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# Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$.

On the other hand the Laplacian of a non-compact manifold may be continuous. For example on $\mathbb{R}$ the spectrum of the second derivative operator is $(-\infty,0]$ (I use the convention that the Laplacian is negative semi-definite).

I was wondering if it is always the case that when a sequence of pointed Riemannian manifold $(M_n,o_n,g_n)$ converges smoothly to a limit Riemannian manifold $(M,o,g)$ then the spectrum of the corresponding Laplace operators also converges. I don't think this is true but a semi-continuity property should hold (and I'm guessing is well known, hence the reference-request tag).

Given a Riemannian manifold $M$ let $-\lambda_1(M)$ be the maximum of the spectrum (i.e. the closest point to $0$). Then $\lambda_1$ satisfies the following "domain monotonicity": If $\Omega$ is an open submanifold of $M$ with boundary then $\lambda_1(\Omega) \ge \lambda_1(M)$ (e.g. $\lambda_1(\mathbb{R})=0$ but $\lambda_1([0,2\pi]) = 1$). If one knows that $\lambda_1(M)$ is the limit of $\lambda_1(\Omega_n)$ over an increasing sequence of bounded submanifolds then this is enough to prove upper semi-continuity of the absolute value of $\lambda_1$ under smooth convergence.

To start I'd like to know if the above reasoning is correct and if there is a good reference. Concretely I'd love to have a good reference for the answers to the following questions:

1. Is $\lambda_1$ upper semi-continuous with respect to smooth convergence?
2. Is there a simple counterexample for continuity?

The question: What about the rest of the spectrum? isn't definite enough for this site so I'll try to make it more concrete.

By the spectral theorem the (unique self-adjoint extension of the) Laplacian on the limit manifold is unitarily conjugate to multiplication by some (non-positive) function $\varphi$ on $L^2(\mu)$ for some $\sigma$-finite measure $\mu$ on a Polish space $X$. Let $\varphi_n,\mu_n$ and $X_n$ be defined similary for the sequence.

Is it true that $\mu(\varphi^{-1}((-a,0])) \ge \limsup \mu_n(\varphi_n^{-1}((-a,0]))$ for all $a > 0$?

Edit: (Hopefully) Fixed some issues (positive vs negative definite Laplacian, first eigenvalue vs maximum of the spectrum) in response to comments.

• Also in question 1, I assume by "first eigenvalue" you mean "top of the spectrum"? Because as you note, when $M$ is non-compact the Laplacian may have no eigenvalues. – Nate Eldredge Aug 23 '14 at 3:28
• For your last question, since $(X,\mu,\varphi)$ is not unique, are you sure that $\mu(\varphi^{-1}((-a,0]))$ is well defined? (If you drop the requirement that $X$ be Polish, it definitely is not.) – Nate Eldredge Aug 23 '14 at 3:31
• Hmm, you're right that $(X,\mu,\varphi)$ is non-unique (e.g. multiply $\mu$ by a constant). However I'm hoping there's some way to normalize correctly. When the spectrum is discrete the right thing to do is clearly to replace $\mu(\varphi^{-1}((-a,0]))$ by the number of eigenvalues in $(-a,0]$. I'm not sure what makes sense in general. Maybe $-\int\varphi 1_{\lbrace \varphi > -a\rbrace} \mathrm{d}\mu$ (i.e. minus the integral of $\varphi$ on the set where it is larger than $-a$)?. – Pablo Lessa Aug 25 '14 at 21:54

This is only a partial answer since it concerns only compact manifolds. There is a result due to T. Shioya http://projecteuclid.org/download/pdf_1/euclid.jmsj/1213023969 a special case of which says that if a sequence of compact $n$-dimensional Riemannian manifolds $M_i$ converges to another compact n-dimensional Riemannian manifold M in the Gromov-Hausdorff sense and such that the sectional curvature is uniformly bounded from below, then for any $k$ the $k$-th eigenvalue of the Laplacian of $M_i$ converges to the $k$-th eigenvalue of the Laplacian of $M$. Notice that smooth convergence implies the above type of convergence.


text:
### Theory:

The pathway travelled by the air in humans
The air that is present in the environment is inhaled by us with the help of the two openings in our nose, known as the nostrils. This air passes to the nasal cavity and reaches the lungs through the windpipe.

The process of inhalation and exhalation
Lungs are found in the chest cavity, which is surrounded by the ribs. The floor of the chest cavity is formed by a muscular sheet known as the diaphragm, during the process of breathing the diaphragm and the ribcage moves.
At the time of inhalation, the ribs move outwards, and the diaphragm moves downwards to increase the space in our chest cavity. Due to this, the air rushes inside, thereby filling the lungs.

At the time of exhalation, the ribs and the diaphragm are brought back to their original positions, i.e., the ribs move inwards, and the diaphragm moves upwards so that the size of the chest cavity is reduced. Due to this, the air is pushed out of the lungs.
We can feel all these various movements that occur during the process of inhalation and exhalation.
The air we inhale contains a mixture of gases in which$$21%$$of oxygen and$$0.04%$$of carbon dioxide is present. The air we exhale includes $$16.4%$$ of oxygen and $$4.4%$$ of carbon dioxide.
The reason behind sneezing
The air we breathe in contains a wide range of unnecessary particles such as smoke, dust and pollens. These particles usually get trapped in the hair found in our nasal cavity during inhalation. Sometimes these particles irritate the lining of the nasal cavity resulting in sneezing. Thus sneezing helps us to get rid of the unwanted particles out of the nasal cavity.
Important!
During sneezing, one should always cover the nose so that others do not inhale the foreign particles which are expelled.
Smoking
It is a harmful activity of humans that damages the lungs. It also causes cancer. Hence, smoking should be avoided.
Reference:

format: plain


text:
Open Access Publications from the University of California

## Quantum oscillations in cuprates and Cooper pairing in half filled Landau level

• Author(s): Wang, Zhiqiang
The observation of quantum oscillations in hole under-doped cuprate is a big breakthrough to reveal its normal state nature. To understand the observed oscillation frequencies, in chapter 2, we consider the normal state to be a Fermi liquid and in a symmetry broken phase, whose order parameter is a novel period$-8$ $d-$density wave. This order gives rise to a complex Fermi surface consisting of not only an electron pocket, which can explain the major observed oscillation frequency $F\sim 530 \, \mathrm{T}$, but also a small hole pocket, which corresponds to a newly predicted slower oscillation. This slower oscillation has received some experimental supports recently.
In chapter 3, we study how superconductivity fluctuations, which exist in the form of random vortices, could affect the normal state quasiparticle quantum oscillation. We find that the Onsager rule, which connects extremal normal state Fermi surface areas to quantum oscillation frequencies, remains intact to an excellent approximation in the mixed-vortex state. We also show that the oscillations of the magnetic field $B$ dependent density of states, $\rho(B)$, ride on top of a field independent background in the high field quantum oscillation regime. This feature appears to agree with the most recent specific heat measurement on $\mathrm{YBa_2Cu_3O_{6+\delta}}$. At lower fields the superconductivity fluctuations are quenched and form an ordered vortex lattice. We show that the density of states follows $\rho(B)\propto \sqrt{B}$ as $B\rightarrow 0$, in agreement with the semiclassical results by Volovik.
In chapter 4, we turn to the Cooper pairing problem of composite fermions in the half-filled Landau level. We apply a new pairing mechanism from repulsive forces to the Halperin-Lee-Read composite fermion liquid. This mechanism takes advantage of the dynamical screening at finite frequency from the finite density composite fermions and makes a net attraction possible. We show that the transition from the composite fermion liquid state to a chiral Cooper pairing state, with odd angular momentum channels, is continuous, in disagreement with the previous conclusion that the transition is discontinuous if the bare interaction is short-ranged. We also construct the phase diagrams for different angular momentum channels $\ell$ and show that the $\ell=1$ channel is quite different from higher channels $\ell\ge 3$. Similar analysis has been carried out for the bilayer Hall system with a total filling fraction $\nu=\frac{1}{2}+\frac{1}{2}$ and it is found that the previously established results remain qualitatively unaltered.
Finally, in chapter 5 we apply the above pairing mechanism to the recently proposed particle-hole symmetric Dirac composite fermion liquid theory for the half-filled Landau level. We find that a continuous transition to different chiral pairing states, with angular momentum channels $|\ell|\ge 1$, is possible. These include the Moore-Read Pfaffian and the anti-Pfaffian state. However, the $\ell=0$ channel particle-hole symmetric pairing state, turns out to be energetically impossible although it is symmetry allowed.


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# number of elements on a finite vector space

I know that there is no vector space over any field $\mathbb{F}$ having precisely $6$ elements. I would like to know if there is a theorem which characterizes those $n's \in \mathbb{N}$ such that there is a vector space having exactly $n$ elements over a field $\mathbb{F}$. For example: $n=p^{k}$ where $p$ is a prime number and $k\in \mathbb{N}$ ($k\neq 0$).

-
Infact, the prime-powers are the only $n's$ for which you can find a vector space with those many elements! –  user21436 Jan 23 '12 at 23:47

I think you can put a proof together from the following ingredients.

If $F$ is a finite field with $q$ elements, then $q$ is a power of a prime.

If $V$ is a finite vector space over a field of $q$ elements, then it has a finite basis, which we'll call $\lbrace\,b_1,b_2,\dots,b_r\,\rbrace$, and every element has a unique expression as a linear combination of basis elements, and the number of such linear combinations is $q^r$.

Are the "ingredients" things you already know?

-
you cannot have vector spaces with finite elements. Even if the field is finite having just finite elements consider a vector space with basis $\{e_1, e_2, e_3,\ldots\}$ where \begin{align} e_1 &= (1,0,0,0,\ldots)\\ e_2 &= (0,1,0,0,\ldots)\\ e_3 &= (0,0,1,0,\ldots)\\ & \vdots \end{align}


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# Matplotlib 3D Scatter Plot with Colorbar

Borrowing from the example on the Matplotlib documentation page and slightly modifying the code,

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt

def randrange(n, vmin, vmax):
return (vmax-vmin)*np.random.rand(n) + vmin

fig = plt.figure()
n = 100
for c, m, zl, zh in [('r', 'o', -50, -25), ('b', '^', -30, -5)]:
xs = randrange(n, 23, 32)
ys = randrange(n, 0, 100)
zs = randrange(n, zl, zh)
cs = randrange(n, 0, 100)
ax.scatter(xs, ys, zs, c=cs, marker=m)
ax.set_xlabel('X Label')
ax.set_ylabel('Y Label')
ax.set_zlabel('Z Label')

plt.show()

Will give a 3D scatter plot with different colors for each point (random colors in this example). What's the correct way to add a colorbar to the figure, since adding in plt.colorbar() or ax.colorbar() doesn't seem to work.

• what is cs in your example? Mar 21, 2019 at 12:48

This produces a colorbar (though possibly not the one you need):

Replace this line:

ax.scatter(xs, ys, zs, c=cs, marker=m)

with

p = ax.scatter(xs, ys, zs, c=cs, marker=m)

then use

fig.colorbar(p)

near the end

Using the above answer did not solve my problem. The colorbar colormap was not linked to the axes (note also the incorrect colorbar limits):

from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure()

data = np.random.rand(3, 100)
x, y, z = data  # for show
c = np.arange(len(x)) / len(x)  # create some colours

p = ax.scatter(x, y, z, c=plt.cm.magma(0.5*c))
ax.set_xlabel('$\psi_1$')
ax.set_ylabel('$\Phi$')
ax.set_zlabel('$\psi_2$')

ax.set_box_aspect([np.ptp(i) for i in data])  # equal aspect ratio

fig.colorbar(p, ax=ax)

The solution (see here also) is to use cmap in ax.scatter:

from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure()

data = np.random.rand(3, 100)
x, y, z = data  # for show
c = np.arange(len(x)) / len(x)  # create some colours

p = ax.scatter(x, y, z, c=0.5*c, cmap=plt.cm.magma)
ax.set_xlabel('$\psi_1$')
ax.set_ylabel('$\Phi$')
ax.set_zlabel('$\psi_2$')

ax.set_box_aspect([np.ptp(i) for i in data])  # equal aspect ratio

fig.colorbar(p, ax=ax)

• You passed the values c (which ranges from 0. to 1.) through the cm.magma function, which altered all the values, and gave the impression of an incorrect range. I would not say that my answer was incorrect, but that your first attempt to change the colormap was incorrect. Your second answer correctly sets the colormap with the cmap argument. Jul 25 at 21:04


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# Converge of Random Variables Using Characteristic Function [duplicate]

Assume $X_n$ Converges in distribution to $X$ and $a_n \rightarrow a$ ,$b_n \rightarrow b$ where $a_n$ and $b_n$ are deterministic sequence,

prove $a_nX_n+b_n \rightarrow aX+b$ using characteristic function?

I have started this question as the following but I am not sure

$$\phi_{a_nX_n+b_n}(t) = E\{e^{it(a_nX_n+b_n)}\} = e^{itb_n} E\{e^{ita_nX_n}\} \rightarrow e^{itb} E\{e^{itaX}\} = \phi_{aX+b}(t)$$

how can I show that $a_nX_n$ converges in distribution to $aX$?

## marked as duplicate by Davide Giraudo, Sasha, Michael Hoppe, Dilip Sarwate, Daniel Fischer♦Feb 26 '14 at 14:53

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• Two random variables converge in distribution if their characteristic functions converge pointwise, and the pointwise limit is continuous at the origin. – fgp Feb 25 '14 at 17:12
• – saz Feb 25 '14 at 19:53


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• 1.
Luleå tekniska universitet.
Luleå tekniska universitet. Luleå tekniska universitet.
Development and validation of a numerical model of flow through embankment dams: comparisons with experimental data and analytical solutions1999In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 35, no 3, p. 395-406Article in journal (Refereed)

The development and validation of a numerical simulation model of the flow through embankment dams is described. The paper focuses on basic verification studies, that is, comparisons with analytical solutions and data from laboratory experiments. Two experimental studies, one dealing with the flow in a Hele-Shaw cell and the other with the flow through a bed of packed glass beads, are also described. Comparisons are carried out with respect to the phreatic surfaces, pressure profiles, seepage levels and discharges. It is concluded that the agreement between experimental, analytical and numerical results is generally satisfactory

• 2.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Darcy's law for flow in a periodic thin porous medium confined between two parallel plates2016In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 115, no 3, p. 473-493Article in journal (Refereed)

We study stationary incompressible fluid flow in a thin periodic porous medium. The medium under consideration is a bounded perforated 3D-domain confined between two parallel plates. The distance between the plates is $$\delta$$, and the perforation consists of $$\varepsilon$$-periodically distributed solid cylinders which connect the plates in perpendicular direction. Both parameters $$\varepsilon$$, $$\delta$$ are assumed to be small in comparison with the planar dimensions of the plates. By constructing asymptotic expansions, three cases are analysed: (1) $$\varepsilon \ll \delta$$, (2) $$\delta /\varepsilon \sim \text {constant}$$ and (3) $$\varepsilon \gg \delta$$. For each case, a permeability tensor is obtained by solving local problems. In the intermediate case, the cell problems are 3D, whereas they are 2D in the other cases, which is a considerable simplification. The dimensional reduction can be used for a wide range of $$\varepsilon$$ and $$\delta$$ with maintained accuracy. This is illustrated by some numerical examples.

• 3.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Investigation of Hydrodynamic Dispersion and Intra-pore Turbulence Effects in Porous Media2019In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634Article in journal (Refereed)

The aim of the present paper is to evaluate and compare the pore level hydrodynamic dispersion and effects of turbulence during flow in porous media. In order to compute these quantities, large eddy simulations of turbulent flow in five unit cells comprised of spherical particles are performed and the results are averaged over the cells. Visualizations of vortical structures reveal that the size of the turbulence structures is of the size of the pores. Investigations furthermore yield that volume-averaged values of the hydrodynamic dispersion are of the same order as the Reynolds stress within the pores. It is also shown that the effect of intra-pore turbulence and hydrodynamic dispersion on the redistribution of macroscopic momentum within the porous medium is negligible compared to Forchheimer term. A discussion is provided on the accuracy of the eddy viscosity hypothesis in the modeling of the volume-averaged intra-pore Reynolds stresses. Finally, the effect of variation in the pore-scale geometry on the turbulence structures and averaged values of hydrodynamic dispersion and Reynolds stress is investigated.

• 4.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
A Subgrid-Scale Model for Turbulent Flow in Porous Media2019In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 129, no 3, p. 619-632Article in journal (Refereed)

Given the analogy between the filtered equations of large eddy simulation and volume-averaged Navier–Stokes equations in porous media, a subgrid-scale model is presented to account for the residual stresses within the porous medium. The proposed model is based on the kinetic energy balance of the filtered velocity field within a pore; hence, when using the model, numerical simulations of the turbulent flow in the pores are not required. The accuracy of the model is validated with available data in the literature on turbulent flow through packed beds and staggered arrangement of square cylinders. The validation yields that the model successfully captures the effect of the pore-scale turbulent motion. The model is then used to study turbulent flow in a wall-bounded porous media to assess its accuracy.

• 5.
Luleå University of Technology, Department of Engineering Sciences and Mathematics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Flow-induced deformations within random packed beds of spheres2014In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 104, no 1, p. 43-56Article in journal (Refereed)

Low Reynolds number flow-induced alterations of permeability of random packing of mono-sized spheres is studied. The number of spheres is several thousands and the porosities ranges between 0.4 and 0.6. The change of permeability is obtained for elastic deformations of the positions of the spheres using either of two methods. Each sphere is elastically attached to single points or the spheres that are connected via an elastic porous network. The system of spheres is divided into smaller volumes with Voronoi diagrams and the flow is derived by usage of a dual stream function. The local saturated flow fields are approximated as for close packed spheres and the overall flow pattern is obtained by minimising the dissipation rate of energy. The results show that the permeability for large random systems increases as a function of velocity and thus the deformation. The alteration is, however, much less than for two-dimensional cases like parallel cylinders. The relative increase in permeability becomes larger as the porosity increases from 0.4 to 0.6.

• 6.
Luleå University of Technology, Department of Engineering Sciences and Mathematics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Civil, Environmental and Natural Resources Engineering, Mining and Geotechnical Engineering.
Fluid flow induced internal erosion within porous media: modelling of the no erosion filter test experiment2011In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 89, no 3, p. 441-457Article in journal (Refereed)

An investigation of the potential to numerically model the no erosion filter test is performed here, where the flow through a large ensemble of particles is considered by applying minimisation of dissipation rate of energy on the ensemble that is discretised with modified Voronoi diagrams and Delaunay triangulation. Low-Reynolds number simulations are applied to each part of the Voronoi diagram using computational fluid dynamics. The mechanical friction between particles is modelled by increasing the effective viscosity for closely spaced particles. Microscopic mechanisms for successful and unsuccessful sealing of filters are obtained. The numerical results agree with previously presented experimental observations by Sherard and Dunnigan. A conformity is that the sealing starts from the end of the channel and continues outwards in the radial direction. The sealing implies that the permeability can be reduced several orders of magnitude during a test.

• 7. Frishfelds, Vilnis
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. University of Latvia.
Lattice gas analysis of liquid front in non-crimp fabrics2010In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 84, no 1, p. 75-93Article in journal (Refereed)

The liquid flow front during impregnation of non-crimp fabrics is considered. Irregularities in fibre bundle architecture lead to generation of bubbles at this front. The velocity of this interface is highly influenced by capillary forces mainly caused by the small fibres inside the bundles. In order to better understand which shapes the liquid front takes up at different conditions, a lattice gas model has been applied. First, the macroscopic properties of the solved gas in the liquid are discussed. Next, bubble inclusions are analyzed as to liquid-gas interface position and concentrations of minor component in each phase. The capillary effects at the fluid front are studied for systems both with and without gaps between the bundles. The flow in the interior of the fibre bundles is scrutinized, as well, by also considering the viscous stresses. The flow through unidirectional fabrics is considered by a one-dimensional model, which suggests that the liquid front inside bundles and gaps moves with the same speed when the liquid front inside the bundle has to catch up with the liquid front in the gap

• 8.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Civil, Environmental and Natural Resources Engineering, Architecture and Water. Luleå University of Technology, Department of Civil, Environmental and Natural Resources Engineering, Architecture and Water.
Longitudinal dispersion coefficient: effects of particle-size distribution2013In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 99, no 1, p. 1-16Article in journal (Refereed)

The effects of particle-size distribution on the longitudinal dispersion coefficient (DL) in packed beds of spherical particles are studied by simulating a tracer column experiment. The packed-bed models consist of uniform and different-sized spherical particles with a ratio of maximum to minimum particle diameter in the range of 1–4. The modified version of Euclidian Voronoi diagrams is used to discretize the system of particles into cells that each contains one sphere. The local flow distribution is derived with the use of Laurent series. The flow pattern at low particle Reynolds number is then obtained by minimization of dissipation rate of energy for the dual stream function. The value of DL is obtained by comparing the effluent curve from large discrete systems of spherical particles to the solution of the one-dimensional advection-dispersion equation. Main results are that at Peclet numbers above 1, increasing the width of the particle-size distribution increases the values of DL in the packed bed. At Peclet numbers below 1, increasing the width of the particle-size distribution slightly lowers DL.

• 9.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Transitional and Turbulent Flow in a Bed of Spheres as Measured with Stereoscopic Particle Image Velocimetry2017In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 117, no 1, p. 45-67Article in journal (Refereed)

Stereoscopic particle image velocimetry has been used to investigate inertia dominated, transitional and turbulent flow in a randomly packed bed of monosized PMMA spheres. By using an index-matched fluid, the bed is optically transparent and measurements can be performed in an arbitrary position within the porous bed. The velocity field observations are carried out for particle Reynolds numbers, (Formula presented.), between 20 and 3220, and the sampling is done at a frequency of 75 Hz. Results show that, in porous media, the dynamics of the flow can vary significantly from pore to pore. At (Formula presented.) around 400 the spatially averaged time fluctuations of total velocity reach a maximum and the spatial variation of the time-averaged total velocity, (Formula presented.) increases up to about the same (Formula presented.) and then it decreases. Also in the studied planes, a considerable amount of the fluid moves in the perpendicular directions to the main flow direction and the time-averaged magnitude of the velocity in the main direction, (Formula presented.), has an averaged minimum of 40% of the magnitude of (Formula presented.) at (Formula presented.) about 400. For (Formula presented.), this ratio is nearly constant and (Formula presented.) is on average a little bit less than 50% of (Formula presented.). The importance of the results for longitudinal and transverse dispersion is discussed.

• 10.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry2017In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 116, no 1, p. 413-431Article in journal (Refereed)

Particle image velocimetry (PIV) has been used to investigate transitional and turbulent flow in a randomly packed bed of mono-sized transparent spheres at particle Reynolds number, (Formula presented.). The refractive index of the liquid is matched with the spheres to provide optical access to the flow within the bed without distortions. Integrated pressure drop data yield that Darcy law is valid at (Formula presented.). The PIV measurements show that the velocity fluctuations increase and that the time-averaged velocity distribution start to change at lower (Formula presented.). The probability for relatively low and high velocities decreases with (Formula presented.) and recirculation zones that appear in inertia dominated flows are suppressed by the turbulent flow at higher (Formula presented.). Hence there is a maximum of recirculation at about (Formula presented.). Finally, statistical analysis of the spatial distribution of time-averaged velocities shows that the velocity distribution is clearly and weakly self-similar with respect to (Formula presented.) for turbulent and laminar flow, respectively

• 11.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Experimental investigation of transitional flow in porous media with usage of a pore doublet model2014In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 101, no 2, p. 333-348Article in journal (Refereed)

The transition from laminar to turbulent flow in porous media is studied with a new method. To mimic inter-connected pores, a simplified geometry is studied consisting of a pipe with a relatively large diameter that is split into two parallel pipes with different diameters. This is a pore-doublet set-up and the pressure drops over the parallel pipes are recorded by pressure transducers for different flow rates. Results show that the flow in the parallel pipes is redistributed when turbulent slugs pass through one of them. The presence of the slugs is revealed by positive skewness in the pressure signals as well as an increase of the standard deviation of the pressure drops and correlation between the pressure drops of the pipes. A frequency analysis of the pressure drops show that lower band frequency pressure variations in one pipe are communicated to the other pipe while higher band frequencies are filtered out.

• 12.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Chalmers University of Technology. Chalmers University of Technology.
Measurement of the permeability tensor of compressed fibre beds2002In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 47, no 3, p. 363-380Article in journal (Refereed)

A new method to measure the permeability tensor of highly compressed fibre beds is developed. The method is based on saturated parallel flow and is evaluated through experiments with various textile materials: press fabrics used in papermaking and fibre reinforcements designed for composites. Since the materials are in the form of sheets, two measuring cells are used, one for the principal in-plane permeabilities and the other for the out-of-plane permeability. A unique feature is that the edge and the bulk flow are measured separately, so that any influence from enhanced or suppressed edge flow may be eliminated. The technique is evaluated with good results in terms of scatter in the measured permeability and the influence of test geometry, pressure, and liquid properties.

• 13. Nordlund, Markus
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
Effect of multi-scale porosity in local permeability modelling of non-crimp fabrics2008In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 73, no 1, p. 109-124Article in journal (Refereed)

The influence of multi-scale porosity of fibre reinforcements on local permeability is investigated, in order to determine the possibility of simplifying permeability models for more efficient permeability calculations. Unit cell models of a biaxial Non-Crimp Fabric are developed and used to investigate, whether or not the porous bundles can be excluded, when modelling the local permeability. Numerical accuracy of calculations is controlled to guarantee the quality of the results and the conclusions drawn from them. It is found that fibre bundles with high fibre density can be excluded from permeability models, while bundles with low fibre volume fractions need to be included. A new method to model the local permeability of multi-scale reinforcements is developed and verified for low fibre density in the bundles. In this method, the effects of the flow inside the fibre bundles are included through modifications of the boundary conditions of a single-scale model representing the interbundle regions. The local permeability of multi-scale reinforcements can, therefore, be calculated by models with simplified fluid domains for all fibre bundle porosities, instead of being calculated by models consisting of the entire multi-scale geometry.

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text:
Information for:

# Build a new form¶

Fast Track to Drupal 8 Coding – Back to intro
Previous lesson - Alter an existing simple form
Next lesson – Add a field to an existing form

## Lesson Goal¶

Build a new form that validates entries (in this example, the title must be at least 15 characters in length) and also provides a submission function.

## Implementation Method¶

Drupal Version Method
Drupal 7 Use hook_menu and drupal_get_form
Drupal 8 Use routing.yml and FormBase class

## Drupal 7 method¶

Create a lotus.info file to register your module, with the following contents:

name = Lotus
description = Build a simple form.
package = Custom
core = 7.x

Register a menu link using hook_menu. To this, add the following code in the lotus.module file:

/**
*/
$items['simple-form'] = array( 'title' => 'Simple Form', 'description' => 'Build a simple form', 'page callback' => 'drupal_get_form', 'page arguments' => array('lotus_simple_form'), 'access arguments' => array('access content'), ); return$items;

}

Create a form by calling drupal_get_form and using Form API. To build the form, add the following code to the lotus.module file:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  /** * Creates a simple form. Page callback. */ function lotus_simple_form($form, &$form_state) { $form['title'] = array( '#type' => 'textfield', '#title' => t('Title'), '#description' => t('Title must be at least 15 characters in length.'), '#required' => TRUE, ); // Group submit handlers in an actions element with a key of "actions" so // that it gets styled correctly, and so that other modules may add actions // to the form. This is not required, but is convention.$form['actions'] = [ '#type' => 'actions', ]; // Add a submit button that handles the submission of the form. $form['actions']['submit'] = [ '#type' => 'submit', '#value' => t('Submit'), ]; return$form; } /** * Validation for title. Title must be at least 15 characters in length. */ function lotus_simple_form_validate($form, &$form_state) { $title =$form_state['values']['title']; if (strlen($title) < 15) { // Set an error for the form element with a key of "title". form_set_error('title', t('The title must be at least 15 characters long.')); } } /** * Display a message after successful form submit. */ function lotus_simple_form_submit($form, &$form_state) {$title = $form_state['values']['title']; drupal_set_message(t('You specified a title of %title.', ['%title' =>$title])); }

## Drupal 8 method¶

1. Create a lotus.info.yml file with the following contents:

name: Lotus
description: Build a basic form.
type: module
core: 8.x

2. Register a route by adding the following code to the lotus.routing.yml file:

lotus.simple_form:
path: '/simple-form'
defaults:
_form:  '\Drupal\lotus\Form\SimpleForm'
_title: 'Simple Form Example'
requirements:
_permission: 'access content'

3. Create a form class extending FormBase to build your form. To do this, add the following code to the lotus/src/Form/SimpleForm.php file:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97  'textfield', '#title' => $this->t('Title'), '#description' =>$this->t('Title must be at least 15 characters in length.'), '#required' => TRUE, ]; // Group submit handlers in an actions element with a key of "actions" so // that it gets styled correctly, and so that other modules may add actions // to the form. This is not required, but is convention. $form['actions'] = [ '#type' => 'actions', ]; // Add a submit button that handles the submission of the form.$form['actions']['submit'] = [ '#type' => 'submit', '#value' => $this->t('Submit'), ]; return$form; } /** * Getter method for Form ID. * * The form ID is used in implementations of hook_form_alter() to allow other * modules to alter the render array built by this form controller. it must * be unique site wide. It normally starts with the providing module's name. * * @return string * The unique ID of the form defined by this class. */ public function getFormId() { return 'lotus_simple_form'; } /** * Implements form validation. * * The validateForm method is the default method called to validate input on * a form. * * @param array $form * The render array of the currently built form. * @param FormStateInterface$form_state * Object describing the current state of the form. */ public function validateForm(array &$form, FormStateInterface$form_state) { $title =$form_state->getValue('title'); if (strlen($title) < 15) { // Set an error for the form element with a key of "title".$form_state->setErrorByName('title', $this->t('The title must be at least 15 characters long.')); } } /** * Implements a form submit handler. * * The submitForm method is the default method called for any submit elements. * * @param array$form * The render array of the currently built form. * @param FormStateInterface $form_state * Object describing the current state of the form. */ public function submitForm(array &$form, FormStateInterface $form_state) { /* * This would normally be replaced by code that actually does something * with the title. */$title = $form_state->getValue('title'); drupal_set_message($this->t('You specified a title of %title.', ['%title' => \$title])); } }


text:
# LOG#125. Basic Neutrinology(X).

The topic today is a fascinant subject in Neutrino Astronomy/Astrophysics/Cosmology. I have talked you in this thread about the cosmic neutrino background ($C\nu B$) and that the young neutrino Astronomy or neutrino telescopes will become more and more important in the future. The reasons are simple:

1st. If we want to study the early Universe, we need some “new” tool to overcome the last scattering surface as a consequence of the Cosmic Microwave Background (CMB). Neutrinos are such a new tool/probe! They only interact weakly with matter and we suspect that there are some important pieces of information related to the quark and lepton “complementarity” hidden in their mixing parameters.

2nd. Due to the GZK effect, we expect that the flux of cosmic rays will suffer a sudden cut-off at about $5\cdot 10^{19}eV=50\cdot 10^{18}eV=50EeV$, or about 8 joules. This Greisen–Zatsepin–Kuzmin limit (GZK limit) is a theoretical upper limit on the energy of cosmic rays, since at some high energy, that can be computed, they would interact with the CMB photons producing a delta particle ($\Delta$) which would spoil the observed cosmic rays flux as its decays would not be detected after “a long trip”. Then, it can only be approached when the cosmic rays travel very long distances (hundreds of million light-years or more). Here you are a typical picture of SuperKamiokande cosmic ray detection:

The limit is at the same order of magnitude as the upper limit for energy at which cosmic rays have experimentally been detected. There are some current experiments that “claim” to have observed this GZK effect, but evidence is not conclusive yet as far as I know. Some experiments claim (circa 2013, July) to have observed it, other experiments claim to have observed events well above the GZK limit. The next generation of cosmic ray experiments will confirm this limit from SM physics or they will show us interesting new physics events!

Inspired by the GZK effect, some people have suggested an indirect way to detect the existence of the cosmic relic neutrinos. Remember, cosmic neutrinos have a temperature about $1.9K$ if the SM is right, and their associated neutrino density now is about $110$ per cubic centimeter per species (neutrino plus antineutrino), or $330$ per cubic centimeter including the 3 flavors! Relic neutrinos are almost everywhere, but they are very, very feeble (neutral and weakly interacting) particles. While detecting the $C\nu B$ temperature is one of the most challenging tests of the standard cosmological model, we can try to detect the existence of these phantom neutrinos using a similar (quantum) trick than the one used in the GZK limit (there the delta particle resonance). If some ultra high energy cosmic ray (likely a neutrino coming from some astrophysical source) hits a “relic neutrino” with energy high enough to produce, say, a Z boson (neutral particle as the neutrino himself), then we should observe a “dip” in the cosmic ray spectrum corresponding to this “Z-burst” event! This mechanism is also called the ZeVatron or the Z-dip. It also shows the deep links between particle physics and Cosmology or Astrophysics. When an ultra-high energy cosmic neutrino collides with a relic anti-neutrino in our galaxy and annihilates to hadrons, this process proceeds via a (virtual) Z-boson:

$\nu_{UHE}+\bar{\nu}_{C\nu B}\longrightarrow Z\longrightarrow \mbox{hadrons}$

The cross section for this process becomes large if the center of mass energy of the neutrino-antineutrino pair is equal to the Z-boson mass (such a peak in the cross section is what we call “resonance” in High Energy physics). Assuming that the relic anti-neutrino is at rest, the energy of the incident cosmic neutrino has to be the quantity:

$\boxed{E_{eV}=\dfrac{m_Z^2}{2m_\nu}=4.2\cdot \left(\dfrac{eV}{m_\nu}\right)\cdot 10^{21}eV=42\left(\dfrac{0.1eV}{m_\nu}\right)\cdot 10^{21}eV}$

$\boxed{E_{ZeV}=4.2\left(\dfrac{eV}{m_\nu}\right)ZeV=42\left(\dfrac{0.1eV}{m_\nu}\right)ZeV}$

In fact, this mechanism based on “neutral resonances” is completely “universal”! Nothing (except some hidden symmetry or similar) can allow the production of (neutral) particles using this cosmic method. For instance, if this argument is true, beyond the Z-burst, we should be able to detect Higgs-dips (Higgs-bursts) or H-dips, since, similarely we could have

$\nu_{UHE}+\bar{\nu}_{C\nu B}\longrightarrow H\longrightarrow \mbox{hadrons}$

or more generally, with some (likely) “dark” particle, we should also expect that

$\nu_{UHE}+\bar{\nu}_{C\nu B}\longrightarrow X\longrightarrow \mbox{hadrons}$

In the H-dip case, taking the measured Higgs mass from the last LHC run (about 126GeV), we get

$\boxed{E_{eV}(H-dip)=\dfrac{m_H^2}{2m_\nu}=7.9\left(\dfrac{eV}{m_\nu}\right)\cdot 10^{21}eV=79\left(\dfrac{0.1eV}{m_\nu}\right)\cdot 10^{21}eV}$

$\boxed{E_{ZeV}(H-dip)=7.9\left(\dfrac{eV}{m_\nu}\right)ZeV=79\left(\dfrac{0.1eV}{m_\nu}\right)ZeV}$

In the arbitrary “dark” or “weakly interacting” particle, we have (in general, with $m_X= x GeV$) the formulae:

$\boxed{E_{eV}(X-dip)=\dfrac{m_X^2}{2m_\nu}=\dfrac{(x GeV)^2}{2m_\nu}=\left(\dfrac{x^2}{2m_\nu}\right)\cdot 10^{18}eV^2=\left(\dfrac{x^2}{2000}\right)\left(\dfrac{1eV}{m_\nu}\right) 10^{21}eV}$
or equivalently
$\boxed{E_{ZeV}(X-dip)=\dfrac{m_X^2}{2m_\nu}=\left(\dfrac{x^2}{200}\right)\left(\dfrac{0.1eV}{m_\nu}\right) ZeV=\left(\dfrac{x^2}{2000}\right)\left(\dfrac{1eV}{m_\nu}\right) ZeV}$

Therefore, cosmic ray neutrino spectroscopy is a very interesting subject yet to come! It can provide:

1st. Evidences for relic neutrinos we expect from the standard cosmological model.

2nd. Evidence for the Higgs boson in astrophysical scenarios from cosmological neutrinos. Now, we know that the Higgs field and the Higgs particle do exist, so it is natural to seek out this H-dips as well!

3rd. Evidence for the additional neutral weakly interacting (and/or “dark”) particles from “unexpected” dips at ZeV (1ZeV=1Zetta electron-volt) or even higher energies! Of course, this is the most interesting part from the viewpoint of new physics searches!

Neutrino telescopes and their associated Astronomy is just rising now! IceCube is its most prominent example…

Moreover, following one of the most interesting things in any research (expect the unexpected and try to explain it!) from the scientific viewpoint, I am quite sure the neutrino astronomy and its interplay with cosmic rays or this class of “neutrino spectroscopy” in the flux of cosmic rays open a very interesting window for the upcoming new physics. Are we ready for it? Maybe…After all, the neutrino mixing parameters are very different (“complementary”?) to the quark mixing parameters. You can observe it in this mass-flavor content plot:

Neutrino oscillations are a purely quantum effect, and thus, they open a really interesting “new channel” in which we can observe the whole Universe. Yes, neutrinos are cool!!! The coolest particles in all over the world! We can not imagine yet what neutrino will show and teach us about the current, past and future of the cosmological evolution.

Remark: When I saw the Fermi line and the claim of the Dark Matter particle “evidence” at about 130 GeV, I wondered if it could be, indeed, a hint of a similar “resonant” process in gamma rays, something like

$\gamma \gamma\longrightarrow H (resonance)$

since the line “peaked” close to the known Higgs-like particle mass ($126GeV\sim 130GeV$). Anyway, this line is controversial and its presence has yet to be proved with enough statistical confidence (5 sigmas are usually required in the particle physics community). Of course, the issue with this resonant hypothesis would be that we should expect that this particle would decay into hadrons leaving some indirect clues of those events.  The Fermi line can indeed have more explanations and/or be a fluke in the data due to a bad modeling or a bad substraction of the background. Time will tell us if the Fermi line is really here as well.

Final (geek) remark: I wonder if the Doctor Who fans remember that the reality bomb of Davros and the Daleks used “Z-neutrinos“!!! I presently do not know if the people who wrote those scripts and imagined the Z-neutrino were aware of the Z-bursts…Or not… LOL The Z-neutrino powered crucible was really interesting…

And the reality bomb concept was really scaring…

However, neutrinos are pretty weakly interacting particles, at least when they have low energy, so we should have not fear them. After all, their future applications will surprise us much more. I am quite sure of it!

See you in my next neutrinological post!

May the Z(X)-burst induced superGZK neutrinos be with you!

# LOG#108. Basic Cosmology (III).

The current Universe has evolved since its early phase of thermal equilibrium until the present state. The departure from thermal equilibrium in the early Universe made a fossil record we can observe at current time!

There are some easy rules for thermal equilibrium. The easiest one, is that coming from the “interaction rate” $\Gamma_{int}$. It can be expressed in the following way:

$\Gamma_{int}>\mbox{Expansion rate H}$

and then, at a given temperature T, we get

$\Gamma_{int}(T)=n(T)\langle \sigma \vert v \vert \sigma \rangle^T$

and where $H\approx \dfrac{T^2}{M_p}$

Remark: If $\Gamma_{int}=aT^n$ $\forall n>2$, then

$N_{int}=\int_t^\infty T_{int}(t')dt'=\dfrac{\Gamma (H)\vert_t}{n-2}<1$

and it implies that a particle interacts less than once after the time $\Gamma =H$.

Moreover, we can understand roughly the so-called decoupling era:

1st. Any interaction mediated by a massless gauge boson provides

$\sigma\sim \dfrac{\alpha^2_X}{s}$ with $s\sim E^2$ and $\alpha_X=\dfrac{g^2_X}{4\pi}$

and this implies that

$\Gamma \sim n\langle \sigma v\rangle\sim T^3\dfrac{\alpha_X^2}{T^2}=\alpha^2_XT$

and

$\dfrac{\Gamma}{H}\sim \alpha_X^2\dfrac{M_p}{T}$

so the equilibrium temperature is found whenever $T\leq \alpha_X^2M_p$!

2nd. Interactions mediated by any massive gauge boson provides

$\sigma\sim G_X^2s$ with $G_X\sim \dfrac{\alpha_X}{m_X^2}$

and this implies that

$\Gamma \sim T^3G_X^2T^2=G_X^2T^5$

and

$\dfrac{\Gamma}{H}\sim G_X^2M_pT^3$

and then

$\left( G_X^2M_p\right)^{-1/3}\leq T\leq m_X\longrightarrow \mbox{Equilibrium temperature (E.T.)}$

Moreover,

$T\leq \left( G_X^2M_p\right)^{-1/3}\sim \left(\dfrac{m_X}{100\mbox{GeV}}\right)^{4/3}\mbox{MeV}\longrightarrow \mbox{Freeze out}$

As a consequence, we can realize that the out-of-equilibrium phenomena in the early and current Universe are very important processes! In particular:

1) They provide the formation of light elements during the Big Bang Nucleosynthesis (BBN), also known as primordial nucleosynthesis, i.e., the formation of the first light elements after the Big Bang (circa 300000 years after the Universe “birth”).

2) They provide the path of recombination of electrons and protons into hydrogen atoms.

3) They imply the  (likely) production of dark matter (or equivalently the presence of some kind of “modified gravity” or/and modified newtonian dynamics).

Boltzmann’s equation for annihilation of particles in equilibrium

There is a beautiful equation that condenses the previous physical process of equilibrium at a given temperature and the particle production it yields. Conceptually speaking, we have

$\begin{pmatrix}\mbox{Boltzmann}\\ \mbox{Equation}\end{pmatrix}:$

$\begin{pmatrix}\mbox{Rate of change}\\ \mbox{in the abundance}\end{pmatrix}=\begin{pmatrix}\mbox{Rate of}\\ \mbox{particle production}\end{pmatrix}-\begin{pmatrix}\mbox{Rate of}\\ \mbox{particle erasing/annihilation}\end{pmatrix}$

Consider a process like

$\mbox{particle type 1}+\mbox{particle type 2}\leftrightarrow \mbox{particle type 3}+\mbox{particle type 4}$

and where the particle 1 is the one we are interested in. Then, we deduce that

$\underbrace{\dfrac{1}{a^3}\dfrac{d(n_1a^3)}{dt}}_\text{change in comoving volume}=\underbrace{\int\dfrac{d^3p_1}{(2\pi)^32E_1}\int\dfrac{d^3p_2}{(2\pi)^32E_2}\int\dfrac{d^3p_3}{(2\pi)^32E_3}\int\dfrac{d^3p_4}{(2\pi)^32E_4}}_\text{phase space invariant}\times A$

where A is certain complicated facter involving “delta functions” of the energies and momenta of the particles 1,2,3,4 and an additional term depending on the statistics of the particle. Explicitly, it takes the form

$A=\left[(2\pi)^4\delta^3(p_1+p_2-p_3-p_4)\delta (E_1+E_2-E_3-E_4)\vert M\vert^2\right]\times S$

with $S=\left[ f_3f_4(1\pm f_1)(1\pm f_2)-f_1f_2(1\pm f_3)(1\pm f_4)\right]$

and where

$f_i=\dfrac{1}{e^{(E_i-\mu_i (t))/T}\pm 1}$

is the Fermi-Dirac (FD, -)/Bose-Einstein (BE,+) distribution. In fact, the above FD/BE factors provide the so-called Pauli blocking/”Bose-Einstein” enhancement effects for the particle production in the processes $3+4\rightarrow 1+2$ and $1+2\rightarrow 3+4$. Indeed, particle physics enter into the game here (see above formulae again) and we assume

$M(1+2\rightarrow 3+4)=M(3+4\rightarrow 1+2)$

Do you recognize the principle of detailed balance in this equation?

We can simplify the assumptions a little bit:

1st. The kinetic equilibrium is taken to be a rapid elastic scattering and we input the FD/BE statistics without loss of generality.

2nd. The annihilation in thermal equilibrium will be calculated from the sum of the chemical potential in any balanced equation.

3rd. Low temperature approximation. Suppose that

$T<< (E-\mu)$

then we obtain the Maxwell-Boltzmann approximation to the FD/BE statistics

$f\approx e^{-(E-\mu)/T}$ and $1+f\approx 1$, so, since $E_1+E_2=E_3+E_4$, we get

$f_3f_4(1\pm f_1)(1\pm f_2)-f_1f_2(1\pm f_3)(1\pm f_4)\approx e^{-(E_1+E_2)/T}\left( e^{\frac{(\mu_3+\mu_4)}{T}}-e^{\frac{(\mu_1+\mu_2)}{T}}\right)-\star$

What is $\star$? After a change of variable

$\mu_i (t)\longrightarrow n_i(t)=g_ie^{\mu_i/T}\int \dfrac{d^3p}{(2\pi)^3}e^{-E_i/T}$

$\star$ is the “equilibrium number density” deducen from the expression

$n_i^{0}\equiv g_i\int {d^3p}{(2\pi)^3}e^{-E_i/T}=\begin{cases}g_i\left(\dfrac{m_iT}{2\pi}\right)^{3/2}e^{-m_i/T},\;\; \mbox{if}\;\; m_i>>T\\ g_i\dfrac{T^3}{\pi^2},\;\;\mbox{if}\;\; m_i<

and it yields

$n_i=e^{\mu_i/T}n_\gamma^{0}$

and then we finally get that

$\star$ equals $e^{-(E_1+E_2)/T}\left[\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\right]$

Now, we can define the thermally averaged cross section

$\langle \sigma v\rangle\equiv \dfrac{1}{n_1^0n_2^0}\int \dfrac{d^3p_1}{(2\pi)^32E_1}\cdots \dfrac{d^3p_4}{(2\pi)^32E_4}e^{-(E_1+E_2)/T}(2\pi)^4\delta^3 (p_1+p_2-p_3-p_4)\times$

$\times \delta (E_1+E_2-E_3-E_4)\vert M\vert^2$

The Boltzmann equation becomes with these conventions

$\dfrac{1}{a^3}\dfrac{d(n_1a^3)}{dt}=n_1^0n_2^0\langle \sigma v\rangle \left(\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\right)$

Remark (I): LHS is similar to $\dfrac{n_1}{t}\sim n_1H$ and the RHS is similar to $n_1n_1\langle \sigma v\rangle$

Remark (II): If the reaction rate is $n_1\langle \sigma v\rangle >> H$, then it provides the chemical equilibrium condition well known in the nuclear statistical equilibrium as the Saha equation, i.e.,

$\dfrac{n_3n_4}{n_3^0n_4^0}-\dfrac{n_1n_2}{n_1^0n_2^0}\approx 0$


text:
# Problem understanding “connectivity” characteristic for the $k$-connected subgraph problem

I am reading this article, and I am having trouble to understand the 11th definition (page 7) about the connectivity characteristic. I do understand the raw definitions, but have no idea what this gives my intuitively.

The definition:

• For each pair of distinct vertices v, u of H: let $Com_{H}(v, u)$ be the set of portal completions which augment H to a graph with k internally vertex-disjoint paths between v and u.
• For each vertex v of H, let $Path_{H}(v)$ be the set of pairs (U, Q) where U is a multiset of P(H) with vertex multiplicities bounded by from above by the vertex degrees in H and Q is a partial g-matching on the complete multigraph on P(H) with g(w) equal to the difference between the degree of w in UH and the multiplicity of w in U such that there are $|U|+|Q|$ internally vertex-disjoint paths in H connecting v with each copy a vertex in U, and connecting each pair of endpoints of each edge in Q, respectively.
• $Path_{H}$ is the set of all Q for which $(\phi, Q)$ is in $Path_{H}(v)$ for an arbitrary vertex v in P(H).
• We will call the triple ($Com_{H}(v, u)$, $Path_{H}(v)$, $Path_{H}$) the $connectivity\ characteristic$ of H.

Context: The paper presenta a PTAS for minimum-cost k-connected subgraph problem. This is done by dissecting the graph into smaller parts, brute-forcing a solution on the small parts, and then connecting them in a way that doesn't increase is stays a $(1+\epsilon)$-approximation. This connectivity characteristic is a property of the subgraphs that is maintained by the algorithm while augmenting the big graph from the smaller parts. A foot note at page 6 states that it models "missing" connectivity, that will be added later by the augmentation of the regions.
Note that there is also (m, r)-blue which is another property that should be maintained. Could be that only both give something useful.
The definition of an (m, r)-blue graph is at page 10. Mainly says that a facet is being crossed at portals, and not too often.

Could someone give me some intuitive explanation about what this characteristic is, and why its helpful?

• Could you please make your question more self-contained by including the offending definition and some context? – Dave Clarke May 7 '13 at 10:48

I think I figured it out.

1. $Com_{H}(v, u)$ are paths leaving the region and connecting through some vertices outside the region.

2. $Path_{H}(v)$ are the paths connecting $v$ to the vertices that are not included in this region.

3. $Path_{H}$ are paths inside the region

If this is true, then the proof of Lemma 3.2 has some typing errors (the 2 a.m. kind). And it should be:

To compute $Com_{H}(v, u)$ where u, u v are non-portal vertices in $H$, for each competition $D$ in $Com_{H}(v, u)$, we insert D and the set difference $D\backslash B$ into $Com_{H \cup B}(v, u)$ (not $H \cup C$).
[The rest is good]


text:
# Why Coulomb chose 1.6x10^-19

1. Mar 7, 2012

### kjamha

Does anyone know why Coulomb chose 1.6x10^-19 for the charge of an electron? (or why he chose 6.25x10^18 electrons for 1 coulomb)

2. Mar 7, 2012

### FroChro

Re: Why 1.6 x 10^-19?

I think your question is ill-posed. I believe that in times of Coulomb they hadn't known that there is some ''atom'' of charge, much less what is its magnitude.

3. Mar 7, 2012

### Staff: Mentor

Re: Why 1.6 x 10^-19?

Coulomb didn't choose anything. Unit of charge (defined as charge transferred by 1 ampere current in 1 second) was named Coulomb in 1881, and Coulomb died in 1806. At the time Coulomb (unit of charge) was defined, electron was yet to be discovered.

4. Mar 7, 2012

### kjamha

Re: Why 1.6 x 10^-19?

Thanks for the history!

But isn't 1 Ampere equal to 1 coulomb per second - so are you saying 1 coulomb per sec in one sec? Can you clarify this for me?

5. Mar 7, 2012

### cepheid

Staff Emeritus
Re: Why 1.6 x 10^-19?

You are concerned, because the definition seems circular (a coulomb is the charge transferred by a current of one ampere in one second, and an ampere is the current produced when one coulomb of charge flows past in one second).

But the definition is not circular, because there is a separate and independent definition of what an ampere is. See my post in this thread:

EDIT: Why don't I just quote my post from that thread here:

6. Mar 7, 2012

### Dickfore

Re: Why 1.6 x 10^-19?

The real question is why the fine structure constant:
$$\alpha \equiv \frac{k \, e^2}{\hbar \, c} = \frac{1}{137.0}$$
where k is Coulomb's constant has the numerical value that it does.

7. Mar 8, 2012

### kjamha

Re: Why 1.6 x 10^-19?

Thanks Cepheid, but now I am wondering why they chose 2e-7 N - seems odd.

8. Mar 8, 2012

### Staff: Mentor

To quote wikipedia:


text:
# Insane integral tricks

1. Nov 3, 2008

### tim_lou

Ingenious integral tricks

In QFT today I learnt some insane tricks in calculating impossible integrals....I figure it'd be a good idea to see if others have similar tricks so we can all learn from each other.

here goes:
to integrate (bounds are assumed to be from negative inf to inf, k^2 means the vector dot product)
edit: missing factors of 2pi

$$\int \frac{d^d k}{(2\pi)^d}\frac{1}{(k^2 + m^2)^n}$$
substitute
$$\frac{1}{(k^2 + m^2)^n}=\frac{1}{\Gamma(n)}\int_0^\infty t^n e^{-t(k^2 + m^2)} dt$$

and get a gaussian in the integral. At the end of the day,
$$\int \frac{d^d k}{(2\pi)^d}\frac{1}{(k^2 + m^2)^n}=\frac{\Gamma(n-d/2)(m^2)^{d/2-n}}{\Gamma(n)(4\pi)^{d/2}}$$

$$\Gamma(n)=(n-1)!$$

Quite crazy eh? I would've never thought of it myself... what about your favorite crazy tricks?

Last edited: Nov 4, 2008
2. Nov 3, 2008

### tim_lou

Another I have to share (I read it in some analysis book...)

take the following integral
$$\binom{m+n+1}{m,n}\int_0^1 u^{m} (1-u)^n du$$
consider the integral as a sum, we see that given m+n+1 particles on (0,1), it sums the probability of finding m particles in (0, u), 1 particle around the point u and n particles in (u, 1), so it must be one. This gives the beta integral for integers m, n!!!
$$\int_0^1 u^{m} (1-u)^n du=\frac{m!n!}{(m+n+1)!}$$

3. Nov 4, 2008

### Gerenuk

Re: Ingenious integral tricks

I wonder if most of these crazy tricks can be found without guessing. This substitution suggests that the initial problem could be simply solved by Laplace transforms?!


text:
# Does invariance of proper time lead to the Lorentz transformation, or vice versa?

• I
Hi, I've seen several explanations for sr on youtube. But they all start off explaining from a different perspective. I was wondering how the fundamental postulates of sr lead to the invariance of proper time between frames, and also what "order" everything is derived in. For example, does the invariance of proper time lead to the lorentz transformation, or vice versa?

Proper time is time measured by the objects own clock, so it doesnt matter in what frame you are in. You are just checking his clocks, you dont care about yours.
It is invariant by definition.

PeroK
A.T.
For example, does the invariance of proper time lead to the lorentz transformation, or vice versa?
The invariance of proper time intervals is a fundamental assumption, otherwise you would get conflicting physical outcomes. The Lorentz transformation follows from the SR postulates, and such basic consistency requirements.

Ibix
You can start from a lot of different places. There isn't a Right One, there's just the one you like.

Einstein started from the principle of relativity and the invariance of the speed of light. From that he derived the Lorentz transforms and the invariance of the interval.

Another way is to start from the principle of relativity and show that this is only consistent with Newtonian physics and relativity, and then simply observe that experimental evidence is consistent with relativity, not Newton.

A third way is to start from the invariant interval and show that the Lorentz transforms preserve it.

A fourth way is to assert the Lorentz transforms and show that the constancy of the speed of light follows from them (this is almost what happened - Lorentz deduced the transforms, but didn't realise they applied beyond electromagnetism).

When you have a self-consistent system, it doesn't really matter where you start your deduction. Since scientists are interested in the real world, we often start backwards - with physical measurements that we hope to explain as outputs of a physical theory. Then we teach the theory we eventually came up with, and show logically and mathematically that the theory predicts the measurements. And we make new predictions for experiments we haven't tried yet and go out and do those to test the theory.

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lomidrevo, m4r35n357 and PeroK
Dale
Mentor
2021 Award
does the invariance of proper time lead to the lorentz transformation, or vice versa?
As @Ibix said, it is perfectly valid to go either way.

From a strictly practical viewpoint, I would say vice versa, since you can easily demonstrate the invariant interval from the LT by a couple of lines of algebra. I think going the other way requires much more ingenuity than that.

As @Ibix said, it is perfectly valid to go either way.

I dont get this. How can you derive LT from invarience of proper time? Proper time is invariant even in newtonian physics or in any other imaginable physics that allows definition of proper time.

TeethWhitener
Gold Member
A third way is to start from the invariant interval and show that the Lorentz transforms preserve it.
This may or may not be what OP was after, but I thought the invariance of proper time holds under all coordinate transformations, not just Lorentz transformations. Wouldn't this be a much larger group than the Lorentz group (something like the general linear group)? You'd have to further assume flat spacetime to get the Lorentz group, right?.

Dale
Mentor
2021 Award
How can you derive LT from invarience of proper time? Proper time is invariant even in newtonian physics
The quantity ##d\tau^2=dt^2-dx^2-dy^2-dz^2## (in units where c=1) is not invariant in Newtonian physics. You can derive the Lorentz transform as the transform that leaves this expression unchanged in form.

This may or may not be what OP was after, but I thought the invariance of proper time holds under all coordinate transformations, not just Lorentz transformations.
I assume that the OP was referring to what I just wrote above

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TeethWhitener
Gold Member
I assume that the OP was referring to what I just wrote above
Right, but that definition of proper time assumes Minkowski spacetime. The general definition
$$\tau=\int{\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}}$$
gives a much larger symmetry group under which ##\tau## is invariant, right? I’m just trying to clear this up in my head. To get the Lorentz group of transformations specifically, if you start with invariance of proper time, you also have to postulate flat spacetime. Is that right?

pervect
Staff Emeritus
Right, but that definition of proper time assumes Minkowski spacetime. The general definition
$$\tau=\int{\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}}$$
gives a much larger symmetry group under which ##\tau## is invariant, right? I’m just trying to clear this up in my head. To get the Lorentz group of transformations specifically, if you start with invariance of proper time, you also have to postulate flat spacetime. Is that right?

My understanding is that in flat space-time , the symmetry associated with the invariance of the Lorentz interval is the Poincare group, and the Lorentz group is a subgroup of the Poincare group. The Poincare group can be regarded as a kind of product (the semi-direct product) of the translational symmetries and the Lorentz symmetries. I'm not terribly familiar with semi-direct products, alas.

See for instance https://en.wikipedia.org/w/index.php?title=Poincaré_group&oldid=842331217

In the context of general geometries, I believe it's standard to say that "the geometry of space-time is locally Lorentzian". MTW says this, for instance. General space-times are not (AFAIK) regarded as being translation invariant, for instance the curvature can change from one point to the next. I don't have a reference on this point though.

A key word here is "locally". The Lorentz group isn't a global symmetry group in non-flat space-times, it's only a local symmetry group.

I'm not sure how to write all this in precise mathematics, which is worrying. It could be that there are some hidden mental errors here because of that lack of rigor. For instance, I'm not sure how to rigorously define "locally". But take it for what it's worth, and maybe someone else will have more comments.

TeethWhitener
Dale
Mentor
2021 Award
that definition of proper time assumes Minkowski spacetime.
Which is the usual assumption for special relativity.

To get the Lorentz group of transformations specifically, if you start with invariance of proper time, you also have to postulate flat spacetime. Is that right?
To get the Lorentz transforms you always have to assume flat spacetime, regardless of which direction you are going. It is for SR so spacetime is flat by definition.

TeethWhitener
PeterDonis
Mentor
The general definition

$$\tau=\int{\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}}$$

gives a much larger symmetry group under which ##\tau## is invariant, right?

Yes, although calling it a "symmetry group" might be misleading. It's the group of diffeomorphisms (with a bunch of technicalities that I don't think we need to delve into here).

Note also that interpreting ##\tau## as a "proper time" assumes that the curve along which the integral is being done is everywhere timelike. Obviously not all curves will meet this requirement. But the arc length along a curve obtained by the integral you give will be invariant under diffeomorphisms regardless of the type of curve it is.

TeethWhitener
PeterDonis
Mentor
I'm not sure how to write all this in precise mathematics

The rigorous way to do it is to define a flat tangent space at each point of the manifold, and write all local tensor equations in that tangent space. Then, to do integrals like the one @TeethWhitener wrote down, where you have to integrate along a curve, you need a connection, i.e., a way of mapping vectors/tensors in the tangent space at one point on the curve to vectors/tensors at other points. GR uses the Levi-Civita metric compatible connection for reasons which are well explained in textbooks like MTW (IIRC Wald goes into this in some detail as well).

I'm not sure how to rigorously define "locally"

Per the above, the rigorous definition is "in the tangent space at some chosen point". The Lorentz group is obviously a symmetry group in the tangent space at any point, since that tangent space is just flat Minkowski spacetime. (Note that, at least as I understand it, the Poincare group, which includes translations, is not a symmetry group on the tangent space in the same sense, since the tangent space is a space of vectors, not points, and there is no obvious meaning to "translating" a vector in a vector space. Or, to put it another way, "translating" a vector would mean changing the point to which it is "attached", and there is no way to do that within a single tangent space, since that tangent space by definition is "attached" to a single point in the manifold--the spacetime. "Translating" would mean moving to some other point, and you can't do that within a single tangent space; you need a connection, as noted above.)

TeethWhitener and Dale
The quantity ##d\tau^2=dt^2-dx^2-dy^2-dz^2## (in units where c=1) is not invariant in Newtonian physics. You can derive the Lorentz transform as the transform that leaves this expression unchanged in form.
Well yes, but this expression leads to proper time only in minkowsky spacetime and for timelike separations. Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame. This definition is as good in newtonian as in minkowski spacetime, even though it is redundant concept in newtonian physics and therefore we dont use it.

Perhaps what you wrote is what OP meant, but if i was OP i would be utterly confused (which i was):)

The rigorous way to do it is to define a flat tangent space at each point of the manifold, and write all local tensor equations in that tangent space. Then, to do integrals like the one @TeethWhitener wrote down, where you have to integrate along a curve, you need a connection, i.e., a way of mapping vectors/tensors in the tangent space at one point on the curve to vectors/tensors at other points. GR uses the Levi-Civita metric compatible connection for reasons which are well explained in textbooks like MTW (IIRC Wald goes into this in some detail as well).

This seems that geometrical proper time defined by given equation doesnt even need spacetime metric, all you need is connection. Or am i reading it wrong? This interests me, since in Newtonian gravity that is exactly the case - we dont have spacetime metric, but we do have connection (accoding to MTW, if i remember correctly). Do you have some idea how would the integral looked in this case without metric? I am little at loss, because i dont see connection present in the integral. You do scalar product of tangent vector to the curve, which gives scalar. You do this at every point along the curve and you get function. Now you integrated this function. In no place i see connection comming in.

Ibix
This may or may not be what OP was after, but I thought the invariance of proper time holds under all coordinate transformations, not just Lorentz transformations. Wouldn't this be a much larger group than the Lorentz group (something like the general linear group)? You'd have to further assume flat spacetime to get the Lorentz group, right?.
I assumed that we were only considering global inertial frames on flat spacetime (i.e. plain-vanilla SR). If you want to consider more general cases (even transforms to non-inertial frames on flat spacetime), then yes, I believe you are correct.

TeethWhitener
TeethWhitener
Gold Member
Ok, thanks everyone.
I assumed that we were only considering global inertial frames on flat spacetime (i.e. plain-vanilla SR). If you want to consider more general cases (even transforms to non-inertial frames on flat spacetime), then yes, I believe you are correct.
Yes, it just occurred to me that assuming flat spacetime by itself isn't even enough, because proper time is invariant in Rindler coordinates as well.

Ok, thanks everyone.

Yes, it just occurred to me that assuming flat spacetime by itself isn't even enough, because proper time is invariant in Rindler coordinates as well.
Spacetime is still flat in Rindler coordinates. It is a chart on Minkowski spacetime.

A.T.
How can you derive LT from invarience of proper time?
Not from proper time invariance alone, but from the 2 SR postulates plus some additional assumptions, one of which is that frames must agree on directly observable quantities like proper time intervals.

PeterDonis
Mentor
Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame.

Not quite. Proper time is time measured by a clock that is carried along with the observer or object. Or, to put it another way, it is arc length along a timelike worldline. It is independent of any choice of frames, and does not even require defining a frame.

Umaxo
Mister T
Gold Member
I dont get this. How can you derive LT from invarience of proper time? Proper time is invariant even in newtonian physics or in any other imaginable physics that allows definition of proper time.

You need other things besides invariance of proper time. The reason this hasn't been addressed is likely because of the way the question was worded. If you want to know all of the things you'd need, in addition to the invariance of proper time, to derive the Lorentz transformation equations; or what's needed in addition to the Lorentz transformation equations to derive invariance of proper time, those are some good questions. They're just different questions from the one asked.

Mister T
Gold Member
Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame.

It's defined as the time that elapses between two events that occur at the same location. So if you are always co-located with a particular clock, then that clock is measuring proper time. Being a relativistic invariant, proper time has the same value in all reference frames, so there is no need to refer to any particular reference frame when defining it

stevendaryl
Staff Emeritus
I dont get this. How can you derive LT from invarience of proper time? Proper time is invariant even in newtonian physics or in any other imaginable physics that allows definition of proper time.

I think people mean the invariance of the expression ##d\tau = \sqrt{dt^2 - \frac{1}{c^2} (dx^2 + dy^2 + dz^2)}##

The transformations that leave the right side invariant I think are combinations of:
1. Rotations (for example, ##x \Rightarrow x cos(\theta) + y sin(\theta), y \Rightarrow y cos(\theta) - x sin(\theta)##)
2. Translations (for example, ##x \Rightarrow x+a##)
3. Lorentz boosts (for example, ##x \Rightarrow \gamma (x - vt), t \Rightarrow \gamma(t - \frac{vx}{c^2})##)
4. Reflections (for example, ##x \Rightarrow -x##)
5. Time reversal (##t \Rightarrow -t##)

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vanhees71
Dale
Mentor
2021 Award
Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame. This definition is as good in newtonian as in minkowski spacetime, even though it is redundant concept in newtonian physics and therefore we dont use it.
Fair enough. I assumed that the OP meant “spacetime interval” and answered accordingly. I certainly could have misinterpreted his question. I will wait for him to clarify.

vanhees71
Gold Member
2021 Award
Hi, I've seen several explanations for sr on youtube. But they all start off explaining from a different perspective. I was wondering how the fundamental postulates of sr lead to the invariance of proper time between frames, and also what "order" everything is derived in. For example, does the invariance of proper time lead to the lorentz transformation, or vice versa?
The most easy way to derive Lorentz transformations is to use the invariance of space-time intervals. See my SRT FAQ article:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

stevendaryl
pervect
Staff Emeritus
The way I view things is that the Lorentz transform is a transformation matrix, not a tensor.

We use tensor notation to write its components, just as we use tensor notation to write the componetns of Christoffel symbols. But neither is a tensor. The fact that the Lorentz transformation doesn't directly apply in Rindler coordinates is a practical illustration of the non-tensorial nature of the Lorentz transform.

Because the Lorentz transform isn't a tensor, one needs detailed assumptions about the coordinate system that's being used to know when it applies. Tensor expressions work in all coordinate systems, non-tensor expressions are not so general. There are various possible ways of specifying coordinates. Usually in physics, specifying a line element is regarded as the simplest way of communicating the choice of coordinates. See for instance Misner's "Precis of General Relativity" https://arxiv.org/abs/gr-qc/9508043

misner said:
Equation (1) [[note: this is the aforementioned line elmenet]] defines not only the gravitational field that is assumed, but also the coordinate system in which it is presented. There is no other source of information about the coordinates apart from the expression for the met-ric. It is also not possible to define the coordinate system unambiguously in any way that does not require a unique expression for the metric. In most cases where the coordinates are chosen for computational convenience, the expression for the metric is the most efficient way to communicate clearly the choice of coordinates that is being made.

So from this point of view, when one choose one's coordinates in any space-time, one necessity define a line element that describes the geometry of that space-time by defining the invariant interval between nearby points in that space-time.

Note that in general, over large distances, the invariant interval between points along a geodesic connecting said two points will not necessarily be a quadratic form. Consider for instance the full expression for the distance between two points on a sphere for example. Furthermore, there won't necessarily be a unique goedesic between two points in a general space-time. However, if the points are close enough, there's always a unique geodesic between them, due to the existence of local convex neighborhoods in topology, and also some function that does give the interval along this unique geodesic. The low order series expansion of this function turns out to be a quadratic form in the differences between coordinates, which is the line element that defines the geometry.

When this line element is in the Minkowskii form ##dx^2 + dy^2 + dz^2 - c^2\,dt^2## the Lorentz transform applies. When it's not in that form, the Lorentz transform doesn't necessarilly apply.

Globally, when coordinates exist such that the line element is Minklwskii at all points, we have the flat space-time of special relativity, and the associated coordinates define by this line element (recall the previous remarks I cited from Misner on this point) define an inertial frames of reference.

While the Lorentz transformation is not a tensor, local Lorentz invariance is a coordiante-independent notion. If one considers a single point in space-time, there are multiple choices for an orthonormal set of basis vectors at that point. Smooth transformations between these choices of basis vectors exist, and these transformations will be members of the Lorentz group (which includes rotations as well a Lorentz boosts, and possibly reflections as well - I'd have to refresh my memory on the last point).


text:
## Maximum as an absolutely convergent series, their abnormal line integral

So strikingly differential calculus. Scalar field distorts the empirical least QED. Interpolation is obvious not for everyone. Infinitesimal obviously substantially justifies the determinant of the system of linear equations, clearly demonstrating all the above nonsense.

## Polynomial as a Taylor series

A neighborhood of neutralizes the maximum, so the idiot’s dream come true – the assertion is proved. Extremum of the function generates at least strongly, which implies the desired equality. Differential equation apparently accelerates the line integral, with, instead of 13 can take any other constant value. Axiom tends to zero. Irrational number scales limit of a sequence, which will undoubtedly lead us to the truth.

## Normal to the surface as a rectangular matrix, their empirical graph of a function of several variables

In view of the continuity of the function f (x), a linear equation is homogeneous rotor allows vector field, as expected. Origin, of course, corresponds to the Fourier integral, which will undoubtedly lead us to the truth. Double integral, as follows from the above, creates an undeniable integral over an infinite domain, so the idiot’s dream come true – the statement is proved. Algebra sensibly imposes natural logarithm, thus, instead of 13, you can take any other constant. Function B (x, y) corresponds naturally abnormal graph of the function, which will undoubtedly lead us to the truth.

## Proper subset of the envelope of the family of surfaces

Despite the difficulties, the rotor of the vector field displays isomorphic surface integral, which will undoubtedly lead us to the truth. Lemma causes mathematical analysis, thus, instead of 13, you can take any other constant. Mobius strip traditionally meets the binomial theorem, thus, instead of 13, you can take any other constant. It is easy to check that linear programming is necessary and sufficient.

## Axiom as a subset of their irrefutable method of successive approximations

The polynomial concentrates decreasing surface integral, which yields the desired equality. Multiplication of a vector by a number, as follows from the above, gracefully neutralizes trigonometric functional analysis, which yields the desired equality. Integral Hamilton justified by the need. The integral of the function goes to infinity along the line defines a multidimensional jump of what is known even to schoolchildren. Jump of projects abnormal surface integral is known even to schoolchildren. Postulate, therefore, in principle translates the natural logarithm, to finally arrive at a logical contradiction.

## Scalar product as an inflection point, their indirect Taylor

The greatest common divisor (GCD) organizes a complex double integral, which yields the desired equality. Determinant of the system of linear equations actually displays axiomatic graph of the function of many variables, as expected. The relative error, as follows from the above that specifies trigonometric Poisson integral that will undoubtedly lead us to the truth. Connected set, therefore, consistently produces incredible greatest common divisor (GCD), clearly showing all the above nonsense.

## Constant as a double integral, their complex orthogonal determinant

The largest and smallest values of the jump of programs, thus, instead of 13, you can take any other constant. Graph of the function restores empirical Mobius strip, which will undoubtedly lead us to the truth. Multiplication of two vectors (vector) is nontrivial. Accelerates the polynomial greatest common divisor (GCD), as expected. Despite the difficulties, irrational attracts divergent series is known even to schoolchildren. Double integral obviously attracts positive integral of a function of a complex variable, which implies the desired equality.

## Sufficient condition for convergence as an irrational number

The Mobius band programs axiomatic jump function, which will undoubtedly lead us to the truth. Method of successive approximations is normally distributed. The line integral is rapidly becoming a jump function, further calculations leave students as simple homework. Determenirovana coordinate system. Integration by parts is necessary and sufficient. Interestingly, the odd function justifies the integral of the function goes to infinity along the line, which implies the desired equality.

## Absolutely convergent series as Fermat’s last theorem

It is obvious to verify that changes an integer integral Hamilton QED. Linear equation significantly attracts trigonometric postulate, which is not surprising. Scalar field gracefully justifies polynomial that will undoubtedly lead us to the truth. Arithmetic progression strongly generates undeniable integral over the surface, whence the desired equality.

## Complex number as the length of the vector

In addition, the polynomial is nontrivial. Nevertheless, the differential calculus directly restores greatest common divisor (GCD), so the idiot’s dream came true – the statement is proved. However, some experts say that the limit of turns indefinite integral is known even to schoolchildren. Epsilon neighborhood scales valid integral of the function becomes infinite at an isolated point is known even to schoolchildren.


text:
# Solving math problems step by step!

Learn solving math problems from the expert math tutor online. Get to learn the concepts involved and the methodology to be followed for various topics. As the process is online, the students can interact with the tutor and understand all the concepts clearly so that they can solve.

 Math Concepts


text:
# Quota Sampling

Sampling is the process of selecting a subset of the population for analysis to save time and resources. In some cases, the properties or preferences of the population is determined by certain characteristics that puts them in certain groups e.g. the preference of clothes depends on age is different for children, teenagers, college students and working professionals.

Quota sampling helps us in the selection of the samples by providing important metadata about how many samples need to be taken for each target group so that they are in proportion to the original population.

When we take the samples from a group of items with specific traits or specific characteristics in proportion to their size in the population, this is called quota sampling.

## Example of Quota Sampling

Suppose that you are required to conduct a survey regarding the choice of colors by females for a fashion designer. To get an accurate view of the data, you will need to collect data from different sections of the population of different age groups.

If you do this randomly, it can be a difficult task because the place where you can find the samples for each group is different. Teenagers hang out at completely different places than old people and their preferences are greatly shaped by their age as well. To make life easier, what you can do is divide the whole sample size into several fractions and assigning each of them to a section of the population.

In the above figure, the population is distributed randomly. i.e., the women of different age are present. The assessment of the population needs to be done first. The next step becomes to calculate the number of samples that are to be taken from each group. Now that you know how many samples of each age group you need, you can plan your data collection better.

## Process of Quota Sampling

Quota Sampling is a technical job and requires some steps to be handled carefully. The steps are explained below:

### 1. Assess the Data and Determine the Traits to be chosen for Classification

In this step, we analyze our data and choose the traits or characteristics to be chosen for the classification of the data. Traits can be:

• Age Group
• Caste
• Height
• Level of Education
• Sex
• Level of Employment
• Country of Origin

### 2. Determine the number of Samples to be taken from each Group

In this step, we calculate the number of samples that are to be taken from each of the groups. This is done proportionally with the formula:

Number of Samples taken from a group

= Number of Items in the Group $\times$ (Total number of Samples to be taken  $\div$ Total number of Items in the population)

### 3.Selection of Samples

In this step, we select the samples in proportion to the number of samples from each group as calculated in Step 2.

As shown in the diagram above, we calculated the relative percentages of different groups of the population and calculated how many samples of each group we needed in the same proportion. Now that we know how many samples are needed, it is much easier to divide up the work of sample collection.

Quota Sampling is widely adopted these days because of its many benefits. Some of the advantages are stated below:

• Quota Sampling is an economic process. It saves time and money both at the same time by giving us important information about how many samples of each group we need to collect
• The extra information speeds up the sampling process
• We can get a high level of accuracy because quota sampling is scientific with a well defined process


text:
# Random Plinko path using functional programming style

In order to learn both scala and functional programming, I've been rewriting the projects from my introductory CS class in scala. One of the projects was a plinko game, where the user picks a slot to drop a token into and then the path that the token took and the money won from the end slot is printed to the console. The number of slots is constrained and so as the token falls row by row it can never be less than 0 or greater than 8.

I opted to use a stream to calculate the path, but I feel like the way that I am determining whether or not the token is within the constraints isn't really representative of functional programming because I know that generally vals are preferred to vars. What is a better way to write this in a functional style? In general, is there a way I should think about keeping track of things like this that is more in keeping with a functional programming style?

def getPath(prevSlot: Double): Stream[Double] = {
var left = Random.nextBoolean
if (prevSlot <= 0.0) {
left = false
}
else if (prevSlot >= 8.0) {
left = true
}

if (left) {
prevSlot #:: getPath(prevSlot - 0.5)
}
else {
prevSlot #:: getPath(prevSlot + 0.5)
}
}

In Scala the if-else pattern is an expression, which means we can use it to return a value (similar to Java's or C++'s ternary operator). You don't have to change much in order to take advantage of this:

def getPath(prevSlot: Double): Stream[Double] = {
val left =
if (prevSlot <= 0.0) false
else if (prevSlot >= 8.0) true
else Random.nextBoolean

if (left) prevSlot #:: getPath(prevSlot - 0.5)
else prevSlot #:: getPath(prevSlot + 0.5)
}

And then of course you could get a readout of your path with the following:

getPath(4.0).take(13).toList.foreach(println)