text:
# Peeter Joot's (OLD) Blog.

• ## Archives

 papasu on PHY450H1S. Relativistic Electr… papasu on Energy term of the Lorentz for… lidiodu on PHY450H1S. Relativistic Electr… lidiodu on PHY450H1S. Relativistic Electr… lidiodu on bivector form of Stokes t…

• 307,770

## PHY450H1S. Relativistic Electrodynamics Lecture 8 (Taught by Prof. Erich Poppitz). Relativistic dynamics.

Posted by peeterjoot on February 2, 2011

Covering chapter 2 material from the text [1].

Covering a bit more of Professor Poppitz’s lecture notes: equation of motion, symmetries, and conserved quantities (energy-momentum 4 vector) from relativistic particle action.

Covering lecture notes pp. 56.1-72: comments on mass, energy, momentum, and massless particles (56.1-58); particles in external fields: Lorentz scalar field (59-62); reminder of a vector field under spatial rotations (63) and a Lorentz vector field (64-65) [Tuesday, Feb. 1]; the action for a relativistic particle in an external 4-vector field (65-66); the equation of motion of a relativistic particle in an external electromagnetic (4-vector) field (67,68,73) [Wednesday, Feb. 2]; mathematical interlude: (69-72): on 3×3 antisymmetric matrices, 3-vectors, and totally antisymmetric 3-index tensor – please read by yourselves, preferably by Wed., Feb. 2 class! (this is important, well also soon need the 4-dimensional generalization)

# Finishing previous arguments on action and proper velocity.

For a free particle, our action is

\begin{aligned}S &= - m c \int ds \\ &= -m c^2 \int dt \sqrt{1 - \frac{\mathbf{v}^2}{c^2}}\end{aligned}

Our Lagrangian is

\begin{aligned}\mathcal{L} = -m c^2 \sqrt{1 - \frac{\mathbf{v}^2}{c^2}} .\end{aligned} \hspace{\stretch{1}}(2.1)

We can also make a non-relativistic velocity approximation

\begin{aligned}\mathcal{L} &= -m c^2 \sqrt{1 - \frac{\mathbf{v}^2}{c^2}} \\ &= -m c^2 \left( 1 - \frac{1}{{2}} \frac{\mathbf{v}^2}{c^2} \right) + O((\mathbf{v}^2/c^2)^2) \\ &\approx \underbrace{-m c^2 }_{constant} + \underbrace{\frac{1}{{2}} m \mathbf{v}^2 }_{\text{Classical Lagrangian for free particle}}.\end{aligned}

It is good to know that we recover the familiar Newtonian case when our velocities are small enough.

Our job is to vary the action between a pair of spacetime points

\begin{aligned}(t_a, \mathbf{x}_a) \rightarrow (t_b, \mathbf{x}_b)\end{aligned} \hspace{\stretch{1}}(2.2)

The equations of motion that result from this variation, or from the Euler-Lagrange equations that one can obtain from this variation, are

\begin{aligned}\frac{d}{dt} ( \gamma \mathbf{v}) = 0\end{aligned} \hspace{\stretch{1}}(2.3)

We argued last time, by evaluating the derivatives of 2.3, and taking dot and cross products with $\mathbf{v}$ that we also have

\begin{aligned}\frac{d\mathbf{v}}{dt} = 0\end{aligned} \hspace{\stretch{1}}(2.4)

Observe that since $d\mathbf{v}/dt = 0$, we also have $d\gamma/dt = 0$

\begin{aligned}\frac{d\gamma}{dt} &=\frac{d}{dt} \frac{1}{{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}} \\ &=\frac{d}{dt} \frac{1}{{\left(1 - \frac{\mathbf{v}^2}{c^2}\right)^{3/2}}} (-1/2) (2) (-\mathbf{v} \cdot \dot{\mathbf{v}}) /c^2 \\ &= 0.\end{aligned}

We can therefore combine the pair of equations (after adjusting both to have dimensions of velocity)

\begin{aligned}\frac{d}{dt} ( \gamma \mathbf{v} ) &= 0 \\ \frac{d}{dt} ( \gamma c ) &= 0,\end{aligned} \hspace{\stretch{1}}(2.5)

into

\begin{aligned}u^i = (u^0, \mathbf{u}).\end{aligned} \hspace{\stretch{1}}(2.7)

Here

\begin{aligned}u^0 &= \gamma c \\ \mathbf{u} &= \gamma \mathbf{v}.\end{aligned} \hspace{\stretch{1}}(2.8)

Since we have $du^i/dt = 0$, pre-multiplying this by $\gamma/c$ does not change the equation, and we have

\begin{aligned}0 = \frac{1}{{c \sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}} \frac{du^i}{dt}.\end{aligned} \hspace{\stretch{1}}(2.10)

This now puts things in a nice invariant form, with no bias towards any specific observer’s time coordinates, and we have for the free particle

\begin{aligned}\frac{d u^i}{ds} = 0.\end{aligned} \hspace{\stretch{1}}(2.11)

# Symmetries of spacetime translation invariance.

The symmetries of $S$ imply conservation laws. Our action has $SO(1,3) \times T^4 =$ Lorentz x spacetime translation $\equiv$ Poincar\'{e} group of symmetries.

Consider quantities conserved due to $T^4$ factor

\begin{aligned}\mathbf{x} &\rightarrow \mathbf{x} + \mathbf{a} \qquad \mbox {wherelatex \mathbf{a}is constant} \\ t &\rightarrow t + \text{constant}\end{aligned} \hspace{\stretch{1}}(3.12)

Observe that the Lagrangian is not a function of $\mathbf{x}$, or $t$ explicitly

\begin{aligned}\mathcal{L}(\mathbf{x}, \mathbf{v}, t) = - m c \sqrt{1 - \frac{\mathbf{v}^2}{c^2}}= \mathcal{L}(\mathbf{v}) .\end{aligned} \hspace{\stretch{1}}(3.14)

A consequence from this, utilizing the Euler-Lagrange equations is that we have a zero for the time derivative of the generalized momentum ${\partial {\mathcal{L}}}/{\partial {\mathbf{v}}}$

\begin{aligned}\frac{d}{dt} \frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} = \frac{\partial {\mathcal{L}}}{\partial {\mathbf{x}}} = 0,\end{aligned} \hspace{\stretch{1}}(3.15)

Let’s calculate that generalized momentum

\begin{aligned}\frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} &=\frac{\partial {}}{\partial {\mathbf{v}}} \left(-m c^2 \sqrt{1 - \frac{\mathbf{v}^2}{c^2}} \right) \\ &=\frac{\partial {}}{\partial {\mathbf{v}}} \left( -m c^2 \frac{(1/2)(-2) \mathbf{v}/c^2}{ \sqrt{1 - \frac{\mathbf{v}^2}{c^2}} } \right) \\ &=m \frac{\mathbf{v}}{ \sqrt{1 - \frac{\mathbf{v}^2}{c^2}} } \\ \end{aligned}

So our generalized momentum is

\begin{aligned}\frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} = m \mathbf{v} \gamma.\end{aligned} \hspace{\stretch{1}}(3.16)

Evaluating the Euler-Lagrange equations above we find

\begin{aligned}0 &= \frac{d}{dt} \left( m \gamma \mathbf{v} \right) \\ &= \frac{d}{dt} \left( m u^{1,2,3} \right) \\ \end{aligned}

Recall that $u^0 = c \gamma$, and that $d\gamma/dt = 0$, so we also have

\begin{aligned}\frac{d}{dt} \left( m u^i \right) = 0\end{aligned} \hspace{\stretch{1}}(3.17)

and again with multiplication by $\gamma/c$ we have a Lorentz invariant relation, mostly a consequence of spacetime translation invariance

\begin{aligned}\frac{d}{ds} \left( m u^i \right) = 0.\end{aligned} \hspace{\stretch{1}}(3.18)

We define this quantity, the invariant quantity (a four vector), as the relativistic momentum

\begin{aligned}p^i = m u^i.\end{aligned} \hspace{\stretch{1}}(3.19)

A relativistic particle is characterizes by a conserved 4 vector quantity $p^i$ with

\begin{aligned}p^0 &= m c \gamma \\ \mathbf{p} &= m \gamma \mathbf{v} \\ p^i &= (p^0, \mathbf{p})\end{aligned} \hspace{\stretch{1}}(3.20)

# Time translation invariance

\begin{aligned}\mathcal{L}(\mathbf{x}, \mathbf{v}, t) = \mathcal{L}(\mathbf{v}) \end{aligned} \hspace{\stretch{1}}(4.23)

However, it helps to consider the more general case

\begin{aligned}\mathcal{L}(\mathbf{x}, \mathbf{v}, t) = \mathcal{L}(\mathbf{x}, \mathbf{v}) \end{aligned} \hspace{\stretch{1}}(4.24)

since we have no explicit time dependence.

\begin{aligned}\frac{d}{dt} \mathcal{L}(\mathbf{v}) &= \frac{\partial {\mathcal{L}}}{\partial {\mathbf{x}}} \cdot \dot{\mathbf{x}} + \frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} \cdot \dot{\mathbf{v}} \\ &= \left( \frac{d}{dt} \frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} \right) \cdot \mathbf{v} + \frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} \cdot \frac{d\mathbf{v}}{dt} \\ &= \frac{d}{dt} \left( \frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} \cdot \mathbf{v}\right)\end{aligned}

Regrouping, to pull all the derivative terms together provides the conservation identity

\begin{aligned}\frac{d}{dt} \left(\frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} \cdot \mathbf{v} - \mathcal{L}\right) = 0.\end{aligned} \hspace{\stretch{1}}(4.25)

This quantity $\frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} \cdot \mathbf{v} - \mathcal{L}$ is usually identified as the Hamiltonian $H$, the energy, but we will call it $E$ here.

In our case, with the relativistic free particle Lagrangian

\begin{aligned}\mathcal{L} = -m c^2 \sqrt{ 1 - \frac{\mathbf{v}^2}{c^2} },\end{aligned} \hspace{\stretch{1}}(4.26)

we have

\begin{aligned}E &= \frac{\partial {\mathcal{L}}}{\partial {\mathbf{v}}} \cdot \mathbf{v} - \mathcal{L} \\ &= \mathbf{v} \cdot \left( m \frac{1}{{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}} \mathbf{v} \right) + m c^2 \sqrt{1 - \frac{\mathbf{v}^2}{c^2}} \\ &= \frac{m \mathbf{v}^2}{\sqrt{ 1 - \frac{\mathbf{v}^2}{c^2}}} + mc^2 \sqrt{ 1 - \frac{\mathbf{v}^2}{c^2}} \\ &= \frac{\mathbf{v}^2 + m c^2 \left(1 - \frac{\mathbf{v}^2}{c^2} \right)}{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}} \\ &= \frac{m c^2 }{\sqrt{ 1 - \frac{\mathbf{v}^2}{c^2} }}\end{aligned}

So we define, for the energy, a conserved quantity under time translation, we have

\begin{aligned}\boxed{E = \gamma m c^2 = \frac{ m c^2 }{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}}\end{aligned} \hspace{\stretch{1}}(4.27)

It is only with the $\mathbf{v} \rightarrow 0$ that we recover the famous tee-shirt expression

\begin{aligned}E = m c^2.\end{aligned} \hspace{\stretch{1}}(4.28)

Since we also know (from the spacetime translation) that $p^0 = m c \gamma = E/c$, we get another conserved quantity for free since $(p^0, \mathbf{p})$ then is also a symmetry (i.e. thus a conserved quantity)

\begin{aligned}p^0 &= m \gamma c = \frac{E}{c} \\ \mathbf{p} &= m \gamma \mathbf{v}\end{aligned}

\begin{aligned}p^i = ( p^0, \mathbf{p} )\end{aligned} \hspace{\stretch{1}}(4.29)

Note that the only “mass” you ever want to talk about is “m”. This is a Lorentz scalar, and we won’t use the old notions that mass changes with velocity or “relativistic mass”.

# Some properties of the four momentum.

We have

\begin{aligned}p^i p_i &= (p^0)^2 - \mathbf{p}^2 \\ &= m c^2 \gamma^2 - m^2 \gamma^2 \mathbf{v}^2 \\ &= m c^2 \gamma^2 \left( 1 - \frac{\mathbf{v}^2}{c^2} \right) \\ &= m^2 c^2\end{aligned}

So we have

\begin{aligned}\boxed{p^i p_i = m^2 c^2}\end{aligned} \hspace{\stretch{1}}(5.30)

We say that the 4-vector $p^i$ represents a particle with mass $m$.

Since four momentum is a conserved quantity we can use this conservation property to study relativistic collisions

PICTURE: two particles colliding with two particles resulting (particles trajectories as arrows)

\begin{aligned}\underbrace{p_1^i + p_2^i}_{\text{four momentum before}} = \underbrace{p_3^i + p_4^i}_{\text{four momentum after}}\end{aligned} \hspace{\stretch{1}}(5.31)

\begin{aligned}\mathbf{p} &= \frac{m \mathbf{v}}{\sqrt{ 1 - \frac{\mathbf{v}^2}{c^2} }} \rightarrow 0 \qquad \mbox{whenlatex m \rightarrow 0} \\ E &= \frac{m c^2}{\sqrt{ 1 – \frac{\mathbf{v}^2}{c^2} }} \rightarrow 0 \qquad \mbox{when $m \rightarrow 0$}\end{aligned} \hspace{\stretch{1}}(5.32)

except when ${\left\lvert{\mathbf{v}}\right\rvert} = c$, where if you take $m \rightarrow 0$ and ${\left\lvert{\mathbf{v}}\right\rvert} = c$ you can get anything (any values) in such a limit (limit does not exist).

However, because

\begin{aligned}\frac{E^2}{c^2} - \mathbf{p}^2 = m^2 c^2 = 0\end{aligned} \hspace{\stretch{1}}(5.34)

when $m \rightarrow 0$, $E$ and $\mathbf{p}$ for a massless particle must obey $E = c {\left\lvert{\mathbf{p}}\right\rvert}$.

Massless particles like photons (and gravitons if/when eventually measured) have lightlike 4 momentum vectors

\begin{aligned}p^i p_i = 0\end{aligned} \hspace{\stretch{1}}(5.35)

Gravity waves haven’t been seen yet, but the LIGO and LISA (extremely large infraferometers) experiments are expected to get some results on this in the near future.

# Where are we?

In the notes there’s a review (see that on one’s own). We’ll also want to eventually deal with the conservation laws in four vector form, since it will illustrate how the electric and magnetic fields have to be transformed. We’ll get to that eventually.

# Interactions

In classical mechanics we have

\begin{aligned}\mathcal{L}_\text{kinetic} = \frac{1}{{2}} m \mathbf{v}^2 \end{aligned} \hspace{\stretch{1}}(7.36)

\begin{aligned}\mathcal{L} = \frac{1}{{2}} m \mathbf{v}^2 - U(\mathbf{r})\end{aligned} \hspace{\stretch{1}}(7.37)

Here $U(\mathbf{r})$ is an external potential.

\begin{aligned}S = S_{\text{free}} + S_{\text{interaction}} = \int dt \frac{1}{{2}} m \mathbf{v}^2 + \int dt (-U(\mathbf{r}, t))\end{aligned} \hspace{\stretch{1}}(7.38)

The quantity $U(\mathbf{r}, t)$ is what we call a potential field.

What’s the simplest invariant field we can have? The simplest possibility is to have a relativistic particle which interacts with an external \underline{Lorentz scalar field}. We’d imagine that this is due to some other particle or some distribution of other fields.

Recall that the scalar field under rotations (reminder)

PICTURE: a point with coordinates in a fixed and a rotated coordinate system

That point is

\begin{aligned}P = (x, y) = (x', y')\end{aligned} \hspace{\stretch{1}}(7.39)

Similarly we can define a scalar quantity (like temperature or the Coulomb potential) is then assigned a value at each point

\begin{aligned}\phi(x, y) = \phi'(x', y')\end{aligned} \hspace{\stretch{1}}(7.40)

The value of this scalar in the $x,y$ coordinates system at point $P$ equals the value of this scalar in the $x',y'$ coordinates system at the same point $P$.

A Lorentz scalar field is like this, but for an event $P = (ct, x) = (ct', x')$ is the same.

So, we’d have

\begin{aligned}\phi(ct, x) = \phi'(ct', x')\end{aligned} \hspace{\stretch{1}}(7.41)

The value of this scalar in the $x,ct$ coordinates system at event $P$ equals the value of this scalar in the $x',ct'$ coordinates system at the same event $P$ in the primed frame.

Our action would then be

\begin{aligned}S = - m c \int ds + g \int ds \phi(x^i)\end{aligned} \hspace{\stretch{1}}(7.42)

Here $g$ is a coupling constant, also called the “charge” of a particle under that scalar field.

Note that unfortunately nature hasn’t provided us with scalar fields that are stable enough to observe in classical interactions

We do however have some scalar particles

\begin{aligned}\pi^0, \pi^\pm, k^0, k^\pm\end{aligned} \hspace{\stretch{1}}(7.43)

These are unstable and short ranged.

The LHC is looking for another unstable short lived scalar field (the Higgs). So we have to unfortunately study a more complicated field, a vector field. We’ll do that next time.

# References

[1] L.D. Landau and E.M. Lifshits. The classical theory of fields. Butterworth-Heinemann, 1980.


text:
# (a) Write the principle of a semiconductor device which is used as a voltage regulator.(b) With the help of a circuit diagram explain its working.(c) Draw its I -V characteristics.

(a) Zener diode is the semiconductor device which is used as a voltage regulator.

In the case of reverse-biased Zener diode as the reverse voltage increases current also increases. But at a particular reverse voltage condition reverse breakdown occurs, at that time current increases sharply and the voltage across Zener diode remains constant. That voltage is known as Zener voltage $(V_{z})$. This is the principle applied when Zener diode is used as a voltage regulator.

(b)

Here $V_{i}-V_{o}= IR$

$\frac{V_{i}-V_{o}}{I}= R$

By Kirchoff's law, $I = I_{z}+I_{l}$

In this case when input voltage increases the current through R and Zener diode increases. This results in the voltage drop across R increases but the voltage across the Zener diode is constant. This is because reverse breakdown condition Zener voltage remains constant even if the current through Zener diode increase. Similarly, if the input voltage decreases current through R and Zener diode decreases. This results voltage drop across R decreases but the voltage drop across Zener diode is constant. Thus, it can act as a voltage regulator.

(c)

## Related Chapters

### Preparation Products

##### Knockout NEET Sept 2020

An exhaustive E-learning program for the complete preparation of NEET..

₹ 15999/- ₹ 6999/-
##### Rank Booster NEET 2020

This course will help student to be better prepared and study in the right direction for NEET..

₹ 9999/- ₹ 4999/-
##### Knockout JEE Main Sept 2020

An exhaustive E-learning program for the complete preparation of JEE Main..

₹ 12999/- ₹ 6999/-
##### Test Series NEET Sept 2020

Take chapter-wise, subject-wise and Complete syllabus mock tests and get in depth analysis of your test..

₹ 4999/- ₹ 2999/-

format: plain


text:
## suneja Group Title lim x^2+x-11=9 x tends to 4 prove tis by formal definition of limit one year ago one year ago

1. Jeffrey_Calderon

that's quite long to type. haha. you can do it dude! :)

2. suneja

sure i can ... wanted to jst chk

3. heedcom

just plug 4 into the equation

4. nickhouraney

@suneja using epsilon and delta right

5. suneja

@nickhouraney ya can u show dat here

6. rerukumash

wats lim anyone pls help

7. suneja

lim is limit

8. nickhouraney

ya i can show u

9. rerukumash

ok where is it used

10. rerukumash

11. nickhouraney

|x-a| |x-4| |f(x)-L|<epsilon | (x^2+x-11) - 9| start from here

12. satellite73

start with $|x^2-x-20|<\epsilon$ and see what you need for $$|x-4|$$ to make it happen

13. suneja

mod (x^2-x-11)-9 < epsilon wenever 0<mod x-4 < delta tis is how v start then go by simplifying left inequality to find a suitable value for delta then verify that choice of delta i hav a prob in verification .. need help der

14. nickhouraney

we basically need to take this | (x^2+x-11) - 9| < epsilon and make it |x-4|<epsilon start by simplifying | (x^2+x-11) - 9| that will turn to | x^2+x-11 - 9| < epsilon

15. suneja

ya then u hav to prove that "choice " of delta wic u get by solving the 1st part like u said can u show me 2nd part of tis prob

16. nickhouraney

for the proof you would just go in reverse basically. if |x-4| < delta then |x-4| < (your choice for epsilon) then show how this will lead to |f(x)-L| < epsilon

17. suneja

@nickhouraney thanks

18. nickhouraney

19. suneja

i've done exactly like but was lil confused in 2nd half of it! neways thanks

20. satellite73

oh yes it is. okay you should start with $|x^2+x-20|<\epsilon$ then factor to get $|(x-4)(x+5)|<\epsilon$ you have control over $$|x-4|$$ so you just need a bound for $$|x+5|$$

21. satellite73

trick is to say that since you are taking the limit as $$x\to 4$$ you can assert that say $$3<x<5$$ so the largest $$|x+5|$$ can be is 10

22. suneja

wat do u mean by " a bound for $\left| x+5 \right|$

23. satellite73

ok lets go slow you get to say how large $$|x-4|$$ can be, that is you get to pick your $$\delta$$ so that $$|x-4|$$ is smaller than $$\delta$$

24. satellite73

now you are looking at the product inside the square root, you have $$|(x-4)(x+5)|$$ and you want to make this smaller than $$\epsilon$$

25. satellite73

but $$|(x-4)(x+5)|=|x-4||x+5|$$ you can make the first term as small as you like, but you cannot make $$|x+5|$$ small

26. satellite73

not square root, absolute value is what i meant

27. satellite73

so the question is, how big can $$|x+5|$$ be? well it can be really really large, but don't forget you are making $$x$$ close to 4

28. satellite73

so you can assert that since $$x$$ is close to 4, it is certainly less that say 5, making $$|x+5|<10$$

29. satellite73

now you want $|(x-4)(x+5)|<\epsilon$ if you make $$\delta=\frac{\epsilon}{10}$$ and $$3<x<5$$ then you know $|(x-4)(x+5)|<10|x-4|<10\times \frac{\epsilon}{10}=\epsilon$

30. suneja

got it ..thanks all

31. suneja

$-\infty \rightarrow +\infty$ is not a bound right?


text:
# Time moves forward, what if you move UP or DOWN? Does time stop?

Time moves forward, what if "you" move UP or DOWN? Does time stop?

I know this is a really "dumb" question but I was thinking about the time continuem that moves from left to right according to the "big bang theory" that set the "arrow of time" into motion. Im sorry if this question seems silly but im really trying to understand. if you move "upwards" in space instead of left to right (like a car driving across the united states") does time stop or stay still? -----> the arrow of time does not make sense in my head. What if you were to move DOWN or move UP without moving side to side? Ahhhh. I know this question sounds crazy and please forgive me if you cant understand what im saying, its hard to describe.

thankyou for anyone who cares to respond, i was just curious as i think about random things like this alot

"Moving with the arrow of time" is just fancy speak for "growing older". As far as we know, we can't grow younger, hence the "arrow". It points "where" we go.

I'm no expert in the subject but, since time is considered as one dimension, wouldn't it mean that you can only move either way of the hypothetical "line of time" and you can't go in other directions (such as up and down) because that would require at least 2 dimensions (such as a plane surface)?
I apologize if I said something incorrect

(Note: This post was edited by DH to conform with the site's copyright rules.)

From http://www.hawking.org.uk/the-beginning-of-time.html
Stephen W. Hawking writes "It seems that Quantum theory, on the other hand, can predict how the universe will begin. Quantum theory introduces a new idea, that of imaginary time. ...".

I googled that and I think that was my question but I guess i should have posted the question in the quantum category because it seems a little wacky.

Thanks for the response !

Last edited by a moderator:
ghwellsjr
Gold Member
I know this is a really "dumb" question but I was thinking about the time continuem that moves from left to right according to the "big bang theory" that set the "arrow of time" into motion. Im sorry if this question seems silly but im really trying to understand. if you move "upwards" in space instead of left to right (like a car driving across the united states") does time stop or stay still? -----> the arrow of time does not make sense in my head. What if you were to move DOWN or move UP without moving side to side? Ahhhh. I know this question sounds crazy and please forgive me if you cant understand what im saying, its hard to describe.

thankyou for anyone who cares to respond, i was just curious as i think about random things like this alot
The arrow of time applies only to the time dimension--it doesn't apply to any of the three spatial dimensions so it won't matter whether you are moving up or down or left or right or in or out--time still marches on in the same direction as always.

By the way, the only reason why there is an expression called the "arrow of time" is because in equations that include time, if we didn't know better, solutions could have time going in either direction, but we are smart enough to know that the negative direction is meaningless. It's kind of like if we had an equation for the area, A, of a square determined from the length of a side, s, like this:

A = s2

and then we wanted to turn it around given the area and we want to determine the length of a side, we would have:

s = √A

But since we know that a square root has both a positive and a negative solution, we are smart enough to know that the negative solution is meaningless.

1 person
I'm sorry I didn't get what you meant, lindsayb026. Anyway I think I heard of that hypothesis before, isn't it the one proposed by Hartle and S.Hawking? Or maybe I remember it wrong..

ZapperZ
Staff Emeritus
This thread requires a lot of valid references of the stuff being discussed. Unless you are able to provide ample support for some of the proposals being discussed, then it will be in violation of the Rules of this forum. If these ideas are backed by peer-reviewed journals, as required in our rules, then please make proper citations to them like you would in writing a paper.

Otherwise, this thread will be closed.

Zz.

PeterDonis
Mentor
I googled that and I think that was my question but I guess i should have posted the question in the quantum category because it seems a little wacky.

It could equally well go there or here, since it involves both quantum mechanics and relativity.

However, I think Hawking was indulging in a little poetic license in the way he described imaginary time. It's not actually a second "direction" of time; it's just a different way of mathematically describing the single direction of time that we observe. Suppose, for example, that we're in flat spacetime, where we would write the metric, using our ordinary "real" time, like this (note that I'm using units where the speed of light is 1):

$$ds^2 = - dt^2 + dx^2 + dy^2 + dz^2$$

Now, we define "imaginary time" ##T## such that ##t = iT##. Then we have ##dt^2 = i^2 dT^2 = - dT^2## and the metric becomes

$$ds^2 = dT^2 + dx^2 + dy^2 + dz^2$$

In other words, the metric now looks like an ordinary Euclidean metric in a 4-dimensional space. But we haven't changed any physics; we've only changed the way we describe the physics in the math. (In quantum field theory, this trick is often called "Wick rotation" to further confuse the uninitiated.) Changing the mathematical description this way makes some scenarios easier to analyze; Hartle and Hawking's proposal was that the Big Bang was one such scenario. However, AFAIK their proposal is still speculative.

1 person
Sorry Zapper, that was stephen hawking hypothesis that i googled on the internet. I dont know what exact website i found it on when it came up in the google search.

he was basically saying that the state of entropy increases overtime due to the second law of thermodynamics or "arrow of time" something that gives direction to time and distinguishes past from the future...and there are three different "arrows" of describing time...thermodynamic (entropy) psychological (where you feel time passing in your life) and cosmological arrow of time which is the direction of time that the universe is expanding rather than contracting, hawking stated that the psychological arrow was determined by the increase of disorder ( a baby growing old....) and only can work along side with cosmological time (big bang universe expanding)

what if the universe stopped expanding? would entropy stop? would time stop? is time just relative ? so confusing.

thanks to all who responded

http://www.hawking.org.uk/the-beginning-of-time.html

Last edited:
PeterDonis
Mentor
what if the universe stopped expanding? would entropy stop? would time stop?

The Hawking lecture you linked to makes clear that the answer to these questions is "no":

"[T]he arrow of time will not reverse. People will continue to get older, even after the universe has begun to contract."

It's also worth noting that our best current model of the universe indicates that it will never recollapse; it will continue expanding indefinitely. However, AFAIK there is still enough uncertainty in the data vs. the model that we can't completely rule out the possibility of a recollapse.

Right, i wanted to see if anyone had any other opinions on it though. It seems he contradicts himself with this topic through some youtube videos ive watched.


text:
Remarks on the Relation between Quantum Dynamics and Fractal Spectra

@article{Barbaroux1997RemarksOT,
title={Remarks on the Relation between Quantum Dynamics and Fractal Spectra},
author={Jean-Marie Barbaroux and Jean-Michel Combes and Raphael Montcho},
journal={Journal of Mathematical Analysis and Applications},
year={1997},
volume={213},
pages={698-722}
}
• Published 15 September 1997
• Mathematics, Physics
• Journal of Mathematical Analysis and Applications
Abstract We investigate the large time behaviour of various solutions of the Schrodinger equation in terms of some local and global dimensions of the spectral measure. We emphasize in particular the role of the Hausdorff and correlation dimensions for the growth exponents of position moments. We also discuss the stability of such exponents under local perturbations.
Quantum diffusion and generalized Rényi dimensions of spectral measures
• Mathematics
• 2000
We estimate the spreading of the solution of the Schrodinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound
Quantum Dynamics and generalized fractal dimensions: an introduction
We review some recent results on quantum motion analysis, and in particular lower bounds for moments in quantum dynamics. The goal of the present exposition is to stress the role played by quantities
UPPER BOUNDS FOR QUANTUM DYNAMICS GOVERNED BY JACOBI MATRICES WITH SELF-SIMILAR SPECTRA
• Mathematics
• 1999
We study a class of one-sided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interval. We analyse dimensional properties of the
FRACTAL DIMENSIONS AND THE PHENOMENON OF INTERMITTENCY IN QUANTUM DYNAMICS
• Mathematics, Physics
• 2001
We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state (t) = e it H and averaged in time between
Quantum Diiusion and Generalized Fractal Dimensions: the Continuous Case L 2
We estimate the spreading of the solution of the Schrr odinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower
On scaling in relation to singular spectra
This paper relates uniform α-Hölder continuity, or α-dimensionality, of spectral measures in an arbitrary interval to the Fourier transform of the measure. This is used to show that scaling exponents
Lower bounds for fractal dimensions of spectral measures of the period doubling Schr\"odinger operator
• Mathematics
• 2020
It is shown that there exits a lower bound $\alpha>0$ to the Hausdorff dimension of the spectral measures of the one-dimensional period doubling substitution Schrodinger operator, and, generically in
Generic properties of purely $\alpha$-Hausdorff continuous operators
• Physics, Mathematics
• 2021
It is shown that the time-average quantum return probability of (Baire) generic states of systems with purely α-Hausdorff continuous spectrum has an oscillating behaviour between a (maximum) fast
Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension
• Physics, Mathematics
• 2003
Abstract: We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schrödinger operators which is based on power-law bounds on transfer matrices. It suffices to have such
Rigorous Spectral Analysis of the Metal–Insulator Transition in a Limit-Periodic Potential
• Mathematics
• 1998
We consider the limit-periodic Jacobi matrices associated with the real Julia sets of fλ(z)=z2−λ for which λ∈[2, ∞) can be seen as the strength of the limit-periodic coefficients. The typical local

References

SHOWING 1-10 OF 27 REFERENCES
Spectral Properties of Quantum Diffusion on Discrete Lattices
The possibility of quantum diffusion over discrete lattices is discussed on the ground of general spectral theory, in connection with the problem of quantal suppression of classical chaos. Given the
Dimension, entropy and Lyapunov exponents
• L. Young
• Mathematics
Ergodic Theory and Dynamical Systems
• 1982
Abstract We consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We
On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions
• Y. Pesin
• Mathematics, Computer Science
• 1993
Different definitions of the correlation dimension are considered and some relationships between them and other characteristics of dimension type such as Hausdorff dimension, box dimension, etc are found.
On the asymptotic properties of quantum dynamics in the presence of a fractal spectrum
• Mathematics
• 1994
On obtient une estimation asymptotique pour la dynamique induite par un groupe unitaire discret sur un espace de Hilbert separable, ayant un spectre purement continu. Elle fournit une borne
Singular continuous spectrum under rank one perturbations and localization for random hamiltonians
• Mathematics
• 1986
We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμAφ, we give necessary and sufficient
The geometry of fractal sets
Preface Introduction Notation 1. Measure and dimension 2. Basic density properties 3. Structure of sets of integral dimension 4. Structure of sets of non-integral dimension 5. Comparable net measures
Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian
It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,Hmn=δm, n+1+δm, n−1+δm, nμ[(n+1)α]−[nα]) where α=(√5−1)/2 and [·] means integer part, is a Cantor set of zero


text:
What effect will AI tools like Codex and CoPilot have on teaching programming (and admissions)

New Artificial Intelligence (AI) programming tools like Codex and CoPilot can (to some extent) generate code in different programming languages from natural language descriptions. Obviously at the moment these AI tools are based on text matching and don't actually have any understanding of the specifications or the code, but can help in producing programs by eliminating some of the "boilerplate" programming tasks. I am concerned that these tools will make learning to program more difficult as they do not encourage students to understand what they are doing, because the can generate the code for the sort of simple pedagogical/andragogical tasks we often use when teaching programming (it is bad enough that there are already solutions for these problems available on-line).

I also wonder if these tools are likely to change the job market in ways that will affect admissions. If these tools take a lot of junior programming jobs, and the job market will be focussed on senior programmers or programmers with specialist skills (e.g. hardware/OS or mathematics), that may limit interest in CS.

Are there any solutions to encouraging students to really understand what they are doing when they can get the computer to generate the code for them?

• Couldn't you ask the same question about languages with built-in functions for high-level operations like sorting? Yet educators still teach students about sorting algorithms. Apr 8 at 14:21
• The difference is that CoPilot is a general tool that will replace problem solving skills, at least for the types of programming task used to help students learn problem solving skills. This is because the data on which they were trained will contain solutions to those problems. Abstraction is an important tool in CS, but there are occasions where it is fine to use an abstraction without understanding how it works and times when it isn't. Sorting is a good example if we have a lot of things that need sorting. Apr 8 at 14:52
• We have to think about why we are teaching people stuff. What is the goal? Apr 8 at 23:19
• Do you know how your compiler works? Apr 10 at 15:05
• as an experiment, I gave a demo of assembler and what the compiler does in the first week of my programming module (making it clear that it wasn't examined, just thought it would be a good way to introduce sequence for the more practically minded students). Ideally I want students to be able to be comfortable with programming both as an abstract activity, but also as a hardware-related activity, according to the needs of the job at hand (I am more at the hardware end). Apr 10 at 16:03

There are two cases actually. One is programming education directed at the vocational market and the other is computer science education. The latter isn't really about programming.

For the first case, if programming skills become obsolete then people will need to take up other things to get a job. They might not be computer related at all. Buggy makers still exist, but it is a niche market.

For computer science education, on the other hand, the curriculum will change. It will still be necessary to produce programs in the early part of the curriculum, but different tools will be used, possibly AI tools. For the later part of the curriculum programming in some high level language will probably still be needed, just to drive the understanding of future academics. Algorithmic thinking will likely still be needed for a couple of generations minimum. Producing better AI tools will require deep understanding, some of it at least related to programming.

There are relatively few places now where 360 assembly language is the first course taught as it was many years ago when that was the only available tool. Now we use higher level languages that permit a more abstract view of a program and we have compilers that translate the higher level constructs directly into machine code. An assembly language is probably taught somewhere, though, perhaps in an architecture course. The same will likely be true as we advance, with the lower level tools replaced by higher level ones, including AI.

So, things will change, but not fundamentally for, I predict, quite a long time. I once met Joel Moses who predicted (around 1973 or so) that FULL AI was only about ten years away. And so it has remained ever since.

All advances make the solution of harder problems more amenable to solution. As AI advances it should be the same.

In truth, I find current AI to be a pretty restricted thing. One can solve certain sorts of problems, but the methods and results are pretty constrained, and often erroneous. It is difficult to trust a lot of AI, even things like neural networks if it can't give a verifiable explanation of how it arrives at its outputs. Too much of it fails that test. In particular, the driving data of a lot of AI is, itself, flawed and in ways we don't understand yet.

A CS education is directed at more than practitioners. It is an exercise in changing the brain. Many bachelors graduates go into programming as a first task, but that isn't why the major exists. The same is true of most other undergraduate degrees.

One test you could use to determine whether a program producing AI has "arrived" is whether it can produce itself. A Quine.

• I think my concern is more that it will be difficult for students to gain understanding and develop algorithmic thinking if they are using tools that do these things for them. AI is quite a good example of this as there are an awful lot of practitioners that are using tools (such as SciKit learn) following recipies that they have seen on blogs to solve problems without having a fundamental understanding of the methods they are using (and likely finding pitfalls without knowing about them). Apr 7 at 14:08
• I don't think programming skills actually will become obsolete (because the AI we have is not going to understand what it is doing), just jobs for junior programmers. We will still need programmers that understand the program, it just isn't clear that an eduction with AI tools widely used will produce them. Apr 7 at 14:25
• Eh? I think I trust you (or me!) to be intelligent, even though I'm sure you can't use your intelligence to produce a copy of your own intelligence; conversely, it's easy to write a quine in most languages, and those quines are not intelligent merely by virtue of being able to reproduce their own source, because that's all they can do. I think your answer would be improved by removing the parenthetical aside. Apr 7 at 21:33
• Well, Codex can almost certainly produce a quine, though not a quine which contains Codex. But I can't produce a quine which contains me, either, so that's an extremely unfair demand! Apr 7 at 21:47
• It is not very hard for a human to produce another human. Animals and plants reproduce themselves pretty well too. Even bacteria do it. It doesn't take intelligence to produce intelligence. This is why I keep saying that it will 'evolve' (literally, to give off) quite suddenly. We didn't try to duplicate bird muscles to make an airplane. Apr 8 at 23:17

I don't think AI code is a serious game-changer for hiring or education in the short term, even giving it the benefit of the doubt in terms of its power, which is still limited. You mention the problem of students plagiarizing solutions available online, but hired humans are likely just as big of a contributor to cheating, and similar to AI in that the cheater need only state the specification to get the code.

Sure, AI assistants are more accessible than tracking down (and potentially paying) a human to complete an assignment, but at the end of the day the outcome is more or less the same, assuming the cheating isn't caught: a student gets a good grade on an assignment, course or exam that they didn't deserve, or a prospective employee gets hired for a job they're unqualified for.

The problem in both student and employee cases is that they've created an unsustainable situation in their immediate future. For the student, later assignments/exams/courses are going to get harder, and if they've never built a solid conceptual foundation, then they'll either have to come clean and work extra hard to fill in knowledge gaps, or they'll have to resort to further cheating to keep up appearances. Same for the employee: they're hired, but it'll likely be a huge struggle if they're unqualified, with disastrous consequences for them and the employer. And these are the "lucky" ones who weren't caught immediately.

The techniques for catching cheaters don't seem significantly different between AI and existing cheating approaches. For example, you can ask students/candidates to explain their code verbally (written responses are easily hired out to a human) and/or watch the solution coded live. Of course, verbal isn't a perfect foil or terribly scalable (I've participated in MOOC classes of over 500 students to a teaching staff of about a dozen), complicated by remote hiring/learning.

Additionally, it's fairly easy to programmatically compare submitted code against the output of the most popular AI tools using Moss. This suggests that human-hired plagiarism might be harder to detect than AI because it's more likely to be unique, unless the hired coder didn't bother to try to make it look unplagiarized or used an AI solution on behalf of the student/candidate.

If, someday, real-world programming tasks can be completed purely with AI assistance and no understanding is needed, there'll likely be new technical problems. Exams and assignments will be less about whether or not a solution can be coded and understood line-by-line and more about whether the right problem is being solved, how the AI tool was guided, fine-tuning parameters, system-level design, and so forth. A lot of these new requirements smell a lot like much high-level data science, deep learning and ML. Higher levels of abstraction entail different problems rather than an elimination of problems altogether. The trouble is that high-level tools make it easy to solve low-level tasks, which is nothing new (for example, godbolt for completing assembly assignments).

A lot of this makes me think of doping in biking, using data science to optimize shot locations in basketball, Stockfish in chess, auto-tune in music, etc. If everyone is cheating, is anyone cheating, or is that just the new rules of the game? Where do we draw the line between using an available powerful tool effectively versus a cheat? Technological innovation can create this turmoil, but unlike bike doping, it's still hard to cheat a whole CS degree, bootcamp or job with AI code. And if it gets to the point where you can, that'll be the time to reassess the game (or find stricter ways to cling to and enforce the old rules).

• I don't think code generated by such AI tools would be considered plagiarism, indeed one of the criticisms of these tools for use in industry is that they sanitise plagiarism from the code that was used to train them. Hiring humans on the other hand would be a very serious offence if caused (likely to cause the students to be expelled). Apr 8 at 7:02
• "A lot of these new requirements smell a lot like much high-level data science, deep learning and ML. " I work in ML, the use of tools without understanding is a big problem there. It isn't true that you can safely use these tools without a fundamental understanding of the maths, Apr 8 at 7:08
• This answer (+1) is also missing the question of how we are going to bridge the gap from where we are with programming to the point where AI can do the programming for us, which isn't going to happen any time soon (the current approach is not likely to produce an AI that understands the specifications or the code). Apr 8 at 7:11
• "The use of tools without understanding..." -- yep, that's a big problem in all high-level domains I suspect, from databases to distributed systems to functional programming to networking... you can never really escape the low-level details popping up and ruining the party sooner or later, so it's best to know them. But my post is assuming that everything with AI code just works to the extent the hype promises. I'm giving it the benefit of the doubt and (hopefully) showing that it's still not (yet) a game-changer given the cheating that's already pretty much rampant. Apr 8 at 16:15
• "everything with AI code just works to the extent the hype promises" I don't think it is promising more than just helping out with the boilerplate programming. I've been doing a bit of reading and it seems that running code through CoPilot may be enough to protect the user against copyright/licence infringement prosecution, and it can generate multiple solutions, so I don't think it will be as easy to deal with CoPilot plagiarism as it is for code cut and pasted from the web. Apr 8 at 16:18

There's another aspect to the impact of these tools on teaching programming that is worth considering, as reported in the paper, 'The Robots Are Coming: Exploring the Implications of OpenAI Codex on Introductory Programming' given at the Australasian Computing Education Conference (ACE) in Feb 2022. That is the potential impact of student use of them to solve standard introductory programming problems.

From the abstract, 'In this work, we explore how Codex performs on typical introductory programming problems. We report its performance on real questions taken from introductory programming exams and compare it to results from students who took these same exams under normal conditions, demonstrating that Codex outscores most students.'

The paper is available as Open Access from the ACM Digital Library.

• Yes, that is my primary concern, thanks for the reference! Apr 8 at 15:46

Isn't this the next step in the natural progress of computer sciences?

New "layers of abstractions" were constantly introduced, over decades.

CS moved on

• from machine instructions to assembly code

• from assembly code to high-level languages like C

• from high-level-languages to 4th level languages like SQL

• from declarative languages to no-code/rapid-application developments environments,

• and now the AI-powered tools like Copilot are here (based on language models and knowledge graphs)

They will just become part of the curriculum.

None of these developments have made any previous body of knowledge obsolete. Neither when they were new, nor when they matured over the years.

Each time there were some people concerned that old-school skills will die out, or this will degrade students capabilities.

(Students' average apperception might still go down, due to other reasons: computer hardware that cannot be tampered with; greater societal inequality; underfunding of teaching and education in general,... all of which are more serious threats that the rise of AI-based programming aids.)

• No, I doubt it. The problem is that these AI tools don't understand the code they produce. If the programmer doesn't either, that is hardly a recipe for reliable good quality software. So how do we teach programmers to understand code if they don't write it (and as typical pedagogical tasks are likely in the training set the AI tools can answer the exercises we currently use to teach understanding). They are basically a quick way to look up/launder code on the WWW. May 1 at 17:54
• I don't think that these AI tools can be looked on as a level of abstraction as it isn't clear what they do or how they do it (or what they are going to do), and they don't necessarily produce good code. May 1 at 18:02
• At each "step" there were always trade-offs between correctness, productivity enhancements/ convenience and performance-losses/inefficiencies/larger overhead (due to being farther away from the bare metal).
– knb
May 1 at 18:10
• is there a correctness trade-off in compiler? May 1 at 19:05
• Some compilers you can tell to be fuzzy,e.g. specify how Typescript 2.0's "null- and undefined-aware types" or how its "control flow based type analysis" should work in detail (for edge cases), Aggressive optimization in gcc compilers (-O4, or -ffloat-store) comes with trade-offs w.r.t thread-safety or numerical accuracy AFAIK. I'm not an expert in this matter.
– knb
May 1 at 21:08

First how will it affect programming outside of education.

We have in the past invented higher level languages, that replace lower level languages. So nothing new. However we will NEVER get to the point that we can just write in English. This is because English is a terrible programming language: too ambiguous.

So what will the AI be good at?

It will get good at solving standard very well specified problems, with little ambiguity. That is the sort of problem that one would see in an academic assessment.

So what does this mean?

It is little different in type to the problem of assigning students to write an assembler-language program, and they writing it in C, compiling it, and handing in the result, Using a calculator, or downloading an answer off of the web.

What to do about it

Adapt: I will leave this for others to consider how. But not we have done it before. There is nothing now here.

• This isn't answering the question, which was how it will affect teaching programming. "It is no different in type to the problem of assigning students to write an assembler-language program, and they writing it in C, compiling it, and handing in the result." this isn't actually true. As I pointed out, it means that the students can produce code that they don't understand. I have already observed this as a result of students being able to manually download code from the WWW, having a tool to do it for them makes that worse. May 1 at 12:51
• BTW I don't think it is a good idea for undergraduate assignments to be very well specified with little ambiguity. That causes the student not to think about the specifications, or question them. I have a forum for students to ask questions about the specifications of the coursework (so all students can see the answers) and I encourage them to look for ambiguity or issues with the specifications. It is part of software development. May 1 at 13:12
• @DikranMarsupial I agree, and have read and agreed to many of your other comments. But I do think there are similarities to the past, but not exactly the same type. However, the conclusion "adapt", I think is probably correct. Sorry to not give any advice on how to do this. May 1 at 21:13
• @crtl-alt-delor a discussion of how it will affect teaching would also be of interest. It is somewhat of a concern that these tools are being developed without any investigation of their effect advantages and disadvantages. May 2 at 10:42


text:
# Multiplying a Curve by a Plane

Is there a field of mathematics that considers multiplying functions in a manner analogous to matrix multiplication? For instance,

1. Let $\mathbf{x}$ is an $n$-dimensional vector such that $x_i=\sin(2\pi \frac {i} {n}))$, for $i=1,\ldots, n$.
2. Let $\mathbf{A}$ be an $n\times n$ matrix where $A_{i,j}=\cos(4\pi \frac {i} {n} - 2\pi \frac {j} {n})$ for each $i,j$.
3. Let $\mathbf{y}=\mathbf{A}\mathbf{x}$ so that $y_i=\frac 1 2 \sin(4\pi \frac {i} {n})$.

If we imagine $n$ going to infinity, then we can define an operator, $\text{prod_fcn}$, such that $\text{prod_fcn}(x(t), A(t, s)) = y(t)$. This example is illustrated graphically below:

Obviously this is just one simple example of a curve being "warped" by a plane. I'm curious to know if this sort of functional operation is a commonly studied thing. If so, any information about this would be appreciated.

• How familiar are you with functional analysis? Much of it deals with linear operators on infinite-dimensional spaces, which is a direct generalization of multiplication of vectors by matrices (not by "planes", of course). Very often, these infinite dimensional linear operators can be approximated by matrix multiplication. Dec 3, 2017 at 21:46
• Not very...my background is in statistics. Good lead though. And of course, the "planes" part was misleading. I just needed the title to sound catchier.
– jjet
Dec 4, 2017 at 2:06

Looks as if you are missing a factor $1/n$?. But otherwise, your matrix sum corresponds to a discrete approximation of the following integral operator: $$L \phi(y) = \int_0^1 k(y,x) \phi(x) \; dx$$ where $\phi(x)=\sin(2\pi x)$ and $k(y,x) = \cos(2\pi(2y-x))$. An operator of the above form and with continuous (even better $C^1$) kernel $k(y,x)$ has very nice properties, in particular may be well approximated by the discrete matrix version as in your example.

More general versions are in $L^2$ where you are dealing with Hilbert-Schmidt integral operators. But you have to be more careful when approximating by a matrix.

One may construct Fredholm determinant for such operators (which in the continuous case, you may easily approximate by your matrix version). This enables to calculate e.g. eigenvalues for the operator.

You may also have a look at Integral transforms.

These are just some examples of the use of the limiting case of your matrix multiplication.

• Thanks, this is helpful. The integral transform is the most obvious match for my problem. And clearly it has tons of applications.
– jjet
Dec 9, 2017 at 19:56


text:
zbMATH — the first resource for mathematics

A real variable method for the Cauchy transform, and analytic capacity. (English) Zbl 0645.30016
Lecture Notes in Mathematics, 1307. Berlin etc.: Springer-Verlag. vi, 133 p. DM 23.00 (1988).
The pioneering work of A. G. Vitushkin in the 1960’s showed that analytic capacity $$\gamma$$ is a fundamental concept in a number of problems concerning uniform approximation in the complex plane. Therefore there has been much interest in estimates of the set function $$\gamma$$ in terms of metric concepts, capacities etc. The purpose of this book is to prove some new results about $$\gamma$$ and the basic tool for obtaining these results is the Cauchy (-Hilbert) transform $H_{\Gamma}f(z)=\pi^{- 1}p.v.\int_{\Gamma}f(\zeta)(\zeta -z)^{-1}| d\zeta |;\;z\in \Gamma,$ where $$\Gamma$$ is a finite union of disjoint smooth arcs. A number of impressive results are presented, both about $$H_{\Gamma}$$ and various metric estimates of $$\gamma$$. The last theorem of the book, a recent result due to Peter Jones and the author, serves to illustrate this: For $$0<\alpha <1$$ define the integral-geometric quantity $$Cr_{\alpha}(\cdot)$$ (the $$\alpha$$-Crofton length) as follows: $Cr_{\alpha}(E)=\lim_{\epsilon \to 0}(\inf \int^{\pi}_{- \pi}\int^{\infty}_{0}| \partial(\cup_{k}D(z_ k,r_ k))\cap L(r,\theta)|^{\alpha} dr d\theta,$ where L(r,$$\theta)$$ is the straight line x cos $$\theta+y \sin\theta =r$$, $$| F|$$ denotes the number of elements of the set F and the inf is taken over all coverings $$\{D(z_ k,r_ k)\}$$ $$n_{k=1}$$ of the compact set E with radii less than $$\epsilon$$. Then for all $$0<\alpha <$$ there exists $$E_{\alpha}$$ such that $\gamma(E_{\alpha}) = 1\text{ and } Cr_{\alpha}(E_{\alpha})=0$ (Theorem F).
Reviewer: B.Øksendal

MSC:
 30C85 Capacity and harmonic measure in the complex plane
Keywords:
Cauchy transforms; analytic capacity
Full Text:


text:
1. This hack ensures that when you log on to a server/pc it will execute a screen session which minimizes the risks of you losing your work in case of network disruptions . Just copy command to your bashrc file and source it.
#Best used when you have to work on a different server via ssh.

[sourcecode language="bash"]if [ "$TERM" != "screen" ]; then screen -xRR fi[/sourcecode] 2. This hack enables mouse scrolling and scrollbar history scrolling. Just copy command to your ~/.screenrc file and restart your shell. [sourcecode language="bash"] # Enable mouse scrolling and scroll bar history scrolling defscrollback 5000 termcapinfo xterm* ti@:te@ [/sourcecode] 3. Ever gotten to a point where, you can't remember a certain command you executed and when you try to grep it from history it suddenly disappeared well, this hack fixes that. Copy and paste following codes to ~/.bashrc and source it. It creates a file named 'command_log' and logs all executed commands as well as timestamp. Nifty!! very Nifty [sourcecode language="bash"] # log every command typed and when if [ -n "${BASH_VERSION}" ]; then
trap "caller >/dev/null || \
printf '%s\\n' \"\$(date '+%Y-%m-%dT%H:%M:%S%z')\ \$(tty) \\${BASH_COMMAND}\" 2>/dev/null >>~/.command_log" DEBUG
fi
[/sourcecode]


text:
# How do you graph the inequality y>(x-3)^2-4?

Plot $y = {\left(x - 3\right)}^{2} - 4 :$
Vertex is $\left(3 , - 4\right)$, and passes through $\left(1 , 0\right) , \left(5 , 0\right)$
It is a parabola, opened upwared. $y > {\left(x - 3\right)}^{2} - 4$ happens inside the shape.


text:
# New Results from T2K and MINOS

There are a couple neutrino papers out today.

T2K has a new result looking for short-baseline electron neutrino disappearance in the near detector. This kind of search is looking for neutrinos oscillating into a non-interacting, or sterile, neutrino. There are various “anomalies” in certain oscillation results that can potentially be resolved by introducing sterile neutrinos, so this is a way to search for them or exclude their existence.

MINOS has a new result on charged-current quasielastic (CCQE) scattering of muon neutrinos on iron using their near detector. They find some differences between their data and current models at low Q2.

# New AMS Dark Matter Results

PRL published a couple new papers from AMS today, including this one, which shows an updated plot of the positron fraction (e+/(e+ + e-)) as a function of energy. I don’t think these have been posted to the arXiv yet, which is unusual for our field. Unfortunately, PRL must be accessed either through a network with access or with an account, which requires money, so if you’re not at a university and aren’t an APS member you might have trouble looking at the paper. In conjunction with the papers being released, there was a seminar at CERN earlier today. I didn’t find out about the seminar until it was too late for me to try to call in remotely.

AMS is a large multipurpose detector based at the International Space Station that studies cosmic rays, which are high energy particles flying around in space. The search for dark matter is one of the main purposes of the experiment, but it can study many things about cosmic rays, such as their composition, energy spectra, directions, etc.

This paper shows the positron fraction up to 500 GeV, which is a bit higher than in previous measurements. This measurement is useful for looking for dark matter annihilation to electron-positron pairs. The expected signal is for the positron fraction to increase at some energy and then peak and drop fairly sharply at an energy of approximately the dark matter mass. Previous measurements from AMS, Fermi, PAMELA and ATIC have all seen an increase in the ratio and have even seen the spectrum seem to level off. This new measurement shows that the leveling off continues, and the spectrum may even be starting to fall in the highest energy bin. However, the measurement is limited by both statistics and systematics at the highest energies, so the apparent decrease in the highest bin is not going to be statistically significant at this point in time. That may change with more events in the future and hopefully a better understanding of the detector and of astrophysical models for the non-dark matter background.

# New Minerva Pion Production Result

Minerva has a new preprint out today on coherent charged pion production in carbon. They measure both types of charged pions in neutrino and antineutrino beams. The measurement looks at energies up to 20 GeV and finds that agreement with the GENIE Monte Carlo generator is not very good. Minerva is a neutrino experiment using the NuMI beamline at Fermilab.

# New ATLAS Higgs Diphoton Measurement

ATLAS has yet another Higgs measurement on the arXiv, this time looking at the diphoton channel. This is the Higgs decay H→2γ.

Higgs couplings to other particles are related to those particles’ masses. Couplings to fermions are proportional to the fermions’  masses and couplings to bosons are proportional to the squares of the masses. The photon, however, is massless, so it does not couple to the Higgs. Then how does this channel even exist? The Higgs couples directly to particles that also couple to photons. Using Feynman diagrams, the Higgs can couple to photons via a loop of some massive particle like a heavy quark. This leads to an indirect Higgs-photon coupling but the diagram (representing the mathematical equations to calculate the decay width) is more complicated. As a result, the Higgs to two photon decay exists, but it is not particularly common. It is still a very useful channel for searches because the energies of photons can be measured very well, allowing for an accurate reconstruction of the Higgs energy, and because the two photon channel is much cleaner than most channels. There aren’t many processes that will result in two high energy photons with a large transverse momentum, so while the signal is small, the background is small as well. A statistically stronger measurement can be obtained from the relatively few events in this channel. Higgs production via gluon fusion is basically the opposite process – just with gluons rather than photons – as this decay,

As with the earlier paper on the four lepton final state, this paper measures the total number of H→2γ events at 7-8 TeV and also tries to separate the sample into various Higgs production channels. Not surprisingly, no significant deviations from Standard Model predictions are found.

# New ATLAS Higgs Paper Posted to ArXiv

A new Higgs measurement from ATLAS is out on the arXiv now. The paper presents the result of a measurement looking for the process H→ZZ*→4ℓ, where Z* is a virtual Z boson, since the Higgs isn’t heavy enough for H→ZZ to be possible.

The four lepton channel is one of the best channels for Higgs precision measurements. While ZZ*→4ℓ represents only a small fraction of ZZ* events, it has a number of advantages over other channels. Detectors can measure electrons and muons with much more accuracy than most particles. Electrons are stable while muons live long enough not to decay in the detector. No energy is carried away by neutral particles like neutrinos. By measuring the leptons, the invariant mass of the Z boson can , and the full invariant mass of the Higgs can be reconstructed for each event. The channel is also cleaner than most (has a good signal to background ratio), with the lepton reconstruction allowing for things like using the reconstructed Z mass from pairs of leptons to aid the selection. The paper also uses subsamples of the total data set to estimate the contributions from several different Higgs production mechanisms (like vector boson fusion and gluon fusion).

# New Dark Matter Result from PandaX

The PandaX group has released their first dark matter result, which can be found here. PandaX is a dual-phase liquid xenon detector, similar to XENON-100 and LUX. The actual result is not particularly interesting, but PandaX is an interesting experiment because it is one of the first particle physics experiments to be hosted in China. The collaboration is mostly from China but includes a few people from institutions in the US.

# Dark Matter Evidence: Galactic Clusters

In the latest entry in my Physics for Non-Physicists series, I will go over some of the evidence for dark matter from measurements of galactic clusters. This post is more of a historical overview of the subject than a review of the most recent literature.

I gave a very brief introduction to dark matter in my previous post on the thermal history of the universe. To recap, dark matter is a form of matter believed to exist that does not interact much with either itself or with the particles of the Standard Model.

As far back as the 1930s (and possibly earlier), the theoretical background and experimental methods of astrophysics had advanced to the point where things like the velocities and luminosities of far-away galaxies could be measured with a reasonable amount of accuracy.

Galactic clusters are a structure containing many galaxies and associated gas in a gravitationally-bound system. In cases where the effects of general relativity can be ignored with little effect (i.e. most cases), gravity follows a Kepler potential: the gravitational potential energy is proportional to inverse distance while the gravitational force is proportional to the square of the inverse distance. For a Kepler potential, it can be shown that the center-of-mass (removing any overall motion) kinetic energy (T, calculated from taking the sum of the kinetic energies of all objects in the cluster) has a particular relationship to the internal gravitational potential energy (U): 2T = -U. This is known as the virial theorem.

Still in the 1930s, it was generally assumed that stars and galaxies follow an approximately constant mass-to-luminosity ratio. That is, when we look at large enough objects, the amount of light has a  known relation to the amount of mass, so by measuring the luminosity, the mass could be estimated. The gravitational potential energy between two objects is -G m1m2/r where G is Newton’s gravitational constant, the m‘s are the masses, and r is the distance between them. From dimensional analysis, the total potential energy of the cluster is

$U = - \frac{xGM^2}{R}$

where M is the total mass, R is an approximate radius (size) of the cluster, and x is a constant of order 1 related to the shape of the cluster (spherical, ellipsoidal, etc). The kinetic energy is the sum of 1/2 mv2 (half of mass times center-of-mass velocity squared) for all the objects in the cluster. This can be estimated from a representative sample of galaxies in the cluster, since this can be rewritten as (1/2)MV2 where again M is the total mass and now V2 is the estimated mean squared velocity. The velocity can’t be directly measured because the distances are so incredibly large that motion cannot be detected. Instead, the velocity along the line-of-sight direction can be determined by measuring the light spectrum (amount of light at different wavelengths). Certain spectral features occur at known wavelengths. Motion along the line of sight causes redshift (motion away from us) or blueshift (motion toward us), which are terms for the Doppler effect when applied to light. The sound version of the Doppler effect is something most people have noticed in everyday life: As fast moving objects emitting sound approach and then recede from us, the pitch (frequency/wavelength) of the sound changes. Returning to the topic at hand, by applying the virial theorem, we can see that the mass is given by

$M = \frac{RV^2}{Gx}$.

In the early 1930s, an astronomer named Fritz Zwicky was studying the properties of the Coma Cluster (shown at the top of this post). Using the above results, he compared the mass as estimated from the virial theorem to the mass as estimated from the luminosity. The result was surprising: What I’ll call the “virial mass” was much larger than the “luminosity mass:” the galaxies in the cluster seemed to be moving much too fast. Zwicky’s 1937 paper in Ap.J. can be found here. An earlier paper was published in 1933, but it’s not in English so I cannot read it.

Zwicky notes that there are several ways to explain the discrepancy between the estimated masses. One interesting way is that on very large scales (the mass-to-luminosity ratio was for nearby star systems), the gravitational mass is dominated by some form of matter that is not emitting light (i.e. is “dark matter”). This is one of the earliest and most influential proposals for a theory of dark matter. It turns out that Zwicky actually overestimated the discrepancy by a significant amount. More modern values, using a much stronger understanding of astrophysics, still show a discrepancy – our evidence for dark matter has also become much more convincing – but it’s more modest than Zwicky’s, which showed the “virial mass” to be around 100 times the “luminosity mass.”


text:
# Maximal Ideal iff Quotient Ring is Division Ring

## Theorem

Let $R$ be a ring with unity.

Let $J$ be an ideal of $R$.

The following are equivalent:

$(1): \quad$ $J$ is a maximal ideal.
$(2): \quad$ The quotient ring $R / J$ is a division ring.

## Proof

By Quotient Ring of Ring with Unity is Ring with Unity, $R / J$ is a ring with unity.


text:
Configuring DCOM for remote invocation

Requires Analytica Enterprise.

You can instantiate and use COM objects on remote computers from Analytica, but for this to work, various Windows network and security options must be configured to allow remote DCOM invocation. Windows access permissions are quite difficult to figure out for most of us. This page provides notes and instructions for getting this to work.

Install Component

The component that you want to install needs to be installed on the remote computer. If you want to refer to it by name in COMCreateObject rather than be CLSID, then you'll also need to have it installed on the computer running Analytica.

Enable DCOM

Allow DCOM must be enabled on both computers. Instructions for this are at: Enable or Disable DCOM. In short, go to Component services/Computers/MyComputer/Properties/Default Properties and select Enable Distributed COM on this computer.

Firewalls on both computers must be turned off or configured to allow DCOM communication.

Remote Account

The component will need to run under some account on the remote computer.

Instantiation

Your Analytica model will instantiate the object on the remote computer (or computers) using:

COMCreateObject(name, server, flags, user, pwd)

An example will be inserted here

Errors

If you encounter the error "Cannot create the COM object...", click the "Wiki help on this error" link in the lower left corner for causes and remedies for the particular error encountered.

Here are some additional pages with useful information and instructions: Resolving "Access Denied" Errors

Note: Numerous parts of instructions found on various web pages don't work in Windows 7 Home edition, particularly instructions for assigning users to groups (such as the "Distributed COM users" group). We have not been able to determine from information found in web searches whether there is some sort of inherent limit on the ability to use DCOM on the Home edition. We have burned many hours/days trying to solve the "Access denied" error, and are now floating the conjecture that it is a limitation of the Home edition.

History

Introduced in Analytica 4.6.


text:
# SymPy 0.5.6¶

30 Oct 2007

• _sage_() methods implemented to convert any SymPy expression to a SAGE expression.
• isympy fixed so that it always tries the local unpacked sympy first (the one in the directory where isympy sits) and only then the system wide installation of sympy (Debian package for example).


text:
## Matrix Differential Equation

1. The problem statement, all variables and given/known data
The questions are in the image

3. The attempt at a solution
My solutions are
V1=3*(1 -2)e-2t+ (-2) (1 -3)e-3t
V2=1*(1 -2)e-2t+ (-1) (1 -3)e-3t

How do I get the X matrix since my solutions are in exponential still.

Thank you for all the help
Attached Thumbnails

 PhysOrg.com science news on PhysOrg.com >> Heat-related deaths in Manhattan projected to rise>> Dire outlook despite global warming 'pause': study>> Sea level influenced tropical climate during the last ice age
 According to your solution, $$X=(v_1\ v_2)=\begin{pmatrix}3e^{-2t}-2e^{-3t}&-6e^{-2t}+6e^{-3t}\\ e^{-2t}-e^{-3t}&-2e^{-2t}+3e^{-3t}\end{pmatrix}.$$

 Quote by Some Pig According to your solution, $$X=(v_1\ v_2)=\begin{pmatrix}3e^{-2t}-2e^{-3t}&-6e^{-2t}+6e^{-3t}\\ e^{-2t}-e^{-3t}&-2e^{-2t}+3e^{-3t}\end{pmatrix}.$$
That means I just plug in t=0 to prove that x(0)=(1 0
0 1) and also use the same method to prove that dx/dt =AX

Recognitions:
Gold Member
Homework Help
Staff Emeritus

## Matrix Differential Equation

You have the columns and rows swapped.

 Quote by vela You have the columns and rows swapped.
Even after multiplying it I get this respective solutions, what do I do next?
Attached Thumbnails

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus You can write v1 as a single vector: $$\vec{v}_1 = \begin{pmatrix} 3e^{-2t} - 2e^{-3t} \\ -6e^{-2t} + 6e^{-3t}\end{pmatrix}$$Do the same for ##\vec{v}_2##.

 Quote by vela You can write v1 as a single vector: $$\vec{v}_1 = \begin{pmatrix} 3e^{-2t} - 2e^{-3t} \\ -6e^{-2t} + 6e^{-3t}\end{pmatrix}$$Do the same for ##\vec{v}_2##.
i get that part but after that what do I do to get just numbers in my 2x2 matrix so that I can prove dx/dt =AX and x(0)= \begin{pmatrix}1 0\\ 0 1\end{pmatrix}

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Like the problem says, the first column of X is v1. Its second column is v2. It's not going to be just numbers. I'm not sure why you think it has to be.

 Quote by vela Like the problem says, the first column of X is v1. Its second column is v2. It's not going to be just numbers. I'm not sure why you think it has to be.
Okay but then if I put t=0 into the x equation, I do not get the identity matrix and how would I verify that dx/dt=AX by just differentiating the X matrix?.

Thank you for all the help

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Show us how you're calculating X when t=0.

 Quote by vela Show us how you're calculating X when t=0.
Okay I got just did a arithmetic error, but how do I verify that dx/dt=AX

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Calculate both sides and show they're equal to each other.
 Okay I will give it a try, thank you very much for all the help.