text:
# Customizing¶

The OpenPNM framework was designed with extensibility in mind. Every user will apply OpenPNM to a unique problem, and will therefore require unique pore scale models, phase properties, algorithms and so on.

There are two ways to customize OpenPNM. The first is to download the source code and hack it. With this approach it is possible to create your own sub-classes, add pore-scale models, define new topology generators, and to add or change OpenPNM methods. The other approach is to install OpenPNM in your Python PATH (using pip install openpnm) as with any other package such as Scipy, and write custom functions in a separate ‘working directory’ rather than in the source code. The second approach is the recommended way for several reasons. It avoids accidental changes to the framework, it allows users to keep their ‘projects’ compartmentalized, and it is much easier for users to contribute their work to OpenPNM project since sections can be merged into the framework cleanly. The second approach will be explained in detail below.

The following discussions assume that all custom files will be stored in a folder called 'my_pnm', that will be the ‘working directory’.

## Creating Custom Models¶

In the working directory, place a file called ‘my_models.py’. This file will be the home of all the custom models that will be created. Models that are in this file can be added to any object using the add_model command. The ‘models’ mechanism in OpenPNM was designed to be as straight-forward as possible, so each model is simply a function definition.

Let’s create a model called ‘surface_roughness’ that calculates the surface area of a pore accounting for surface roughness, that accepts a single ‘roughness parameter’ and is a function of pore size. Start by writing a function in the ‘my_models.py’ file that simply accepts the ‘roughness_parameter’ and returns it:

def surface_roughness(roughness_parameter):
return roughness_parameter

Now, we can see this model in action by creating a script in the working directory as follows:

import my_models
a = my_models.surface_roughness(roughness_parameter=2.2)
print(a)
2.2

The next step is to have this model calculate something useful. Assume that surface area due to roughness scales with the projected or smooth surface area as some function, which means this function will need access to the ‘pore.diameter’ information, which is stored on Geometry objects. Thus the relevant Geometry object must be sent as an argument. OpenPNM assumes object for which the data is being calculated is passed in as target, which makes all function calls general:

def surface_roughness(target, roughness_parameter):
P_diam = target['pore.diameter']
projected_area = 4*3.14159*(P_diam/2)**2
rough_area = projected_area**roughness_parameter
return rough_area

We can now update our script:

import openpnm as op
import scipy as sp
import my_models

#Generate a simple Cubic Network
pn = op.network.Cubic(shape=[3,3,3])

#Generate an 'empty' Geometry with no properties
geom = op.geometry.GenericGeometry(network=pn,pores=pn.pores(),throats=pn.throats())

#Assign random pores diameters between 0 and 40
geom['pore.diameter'] = sp.rand(pn.Np,)*40

#Assign model to geometry
model=my_models.surface_roughness,
roughness_parameter=2.2)

The same approach can be used to create models for pore-scale Physics or for calculating fluid properties that are not included with OpenPNM.

## Using non-Default Property Names¶

In the ‘surface_roughness’ example above, the function assumed that pore diameter data would be found under the ‘pore.diameter’ dictionary key. If for some reason, there were multiple different definitions of ‘pore diameter’, then they might be stored as ‘pore.diameter_inscribed’, and ‘pore.diameter_hydraulic’, etc. To allow the ‘surface_roughness’ function to be applied to any arbitrary pore diameter, it should be rewritten as:

def surface_roughness(target, roughness_parameter, pore_diameter='pore.diameter'):
P_diam = target[pore_diameter]
projected_area = 4*3.14159*(P_diam/2)**2
rough_area = projected_area**roughness_parameter
return rough_area

Note that pore_diameter is now an argument name, which defaults to ‘pore.diameter’. Different pore diameters can be specified when calling add_model:

#Assign model to geometry
model=my_models.surface_roughness,
pore_diameter = 'pore.diameter_inscribed',
roughness_parameter=2.2)

All of the models provide with OpenPNM allow for this sort of non-default argument names, and it will make your custom models more general if you follow this practice.

## Creating a Basic Subclass: Predefining Pore-Scale Models¶

Another way to customize OpenPNM is to create custom subclasses. This approach is best if you wish to apply several pore-scale models to a single object, since it let’s you collect them all into one place. This is done by starting with one of OpenPNMs existing Generic classes and adding a suite of pore-scale model definitions in the init stage. This is how OpenPNM classes such as Water, and StickAndBall operate. They do not actually overload or add any method, but just act as a collection of pre-set pore-scale models.

For example, let’s create a custom Phase object for an oil with temperature dependent viscosity and density. The following class definition can be added to 'my_classes.py' in the same directory as 'my_models.py'.

from openpnm.phases import GenericPhase

class Oil(GenericPhase):
def __init__(self, **kwargs):
super().__init__(**kwargs)  # This is a python thing, and calls the init of the parent class

model=op.models.misc.polynomial,
a=[10000, -111, 2],
prop='pore.temperature')

model=op.models.misc.linear,
prop='pore.temperature',
m=2200, b=-20)

The first of the above two models creates a property called ‘pore.viscosity’, which using a polynomial function to describe the dependence on temperature (indicated by the prop argument). The second model is similar but using a linear fitting to describe the density. The values used for the coefficients of these two models will dicatate the final physical properties of the Phase, so this is now a custom phase.

Thus when you change the values of ‘pore.temperature’ on an instance of Oil, then call regenerate_models, these two models will be run and will look at the current value in ‘pore.temperture’.

## Creating Customized Networks¶

Unlike Geometry, Phase and Physics objects, a Network object requires more than a collection models. In fact, Networks typically have no models. Creating a custom Network type is all about defining the locations of the pores, and the connections between the throats. Consider the following basic graph:

4 ----- 3
| \   / |
5 - 0   |
|     \ |
2 - 6 - 1

Implementing this as a ‘custom’ Network can be done as by noting that the pore coordinates are:

coords = [[1, 1, 0],
[2, 0, 0],
[0, 0, 0],
[2, 2, 0],
[0, 2, 0],
[0, 1, 0],
[1, 0, 0]]

And the connections are:

conns = [[0, 1],
[0, 3],
[0, 4],
[0, 5],
[1, 3],
[1, 6],
[2, 5],
[2, 6],
[3, 4],
[4, 5]]

A custom Network can be created by passing these two arrays in to a GenericNetwork class:

pn = op.network.GenericNetwork(coords=coords, conns=conns)

The GenericNetwork class is able to handle topologies of any sort, so you can be as imaginitive as you wish when defining the network. There are several rules and assumptions about how the conns and coords data must be formatted, which are descibed in Representing Topology

To create an actual Network subclass, you can add the following to your 'my_networks.py' file:

from openpnm.network import GenericNetwork

class MyNetwork(GenericNetwork):
def __init__(self, **kwargs):
# Place pore locations and throat connecting generation code here
super().__init__(conns=conns, coords=coords, **kwargs)

## Creating Customized Algorithms¶

Algorithms can also be customized as described above. The GenericAlgorithm has a few additional methods that are meant to be implemented by subclasses, such as return and reset. The intention of this method is to send the pertinent results of a calculation ‘out’ of the Algorithm object and to the correct object in the simulation. This step is handy, but is not actually necessary. One can of course manually transfer data from an Algorithm to a Phase, for instance with:


text:
# Homework Help: The angle between two horizontal forces?

1. Oct 17, 2006

### zcm5000

I need help with this problem (my work is at the bottom):

http://img96.imageshack.us/img96/9823/physicsquestiondd9.jpg [Broken]

Here is what I tried:

$$a_x$$ from graph = 3 m/s^2

$$Fnetx$$ = m* $$a_x$$

$$F_n_e_t_x$$ = 3.9 N

$$F_n_e_t_x$$ = -3 + $$F_2*\cos(\theta)$$

6.9 = 9.0*$$\cos(\theta)$$

$$\cos(\theta)$$ = 0.7667
$$\theta$$ = 39.9 degrees from x-axis
degrees between forces = 180+39.9 = 219.9 degrees

This is wrong and I've been trying to figure out what to do for a long time so I'm hoping someone can help me out please? Thanks

Last edited by a moderator: May 2, 2017
2. Oct 17, 2006

### BishopUser

Typically you use the smaller angle between two things. Have you tried that?

3. Oct 17, 2006

If the force F1 points in the negative x direction, and F2 has an angle of 39.9 with the positive x direction, then the angle between them is 90 + (90 - 39.9) = 180 - 39.9 = 140.1 degree, unless I missed something.

4. Oct 17, 2006

### zcm5000

Yes, I believe that is right. However, I tried inputting 140 and it is incorrect. I am unsure if $$F_1$$ is actually going at 180 degrees. It only says it is going in the negative x direction? Also, I am unsure if I am setting up the equation correctly, and also if I am supposed to use $$F_2\cos(\theta)$$.

Can anyone confirm or deny this?

Last edited: Oct 17, 2006
5. Oct 17, 2006

Using $$F_{2} \cos(\theta)$$ is correct, since $$\vec{F}_{2}$$ makes some angle $$\theta$$ with the x axis. Further on, $$\vec{F}_{1}$$ is not 'going' at 180 degrees. It is a simple calculation of the angle between the two force vectors, and 180 arises in it.

6. Oct 17, 2006

### zcm5000

Well I am not sure what I did wrong because neither 140 or 220 degrees is correct. Is the equation supposed to be $$F_n_e_t_x$$ = $$-F_1$$ + $$F_2\cos(\theta)$$? Hopefully someone can show me where I went wrong

Last edited: Oct 17, 2006
7. Oct 17, 2006

### civil_dude

I also get that the angle between them is 140.06 degrees.

8. Oct 17, 2006

### zcm5000

Thanks guys, the answer WAS 140 degrees :rofl: . I thought I had tried that answer before I tried 220, but I only tried 220 :grumpy: Thanks for making my physics experience a little better

At least I understand why I got that answer now

format: plain


text:
# [FOM] Correction about claim on RT(2)

Andreas Weiermann weiermann at math.uu.nl
Sun Oct 10 12:32:16 EDT 2004

There has been a subtle error in my proof concerning my claim
What can be said is now the following.

Let DENSE(2,3) be the principle:

For every $n$ there is an $y$ such that
$[0,y]$ is $n$-dense(2,3).

Here a $0$-dense(2,3) set is by definition a finite set
satisfying $card(X)\geq min(X)$.
A finite set $X$ is $n+1$-dense(2,3) if for every
$F:[X]^2\to 3$ there exists a monochromatic
set $Y\subseteq X$ such that $Y$ is $n$-dense(2,3).

Theorem: PRA does not prove DENSE(2,3)

The reason is that DENSE(2,3) proves over RCA_0
the totality of the Ackermann function.
(It seems that the proof also applies to
the case of two colors.)

I thought that WKL_0+RT^2_2 suffices for a routine proof of
DENSE(2,3) but there was a mistake
(and perhaps the claim is just false).
In any case DENSE(2,3) is provable in RCA_0 plus the
one consistency (uniform reflection principle) for WKL_0+RT^2_2.
A proof is contained in the Friedman, McAloon and Simpson
paper from the Patras Logic Symposium Proceedings 1982.

Best regards,
Andreas Weiermann.


text:
# Fitting distribution in histogram using Python

I was surprised that I couldn't found this piece of code somewhere.

What I basically wanted was to fit some theoretical distribution to my graph. If you are lucky, you should see something like this:

from scipy import stats
import numpy as np
import matplotlib.pylab as plt

# create some normal random noisy data
ser = 50*np.random.rand() * np.random.normal(10, 10, 100) + 20

# plot normed histogram
plt.hist(ser, normed=True)

# find minimum and maximum of xticks, so we know
# where we should compute theoretical distribution
xt = plt.xticks()[0]
xmin, xmax = min(xt), max(xt)
lnspc = np.linspace(xmin, xmax, len(ser))

# lets try the normal distribution first
m, s = stats.norm.fit(ser) # get mean and standard deviation
pdf_g = stats.norm.pdf(lnspc, m, s) # now get theoretical values in our interval
plt.plot(lnspc, pdf_g, label="Norm") # plot it

# exactly same as above
ag,bg,cg = stats.gamma.fit(ser)
pdf_gamma = stats.gamma.pdf(lnspc, ag, bg,cg)
plt.plot(lnspc, pdf_gamma, label="Gamma")

# guess what :)
ab,bb,cb,db = stats.beta.fit(ser)
pdf_beta = stats.beta.pdf(lnspc, ab, bb,cb, db)
plt.plot(lnspc, pdf_beta, label="Beta")

plt.show()


text:
## anonymous 4 years ago Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this. 10x+5y=52.5 30x-7y=80.5

1. anonymous

multiply the first by -3, add to the second

2. anonymous

ok so 10x*-3

3. anonymous

ok now is the second part or the whole thing

4. anonymous

ok i messed up yet again so let me start afresh

5. anonymous

ok great

6. anonymous

7. anonymous

$10x+5y=52.5$ $30x-7y=80.5$ multiply top by -3 get $-30x-15y=-157.5$ $30x-7y=80.5$

8. anonymous

add the two equations together. the x's add up to zero and you get one equation in y, namely $-22y=-77$ solve for y and get $y=\frac{77}{22}=\frac{7}{2}=3.5$

9. anonymous

now that we know y, replace y by 3.5 in either equation and solve for x

10. anonymous

$10x+5y=52.5$ $10x+5\times 3.5=52.5$ $10x+17.5=52.5$ $10x=35$ $x=3.5$

11. anonymous

so both x and y are 3.5 it doesn't usually work out that way

12. anonymous

ok got it now


text:
# Combined distribution of beta and uniform variables

Given

$$X \sim \text{Beta}(\alpha,\beta)$$ (where $\alpha=\beta$, if that helps) and

$$\theta \sim \text{Uniform}(0, \pi/2).$$

I'm trying to find a formula for $P(Y)$ (or even the cdf) of

$$Y = X + (1-2X)\cos(\theta)$$

on the domain $(0,1)$.

I know from here that given $C = \cos(\theta),$ its PDF is

$$P(C) = \frac{2}{\pi\sqrt{1-C^2}}$$

and of course the pdf of a beta distribution is

$$P(X) = \frac{X^{\alpha-1}(1-X)^{\beta-1}}{\mathrm{Beta}(\alpha, \beta)}.$$

But combining them is getting beyond my skills as an engineer.

EDIT: As it seems a closed form is not possible for this, could someone please show me how to formulate the integral to calculate this sort of PDF? There may be some way I can reformulate the problem to be more solution-friendly if I could wrap my head around how compound distributions of this type are built mathematically.

• I can obtain analytic solutions for the PDF when $\alpha=\beta$ are integral, but they are messy and get exponentially messier as $\alpha$ grows, suggesting there won't be any nice formula. Therefore they would be of little use for analysis. If you just need the distribution for particular $\alpha,\beta$ then numerical integration or even simulation will work well. If you need to analyze this distribution to obtain general properties, then you will have to work with the integrals without directly evaluating them. – whuber Jan 18 '18 at 15:26
• An integral solution would be worth an answer for me, unless someone else has some brilliant closed form, whether it's for $\alpha = \beta$ or not. I'm also just interested in the process for formulating the integral. – Daniel F Jan 19 '18 at 5:57


text:
# Axions and cosmic rays

• Published on
25-Aug-2016

• View
212

0

Transcript

• Theoretical and Mathematical Physics, 170(2): 249262 (2012)

AXIONS AND COSMIC RAYS

D. Espriu and A. Renau

We investigate the propagation of a charged particle in a spatially constant but time-dependent pseudo-

scalar background. Physically, this pseudoscalar background could be provided by a relic axion density.

The background leads to an explicit breaking of Lorentz invariance; processes such as p p or e eare consequently possible under some kinematic constraints. The phenomenon is described by the QED

Lagrangian extended with a ChernSimons term that contains a four-vector characterizing the breaking

of Lorentz invariance induced by the time-dependent background. While the induced radiation (similar to

the Cherenkov eect) is too small to inuence the propagation of cosmic rays signicantly, the hypothetical

detection of the photons radiated by high-energy cosmic rays via this mechanism would provide an indirect

way to verify the cosmological relevance of axions. We discuss the order of magnitude of the eect.

Keywords: axion, high-energy cosmic ray, galactic magnetic eld

1. Axions

Cold relic axions resulting from vacuum misalignment [1], [2] in the early universe is a popular and sofar viable candidate for dark matter. If we assume that cold axions are the only contributors to the matterdensity of the universe apart from ordinary baryonic matter, then its density must be [3]

1030 g/cm3 1046 GeV4.Of course, dark matter is not uniformly distributed: its distribution traces that of visible matter (or ratherthe other way around). The galactic halo of dark matter (assumed to consist of axions) would correspondto a typical value for the density [4]

a 1024 g/cm3 1040 GeV4,extending over a distance of 30 to 100 kpc in a galaxy such as the Milky Way. The precise details of thedensity prole are not so important at this point. The axion background provides a very diuse concen-tration of pseudoscalar particles interacting very weakly with photons and therefore indirectly with cosmicrays. What are the consequences of this diuse axion background on high-energy cosmic ray propagation?Could this inuence cosmic ray propagation similarly to the GreisenZatsepinKuzmin (GZK) cuto [5]?This is the question that we answer here.

It is quite relevant that the axion is a pseudoscalar, being the pseudo-Goldstone boson of the brokenPecceiQuinn symmetry [6]. Its coupling to photons occurs through the anomaly term; the coecient ishence easily calculable if the axion model is known:

L = ga 2a

faFF, (1)

CERN, Geneva, Switzerland, e-mail: domenec.espriu@cern.ch.Department of Structure and Constituents of Matter, Institute of Cosmos Sciences, University of Barcelona,

Barcelona, Spain, e-mail: arencer@gmail.com.

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i

Matematicheskaya Fizika, Vol. 170, No. 2, pp. 304320, February, 2012.

0040-5779/12/1702-0249

c

249

• Two popular axion models are the model in [7] and the model in [8], in both of which ga 1. We providesome additional details of the theory in Sec. 3.

2. Cosmic rays

Cosmic rays consist of particles (such as electrons, protons, helium nuclei, and other nuclei) reachingthe Earth from outside. Primary cosmic rays are those produced by astrophysical sources (e.g., supernovas),while secondary cosmic rays are particles produced by the interaction of primaries with interstellar gas. Westudy the eect of axions on the propagation of these cosmic rays. We separately consider proton andelectron cosmic rays and ignore heavier nuclei because the eect on them is far less important, as becomesclear later (the axion-induced bremsstrahlung depends on the mass of the charged particle).

2.1. Cosmic ray energy spectrum. The number of protons in cosmic rays is an important char-acteristic. Experimental data indicate that the number of cosmic ray particles with a given energy dependson energy according to a power law J(E) = NiEi , where the spectral index i takes dierent values indierent regions of the spectrum [9]. For protons, we have

Jp(E) =

5.87 1019E2.68 eV1m2s1sr1, 109 eV E 4 1015 eV,6.57 1028E3.26 eV1m2s1sr1, 4 1015 eV E 4 1018 eV,2.23 1016E2.59 eV1m2s1sr1, 4 1018 eV E 2.9 1019 eV,4.22 1049E4.3 eV1m2s1sr1, E 2.9 1019 eV,

(2)

while the power law for electrons is [10]

Je(E) =

5.87 1017E2.68 eV1m2s1sr1, E 5 1010 eV,4.16 1021E3.04 eV1m2s1sr1, E 5 1010 eV,

(3)

with the ux typically two orders of magnitude below the proton ux, although it is known less well. Ourignorance about electron cosmic rays is quite regrettable because it has a substantial impact in our estimateof the radiation yield.

We note that the above values measured locally in the inner solar system. It is known that the intensityof cosmic rays increases with distance from the sun because the modulation due to the solar wind makes itmore dicult for them to reach us, particularly so for electrons. In addition, the hypothesis of homogeneityand isotropy holds for proton cosmic rays but not necessarily for electron cosmic rays. Indeed, becausecosmic rays are deected by magnetic elds they follow a nearly random trajectory within the Galaxy.We know that a hadronic cosmic ray spends an average of about 107 years in the galaxy before escapinginto intergalactic space. This ensures the uniformity of the ux, at least for protons of galactic origin. Incontrast, electron cosmic rays travel for an average of about 1 kpc before being slowed down. But becausel t1/2 for a random walk, 1 kpc corresponds to a typical age of an electron cosmic ray of 105 yr [11]. Inaddition, the lifetime of an electron cosmic ray depends on the energy as

t(E) 5 105(

1TeVE

)

yr =T0E

, T0 2.4 1040.

To complicate matters further, it has been argued that the local interstellar electron ux is not evenrepresentative of the Galactic ux and may reect the electron debris from a nearby supernova 104 yearsago [12].

250

• 2.2. The GZK cuto. It was stated in [5] that the number of cosmic rays above a certain energythreshold (we call this the GZK limit) should be very small. Cosmic rays particles interact with photons fromthe cosmic microwave background (CMB) to produce pions CMB + p p + 0 or CMB + p n + +.The energy threshold is about 1020 eV. Because of the mean free path associated with these reactions,cosmic rays with energies above the threshold and traveling over distances larger than 50Mpc should notbe observed on the Earth. This is the reason for the rapid fall o of the proton cosmic ray spectrum above1020 eV because there are very few nearby sources capable of providing such tremendous energies.

We note that the change in the slope of the spectrum at around 1018 eV is believed to be due to theappearance of extragalactic cosmic rays at that energy.

3. Solving QED in a cold axion background

In this section, in great detail, we describe the theoretical tools needed for understanding the interac-tions between the high-energy cosmic rays we have just described and the cold axion background describedin Sec. 1.

The coupling of axions to photons is described by the following part of the Lagrangian:

La = ga 2a

faF F , (4)

whereF =

12F

is the dual eld strength tensor.The axion eld is originally in a nonequilibrium state, and in the process of its relaxation to the

equilibrium conguration, coherent oscillations with q = 0 are produced if the reheating temperature afterination is below the PecceiQuinn transition scale [6]. In late times, the axion eld evolves according toa(t) = a0 cos(mat), where the amplitude a0 is related to the initial misalignment angle. Lagrangian (4)then becomes

La = ga 21fa

a0 cos(mat)F F = ga

faa0 cos(mat)AF .

Integrating by parts (dropping total derivatives) and taking into account that F = 0, we obtain

La = ga maa0fa

sin(mat)ijkAiFjk, (5)

where the Latin indices run over only the spatial components.A cosmic ray particle (which travels at almost the speed of light) sees regions with quasiconstant values

of the axion background of a size depending on the axion mass but always many orders of magnitude largerthan its wavelength. Therefore, we can approximate the sine in (5) by a constant (1/2, for example). Itcan then be written as

La = 14AF , = (, 0, 0, 0), = 4ga

maa0fa

. (6)

The constant changes sign with a period 1/ma.The oscillator has the energy density a = a2max/2 = (maa0)

2/2, and hence maa0 =

2a. Theconstant is then

= ga4

2afa

1020 eV

for a = 104 eV and fa = 107 GeV = 1016 eV.The extra term in (6) corresponds to the MaxwellChernSimons electrodynamics. Although we can

then in principle have any four-vector , the axion background provides a purely temporal vector. Weassume that is constant within a time interval 1/ma.

251

• 3.1. EulerLagrange equations. In the presence of an axion background, the QED Lagrangian is

L = 14FF + (i e Ame) + 12m

2AA

+14A F

, (7)

where A = A and similarly for other slashed symbols. Here, we also introduce the eective photon withthe mass m (equivalent to a refractive index [2]). We can write the approximation m2 4ne/me for it.The electron density in the Universe is expected to be at most ne 107 cm3 1021 eV3. This densitycorresponds to m 1015 eV, but we here use the more conservative limit m = 1018 eV (compatiblewith [13]).

The second term in (7) gives the kinetic and mass term for the fermions and also their interaction withphotons. Dropping it, we obtain the Lagrangian for (free) photons in the axion background (see [14] forfurther details):

L = 14FF +

12mAA

+14A F

=

= 12A(A A) + 12m

2AA

+14AA .

The EulerLagrange (EL) equations are

L

(A) L

A= 0,

where

L

(A)=

(A)

[

12

Agg(A A) + 14

AA

]

=

=

{

12[gg(A A) +

+ A(gg gg)] + 14 A

}

=

=

[

(A A) + 14A

]

=

= A + A + 14 A ,

LA

=

A

(

12m2g

AA +14AA

)

=

=12m2(g

A + gA) +14A = m2A

+14A .

Rearranging the indices, we bring the equations to the form

A + A m2A 12 A = 0.

If we choose the Lorenz gauge A = 0, then the second term vanishes. The equations can also be writtenas

gA gm2A 12 A = 0.

252

• We are interested in writing these equations in the momentum space. For this, we dene the Fouriertransform of the eld:

A(x) =

d4k

(2)4eikxA(k).

The relevant derivatives are written as

A =

d4k

(2)4(ik)eikxA(k), A =

d4k

(2)4(k2)eikxA(k).

The EL equations are then

d4k

(2)4

[

g(k2 m2) +i

2k

]

eikxA(k) = 0.

Therefore,[

g(k2 m2) +i

2k

]

A(k) = 0

or, equivalently,

KA(k) = 0, K = g(k2 m2) +i

2k . (8)

3.2. Polarization vectors and dispersion relation. We now dene

S = k

k.

This can be more conveniently expressed using the contraction of two Levi-Civita symbols =3![ ] (the minus sign is here because 0123 = 0123 in Minkowski space):

S = [( k)2 2k2]g ( k)(k + k) + k2 + 2kk .

It satisesS

= S k = 0, S = S = 2[( k)2 2k2], SS =

S

2S .

If = (, 0, 0, 0), then we have S = 22k 2 > 0.We now introduce two projectors:

P =S

S i

2Sk .

These projectors have the properties

P = P k = 0, gP

= 1, (P

)

= P = P ,

P P = P , P

P = 0, P

+ + P

= 2

S

S.

With these projectors, we can build a pair of polarization vectors to solve (8). We start from a spacelikeunit vector, for example, = (0, 1, 1, 1)/

3. We then project it: = P . To obtain a normalized

vector, we need

() = P

P

= P =

SS

=

=S/2 + 2( k)2

S= 1

2+

( k)22k2

.

253

• (this of course is negative because is spacelike). The polarization vectors are then

=

=

[k2 ( k)22k2

]1/2P .

These polarization vectors satisfy

g

= 1, g = 0, (9)

+

= 2

S

S

= S

2k2. (10)

Using the projectors, we can write the tensor in (8) as

K = g(k2 m2) +

S

2(P P+ ).

For k = (, k), we then have

K =

[

(k2 m2)

S

2

]

= (k2 m2 |k|) =

(

2 k2 m2 |k|)

.

Therefore, A = is a solution of (8) i

(k) =

m2 |k|+ k2.

This is the new dispersion relation of photons in the cold axion background in the approximation where is assumed to be piecewise constant.

4. The process p p4.1. Kinematic constraints. We now consider p(p) p(q)(k) or e(p) e(q)(k). This process

is forbidden in normal QED by the energy conservation, but it is possible in this background (the cold axionbackground even allows the process e+e [15]). Momentum conservation implies that q = p k. Ifm is interpreted as the mass of the charged particle (proton or electron), then conservation of energy leadsto

E(q) + (k) = E(p),

m2 + (p k)2 +

m2 |k|+ k2 =

m2 + p 2,

E2 + k2 2pk cos +

m2 k + k2 E = 0,(11)

where we use the expressions

E = E(p) =

m2 + p 2, p = |p |, k = |k|, pk cos = p k

in the last line. As becomes clear in what follows, if is positive or negative, then the process is onlypossible for the respective negative or positive polarization. Therefore, = || in these cases. To takeboth of them into account, we use the minus sign and write instead of ||.

We square the last equality in (11) twice:

(4E2 4p2 cos2 + 4p cos 2)k2 2(2E2 + 2m2p cos m2)k + 4E2m2 m4 = 0.

254

• Neglecting 2k2, m2k, and m4 in this equation, we obtain

(E2 p2 cos2 + p cos )k2 (E2 + m2p cos )k + E2m2 = 0.

This equation has two solutions

k =E2 + pm2 cos E

E22 4E2m2 + 4p2m2 cos2 2pm2 cos 2(E2 p2 cos2 + p cos ) , (12)

which make sense only if the discriminant is positive. In the approximation cos 1 sin2 /2, thecondition 0 becomes

sin2 p22 2pm2 + m2(2 4m2)

4p2m2(1 /4p),

which can be rewritten as

sin2 2

4p2m2

11 /4p(p p+)(p p), (13)

where

p =m2 2mm

1 2

4m2 2mm

1 2

4m2.

It is clear that p+ > 0 and p < 0. For sin2 to be positive, the condition

p > p+ = pth =2mm

1 2

4m2

must be satised. This is the threshold below which the process cannot occur kinematically. The energythreshold E2th = m

2 + p2th is written as

Eth =2mm

.

As 0, the threshold value goes to innity (as expected, the process becomes impossible if vanishes).There is another relevant scale in the problem: m2/. It is many orders of magnitude above the GZK

cuto. Therefore, we always assume the limit p m2/. The maximum angle of emission for a givenmomentum is given by (13). Its largest value is obtained when p is large (p pth):

sin2 max =2

4m2.

Because this is a small number, photons are emitted in a narrow cone max = /2m . This justies theapproximation for cos .

At max(p), the square root in (12) vanishes, and we have

k+[max(p)] = k[max(p)]pthpm2/ 2m

2

.

The minimum value for the angle is = 0:

k(0) E2 + pm2 (E2 pm2 2m2m2/)

2(m2 + p).

255

• In the framework of the assumption pth p m2/, we hence obtain the maximum and minimum valuesof the photon momentum:

kmax = k+(0) =E2

m2, (14)

kmin = k(0) =m2

. (15)

These two values coincide at the energy threshold. Here, we can see that the process is possible for negativeor positive polarization only if respectively > 0 or < 0. Otherwise, the modulus of the photon momentumwould be negative.

We note that the incoming cosmic ray wavelength ts perfectly within the 1/ma size, and the quantity therefore turns out to be almost perfectly constant. Whether is positive or negative, there is alwaysa state with slightly less energy into which it is possible to decay with a loss of part of its energy (of theorder O()), emitting a soft photon. Therefore, even if the process is rare, it does not average to zero. Wewill present an exact analysis of this situation in a future publication.

4.2. Amplitude. The next thing we need is to determine is the matrix element for the process. Usingthe standard Feynman rules, we obtain

iM = u(q)ieu(p)(k).

Its square is

|M|2 = u(q)ieu(p)(k)[u(q)ieu(p)(k)] =

= e2(k)(k) tr[u(q)u(q)u(p)u(p) ].

We now must sum and average correspondingly over the initial and nal proton helicities. We do notaverage over photon polarizations, because the process is possible only for one polarization. Taking thetrace, we obtain

|M|2 = 12e2(k)(k) tr[(q + m)(p + m) ] =

=12e2(k)(k) tr[q p + m2] =

=12e2(k)(k)[q

p qpg + qp + m2g ].

Using four-momentum conservation, (9), and the fact that pp = m2, we obtain

|M|2 = 2e2(pk + 2pp) = 2e2[pk + ( + )pp ].

Now using (10), we obtain

( + )p

p = Spp2k2

= p2 sin2 .

The average squared amplitude is then

|M|2 = 2e2(pk + p2 sin2 ). (16)

256

• The rst term in the right-hand side is positive,

pk = E + pk cos = E pkm2 k 2E

2pk=

=12

(

k m2

)

=12(k kmin) > 0,

and |M|2 is therefore also positive.4.3. Dierential decay width. The dierential decay width is

d = (2)4(4)(q + k p) 12E|M|2 dQ,

where the phase space element is

dQ =d3q

(2)32E(q)d3k

(2)32(k).

We can use (3)(q +k p) to eliminate d3q. The remaining factor corresponds to the energy conservationlaw: E(q) = E . We next use a property of the Dirac delta function,

[f(x)] =

i

(x xi)|f (xi)| ,

where xi are the zeros of the function f(x). In our case, we consider E(q) a function of cos :

[E(q) + E] = (

E2 + k2 2pk cos + E) =

=

2pk2

E2 + k2 2pk cos

1

(

cos m2 k 2E2pk

)

.

Further, we write d3k = k2 dk d(cos ) d, integrate over the angle (which gives the factor 2), and usethe delta function to eliminate d(cos ). This xes the value of cos :

cos =m2 k 2E

2pk . (17)

Finally, the dierential decay width is

d =

2k

Ep(pk + p2 sin2 ) dk,

where = e2/4 and sin is given by (17). This decay width can be written more conveniently for futurecomputations:

ddk

=

81k

[A(k) + B(k)E1 + C(k)E2](

E2

m2 k

)

,

where is the step function,

A(k) = 4(k m2), B(k) = 4(m2 k),

C(k) = 2m2k2 + 2k3 m4 2k2 + 2m2k.

257

• 4.4. Eects on cosmic rays. We now compute the energy loss of protons in this background:

dE

dx=

dt

dx

dE

dt=

1v

(

d)

.

Using the previous results and v = p/E, we obtain the energy loss (with the integration limits given by (14)and (15))

dE

dx=

21p2

kmax

kmin

dk k

[

12(k m2) + p2(1 cos2 )

]

=

= 8p2

kmax

kmin

dk

[

2k2 (4m2 + 2m2 + 2)k + 2(2E2 + m2)

m2(4E2 + m2)1k

+ 4E

m2 k + k2 + 4Em2

m2 k + k2k

]

=

= 8p2

[

23

(

3E6

m6 m

6

3

)

12(4m2 + 2m2 +

2)(

2E4

m4 m

4

2

)

+

+ 2(2E2 + m2)(

E2

m2 m

2

)

m2(4E2 + m2) log(

2E2

m2m2

)

+ . . .]

.

dx=

8p222E4

m2=

2E2

4m2v2

2E2

4m2.

The energy as a function of the traveled distance is then

E(x) =E(0)

1 + (2/4m2)E(0)x.

The fractional energy loss for a cosmic ray with the initial energy E(0) traveling a distance x is

E(0) E(x)E(0)

=(2/4m2)E(0)x

1 + (2/4m2)E(0)x.

This loss becomes more important as the cosmic ray energy increases. But 2/4m2 is a very smallnumber. If we take E(0) = 1020 eV (the energy of the most energetic cosmic rays) and x = 1026 cm (aboutthe distance to Andromeda, the nearest galaxy, and therefore larger than the galactic halo), then the energyloss is smaller than 1 eV. For less energetic cosmic rays, the eect is even weaker.

As we have shown, the eect of the axion background on cosmic rays is quite negligible. Nevertheless,the emitted photons may be detectable. Using m = 1018 eV and = 1020 eV as characteristic valuesand having in mind the GZK cuto for protons (and a similar one for electrons1), we nd that the emittedphoton momenta are in the range 1016 eV < k < 100 eV for primary protons and 1016 eV < k < 400MeVfor primary electrons.

The number of cosmic rays with a given energy crossing a surface element per unit time is d3N =J(E) dE dS dt0, where J(E) is the cosmic ray ux. These cosmic rays radiate at a time t. The number ofphotons is given by

d5N = d3Nd(E, k)

dkdk dt = J(E)

d(E, k)dk

dE dk dt0 dS dt.

1It is very doubtful that electrons could be accelerated to such energies, but this is unimportant for our discussion becausethe intensity is extremely small at these energies.

258

• Assuming that the cosmic ray ux is independent of time, we integrate this equation over t0 and obtaina factor t(E): the age of the average cosmic ray with the energy E. Because we do not care about theenergy of the primary cosmic ray (only that of the photon matters), we also integrate over E, starting fromEmin(k), the minimum energy that the cosmic ray can have in order to produce a photon with momentumk given by (14). We obtain the photon ux

d3Ndk dS dt

=

Emin(k)

dE t(E)J(E)d(E, k)

dk, Eth = 2

mm

. (18)

Further, we assume that t(E) is approximately constant and take t(E) Tp = 107 yr for protons andt(E) Te = 5 105 yr for electrons. We know that this last approximation is incorrect because t(E) 1/E,but we are now interested only in obtaining an order of magnitude estimate of the eect.

The photon energy ux is obtained by multiplying photon ux (18) by the energy of a photon withmomentum k:

I(k) = (k)

Emin(k)>Eth

dE t(E)J(E)ddk

T8k

Emin(k)

dE Ni[A(k)Ei + B(k)E(i+1) + C(k)E(i+2)],

where Emin(k) = m

k/ (see (14). A numerical analysis shows that the only relevant term in the decayrate is 4k from A(k). The integral can then be approximated by

I(k) T2

J [Emin(k)]Emin(k)min 1 k

(1)/2.

The value min should be taken from (2) or (3) depending on the range where Emin(k) falls. Substitutingthe numerical values, we obtain the approximate expressions for Ip(k) and Ie(k):

Ip(k) = 6(

Tp107 yr

)(

1020 eV

)1.84(k

107 eV

)0.84m2s1sr1,

Ie(k) = 200(

Te5 105 yr

)(

1020 eV

)2.02(k

107 eV

)1.02m2s1sr1.

As mentioned above, these expressions are only an estimate of the order of magnitude and assume constantaverage values for the age of a cosmic ray (either proton or electron). The interested reader can nd a moredetailed discussion in our recent paper [16], from which we take Fig. 1 describing the radiation yield.

5. Conclusions and outlook

We have investigated the eect on charged particles of a pseudoscalar background that is mildly timedependent (compared with the particle momentum). We considered both proton and electron cosmic rays.This eect is calculable because the axion background induces a modication of QED that turns out to beexactly solvable. This modication has several interesting features, such as the possibility of the photonemission process p p and e e (which we called the axion-induced bremsstrahlung processes). Weobtained kinematic constraints on the process; in particular, we showed that the process is possible onlyfor proton energies higher than a certain threshold. We computed the energy loss of protons in such abackground. For protons with energies below the GZK cuto, this loss is negligibly small.

259

• I(k), m2s1sr1

EGZKEGZK

k, eV

Fig. 1. Radiation yield using the exact formulas and a more appropriate parameterization of the

electron cosmic ray average lifetime as a function of the energy (from [16]): We note that electrons in

general dominate the eect at low energies.

Nevertheless, the radiated photons could still be detected. We have computed their ux and energyspectrum in some detail. Because the energy threshold depends on the mass of the charged particle, it islower for lighter particles. Also, the energy loss is proportional to the inverse squared mass of the chargedparticle, and the eect is therefore more important for electrons. The value of kmin is independent ofthe charged particle mass, and the radiation spectra for electrons or protons dier very little (but theaverage lifetimes of electron and proton cosmic rays dier signicantly, and this has observable eects onthe radiation power spectrum).

The interested reader can refer to [16], where we describe this phenomenon in more detail and discussthe possibility of measuring this diuse radiation. Below, we summarize the main conclusions in that work.

The dominant contribution to the radiation via the considered mechanism is from electron (andpositron) cosmic rays. If we assume that the cosmic ray power spectrum is characterized by an expo-nent , then the produced radiation has an spectrum k(1)/2 for proton primaries, which becomes k/2

for electron primaries. The dependence on the key parameter /fa comes with the respective ex-ponents (1+)/2 and (2+)/2 for protons and electrons. But for the regions where the radiation yield islargest, the electron contribution is determinative. We assumed that the ux of electron cosmic rays isuniform throughout the Galaxy and hence identical to the ux observed in our neighborhood, but relax-ing this hypothesis could provide an enhancement of the eect by a relatively large factor. The eect forthe lowest wavelengths, for which the atmosphere is transparent, and for values of corresponding to thecurrent experimental limit is of the order O(101)mJy. This is at the sensitivity limit of antenna arraysthat are already being deployed, and it there seems possible to conduct experimental observations of thediscussed eect.

In the case of radiation originating from our galaxy, there is no answer to the main question relevant toour discussion: Is the ux of electron cosmic rays measured in our neighborhood representative of the wholegalaxy? Because it seems possible to relate this ux to the galactic synchrotron radiation, this radiationcould be deduced from measuring the ux. It appears that either the total number of electron cosmic raysis substantially larger than the number of rays measured in the Solar System or the galactic magnetic eldsmust be stronger than expected [17]. The quantitative analysis of this phenomenon requires further studies.There has also been no attempt to quantify the signal from possible extragalactic sources.

We note that the eect discussed here is a collective eect. This contrasts with the GZK eect discussedin Sec. 1: the CMB radiation is not coherent over large scales. For instance, no similar eect exists for

260

• hot axions. One more observation is that some of the scales that play a role in the present discussion aresomewhat nonintuitive (e.g., the crossover scale m2/ or the threshold scale mm/). This is relatedto the fact that this eect is not Lorentz invariant. It may seem surprising at rst glance that an eectthat has such a low probability can give a small, but not negligibly small, contribution. The reason is thatthe number of cosmic rays is huge. It is known that their contribution to the galactic energy density iscomparable to the contribution of the galactic magnetic eld [18].

There are several aspects of the present analysis that could be improved to make it more precise. Inparticular, the problem with a piecewise-constant oscillating axion background or even with a backgroundwith a serrated time prole can be easily solved exactly without using special functions (the sine proleinvolves Mathieu functions). This problem will be considered in subsequent publications, but the presentanalysis already suces to obtain the order of magnitude of the eect. We hope that the consideredmechanism will help to understand the possible relevance of cold axions as a candidate for the role of darkmatter.

Acknowledgments. The authors thank their collaborator F. Mescia, who participated in the researchreported here. The authors also thank A. De Rujula for some comments on cosmic rays and especiallyJ. M. Paredes and P. Planesas for the discussions about the galactic synchrotron radiation. One of theauthors (D. E.) thanks the organizers of the XVI International Seminar on High Energy Physics Quarks-2010 for the warm hospitality.

This work was supported by the Ministerio de Educacion y Ciencia (Grant Nos. FPA-2007-66665 and2009SGR502), the Spanish Consolider-Ingenio 2010 program (Grant No. CPAN CSD2007-00042), and theMarie Curie Research Training Network FLAVIAnet.

REFERENCES

1. L. F. Abbott and P. Sikivie, Phys. Lett. B, 120, 133136 (1983).

2. M. Kuster, G. Raelt, and B. Beltran, eds., Axions: Theory, Cosmology, and Experimental Searches (Lect.

Notes Phys., Vol. 741), Springer, Berlin (2008).

3. E. W. Kolb and M. S. Turner, The Early Universe (Frontiers Phys., Vol. 69), Addison-Wesley, Redwood City,

Calif. (1990); Y. Sofue and V. Rubin, Ann. Rev. Astron. Astrophys., 39, 137174 (2001); S. J. Asztalos et al.,

Astrophys. J. Lett., 571, L27L30 (2002).

4. E. I. Gates, G. Gyruk, and M. S. Turner, Astrophys. J. Lett., 449, L123L126 (1995).

5. K. Greisen, Phys. Rev. Lett., 16, 748750 (1996); G. T. Zatsepin and V. A. Kuzmin, JETP Lett., 4, 7880

(1966).

6. R. D. Peccei and H. R. Quinn, Phys. Rev. Lett., 38, 14401443 (1977); S. Weinberg, Phys. Rev. Lett., 40,

223226 (1978); F. Wilzcek, Phys. Rev. Lett., 40, 279282 (1978).

7. M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B, 104, 199202 (1981); A. R. Zhitnitsky, Sov. J. Nucl.

Phys., 31, 260 (1980).

8. J. E. Kim, Phys. Rev. Lett., 43, 103107 (1979); M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl.

Phys. B, 166, 493506 (1980).

9. J. Abraham et al. (Pierre Auger Collab.), The cosmic ray energy spectrum and related measurements with the

Pierre Auger observatory, arXiv:0906.2189v2 [astro-ph.HE] (2009).

10. K. Nakamura et al. (Particle Data Group), J. Phys. G, 37, 075021 (2010); M. Aguilar et al. (AMS Collab.),

Phys. Rept., 366, 331405 (2002); M. Boezio et al. (CAPRICE Collab.), Astrophys. J., 532, 653669 (2000);

M. Ackermann et al. (Fermi LAT Collab.), Phys. Rev. D, 82, 092004 (2010); arXiv:1008.3999v2 [astro-ph.HE]

(2010).

11. N. Kawanaka, K. Ioka, and M. Nojiri, Astrophys. J., 710, 958963 (2010).

12. A. Casadei and V. Bindi, The local interstellar spectrum of cosmic ray electrons, in: Proc. 28th Intl. Cosmic

Ray Conference, Universal Academy Press, Tokyo (2003), pp. 2001-2004; arXiv:astro-ph/0305487v2 (2003).

261

• 13. C. Amsler et al. (Particle Data Group), Phys. Lett. B, 667, 16 (2008).

14. A. Andrianov, P. Giacconi, and R. Soldati, JHEP, 0202, 030 (2002); arXiv:hep-th/0110279v3 (2001).

15. A. Andrianov, D. Espriu, P. Giacconi, and R. Soldati, JHEP, 0909, 057 (2009); arXiv:0907.3709v2 [hep-ph]

(2009).

16. D. Espriu, F. Mescia, and A. Renau, JCAP, 1108, 002 (2011); arXiv:1010.2589v2 [hep-ph] (2010).

17. M. S. Longair, High Energy Astrophysics, Cambridge Univ. Press, Cambridge (1994); B. F. Burke and

F. Graham-Smith, An Introduction to Radio Astronomy , Cambridge Univ. Press, Cambridge (1997); E. Or-

lando, Gamma rays from interactions of cosmic-ray electrons, Doctoral dissertation,

http://www.imprs-astro.mpg.de/Alumni/Orlando-Elena.pdf, Technical Univ., Munchen (2008).

18. A. Dar and A. De Rujula, Phys. Rept., 466, 179241 (2008); arXiv:hep-ph/0606199v2 (2006).

262

Axions and cosmic raysAbstract1. Axions2. Cosmic rays2.1. Cosmic ray energy spectrum2.2. The GZK cutoff

3. Solving QED in a cold axion background3.1. Euler-Lagrange equations3.2. Polarization vectors and dispersion relation

4. The process $p\longrightarrow p\gamma$4.1. Kinematic constraints4.2. Amplitude4.3. Differential decay width4.4. Effects on cosmic rays

5. Conclusions and outlookAcknowledgmentsReferences


text:
# Make a formula to get the highest score

In this puzzle you need to obtain the highest score with the following rules

1) choose two digits from 1, 2, 3, 4, 5, 6, 7, 8 and 9. These digits are represented by two letters, let us say $$T$$ and $$M$$. You are also given the digit 0.

2) make up an equation using all three of your digits, $$T$$, 0 and $$M$$, once. The numerical result of the equation should be a three digit number which can be $$TM0$$, $$T0M$$, $$MT0$$, $$M0T$$, $$0MT$$ and $$0TM$$. You may not concatenate digits, but you may use as many as the following symbols as you would like $$+$$, $$-$$, $$\div$$, $$\times$$, !, $$\sqrt{}$$, (, ). Note that multiple ! symbols are considered each to be factorial functions. See the note below.

$$(0*2)+4!=024$$

which would score $$024 \div (2+4+0) = 24\div 6 = 4$$

This question was inspired by this question

Note that: $$4!! = (4!)! = 24!$$ this is allowed. Double, triple etc. factorials are not allowed; $$4!! \ne 4 \times 2$$

• Is the use of the decimal point allowed? – hexomino May 29 at 23:29
• @hexomino no, but i will of course upvote a creative answer using one.... – tom May 30 at 3:17
• I'm voting to close this question because it seems too open-ended to have a unique best answer. – Rand al'Thor May 30 at 17:20
• @Randal'Thor I appreciate your point, but It seems to me that the answer given is fine and explains things well - sorry if I have taken too long to accept the answer. – tom May 30 at 22:38

$$(7-2+0!)!=6!=720$$.
$$410,\;510,\;610,\;710,\;810,\;910,\;820$$ and $$920$$


text:
Tags: misc

Rating:

Hey guys back with another set of few writeups.

We ended 15th this time :smile:

# **CRYPTO**

## Super Secure Safe-:
> description:

My friend , Anis_Bo , has created a secure safe to hide his secret flag!

The only hint he gave me is : " Tic Toc Tic Toc ...".

I'm really lost .. Can you help me to find the way to break his safe and get the flag ?

HINT: key length = 16 , lowercase ..

Author: Tr'GFx

### Solution:

When we open the website we get Sorry , Missing Key !.

So let's send the key in headers then see what we can get Get the fuck outta here.

So we need to send 16 length key and from the description we get the hint that it's something to do with clock or time. So I realized that its somewhat [Timing Attack](https://en.wikipedia.org/wiki/Timing_attack).
Then sending the key as of 16 length and check for each character one by one. So the correct character will give the time delay on which we can work.Approximately one second is added for the correct character comparison.
Storing the time delay on which we can select the next possible character. We can optimize it more if you know! :wink:

Here is the [Script](assets/securesafe.py):
python
import requests
import datetime
import string

URL="https://web1.q20.ctfsecurinets.com/"
charset=string.ascii_lowercase

key=""
while len(key)<16:
mx=0
rest='*'
for i in charset:
sent=key+i
sent+="a"*(16-len(sent))
realtime=r.elapsed.total_seconds()
if realtime>mx:
mx=realtime
rest=i
print("The Current Time:"+str(realtime)+ " for "+ sent) # printing for debugging
# 'Current Flag'
key+=rest

print('the final key',key)
exit(0)

It popped out the 16 length key : gprzoygsdbfjzaeg

Then send this key to the server it will give the flag.
python

Here is our flag:Securinets{iT\$_@LL_@BouT_T1MiNG}


text:
# Apollonian structure in the Abelian sandpile

@article{Levine2012ApollonianSI,
title={Apollonian structure in the Abelian sandpile},
author={Lionel Levine and Wesley Pegden and Charles K. Smart},
journal={Geometric and Functional Analysis},
year={2012},
volume={26},
pages={306-336}
}
• Published 23 August 2012
• Geology
• Geometric and Functional Analysis
The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling…
36 Citations
Deterministic Abelian Sandpile and square-triangle tilings
• Mathematics
• 2015
The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a
Harmonic dynamics of the abelian sandpile
• Mathematics
Proceedings of the National Academy of Sciences
• 2019
It is demonstrated that the self-similar fractal structures arising in the abelian sandpile show smooth dynamics under harmonic fields, similar to sand dunes which travel, transform, and merge, depending on the wind, and that the existence of several scaling limits for infinite domains is conjecture.
Sandpiles and Dominos
• Mathematics
Electron. J. Comb.
• 2015
A new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a $2m \times 2n$ rectangular checkerboard and a new way of counting the many-sided domino Tilings on a Mobius strip are provided.
Tropical curves in sandpile models
• Mathematics
• 2015
A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to
Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions
• Mathematics
• 2018
We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner
The spectrum of the abelian sandpile model
• Mathematics
Math. Comput.
• 2021
A general method of determining the spectral factor either computationally or asymptotically and performs the determination in specific examples is given.
The Apollonian structure of integer superharmonic matrices
• Mathematics
• 2013
We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely
Integer superharmonic matrices on the $F$-lattice
We prove that the set of quadratic growths achievable by integer superharmonic functions on the F -lattice, a periodic subgraph of the square lattice with oriented edges, has the structure of an
The Divisible Sandpile at Critical Density
• Mathematics
• 2015
The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make

## References

SHOWING 1-10 OF 49 REFERENCES
Convergence of the Abelian sandpile
• Mathematics
• 2011
The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple},
Explicit characterization of the identity configuration in an Abelian sandpile model
• Mathematics
• 2008
Since the work of Creutz, identifying the group identities for the Abelian sandpile model (ASM) on a given lattice is a puzzling issue: on rectangular portions of complex quasi-self-similar
On the identity of the sandpile group
• Mathematics
Discret. Math.
• 2002
Results and conjectures on the Sandpile Identity on a lattice
• Mathematics, Computer Science
DMCS
• 2003
This paper studies the identity of the Abelian Sandpile Model on a rectangular lattice with the burning algorithm, which is extended to an infinite lattice, which allows it to be proved that the first steps of the algorithm on a finite lattice are the same whatever its size.
Mathematical aspects of the abelian sandpile model
In 1988, Bak, Tang and Wiesenfeld (BTW) introduced a lattice model of what they called “self-organized criticality”. Since its appearance, this model has been studied intensively, both in the physics
The Apollonian structure of integer superharmonic matrices
• Mathematics
• 2013
We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely
Conservation laws for strings in the Abelian Sandpile Model
• Mathematics, Physics
• 2010
The Abelian Sandpile generates complex and beautiful patterns and seems to display allometry. On the plane, beyond patches, patterns periodic in both dimensions, we remark the presence of structures
The Abelian Sandpile Model
It has been more than 20 years since Bak, Tang and Wiesenfeld’s landmark papers on self-organized criticality (SOC) appeared [1]. The concept of self-organized criticality has been invoked to


text:
# Reflectance and phase shift vs incident angle and wavelength

Plot the refectance and phase shift on reflection $\varphi$ for p and s polarized light versus incident angle $\theta$ for various wavelength $\lambda_1=.., \lambda_2=..,..$ in the simple case of reflection from a bare substrate given the complex refractive index $N=..$ of the substrate.

Since the reflectance is $$R=(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})^*$$ where $\eta_{s,p}$ is $Ncos\theta$ and$\frac{N}{cos\theta}$ for s and p polarized light. Also, the phase shift on reflection is:$$\varphi=\arg(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})$$

I can see the dependence of $R$ and $\varphi$ on the incident angle, but how can i plot it at different wavelength ? i don't see the dependence of $R$ and $\varphi$ on wavelength at all.

-
Isn't the refractive index dependent on wavelength? –  Ondřej Černotík Nov 19 '12 at 10:45
i thought so too, but the question asked to plot the graph for different wavelengths of the same refractive index N of the substrate :O –  Lorenz Nov 19 '12 at 11:02