OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Magnons
## Magnons and magnon dispersion

Magnons are defined as collective excitations of a magnetic lattice that possesses long range magnetic order; more specifically, a single magnon excitation corresponds to the change by one unit of the magnetic moment of the lattice or system. This non-local nature of magnons is the cause of the experimentally observed dispersive behavior, that is a non-linear or non-constant relation between the transferred momentum and the transferred energy of the magnetic lattice upon its excitation. Typical values for the peak(s) of the transferred energy are on the order of 0.5 eV to 2 eV.
Two- and mutiple- magnon dispersion phenomena have been reported, and were recently employed to explain the nonlinear dispersion behavior of both crystalline and non-crystalline systems with long range ordering compared with the atomic scale.

### Experimental observation of the non-linear magnon dispersion

Experimentally, magnon dispersions have been detected by resonant microwave absorption in external magnetic fields (that is, by Spin-Wave resonance excitation (SWR) and ferromagnetic resonance (FMR)) for ferromagnetic metallic glasses at ambient temperatures. Several neutron inelastic, as well as Cu $K_{\alpha }$ edge resonant inelastic X-ray scattering (RIXS), spectra were also reported for crystaline materials such as the undoped antiferromagnetic cuprates below 20 K.

### Applications

Such measurements and corresponding theories are of significant interest for an improved understanding of high temperature superconductivity; upon doping ( for example with Ytrium or Lanthanum, and Barium) the long-range ordering in a antiferromagnetic lattice-- that was present in certain undoped copper oxide insulators-- becomes frustrated, thus leading to short range antiferromagnetic fluctuations, symmetry breaking and high temperature superconductivity.
# PlanetPhysics/Lorentz Force Law

For given electric and magnetic fields, the force on a particle with charge $q$ is determined by multiplying the charge by the Electric Field plus the cross product of the particle's velocity with the magnetic field. This relationship is known as the Lorentz force law
$\mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$
# PlanetPhysics/M Theory
## M-theory

In mathematical physics, M-theory is a recent extension of five superstring theories, or a relatively recent theory of everything that includes the time evolution through the Big-Bang event that created our Universe. It replaces the older string theory in 10 dimensions with a P-brane theory in 11 dimensions, one time dimension and 10 spatial dimensions. The Big-Bang is hypothesized to occur in M-theory as a result of the collision of two parallel universe P-branes, each with ten spatial dimensions and one temporal dimension. Heterogeneities in the new Universe formed after the collision would be explained by wave-like fluctuations of the two colliding branes. Moreover, a unification of five previous 10D string theories would occur in this M-theory, or Master theory. Its equations may be then also unified with previous supergravity theory in 11 dimensions.
It has been said that: "The final mathematical formulation of M-theory will have to make contact with the theory of vector bundles, K-theory and noncommutative geometry."
# PlanetPhysics/Magnetic Susceptibility

In Electromagnetism, the volume magnetic susceptibility , represented by the symbol $\chi _{v}$ is defined by the following equation
${\vec {M}}=\chi _{v}{\vec {H}},$ where in SI units ${\vec {M}}$ is the magnetization of the material (defined as the magnetic dipole moment per unit volume, measured in amperes per meter), and H is the strength of the magnetic field ${\vec {H}}$, also measured in amperes per meter.
On the other hand, the magnetic induction ${\vec {B}}$ is related to ${\vec {H}}$ by the equation
${\vec {B}}\ =\ \mu _{0}({\vec {H}}+{\vec {M}})\ =\ \mu _{0}(1+\chi _{v}){\vec {H}}\ =\ \mu {\vec {H}},$ where $\mu _{0}$ is the magnetic constant, and $\ (1+\chi _{v})$ is the relative permeability of the material.
Note that the magnetic susceptibility $\chi _{v}$ and the magnetic permeability $\mu$ of a material are related as follows:
$\mu =\mu _{0}(1+\chi _{v})\,.$
There are two other measures of susceptibility, the mass magnetic susceptibility , $\chi _{g}$ or $\chi _{m}$, and the molar magnetic susceptibility , $\chi _{mol}:$
$\chi _{=mass=}=\chi _{v}/\rho ,$ $\chi _{mol}\,=\,M\chi _{m}=M\chi _{v}/\rho ,$
where $\rho$ is the density and M is the molar mass.

#### Susceptibility Sign convention

If $\chi$ is positive, then $(1+\chi _{v})>1$ (or, in cgs units, $(1+4\pi \chi _{v})>1)$ and the material can be paramagnetic, ferromagnetic, ferrimagnetic, or anti-ferromagnetic; then, the magnetic field inside the material is strengthened by the presence of the material, that is, the magnetization value is greater than the external H-value.
On the other hand there are certain materials--called diamagnetic -- for which $\chi$ negative, and thus 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 (1+Ï‡v) < 1} (in SI units).

### Magnetic Susceptibility Tensor, χ

The magnetic susceptibility of most crystals (that are anisotropic) cannot be represented only by a scalar, but it is instead representable by a tensor $\chi$ . Then, the crystal magnetization ${\vec {M}}$ is dependent upon the orientation of the sample and can have non-zero values along directions other than that of the applied magnetic field ${\vec {H}}$. Note that even non-crystalline materials may have a residual anisotropy, and thus require a similar treatment.
In all such magnetically anisotropic materials, the volume magnetic susceptibility tensor is then defined as follows:
$M_{i}=\chi _{ij}H_{j},$
where $i$ and $j$ refer to the directions (such as, for example, x, y, z in Cartesian coordinates) of, respectively, the applied magnetic field and the magnetization of the material. This rank 2 tensor (of dimension (3,3)) relates the component of the magnetization in the $i$-th direction, $M_{i}$ to the component $H_{j}$ of the external magnetic field applied along the $j$-th direction.
# PlanetPhysics/Locally Compact Hausdorff Spaces
## Locally compact Hausdorff spaces
### Definition

A locally compact Hausdorff space $H_{LC}$ is a locally compact topological space $(X_{LC},\tau )$ with $\tau$ being a Hausdorff topology, that is, if given any distinct points $x,y\in X_{LC}$, there exist disjoint sets $U,V\in \tau$ such that, $U\cap V=\emptyset$ (that is, open sets), and with $x$ and $y$ satisfying the conditions that $x\in U$ and $y\in V$.

### Remark

An important, related concept to the locally compact Hausdorff space is that of a locally compact (topological) groupoid, which is a major concept for realizing extended quantum symmetries in terms of quantum groupoid representations in: Quantum Algebraic Topology (QAT), topological QFT (TQFT), algebraic QFT (AQFT), axiomatic QFT, QCG, and quantum gravity (QG). This has also prompted the relatively recent development of the concepts of homotopy 2-groupoid and homotopy double groupoid of a Hausdorff space. It would be interesting to have also axiomatic definitions of these two important algebraic topology concepts that are consistent with the T2 axiom.
# PlanetPhysics/Loop Example of Biot Savart Law

Here we will examine two examples of the Biot-Savart law, one simple and the other more challenging. To begin we will find the magnetic field at the center of a current carrying loop as shown in figure 1
\begin{figure} \includegraphics[scale=.8]{CurrentLoop.eps} \vspace{10 pt} \caption{Figure 1: Current Loop} \end{figure}
this setup is the same as the quater loop example of Biot-Savart law where we have
$d{\mathbf {l} }\times {\hat {\mathbf {r} }}=dl\,sin(\pi /2)$ $dl=rd\theta$
giving us a similar integral, except from 0 to 2 $\pi$ and the lack of a minus sign since the current is going in the opposite direction so the magnetic field will be out of the web browser
${\mathbf {B} }={\hat {\mathbf {k} }}{\frac {\mu _{0}I}{4\pi r}}\int _{0}^{2\pi }\,d\theta$
taking the integral gives the magnetic field at the center of the loop
${\mathbf {B} }={\frac {\mu _{0}I{\hat {\mathbf {k} }}}{2r}}$
The second more challenging example is the magnetic field at a point z above the loop as shown in figure 2
\begin{figure} \includegraphics[scale=.8]{AxisCurrentLoop.eps} \vspace{10 pt} \caption{Figure 2: Current Loop} \end{figure}
The not so obvious hint is the direction of $d{\mathbf {B} }$. The cross product of ${\mathbf {\hat {r}} }$ with $d{\mathbf {l} }$ leads to a vector perpendicular to both of them and as you go around the loop, $d{\mathbf {B} }$ will always be off the z axis by an angle $\phi$. This makes all the horizontal components of $d{\mathbf {B} }$ cancel leaving just the vertical so
$d{\mathbf {l} }\times {\mathbf {\hat {r}} }=dl\,cos(\phi ){\hat {\mathbf {k} }}$
once again the differential is given as $dl=Rd\theta$, so the integral to get the magnetic field is
${\mathbf {B} }={\hat {\mathbf {k} }}{\frac {\mu _{0}IR}{4\pi r^{2}}}\int _{0}^{2\pi }\,cos(\phi )\,d\theta$
From the geometry of the problem we see that
$r^{2}=R^{2}+z^{2}$ $cos(\phi )={\frac {R}{r}}$
this leads to
$r={\sqrt {R^{2}+z^{2}}}$ $cos(\phi )={\frac {R}{\sqrt {R^{2}+z^{2}}}}$
substituting these relations into the integral
${\mathbf {B} }={\hat {\mathbf {k} }}{\frac {\mu _{0}IR^{2}}{4\pi (R^{2}+z^{2})^{\frac {3}{2}}}}\int _{0}^{2\pi }\,d\theta$
Finally, taking the integral gives us the magnetic field
${\mathbf {B} }={\frac {\mu _{0}IR^{2}\,{\hat {\mathbf {k} }}}{2(R^{2}+z^{2})^{\frac {3}{2}}}}$
# PlanetPhysics/Maple for Planet Physics

\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle [{\it poincare},{\it generate_ic},{\it zoom},{\it hamilton_eqs}\\ \mbox]} } \end{maplelatex}\begin{maplelatex}\begin{Maple Heading 1}{The Toda Hamiltonian}\end{Maple Heading 1} \end{maplelatex} \begin{maplelatex}\begin{Maple Normal}{Reference: A.J. Lichtenberg and M.A. Lieberman, "regular and Stochastic motion", Applied Mathematical Sciences 38 (New York: Springer Verlag, 1994).}\end{Maple Normal} H := 1/2*(p1\symbol{94 2 + p2\symbol{94}2) + 1/24*(exp(2*q2+2*sqrt(3)*q1) + exp(2*q2-2*sqrt(3)*q1) + exp(-4*q2))-1/8;}\end{maplelatex} \mapleresult \begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle H\, := \,1/2\,{{\it p1}}^{2}+1/2\,{{\it p2}}^{2}+1/24\,{e^{2\,{\it q2}+2\,\sqrt {3}{\it q1}}}\\ \mbox+1/24\,{e^{2\,{\it q2}-2\,\sqrt {3}{\it q1}}}+1/24\,{e^{-4\,{\it q2}}}-1/8} } \end{maplelatex}\begin{mapleinput} \mapleinline{active}{1d}{H, t=-150..150, \{[0,.1,1.4,.1,0]\ :}} \end{mapleinput} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}{poincare( } \end{mapleinput}
\mapleresult \begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle {\it _____________________________________________________}} } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle \mbox {{\tt `H = .99005020`}},\,\mbox {{\tt ` Initial conditions:`}},\,t=0,\,{\it p1}= 0.1,\,{\it p2}= 1.4,\,{\it q1}= 0.1,\,{\it q2}=0} } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle \mbox {{\tt `Number of points found crossing the (p1,q1) plane: 127`}}} } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle \mbox {{\tt `Maximum H deviation : .5740000000e-5 \ \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ <math>{\it _____________________________________________________}} } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ Failed to parse (syntax error): {\displaystyle \mbox {{\tt `Time consumed: 12 seconds`}}} } \end{maplelatex}
\medskip \begin{figure} \includegraphics{poly.eps} \end{figure} \medskip
\begin{maplelatex}\begin{Maple Normal}{Figure 1.a. shows a 2-D surface-of-section (2PS) over the q2=0 plane, with 127 intersection points lying on smooth curves. }\end{Maple Normal} \end{maplelatex}\begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}{poincare( } \end{mapleinput}
# Refresher/csharp/Introduction
### Presentation

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* This is a refresher course for those who need to review basic knowledge of C# (csharp).
# Refresher/dotnet/Basic Concepts
### Presentation

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# Refrigerants/R134a

r134a :
| F° | psi | info | F° | psi |
| 16 | 15 | | 100 | 124 |
| 18 | 17 | | 102 | 129 |
| 20 | 18 | | 104 | 133 |
| 22 | 19 | oil separates | 106 | 137 |
| 24 | 21 | | 108 | 142 |
| 26 | 22 | condenser low end | 110 | 147 |
| 28 | 24 | | 112 | 152 |
# Refrigeration and air conditioning

* Please help develop and classify this resource
* Learn how you can develop this resource to teach participants about Refrigeration and air conditioning. Choose a subject, education level, and resource type to classify the resource by. You may learn more about the topic to help you develop and classify the resource from:
Refrigeration and Air Conditioning requires the movement of energy from a lower energy(ie. Lower Temperature) reservoir to a higher energy reservoir. This Process cannot occur without additional energy being supplied to the system. One common system used is the vapour-compression refrigeration cycle.

### Vapour-compression refrigeration cycle

This Cycle consists of an evaporator, compressor, condenser and an expansion valve. The Refrigerant moves in this order through the cycle. In order to absorb energy from the low temperature reservior the Refrigerant must be below the temperature of the low temperature reservior when it enters the evaporator.

### Evaporator

Facilitates the movement of energy from the low temperature reservior to the Refrigerant. This results in a temperature increase of the Refrigerant and a temperature decrease in the room or fridge.

### Compressor

The system is able to work because of the energy input to the compressor (W$_{net,in}$). The Compressor does work on the refrigerant to compress it. As a result of compression the refrigerants pressure and temperature increase (to a value higher than the high temperature reservior).

### Condenser

The condenser facilitates the movement of energy from the refrigerant to the high temperature reservior. This results in the refrigerants temperature decreasing.

### Expansion Valve

The Expansion Valve reduces the pressure and temperature via the effects of throttling.

### Readings
#### Wikipedia

* Air conditioning
* Heat pump (can both heat and cool)
# Refresher/dotnet/Topic List
### Presentation

| Introduction | Topic List | Basic Concepts | This presentation was made possible by Wikiversal/WikiverSuite, which is made possible by your donation. Thank you for your support. | Basic Concepts |
* Basic Concepts
* Basic Classes
* Windows Services
# Refresher/dotnet/Windows Services
### Presentation

| Basic Classes | Windows Services | List of Topics | This presentation was made possible by Wikiversal/WikiverSuite, which is made possible by your donation. Thank you for your support. | List of Topics |
* Windows services: background applications
* Windows service creation: add new windows service project, set properties, write the logic for OnStart and OnStop, override default methods as needed, add installer, build project, install and test or run
# Refresher/csharp/Topic List
### Presentation

| Introduction | Topic List | Basic Concepts | This presentation was made possible by Wikiversal/WikiverSuite, which is made possible by your donation. Thank you for your support. | Basic Concepts |
* Basic Concepts
* Basic Classes
* Basic Limitation
* Basic Operations
* Authentication
* Database
* Accessibility Modifiers
# Refresher/dotnet/Introduction
### Presentation

| Introduction | Topic List | List of Topics | This presentation was made possible by Wikiversal/WikiverSuite, which is made possible by your donation. Thank you for your support. | Topic List |
* This is a refresher course for those who need to review basic knowledge of .NET (dotnet).
# Refresher/csharp/Singleton
### Presentation

| Delegates How and When | Singleton | Topic 10 | List of Topics | This presentation was made possible by Wikiversal/WikiverSuite, which is made possible by your donation. Thank you for your support. | Topic 10 |
* Singleton design pattern: interface allowing a class to enforce single instantiation
* C# singleton example: http://www.dotnetperls.com/singleton
# Refresher/dotnet/Basic Classes
### Presentation

| Basic Concepts | Basic Classes | Windows Services | List of Topics | This presentation was made possible by Wikiversal/WikiverSuite, which is made possible by your donation. Thank you for your support. | Windows Services |
* (coming soon)
# PlanetPhysics/Noncommutative Geometry 2
### Non-commutative Geometry and Non-Abelian Algebraic Topology

Noncommutative `geometry' utilizes non-Abelian (or nonabelian) methods for quantization of spaces through `deformation' to non-commutative 'spaces' (in fact non-commutative algebraic structures, or algebras of functions). therefore, it can be considered a subfield of non-Abelian algebraic topology (NAAT).
An alternative meaning is often given to Noncommutative Geometry (viz . A Connes et al.) : that is, as a non-commutative `geometric' approach-- in the relativistic sense -- to quantum gravity. This approach is therefore relevant to Non-Abelian Quantum Algebraic Topology (NA-QAT, or NAQAT).
A specific example due to A. Connes is the convolution $C^{*}$-algebra of (discrete) groups; other examples are non-commutative $C^{*}$-algebras of operators defined on Hilbert spaces of quantum operators and states. (Please see also the other PM entries on $C^{*}$-algebra and noncommutative topology.)

### Recent Developments in NCG

* The Royal Swedish Academy of Sciences has awarded the 2001 Crafoord Prize in mathematics to Professor Alain Connes of the Institut des Hautes Études Scientifiques (IHES) and the Collége de France, Paris, for his penetrating work on the theory of... (quantum)... operator algebras and for having been a founder of noncommutative geometry . (Crafoord Prize in 2001 in Noncommutative Geometry and Quantum Operator Algebras). Professor Alain Connes is also the 1983 recipient of the fields Medal. The following is a concise quote of his work from the Crafoord Prize announcement in 2001: Noncommutative geometry is a new field of mathematics, and Connes played a decisive role in its creation. His work has also provided powerful new methods for treating renormalization theory and the standard model of quantum and particle physics...(SUSY)... He has demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann Hypothesis.
* The Crafoord Prize prize consisted of a gold medal and US dollars 500,000. The Anna-Greta and Holger Crafoord Foundation was established in 1980 for promoting basic research in mathematics, astronomy, the biosciences (particularly ecology), the geosciences, and polyarthritis (joint rheumatism). Previous ('Nobel style'), Crafoord Laureates in Mathematics were: Vladimir I. Arnold and Louis Nirenberg in 1982, Alexandre Grothendieck (who publicly declined the prize) and Pierre Deligne--who accepted the prize in 1988, and Simon Donaldson and Shing-Tung Yau (1994).
# PlanetPhysics/Nuclear C Algebra

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### Examples of nuclear C*-algebras

* All commutative C*-algebras and all finite-dimensional C*-algebras
* group C*-algebras of amenable groups
* Crossed products of strongly amenable C*-algebras by amenable discrete groups,
* type $1$ C*-algebras.

### Exact C*-algebra

In general terms, a $C^{*}$-algebra is exact if it is isomorphic with a $C^{*}$-subalgebra of some nuclear $C^{*}$-algebra. The precise definition of an exact $C^{*}$-algebra follows.
Let $M_{n}$ be a matrix space, let ${\mathcal {A}}$ be a general operator space, and also let $\mathbb {C}$ be a C*-algebra. A $C^{*}$-algebra $\mathbb {C}$ is exact if it is `finitely representable' in $M_{n}$, that is, if for every finite dimensional subspace $E$ in ${\mathcal {A}}$ and quantity $epsilon>0$, there exists a subspace $F$ of some $M_{n}$, and also a linear isomorphism $T:E\to F$ such that the $cb$-norm $|T|_{cb}|T^{-1}|_{cb}<1+epsilon.$

### Counter-example

The group C*-algebras for the free groups on two or more generators are not nuclear. Furthermore, a $C^{*}$ -subalgebra of a nuclear C*-algebra need not be nuclear.
# PlanetPhysics/Nutest

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non-Abelian algebraic topology and noncommutative geometry{ http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}. The latter two fields have already made an impact on quantum theories that seek a new setting beyond SUSY--the Standard Model of modern physics. Moreover, it is also possible to consider in this topic novel, possible approaches to relativity theories, especially to general relativity on spacetimes that are more general than pseudo- or quasi- Riemannian `spaces'. Furthermore, other theoretical physics developments may expand specific Anabelian Geometry applications to quantum geometry and Quantum Algebraic Topology.

## Anabelian Geometry

The area of mathematics called Anabelian Geometry (AAG) began with Alexander Grothendieck's introduction of the term in his seminal and influential work "Esquisse d'un Programme" $[1]$ produced in 1980. The basic setting of his anabelian geometry is that of the algebraic fundamental group ${\mathcal {G}}$ of an algebraic variety $X$ (which is a basic concept in Algebraic Geometry), and also possibly a more generally defined, but related, geometric object. The algebraic fundamental group , ${\mathcal {G}}$, in this case determines how the algebraic variety $X$ can be mapped into, or linked to, another geometric object $Y$, assuming that ${\mathcal {G}}$ is non-Abelian or noncommutative. This specific approach differs significantly, of course, from that of Noncommutative Geometry introduced by Alain Connes. It also differs from the main-stream nonabelian algebraic topology (NAAT)'s generalized approach to topology in terms of groupoids and fundamental groupoids of a topological space (that generalize the concept of fundamental space), as well as from that of higher dimensional algebra (HDA). Thus, the fundamental anabelian question posed by Grothendieck was, and is: "how much information about the isomorphism class of the variety $X$ is contained in the knowledge of the etale fundamental group?" (on p. 2 in $http://www.math.jussieu.fr/~leila/SchnepsLM.pdf$ ).