Journal of Combinatorial Algebra 1 Discrete Mathematics 1 Indian Journal of Pure & Applied Mathematics 1 Journal d’Analyse Mathématique 1 Rocky Mountain Journal of Mathematics 1 Acta Mathematica 1 Algebra Universalis 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 1 Bulletin de la Société Mathématique de France 1 Collectanea Mathematica 1 Functional Analysis and its Applications 1 Hokkaido Mathematical Journal 1 Illinois Journal of Mathematics 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Computational and Applied Mathematics 1 Journal of Geometry 1 Journal of Graph Theory 1 Journal of the Korean Mathematical Society 1 Journal of the London Mathematical Society. Second Series 1 Mathematica Scandinavica 1 Mathematika 1 Pacific Journal of Mathematics 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Combinatorica 1 Physica D 1 Graphs and Combinatorics 1 $$K$$-Theory 1 Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni 1 Differential Geometry and its Applications ...and 32 more Serials
all top 5

#### Cited in 39 Fields

 272 Manifolds and cell complexes (57-XX) 215 Group theory and generalizations (20-XX) 126 Algebraic topology (55-XX) 95 Algebraic geometry (14-XX) 76 Convex and discrete geometry (52-XX) 74 Combinatorics (05-XX) 66 Differential geometry (53-XX) 63 Topological groups, Lie groups (22-XX) 39 Commutative algebra (13-XX) 29 Geometry (51-XX) 26 Functional analysis (46-XX) 25 Several complex variables and analytic spaces (32-XX) 18 Dynamical systems and ergodic theory (37-XX) 18 Global analysis, analysis on manifolds (58-XX) 17 Nonassociative rings and algebras (17-XX) 17 General topology (54-XX) 16 Associative rings and algebras (16-XX) 16 Abstract harmonic analysis (43-XX) 14 $$K$$-theory (19-XX) 11 Category theory; homological algebra (18-XX) 4 Order, lattices, ordered algebraic structures (06-XX) 4 Biology and other natural sciences (92-XX) 3 History and biography (01-XX) 3 Number theory (11-XX) 3 Operations research, mathematical programming (90-XX) 2 Operator theory (47-XX) 2 Quantum theory (81-XX) 1 Real functions (26-XX) 1 Measure and integration (28-XX) 1 Potential theory (31-XX) 1 Partial differential equations (35-XX) 1 Approximations and expansions (41-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Probability theory and stochastic processes (60-XX) 1 Computer science (68-XX) 1 Mechanics of particles and systems (70-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Relativity and gravitational theory (83-XX) 1 Information and communication theory, circuits (94-XX)


text:
Why is $L^3$ weaker than $L^2$?

Someone told me today that if I can show $\Vert A_n-B_n\Vert_3\to 0$ as $n\to \infty$, then claiming $A=B$ as $n\to \infty$ (where $A$ and $B$ are the respective limits of $A_n$ and $B_n$) is a weaker claim than if I were to show that $\Vert A_n-B_n\Vert_2\to 0$ (which in turn is weaker than if I were to show $\Vert A_n-B_n\Vert_1\to 0$). Why is this so?

-
That would depend on what $A_n,B_n$ would be. –  tomasz Oct 29 '12 at 2:03
@tomasz I don't follow... if need be, ignore $A_n$ and $B_n$ and just consider $\Vert A-B\Vert_3$ (and similarly, 2 and 1) –  fkmasö Oct 29 '12 at 2:21
Presumably, you are getting at this: math.stackexchange.com/questions/21460/… –  JavaMan Oct 29 '12 at 17:35

The answer pretty much depends on the measure space $(X,\Sigma,\mu)$ that you are working with. There is the following well-known result (Rudin poses it as an exercise in his Real and Complex Analysis):

Let $1 \leq p < q \leq \infty$. Then

1. ${L^{q}}(X,\Sigma,\mu) \subseteq {L^{p}}(X,\Sigma,\mu)$ iff $X$ does not have measurable subsets of arbitrarily large (but non-infinite) measure.

2. ${L^{q}}(X,\Sigma,\mu) \supseteq {L^{p}}(X,\Sigma,\mu)$ iff $X$ does not have measurable subsets of arbitrarily small (but non-zero) measure.

If $\mu(X) < \infty$, in which case (1) holds, we even have the following relationship between the norms $\| \cdot \|_{p}$ and $\| \cdot \|_{q}$: $$\forall \, \text{measurable functions  f  on  X }: \quad \| f \|_{p} \leq [\mu(X)]^{\frac{1}{p} - \frac{1}{q}} \cdot \| f \|_{q}.$$ This yields the following sequence of implications: $$\lim_{n \rightarrow \infty} \| f_{n} \|_{3} = 0 \quad \Longrightarrow \quad \lim_{n \rightarrow \infty} \| f_{n} \|_{2} = 0 \quad \Longrightarrow \quad \lim_{n \rightarrow \infty} \| f_{n} \|_{1} = 0.$$

I do not know what the conditions on $(X,\Sigma,\mu)$ have to be in order to obtain the reverse of the above sequence of implications, namely, $$\lim_{n \rightarrow \infty} \| f_{n} \|_{1} = 0 \quad \Longrightarrow \quad \lim_{n \rightarrow \infty} \| f_{n} \|_{2} = 0 \quad \Longrightarrow \quad \lim_{n \rightarrow \infty} \| f_{n} \|_{3} = 0.$$ This is the case that you are interested in. There may be results in interpolation theory (similar to the Riesz-Thorin Interpolation Theorem) that would give you precisely what you need.

-


text:
Show that $\sup(S)=\inf(T)$.

Suppose that $S$ is a nonempty subset of $\mathbb{R}$ and $S$ is bounded above. Also suppose that $T=\{x\in\mathbb{R}:x$ is an upper bound of $S\}$ Show that $\sup(S)=\inf(T)$.

My proof: Since $S$ is nonempty bounded above subset of $\mathbb{R}$, $\sup(S)$ exists. Let $\sup(S)=\alpha$. It is clear that $\alpha\in T$ because $\alpha$ is an upper bound of $S$. Also, $\alpha\leq x$ for all $x\in T$ because $\alpha$ is a supremum. Since $\alpha\leq x$ and $\alpha\in T$, we can conclude that $\inf(T)=\alpha=\sup(S)$ as required.

Is this proof ok?

• Your proof is fine. Mar 9, 2018 at 9:17
• The ideas are all there, it seems to me. Two points you might want to go into more depth (since this is apparently an introductory course). Clarify how $\alpha \leq x$ because "$\alpha$ is a supremum". Also clarify how $\alpha \leq x$ and $\alpha \in T$ implies that $\alpha = \inf T$. Mar 9, 2018 at 9:18
• You should use \mathbb{R} to get the correct symbol for the real numbers Mar 9, 2018 at 9:28

Your work is fine, rephrasing a bit:

$S \subset \mathbb{R}$, $S$ nonempty, is bounded above : $\sup(S)$ exists.

$T \subset \mathbb{R}$, $T$ nonempty (why?), is bounded below by $s \in S$: $\inf(T)$ exists.

$\sup(S)$ is the least upper bound of $S$, i.e. an upper bound, hence $\sup(S) \in T.$

Assume there is an $x \in T$ with $x \lt \sup(S)$.

Since $x$ is an upper bound of $S$ : contradiction, $\sup(S)$ is the least upper bound of $S$.

Hence for all $x \in T$: $x \ge \sup(S)$,

and since $\sup(S) \in T$, we have

$\inf(T)=\sup(S).$


text:
# Minecraft TestFor command help

I am using the testifier command, /testfor @a[x=-436,y=9,z=-758], but instead of testing for a player at those coordinates, it is testing everywhere, and finding me. Any help?

Your command does not work as supplying a x, y and z modifier only changes the location of which the command block detects from.

If you'd like to detect for a player at that location, add the r (radius) modifier to your selector.
By default, the radius (if one is not specified) is infinite (also called 'none').

You'll want to use:

/testfor @a[x=-436,y=9,z=-758,r=1]

Try to use execute:

execute @a -436 9 758 [command you want to happen]

This command makes a player execute the command you put when they go to the location you entered [-436 9 758].


text:
# CKEDITOR.feature

class since 4.1.0 abstract

Interface that may be automatically implemented by any instance of any class which has at least the name property and can be meant as an editor feature.

For example:

• "Bold" command, button, and keystroke – it does not mean exactly <strong> or <b> but just the ability to create bold text.
• "Format" drop-down list – it also does not imply any HTML tag.
• "Link" command, button, and keystroke.
• "Image" command, button, and dialog window.

Thus most often a feature is an instance of one of the following classes:

None of them have a name property explicitly defined, but it is set by CKEDITOR.editor.addCommand and CKEDITOR.ui.add.

During editor initialization all features that the editor should activate should be passed to CKEDITOR.editor.addFeature (shorthand for CKEDITOR.filter.addFeature).

This method checks if a feature can be activated (see requiredContent) and if yes, then it registers allowed content rules required by this feature (see allowedContent) along with two kinds of transformations: contentForms and contentTransformations.

By default all buttons that are included in toolbar layout configuration are checked and registered with CKEDITOR.editor.addFeature, all styles available in the 'Format' and 'Styles' drop-down lists are checked and registered too and so on.

## Properties

• ### allowedContent : allowedContentRulesCKEDITOR.feature#allowedContent

HTML code that can be generated by this feature.

For example a basic image feature (image button displaying the image dialog window) may allow 'img[!src,alt,width,height]'.

During the feature activation this value is passed to CKEDITOR.filter.allow.

Defaults to null

• ### contentForms : ObjectCKEDITOR.feature#contentForms

Feature content forms to be registered in the CKEDITOR.editor.filter during the feature activation.

Defaults to null

• ### contentTransformations : ObjectCKEDITOR.feature#contentTransformations

Transformations (usually for content generated by this feature, but not necessarily) that will be registered in the CKEDITOR.editor.filter during the feature activation.

Defaults to null

• ### name : StringCKEDITOR.feature#name

The name of the feature.

It is used for example to identify which CKEDITOR.filter.allowedContent rule was added for which feature.

• ### requiredContent : contentRuleCKEDITOR.feature#requiredContent

Minimal HTML code that this feature must be allowed to generate in order to work.

For example a basic image feature (image button displaying the image dialog window) needs 'img[src,alt]' in order to be activated.

During the feature validation this value is passed to CKEDITOR.filter.check.

If this value is not provided, a feature will be always activated.

Defaults to null

## Methods

• ### toFeature() → featureCKEDITOR.feature#toFeature

Returns a feature that this feature needs to register.

In some cases, during activation, one feature may need to register another feature. For example a CKEDITOR.ui.button often registers a related command. See CKEDITOR.ui.button.toFeature.

This method is executed when a feature is passed to the CKEDITOR.editor.addFeature.

#### Returns

feature


text:
# Understanding sine, cosine, and tangent in the unit circle

In the following diagram I understand how to use angle $$\theta$$ to find cosine and sine. However, I'm having a hard time visualizing how to arrive at tangent. Furthermore, is it true that in all right triangle trig ratios we always need to use one of the non-right angles?

• how did you find cosine or sine? – Nosrati Oct 8 '18 at 17:22
• You may (or may not) find my document on this matter helpful: "(Almost) Everything You Need to Remember about Trig, in One Simple Diagram" (PDF). (It's currently undergoing revisions, so please pardon some rough spots.) – Blue Oct 8 '18 at 18:14
• Nosrati - sin $\theta$ is just opposite over hypotenuse. Since the hypotnuse is always 1 in the unit circle sin $\theta$ will equal the height of the triangle and Y coordinate on the circle. I will now read the answers for finding tangent $\theta$ – user27343 Oct 9 '18 at 4:07

If you pay attention, the smaller left right triangle is similar to the larger right triangle adjacent to it as they are both right triangles with angles of measure $$\frac{\pi}{2}$$, $$\theta$$, and $$(\frac{\pi}{2}-\theta)$$.

When two triangles are similar, the ratio of their corresponding sides will be equivalent.

$$\frac{\cos\theta}{1}$$ In the smaller right triangle, $$\cos\theta$$ is opposite the $$(\frac{\pi}{2}-\theta)$$ angle and $$1$$ is the hypotenuse.

In the larger right triangle, $$\sin\theta$$ is opposite to the $$(\frac{\pi}{2}-\theta)$$ angle and $$x$$ (unknown variable) is the hypotenuse.

$$\frac{\sin\theta}{x}$$

Using the rules of similarity, we can say the two ratios are equivalent.

$$\frac{\cos\theta}{1} = \frac{\sin\theta}{x}$$ $$x\cos\theta = \sin\theta \implies x = \frac{\sin\theta}{\cos\theta} =\implies \boxed{x = \tan\theta}$$

You can check this page out for $$\csc\theta$$, $$\sec\theta$$, and $$\cot\theta$$: graphical representation of trig functions.

• Thank you. The link was very helpful as well. – user27343 Oct 9 '18 at 4:22
• No problem. :-) – KM101 Oct 9 '18 at 12:23
• In the first and third quadrants, $$\tan(\theta)$$ is the length from $$(\cos(\theta),\sin(\theta))$$ to the $$x$$ axis along the line tangent to the circle at $$(\cos(\theta),\sin(\theta))$$. In the second and fourth quadrants the situation is essentially the same but we use the opposite sign.
• One can think of $$\cos(\pi/2)$$ and $$\sin(\pi/2)$$ from the triangle point of view as the ratios of a degenerate triangle with angles $$(\pi/2,\pi/2,0)$$, which is really just a line segment. Now you identify one of the $$\pi/2$$'s as the "right" angle and the other as the "acute" angle and measure ratios relative to that "acute" angle.

The angle between the red and blue lines is $$\theta$$. So in that triangle, $$\cos\theta=\frac{\rm red}{\rm blue}$$ Or$${\rm blue}=\frac{\rm red}{\cos\theta}=\frac{\sin\theta}{\cos\theta}=\tan\theta$$

• There are a bunch of minor errors here. – Ian Oct 8 '18 at 17:24
• Please feel free to edit the response, to make it more accurate/clear. – Andrei Oct 8 '18 at 17:25

Commit this SohCahToa Table to memory so that you can always write it out:

For the $$sin$$ and $$cos$$ in the unit circle you can think of putting the values in the table over a big $$1$$ for the hypotenuse.

Example: Think of

$$sin(\frac{\pi}{6})= \frac{{\frac{\sqrt1}{2}}}{1}$$

$$cos(\frac{\pi}{6})= \frac{{\frac{\sqrt3}{2}}}{1}$$

and the unit circle placement of $$\left(cos(\frac{\pi}{6}),sin(\frac{\pi}{6})\right)$$.