calculation puzzle - Magic: the Gathering - Challenge #6.b: It's Gonna Cost You

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Background

Alright I'm going to try a more clear approach to my last attempted puzzle. Given the mana costs given below, name cards with those exact mana costs to use in a solution that wins this turn (the card chosen must match the exact mana cost printed on the card so e.g. RR can be a Wild Guess but not a Rolling Thunder or a kicked Shattering Spree). Once you decide what card is associated with a particular cost, it must stay the same card in all zones; all usual rules apply to the card once chosen so if you decide that 3B is a Heartstabber Mosquito, you can pay kicker as though it were that card. You can play any card face down as a basic land that makes any color mana (still only one land per turn)

Some addition rules:

• The same spell may not be named for multiple cards (e.g. even if you're only casting one copy of Dark Confidant, no other 1B card may be named as Dark Confidant)


• The same spell may not be cast more than once


• The Legacy banned list and rules should be observed (sorry Demonic Tutor)

Puzzle Setup

It's turn 1 and you're on the play. Let's assume that (magically) your opponent has no cards in hand. Win this turn with the rules given above and the mana costs given below. Both boards and graveyards are empty. Your solution must be a guaranteed win, not involving your opponent self-destructing (e.g. they choose 20 on a Choice of Damanations) or random chance (you Hymn to Tourach yourself and hit the right 2 cards). If multiple solutions are achieved, the one that does the most damage wins.

Your hand:

1B x 2
1
BBB
B x 3

Your library: (top to bottom)
B
1G
1WW
R
U

BB
G

Your opponent's library:
8GGG x 60

general relativity - The einbein in the action of a relativistic massive point particles

The action of a relativistic massive point particle moving in space-time is

$$S=-m\int d\tau \sqrt{g _{\nu \rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau}}$$

[with Minkowski sign convention $(+,-,-,-)$]. Because of the the square root in the action, one introduces the einbein field $e$ and the action becomes

$$S=1/2\int d\tau(e^{-1}\dot{X}^{2}-em^{2}).$$

I understand this but I can't come up with the expression of the action with the einbein field myself.

fluid dynamics - Pressure in speed of efflux

While describing speed of efflux, using Bernoulli equation, we use value of $P$ at both openings as atmospheric pressure. Why don't we use $P_\text{atmosphere}+\rho gh$ for $P$ at bottom opening?

everyday life - Why is concrete dry around cracks?

On a warm and partly cloudy day this summer, I was sitting in my office and suddenly heard rain on the roof. I went outside to shut my car windows. By the time I did that and was walking back into the building, the rain had stopped. Such short showers from a small single passing cloud are not unusual here in New England.

The interesting part is that on the way back I noticed a strange pattern of wetness on a concrete area against the building. I went back to my office, grabbed the office point-and-shoot camera, and took this picture:

This was looking from the top of a staircase down onto the concrete area. This area is around 10x10 meters, maybe a bit less.

The question is, why does the concrete look dryer around the cracks? The effect became less evident in the minute or two it took me to go back to my office, grab the camera, come back out, and take this picture. But, you can still see light-colored streaks in the pattern of darkness due to wetness. It may not be evident in all cases in the picture, but these streaks all followed cracks in the concrete. Why?

Other facts that may be relevant:

1. The rain lasted only 2-3 minutes.



2. The rain came in large drops, definitely not "misty".


3. There was never enough rain for water to flow on the ground.


4. Temperature was probably in the 80s °F the whole time.


5. It had been on/off sunny previously.


6. The picture was taken probably about 5 minutes after first hearing raindrops.


7. The concrete was already noticably drier from when I first saw it until the picture was taken. The darker wet areas had also diffused out more.


8. The whole area has a gentle slope towards the drain in the upper right corner of the picture. Note that this phenomenon seems to be independent of the orientation of the crack.

homework and exercises - Showing $K_pm$ are raising/lowering operators

In this post, I have the following operators defined: $$K_1=\frac 14(p^2-q^2)$$ $$K_2=\frac 14 (pq+qp)$$ $$J_3 = \frac 14 (p^2+q^2)$$ I am given $ J_3|m\rangle = m|m\rangle$ and asked to show that $K_\pm \equiv K_1 \pm i K_2$ are ladder operators.

My approach (raising operator): $$K_+|m\rangle=K_1|m\rangle+iK_2|m\rangle$$ $$=K_1|m\rangle+[J_3,K1]|m\rangle$$ $$=K_1|m\rangle+ (J_3K_1-K_1J_3)|m\rangle$$ $$=K_1|m\rangle+J_3K_1|m\rangle-K_1m|m\rangle$$

First off, I'm unsure if this is the correct approach, and then I'm also lost on what to do next.

Answer

Here's the basic idea behind ladder operators in a bit of generality.

Let's say that I have a self-adjoint operator $J$ on the Hilbert space $\mathcal H$ of a given system, and suppose that $\{|m\rangle\}$ were an orthonormal basis for $\mathcal H$ consisting of eigenvectors of $J$, namely \begin{align} J|m\rangle = m|m\rangle. \end{align} Now, suppose you were to also find an operator $O_+$ that has the following commutation relation with $J$: \begin{align} [J,O_+] = cO_+ \tag{$\star$} \end{align} for some number $c$, then notice that an interesting thing happens when we apply $O_+$ to the states $|m\rangle$; \begin{align} J(O_+|m\rangle) &= (O_+ J +[J,O_+])|m\rangle\\ &= (O_+J+cO_+)|m\rangle \\ &= (m+c)(O_+|m\rangle). \end{align} In other words, $O_+|m\rangle$ is an eigenvector of $J$ with eigenvalue $m+c$, so $O_+$ raises the eigenvalues of a given state by $c$.

In your case, $J_3$ is analogous to $J$, and you simply need to show that $K_\pm$ have commutation relations with $J_3$ that are analogous to $(\star)$.

astrophysics - Can there be Electron and/or Proton Stars?

1. 
   

   What happens to all of the electrons and protons in the material of a neutron star?

   
   


2. 
   

   Could there ever be an electron star or a proton star?

   
   
   

Answer

If a dense, spherical star were made of uniformly charged matter, there'd be an attractive gravitational force and a repulsive electrical force. These would balance for a very small net charge: $$ dF = \frac1{r^2}\left( - GM_\text{inside} dm + \frac1{4\pi\epsilon_0}Q_\text{inside} dq \right) $$ which balances if $$ \frac{dq}{dm} = \frac{Q_\text{inside}}{M_\text{inside}} = \sqrt{G\cdot 4\pi\epsilon_0} \approx 10^{-18} \frac{e}{\mathrm{GeV}/c^2}. $$ This is approximately one extra fundamental charge per $10^{18}$ nucleons, or a million extra charges per mole — not much. Any more charge than this and the star would be unbound and fly apart.

What actually happens is that the protons and electrons undergo electron capture to produce neutrons and electron-type neutrinos.

lagrangian formalism - Noether's theorem: meaning of transformation of coordinates

I have a question regarding Noether's theorem. In our introductory QFT class (which is based on the book by Michele Maggiore) we have derived the Noether currents in the same form as displayed in this post: Question about Noether theorem In this formula, there are contributions from two different kinds of transformations: a transformation of the field alone and a transformation of the coordinates.

My problem is: I don't understand the meaning of the transformation of coordinates. I have tried to understand the derivation from different QFT books (and I haven't found the same derivation twice, which doesn't make it easier) in the hope that I then would better understand the premises, but unfortunately I have not succeeded so far.

Also Peskin/Schröder for example only discuss transformations of fields and don't mention the transformation of coordinates at all. Poincare symmetry, which is in most books treated as a transformation of the coordinates, can be treated also as a transformation of fields, as shown in the answer to the following question for pure translations: Noether's Theorem: Foundations. Like the guy who asked that question, I think that the coordinates entering the action are only dummy variables. So what is then the meaning of the coordinate transformations in the prevalent formulation of Noether's theorem? Maybe someone can give a concrete example to illustrate the idea.

Answer

I think that after 1,5 years I can finally appreciate Qmechanic's answer. Let me try to formulate what I think would have been the ideal answer on my question and correct me if I am wrong. I'm using the symbols as defined in Weinberg's Quantum Theory of Fields.

A field is a function $\Psi: M \rightarrow N$, where $M$ is Minkowski space (which we call for convenience the horizontal space) and $N$ is the target space of the fields (which we call for convenience the vertical space). $N$ can e.g. be the space of scalars, vectors, Dirac spinors, antisymmetric tensors, etc. In order to define an infinitesimal transformation $X \rightarrow X$ (where $X$ is the space of fields), we can consider separate infinitesimal transformations in the horizontal and vertical space:

a) horizontal transformation: Transformation of the type $h:M\rightarrow M$. An example is $x \mapsto x + \omega x$ (if $\omega$ with both indices up or down is antisymmetric, this is an infinitesimal Lorentz transformation).

b) vertical transformation: Transformation of the type $v:N \rightarrow N$. An example is $\Psi \mapsto \Psi + \frac{i}{2}\omega^{\mu\nu} \mathcal J_{\mu\nu} \Psi$, where $J_{\mu\nu}:N\rightarrow N$ are the infinitesimal generators in the irreducible representation of the Lorentz group under which the elements of $N$ transform. $\omega$ is defined as above.

We can now define the transformation of fields $X \rightarrow X$ by combining a horizontal and a vertical transformation: $\Psi \mapsto \Psi'$ with $\Psi'(h(x)) := v(\Psi(x))$. The action is only a functional of the fields themselves, but as we see the transformed field depends on both a vertical and a horizontal transformation.