OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Geiger's Method

Geiger's method is an iterative procedure using Gauss-Newton optimization to determine the location of an earthquake, or seismic event. Originally his method was developed to obtain the origin time and Epicentre but it is easily extended to include the Focal Depth for Hypocentre determination.
Given a set of $M$ arrival times $t_{i}$ find the origin time $t_{0}$ and the hypocentre in cartesian coordinatios $(x_{0},y_{0},z_{0})$ which minimize
the objective function
   $F(\mathbf {X} )=\sum _{i=1}^{M}r_{i}^{2}.$
Here, $r_{i}$ is the difference between observed and calculated arrival times
   $r_{i}=t_{i}-t_{0}-T_{i},$
and the unknown parameter vector is
   $\mathbf {X} =(t_{0},x_{0},y_{0},z_{0})^{\mathrm {T} }$
In matrix form (1) becomes
   $F(\mathbf {X} )=\mathbf {r} ^{\mathrm {T} }\mathbf {r}$
The Gauss--Newton procedure requires an initial guess of the sought parameters, denoted here as
   $\mathbf {X} ^{*}=(t_{0}^{*},x_{0}^{*},y_{0}^{*},z_{0}^{*})^{\mathbf {T} },$
which are then used to calculate the adjustment vector
   $\delta \mathbf {X} =(\delta t_{0},\delta x_{0},\delta y_{0},\delta z_{0})^{\mathrm {T} }$
in
   $(1)\mathbf {A} ^{\mathrm {T} }\mathbf {A} \delta \mathbf {X} =-\mathbf {A} ^{\mathrm {T} }\mathbf {r} .$
The Jacobian matrix $\mathbf {A}$ is defined as
   $\mathbf {A} =\left({\begin{matrix}\partial r_{1}/\partial t_{0}&\partial r_{1}/\partial x_{0}&\partial r_{1}/\partial y_{0}&\partial r_{1}/\partial z_{0}\\\partial r_{2}/\partial t_{0}&\partial r_{2}/\partial x_{0}&\partial r_{2}/\partial y_{0}&\partial r_{2}/\partial z_{0}\\\vdots &\vdots &\vdots &\vdots \\\partial r_{M}/\partial t_{0}&\partial r_{M}/\partial x_{0}&\partial r_{M}/\partial y_{0}&\partial r_{M}/\partial z_{0}\\\end{matrix}}\right).$
The partial derivatives are evaluated at the initial guess, or trial vector, $\mathbf {X} ^{*}$. Equation (45) can be rewritten as
   $(2)\mathbf {G} \delta \mathbf {X} =\mathbf {g} .$
Using (46) and an initial guess $\mathbf {X} ^{*}$ an adjustment vector can be calculated. The initial guess can then be updated $\mathbf {X} ^{*}+\delta \mathbf {X}$ and used as the inital guess in the next run of the algorithm. In this manner the sought parameters $\mathbf {X}$ can be determined to some tolerance.
# PlanetPhysics/Gauss's Law
### Introduction

Gauss's law, one of Maxwell's equations, gives the relation between the electric or gravitational flux flowing out a closed surface and, respectively, the Electric Charge or mass enclosed in the surface. It is applicable whenever the inverse-square law holds, the most prominent examples being electrostatics and Newtonian gravitation.
If the system in question lacks symmetry, then Gauss's law is inapplicable, and integration using Coulomb's law is necessary.

### Definition (Integral form)

In its integral form, Gauss's law is \begin{displaymath} \Phi = \oint_S \vec{E} \cdot \,\vec{dA} = \frac{1}{\epsilon_0}\int_V \,dV = \frac{q_{enc}}{\epsilon_0} \end{displaymath} where $\Phi$ is electric flux, $S$ is some closed surface with outward normals, ${\vec {E}}$ is the Electric Field, ${\vec {dA}}$ is a differential area element, $\epsilon _{0}$ is the permittivity of free space, $q_{enc}$ is the charge enclosed by $S$, and $V$ is the volume enclosed by $S$.

### Definition (Differential form)

In its differential form, Gauss's law is \begin{displaymath} \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \end{displaymath} where $\nabla$ is the divergence operator, and $\rho$ is the charge density.

### Gauss's Law with Electric Displacement

When dielectrics or other polarizable media enter the system, we must modify Gauss's law accordingly. However, we rescind the mathematical perfection of the above formulation of Gauss's law in favor of a more accurate approximation of the real world.
Polarizable media can contain two types of charge - free and bound. Free charge can move around, while bound charge results from the induced dipoles within the dielectric. Replacing the electric field with the electric displacement field, and the charge density with, specifically, the free charge density, we have a new form of Gauss's Law: \begin{displaymath} \nabla \cdot \vec{D} = \rho_{\mathrm{free}} \end{displaymath}
# PlanetPhysics/Gelfand Tornheim Theorem

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}.}\, Any normed field is isomorphic either to the field $\mathbb {R}$ of real numbers or to the field $\mathbb {C}$ of complex numbers.\\
The normed field means here a field $K$ having a subfield $R$ isomorphic to $\mathbb {R}$ and satisfying the following: \, There is a mapping $\|\cdot \|$ from $K$ to the set of non-negative reals such that
* $\|a\|=0$\, if and only if\, $a=0$,
* $\|ab\|\leqq \|a\|\cdot \|b\|$,
* $\|a+b\|\leqq \|a\|+\|b\|$,
* $\|ab\|=|a|\cdot \|b\|$\, when\, $a\in R$\, and\, $b\in K$.
Using the Gelfand--Tornheim theorem, it can be shown that the only fields with archimedean valuation are isomorphic to subfields of $\mathbb {C}$ and that the valuation is the usual absolute value (the complex modulus) or some positive power of the absolute value.
# PlanetPhysics/GRE Physics Study Guide

This article is aimed at helping students prepare for the GRE Phyics subject test. It will contain helpful hints and tips along with a jumping off spot for exploring the subjcects that will be on the test. please contribute your own ideas with the forum or through corrections/addendum.
Since 1997, the percentage of allocated test questions does not appear to have changed and is given in the below table.
| Subject | Percentage |
| Classical mechanics | 20 |
| Electromagnetism | 18 |
| Optics and wave phenomena | 9 |
| Thermodynamics and statistical mechanics | 10 |
| quantum mechanics | 12 |
| Atomic Physics | 10 |
| special relativity | 6 |
| Laboratory Methods | 6 |
| Spceialized Topics | 9 |

### Classical Mechanics

- Newton's laws of motion \\ - friction \\ - collisions \\ - Oscillations \\ - Uniform Circular motion \\
examples:
- example constant acceleration and speed of sound \\ - 1D example of the relation between force and potential energy

### Electromagnetism

- Circuits \\ - Induction \\ - Electric Field \\ - Electric Potential \\ -

### Thermodynamics and Statistical Mechanics

- Specific heat \\ - conduction \\ - Heat and work \\ - Mean Free Path \\

### Atomic Physics

- The Bohr Model \\ - atomic spectra \\ - Zeeman effect \\
# PlanetPhysics/Galilean System of Co Ordinates
### The Galilean System of Co-ordinates

From Relativity: The Special and General Theory by Albert Einstein As is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the law of inertia, can be stated thus: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line. This law not only says something about the motion of the bodies, but it also indicates the reference-bodies or systems of coordinates, permissible in mechanics, which can be used in mechanical description. The visible fixed stars are bodies for which the law of inertia certainly holds to a high degree of approximation. Now if we use a system of co-ordinates which is rigidly attached to the earth, then, relative to this system, every fixed star describes a circle of immense radius in the course of an astronomical day, a result which is opposed to the statement of the law of inertia. So that if we adhere to this law we must refer these motions only to systems of coordinates relative to which the fixed stars do not move in a circle. A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a ``Galileian system of co-ordinates." The laws of the mechanics of Galflei-Newton can be regarded as valid only for a Galileian system of co-ordinates.
# KVM/qemu-img/qemu-img info

qemu-img info IMAGE_NAME
Notes:
* Encrypted information will be available only if the image is encrypted (The use of encryption in qcow and qcow2 images is considered to be flawed by modern cryptography standards)
* Snapshot list: A list of all internal snapshots
# K^n/Linear form/Example

A linear form on $K^{n}$ is of the form
    $K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}x_{i},$
for a tuple $\left(a_{1},\,\ldots ,\,a_{n}\right)$. The projections
    $p_{j}\colon K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto x_{j},$
are the easiest linear forms.
The zero mapping to $K$ is also a linear form, called the zero form.
# KVM/qemu
## Packages

* qemu-kvm-rhev or qemu-kvm
* qemu-kvm-common
* qemu-img-rhev (RHEL) or qemu-utils (Ubuntu). Commands included: qemu-img, qemu-io and qemu-nbd
* qemu-guest-agent, required for virsh domfsfreeze
* qemu-system-x86, binary included: qemu-system-x86_64

## Configuration

* /etc/libvirt/qemu/
* /etc/libvirt/qemu.conf
* /var/run/libvirt/qemu

## Logs

/var/log/libvirt/qemu/
    qemu-system-x86_64: failed to initialize KVM: Cannot allocate memory

## Basic Commands

* qemu-img info -U /srv/libvirt/YOUR_IMAGE_NAME.qcow2
* qemu-img snapshot
# KVM/virt-sysprep

virt-sysprep command included in libguestfs-tools package reset, unconfigure or customize a virtual machine so clones can be made.
WARNING: Using virt-sysprep on live virtual machines, or concurrently with other disk editing tools, can be dangerous, potentially causing disk corruption. Virtual Machine must be shut down.

## Command examples

* List operations:
          virt-sysprep --list-operations
* Performs a read-only "dry run" sysprep operation:
          virt-sysprep --dry-run TEST_VM_TO_RESET
* To reset everything:
          virt-sysprep -d TEST_VM_TO_RESET
* To Change name and force to keep user account MY_USERNAME_TO_KEEP
          virt-sysprep -d testvm --hostname testvm --enable user-account --keep-user-accounts MY_USERNAME_TO_KEEP
# Polynomial ring/Ordered field/Ordered quotient field/Not archimedean/Exercise

Let $K$ be an ordered field, let $K[X]$ be the polynomial ring over $K$ and set
    $Q=K(X)\,,$
the field of rational functions over $K$. Show, using exercise, that $Q$ can be made into an ordered field, which is not an archimedean ordered field.
# Polynomial ring/Field/Zeroes/Introduction/Section

A zero of a polynomial $P$ is an element $a\in K$ such that $P(a)=0$. A polynomial does not necessarily have zeroes, and this depends also on the base field. The polynomial $X^{2}+1$ has no real zero, but it has the complex zeroes ${\mathrm {i} }$ and $-{\mathrm {i} }$. As an element in $\mathbb {R} [X]$, the polynomial $X^{2}+1$ can not be written as a product of simpler polynomials. However, in $\mathbb {C} [X]$, it has the factor decomposition
    $X^{2}+1=(X-{\mathrm {i} })(X+{\mathrm {i} })\,.$

## Remark

Let $K$ be a field, let $K[X]$ be the polynomial ring over $K$ and $a\in K$. Then the evaluation mapping
    $K[X]\longrightarrow K,P\longmapsto P(a),$
is $K$-linear. Moreover, we have
    $(PQ)(a)=P(a)Q(a)\,,$
see exercise.

## Lemma

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let $P\in K[X]$ be a polynomial and $a\in K$. Then $a$ is a zero
$P$
$P$
$X-a$

### Proof

If $P$ is a multiple of $X-a$, then we can write
    $P=(X-a)Q\,$
with another polynomial $Q$. Inserting $a$ yields
    $P(a)=(a-a)Q(a)=0\,.$
In general, there exists, due to fact, a representation
    $P=(X-a)Q+R\,,$
where either $R=0$ or the degree of $R$ is $0$, so in both cases $R$ is a constant. Inserting $a$ yields
    $P(a)=R\,.$
So if $P(a)=0$ holds, then the remainder must be $R=0$, and this means $P=(X-a)Q$.
$\Box$

## Corollary

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let $P\in K[X]$ be a polynomial ($\neq 0$) of degree
$d$
$P$
$d$

### Proof

We prove the statement by induction over $d$. For $d=0,1$ the statement holds. So suppose that $d\geq 2$ and that the statement is already proven for smaller degrees. Let $a$ be a zero of $P$ (if $P$ does not have a zero at all, we are done anyway). Hence, $P=Q(X-a)$ by fact and the degree of $Q$ is $d-1$, so we can apply to $Q$ the induction hypothesis. The polynomial $Q$ has at most $d-1$ zeroes. For $b\in K$ we have $P(b)=Q(b)(b-a)$. This can be zero, due to fact, only if one factor is $0$, so the zeroes of $P$ are $a$ or a zero of $Q$. Hence, there are at most $d$ zeroes of $P$.
$\Box$
# Polynomial ring/K/Division/Definition

Let $K$ be a field. We say that a polynomial $T\in K[X]$ divides a polynomial $P\in K[X]$, if there exists a polynomial $Q\in K[X]$ such that
    $P=TQ\,.$
# Polynomial ring/Ordered field/Positivity/Exercise

Let $K$ be an ordered field and let $R=K[X]$ be the
polynomial ring over $K$. Let
    $P={\left\{F\in K[X]\mid {\text{The leading coefficient of }}F{\text{ is positive}}\right\}}\,.$
Show that $P$ fulfils the following three properties.
* Either $F\in P$ or $-F\in P$ or $F=0$.
* If $F,G\in P$, then also $F+G\in P$.
* If $F,G\in P$, then also $F\cdot G\in P$.
# Polynomial ring/Quotient field/Field/Exercise

Let $K[X]$ be the polynomial ring over a field $K$. Show that the set
    ${\left\{{\frac {P}{Q}}\mid P,Q\in K[X],\,Q\neq 0\right\}}$
with a suitable addition and multiplication is a field, where two fractions ${\frac {P}{Q}}$ and ${\frac {P'}{Q'}}$ are considered to be equal if $PQ'=P'Q$.
# Polynomial ring/Field/Zeroes/Number/Fact/Proof

We prove the statement by induction over $d$. For $d=0,1$ the statement holds. So suppose that $d\geq 2$ and that the statement is already proven for smaller degrees. Let $a$ be a zero of $P$ (if $P$ does not have a zero at all, we are done anyway). Hence, $P=Q(X-a)$ by fact and the degree of $Q$ is $d-1$, so we can apply to $Q$ the induction hypothesis. The polynomial $Q$ has at most $d-1$ zeroes. For $b\in K$ we have $P(b)=Q(b)(b-a)$. This can be zero, due to fact, only if one factor is $0$, so the zeroes of $P$ are $a$ or a zero of $Q$. Hence, there are at most $d$ zeroes of $P$.
# Polynomial ring/Field/Zero/Linear factor/Fact

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let $P\in K[X]$ be a polynomial and $a\in K$.
$a$
zero
$P$
$P$
$X-a$
# Polynomial ring/Field/Zero/Linear factor/Fact/Proof

If $P$ is a multiple of $X-a$, then we can write
    $P=(X-a)Q\,$
with another polynomial $Q$. Inserting $a$ yields
    $P(a)=(a-a)Q(a)=0\,.$
In general, there exists, due to fact, a representation
    $P=(X-a)Q+R\,,$
where either $R=0$ or the degree of $R$ is $0$, so in both cases $R$ is a constant. Inserting $a$ yields
    $P(a)=R\,.$
So if $P(a)=0$ holds, then the remainder must be $R=0$, and this means $P=(X-a)Q$.
# Polynomial ring/Field/Zeroes/Number/Fact

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let $P\in K[X]$ be a polynomial ($\neq 0$) of degree $d$.
$P$
$d$
# Polynomial ring/Formal partial derivatives/Commute/Exercise

Prove for the polynomial ring $K[X_{1},\ldots ,X_{n}]$ over an arbitrary field $K$ that the formal partial derivatives commute.
# PlanetPhysics/Category of Hilbert Spaces

The category ${\mathcal {H}}ilb_{f}$ of finite-dimensional Hilbert spaces is defined as the category whose objects are all finite-dimensional Hilbert spaces ${\mathcal {H}}_{f}$, and whose morphisms are linear maps between ${\mathcal {H}}_{f}$ spaces. The isomorphisms in ${\mathcal {H}}ilb_{f}$ are all isometric isomorphisms.
Furthermore, one also has the following, general definition for any Hilbert space.
The category Failed to parse (syntax error): {\displaystyle \mathcal{Hilb} of Hilbert spaces} is defined as the category whose objects are all Hilbert spaces ${\mathcal {H}}$, and whose morphisms are linear maps between ${\mathcal {H}}$ spaces. The isomorphisms in ${\mathcal {H}}ilb$ are all isometric isomorphisms.
The category of ${\mathcal {H}}ilb$ Hilbert spaces has direct sums and is a Cartesian category.
# PlanetPhysics/Category of Molecular Sets 2
### Molecular Sets and Representations of Chemical Reactions

The uni-molecular chemical reaction is represented by the natural transformations $\eta :h^{A}\longrightarrow h^{B},$ as specified by the following commutative diagram:
   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}} }
with the states of the molecular sets $A_{u}=a_{1},\ldots ,a_{n}$ and $B_{u}=b_{1},\ldots b_{n}$ being represented by certain endomorphisms in $Hom(A,A)$ and $Hom(B,B)$, respectively. In general, molecular sets $M_{S}$ are defined as finite sets whose elements are `molecules' defined in terms of their molecular observables that are specified below. molecular class variables, or $m.c.v$'s are defined as families of molecular sets $[M_{S}]_{i\in I}$, with $I$ being an indexing set, or class, defining the range of molecular variation of the $m.c.v$ ; most applications require that $I$ is a proper, finite set, (i.e., without any sub-classes). A morphism $M_{t}:M_{S}\to M_{S}$ of molecular sets, with $t\in T$ being real time values, is defined as a time-dependent mapping or function $M_{S}(t)$ also called a molecular transformation, $M_{t}$.
An $m.c.v.$ observable of $B$, characterizing the products of chemical type "B" of a chemical reaction is defined as a morphism:
$\gamma :Hom(B,B)\longrightarrow \Re ,$ where $\Re$ is the set or field of real numbers. This mcv-observable is subject to the following commutativity conditions:
   Failed to parse (unknown function "\def"): {\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},} }
with $c:A_{u}^{*}\longrightarrow B_{u}^{*}$, and $A_{u}^{*}$, $B_{u}^{*}$ being, respectively, specially prepared fields of states of the molecular sets $A_{u}$, and $B_{u}$ within a measurement uncertainty range, $\Delta$, which is determined by Heisenberg's uncertainty relation, or the commutator of the observable operators involved, such as $[A^{*},B^{*}]$, associated with the observable $A$ of molecular set $A_{u}$, and respectively, with the obssevable $B$ of molecular set $B_{u}$, in the case of a molecular set $A_{u}$ interacting with molecular set $B_{u}$.
With these concepts and preliminary data one can now define the category of molecular sets and their transformations as follows.

### Category of molecular sets and their transformations

The category of molecular sets is defined as the category $C_{M}$ whose objects are molecular sets $M_{S}$ and whose morphisms are molecular transformations $M_{t}$.
This is a mathematical representation of chemical reaction systems in terms of molecular sets that vary with time (or $msv$'s), and their transformations as a result of diffusion, collisions, and chemical reactions.
# PlanetPhysics/Category of Molecular Sets 3
### Molecular sets and representations of chemical reactions

A uni-molecular chemical reaction is represented by the natural transformations $\eta :h^{A}\longrightarrow h^{B}$, through the following commutative diagram:
   Failed to parse (unknown function "\def"): {\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}} }
with the states of molecular sets $A_{u}=a_{1},\ldots ,a_{n}$ and $B_{u}=b_{1},\ldots b_{n}$ being represented by certain endomorphisms in $Hom(A,A)$ and $Hom(B,B)$, respectively. In general, molecular sets $M_{S}$ are defined as finite sets whose elements are molecules defined in terms of their molecular observables that are specified next. One need to define first the concept of a molecular class variable. A molecular class variable , or $m.c.v$ is defined as a family of molecular sets $[M_{S}]_{i\in I},$ with $I$ being an indexing set, or class, defining the molecular range of variation of the $m.c.v$. Most applications in Physics, Chemistry or Biochemistry require that $I$ is a finite set, (that is, without any sub-classes). A homomorphism of molecular sets $M_{t}:M_{S}\to M_{S}$, with $t\in T$ being real time values, is defined as a time-dependent mapping or function $M_{S}(t)$ also called a $M_{t}$ molecular transformation .
An $m.c.v.$ observable of $B$, characterizing the products of chemical type "B" of a chemical reaction is defined as a morphism:
$\gamma :Hom(B,B)\longrightarrow \Re ,$ where $\Re$ is the set or field of real numbers. This mcv-observable is subject to the following commutativity conditions:
   Failed to parse (unknown function "\def"): {\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},} }
with $c:A_{u}^{*}\longrightarrow B_{u}^{*}$, and $A_{u}^{*}$, $B_{u}^{*}$ being, respectively, specially prepared fields of states of the molecular sets $A_{u}$, and $B_{u}$ within a measurement uncertainty range, $\Delta$, which is determined by Heisenberg's uncertainty relation, or the commutator of the observable operators involved, such as $[A^{*},B^{*}]$, associated with the observable $A$ of molecular set $A_{u}$, and respectively, with the obssevable $B$ of molecular set $B_{u}$, in the case of a molecular set $A_{u}$ interacting with molecular set $B_{u}$.
With these concepts and preliminary data one can now define the category of molecular sets and their transformations as follows.

### Category of molecular sets and their transformations

The category of molecular sets is defined as the category $C_{M}$ whose objects are molecular sets $M_{S}$ and whose morphisms are molecular transformations $M_{t}$.
This is a mathematical representation of chemical reaction systems in terms of molecular sets that vary with time (or $msv$'s), and their transformations as a result of diffusion, collisions, and chemical reactions.