OpenLLM-France/wikimedia huggingface dataset:
# Materials Science and Engineering/List of Topics/Thermodynamics/Equations of State

In physics and thermodynamics, an equation of state is a relation between state variables.[1] More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars.

## Overview

The most prominent use of an equation of state is to predict the state of gases and liquids. One of the simplest equations of state for this purpose is the ideal gas law, which is roughly accurate for gases at low pressures and high temperatures. However, this equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid. Therefore, a number of much more accurate equations of state have been developed for gases and liquids. At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions.
In addition to predicting the behavior of gases and liquids, there are also equations of state for predicting the volume of solids, including the transition of solids from one crystalline state to another. There are equations that model the interior of stars, including neutron stars. A related concept is the perfect fluid equation of state used in cosmology.
# Materials Science and Engineering/Magnetic Materials
## Topics

* Giant Magneto-Resistive Tunnelling Effect

## Questions

* Saturation Magnetization Near Curie Temperature
# Materials Science and Engineering/Magnetic Materials/Giant Magneto-Resistive Tunnelling Effect

The development of devices that sense and store information is driven by the demand for more capable computers. The utilization of the phenomenon of giant magneto-resistance (GMR) has led to the creation of more sensitive sensors that read bits coded on magnetized regions of disks. An understanding of GMR has led to a large increase in information storage capacity. The phenomenon of GMR has also been of interest to physicists.
The magneto-resistance effect (MR effect) describes the change in the electrical resistance of a material due to the application of a magnetic field. Research advances related to electron spin, tunneling effects, and production of ultra-thin layers have made possible the design of electrical devices based on quantum mechanical effects of the electron. The giant magneto-resistance effect is dependent on spin properties of electrons. Several other magneto-resistive effects have been discovered, and they are classified under XMR effects or XMR technology.
An example of a structure where the giant magneto-resistive effect is observed consists of a layer of copper, a normal metal, between layers of cobalt, a ferromagnetic material. When the magnetic moments of the ferromagnetic layers are parallel, there is less resistance to the flow of current. In current devices, the direction of magnetization of one layer is typically fixed in one direction, while the other is determined by an external field. The difference of resistance between parallel and anti-parallel configurations is 10-15% in current devices.
The effect of GMR is similar to phenomenon of polarization. In a polarization experiment, light cannot pass through if two polarizers are perpendicular to one another. Similarly, one magnetic layer may allow electrons of one type of spin to pass, and, if the second structure is aligned in the same direction, current can easily pass through. If the second layer is misaligned, neither spin channel can pass through the structure easily, and the electrical resistance is high.
Early studies in which GMR was first observed may have been published in 1987. The patent of the effect is owned by Peter Grunberg, who led a team at the Julich Research Centre and observed the effect in trilayers of Fe/Cr/Fe. The effect was simultaneously and independently observed in multilayers of Fe/Cr.
Three types of GMR are multilayer, granular, and spin-valve GMR. The effect was first observed in multilayer configurations and the effect in granular GMR is not as large as multilayer GMR. Spin-valve GMR is the most useful industrially, and the magnetic recording industry is researching structures that are very sensitive to magnetic fields. IBM and Seagate develop high density disk and tape playback heads. Additional applications include avionic compasses, swipe-card readers, wheel rotation sensors in ABS brakes, and current sensors for use in safety powerbreakers and electricity meters.
# Materials Science and Engineering/List of Topics/Thermodynamics
## Thermodynamics

Zeroth Law of Thermodynamics
Equations of State
Second Law of Thermodynamics
Third Law of Thermodynamics
Chemical Equilibrium
Phase Equilibrium
Electrochemical Equilibrium
Ionic Equilibria
# Materials Science and Engineering/List of Topics/Thermodynamics/Chemical Equilibrium

In a chemical process, chemical equilibrium is the state in which the chemical activities or concentrations of the reactants and products have no net change over time. Usually, this would be the state that results when the forward chemical process proceeds at the same rate as their reverse reaction. The reaction rates of the forward and reverse reactions are generally not zero but, being equal, there are no net changes in any of the reactant or product concentrations. This process is known as dynamic equilibrium.
# Materials Science and Engineering/List of Topics/Thermodynamics/Third Law of Thermodynamics

The third law of thermodynamics is an axiom of nature regarding entropy and the impossibility of reaching absolute zero of temperature. The most common enunciation of third law of thermodynamics is:
It can be concluded as 'If T=0K, then S=0' where T is the temperature of a closed system and S is the entropy of the system.
The essence of the postulate is that the entropy of the given system near absolute zero depends only on the temperature (i.e. tends to a constant independently of the other parameters).

## Overview

In simple terms, the Third Law states that the entropy of a pure substance approaches zero as the absolute temperature approaches zero. This law provides an absolute reference point for the determination of entropy. The entropy determined relative to this point is the absolute entropy.
A special case of this is systems with a unique ground state, such as most crystal lattices. The entropy of a perfect crystal lattice as defined by Nernst's theorem is zero (if its ground state is singular and unique, whereby log(1) = 0). An example of a system which does not have a unique ground state is one containing half-integer spins, for which time-reversal symmetry gives two degenerate ground states. Of course, this entropy is generally considered to be negligible on a macroscopic scale. Additionally, other exotic systems are known that exhibit geometrical frustration, where the structure of the crystal lattice prevents the emergence of a unique ground state.
Real crystals with frozen defects obey this same law, so long as one considers a particular defect configuration to be fixed. The defects would not be present in thermal equilibrium, so if one considers a collection of different possible defects, the collection would have some entropy, but not actually have a temperature. Such considerations become more interesting and problematic in considering various forms of glass, since glasses have large collections of nearly degenerate states, in which they become trapped out of equilibrium.
Another application of the third law is with respect to the magnetic moments of a material. Paramagnetic materials (moments random) will order as T approaches 0 K. They may order in a ferromagnetic sense, with all moments parallel to each other, or they may order in an antiferromagnetic sense, with all moments antiparallel to each other.
Yet another application of the third law is the fact that at 0 K no solid solutions should exist. Phases in equilibrium at 0 K should either be pure elements or atomically ordered phases.
# Materials Science and Engineering/List of Topics/Thermodynamics/Zeroth Law of Thermodynamics

The zeroth law of thermodynamics is a generalized statement about bodies in contact at thermal equilibrium and is the basis for the concept of temperature. The most common enunciation of the zeroth law of thermodynamics is:
"If two thermodynamic systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other."
In other words, the zeroth law says that if considered a mathematical binary relation, thermal equilibrium is transitive.

## Temperature and the zeroth law

It is often claimed, for instance by Max Planck in his influential textbook on thermodynamics, that this law proves that we can define a temperature function, or more informally, that we can 'construct a thermometer'. Whether this is true is a subject in the philosophy of thermal and statistical physics.
In the space of thermodynamic parameters, zones of constant temperature will form a surface, which provides a natural order of nearby surfaces. It is then simple to construct a global temperature function that provides a continuous ordering of states. Note that the dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters (thus, for an ideal gas described with 3 thermodynamic parameter P, V and n, they are 2D surfaces). The temperature so defined may indeed not look like the Celsius temperature scale, but it is a temperature function.
For example, if two systems of ideal gas are in equilibrium, then $P1V1/N1=P2V2/N2$ where $P_{i}$ is the pressure in the ith system, $V_{i}$ is the volume, and $N_{i}$ is the 'amount' (in moles, or simply number of atoms) of gas.
The surface $PV/N={\mbox{const}}$ defines surfaces of equal temperature, and the obvious (but not only) way to label them is to define T so that $PV/N=RT$ where R is some constant. These systems can now be used as a thermometer to calibrate other systems.
# Real sequence/Decreasing/Definition

A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called decreasing, if $x_{n+1}\leq x_{n}$ holds for all $n\in \mathbb {N}$.
# Real sequence/Positive/Divergent to infinity and inverse null sequence/Exercise

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence with $x_{n}>0$ for all $n\in \mathbb {N}$. Prove that the sequence diverges to $+\infty$ if and only if the sequence ${\left({\frac {1}{x_{n}}}\right)}_{n\in \mathbb {N} }$ converges to $0$.
# Real sequence/Difference of square roots/Exercise

Decide whether the sequence
    $a_{n}={\sqrt {n+1}}-{\sqrt {n}}\,$
converges and in case determine the limit.
# Real sequence/Infinity divergent subsequence/Exercise

Give an example of a real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$, such that it contains a subsequence that diverges to $+\infty$ and also a subsequence that diverges to $-\infty$.
# Real sequence/Not convergent/Convergent subsequence/Exercise

Give an example of a real sequence, that does not converge, but it contains a convergent subsequence.
# Real sequence/Tending to infinity/+/Definition

$+\infty$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is said to tend to $+\infty$, if for every $s\in \mathbb {R}$, there exists some $N\in \mathbb {N}$, such that
    $x_{n}\geq s{\text{ holds for all }}n\geq N.$
# Real sequence/Increasing/Definition

A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called increasing, if $x_{n+1}\geq x_{n}$ holds for all $n\in \mathbb {N}$.
# Real sequence/Tending to infinity/-/Definition

$-\infty$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is said to tend to $-\infty$, if for every $s\in \mathbb {R}$, there exists some $N\in \mathbb {N}$. such that
    $x_{n}\leq s{\text{ holds for all }}n\geq N.$
# Real sequence/Unit fractions/Continuous and convergent/Function limit/Exercise

Let
    $T={\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\subseteq \mathbb {R} \,$
be the set of the unit fractions and let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote a real sequence. Let $b\in \mathbb {R}$ and $D=T\cup \{0\}$. Show that the following properties are equivalent.
* The sequence converges to $b$.
* The function given by has a limit $\operatorname {lim} _{x\rightarrow 0}\,f(x)=b$.
    $f\colon T\longrightarrow \mathbb {R}$
    $f{\left({\frac {1}{n}}\right)}=x_{n}\,$
* The function given by and ${\tilde {f}}(0)=b$ is continuous.
    ${\tilde {f}}\colon D\longrightarrow \mathbb {R}$
    ${\tilde {f}}{\left({\frac {1}{n}}\right)}=x_{n}\,$
# Real sequence/Fraction with roots/1/Exercise

Decide whether the real sequence
    $x_{n}={\frac {5n^{\frac {3}{2}}+4n^{\frac {4}{3}}+n}{7n^{\frac {5}{3}}+6n^{\frac {3}{2}}}}\,$
(with $n\geq 1$) converges in $\mathbb {R}$, and determine, if applicable, the limit.
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/Convergent sequences/Rules/Fact/Factreferencenumber

Lemma 8.1
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/Elementary properties/3/Fact/Factreferencenumber

Lemma 5.2
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/Elementary properties/3/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 5.5
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/a \geq b positive/Ratio/Exercise/Exercisereferencenumber

Exercise 5.11
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/x negative/Inverse/Exercise/Exercisereferencenumber

Exercise 5.37
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/x positive/Inverse/Exercise/Exercisereferencenumber

Exercise 5.8
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/a \geq b positive/Inverse/Exercise/Exercisereferencenumber

Exercise 5.10
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Ordered field/x \geq 1/Inverse/Exercise/Exercisereferencenumber

Exercise 5.9
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I

This is the English version of the material (lecture notes, exercises) of a mathematical course taught at the University of Osnabrück. So this is not an online-course, only the material for a real university course. The material is organized in a modular way (e.g. facts, proofs, examples, exercises are in separate files) so that it can be used also in other courses. The mathematical material is quite standard for a university beginners class, and the course has been taught several times. The English version is a direct translation. Suggestions are welcome.
This mathematical course is for beginners at the University of Osnabrück in the subjects Computer sciences, Cognitive sciences, Physics and System sciences. It is an introduction to advanced mathematics on the academic level. It covers logical foundation, sets, mappings, algebraic structures like fields and polynomials, the basics of analysis like sequences, continuity, differentiability, primitive functions and the basics of linear algebra like vector spaces, bases, linear maps, eigenvalues. Enjoy!
* List of definitions
* Important theorems
# PlanetPhysics/Thin Equivalence Relation
## Thin equivalence relation

Let $a,a':x\simeq y$ be paths in $X$. Then $a$ is thinly equivalent to $a'$, denoted $a\sim _{T}a'$, if there is a thin relative homotopy between $a$ and $a'$.
We note that $\sim _{T}$ is an equivalence relation, see . We use $\langle a\rangle :x\simeq y$ to denote the $\sim _{T}$ class of a path $a:x\simeq y$ and call $\langle a\rangle$ the {\it semitrack} of $a$. The groupoid structure of ${\boldsymbol {\rho }}_{1}^{\square }(X)$ is induced by concatenation, +, of paths. Here one makes use of the fact that if </math> a: x \simeq x', \ a' : x' \simeq x, \ a : x \simeq x' Failed to parse (unknown function "\begin{matrix}"): {\displaystyle are paths then there are canonical thin relative homotopies <center><math> \begin{matrix}{r} (a+a') + a'' \simeq a+ (a' +a'') : x \simeq x''' \ ({\it rescale}) \\ a+e_{x'} \simeq a:x \simeq x' ; \ e_{x} + a \simeq a: x \simeq x' \ ({\it dilation}) \\ a+(-a) \simeq e_{x} : x \simeq x \ ({\it cancellation}). \end{matrix} }
The source and target maps of ${\boldsymbol {\rho }}_{1}^{\square }(X)$ are given by Failed to parse (unknown function "\enskip"): {\displaystyle \partial^{-}_{1} \langle a\rangle =x,\enskip \partial^{+}_{1} \langle a\rangle =y,}
if $\langle a\rangle :x\simeq y$ is a semitrack. Identities and inverses are given by $\varepsilon (x)=\langle e_{x}\rangle \quad \mathrm {resp.} -\langle a\rangle =\langle -a\rangle .$
# PlanetPhysics/Thermodynamics

\setcounter{section}{1}
Thermodynamics is a phenomenological description of equilibrium properties of macroscopic systems.
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As a phenomenological description, it is based on a number of empirical observations which are summarized by the laws of thermodynamics. A coherent logical and mathematical structure is then constructed on the basis of these observations, which leads to a variety of useful concepts, and to testable relationships among various quantities. The laws of thermodynamics can only be justified by a more fundamental(microscopic) theory of nature. For example, statistical mechanics attempts to obtain these laws starting from classical or quantum mechanical equations for the evolution of collections of particles.
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A system under study is said to be in equilibrium when its properties do not change appreciably with time over the intervals of interest(observation times). The dependence on the observation time makes the concept of equilibrium subjective. For example, window glass is in equilibrium as a solid over many decades, but flows like a fluid over time scales of millennia. At the other extreme, it is perfectly legitimate to consider the equilibrium between matter and radiation in the early Universe during the first minutes of the big bang.
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The macroscopic system in equilibrium is characterized by a number of thermodynamic coordinates or state functions. Some common examples of such coordinates are pressure and volume (for a fluid), surface tension and area (for a film), tension and length (for a wire), Electric Field and Polarization(for a dielectric), $\dots$. A closed system is an idealization similar to a point particle in mechanics in that it is assumed to be completely isolated by adiabatic walls that donâ€™t allow any exchange of heat with the surroundings. By contrast, diathermic walls allow heat exchange for an open system. In addition to the above mechanical coordinates, the laws of thermodynamics imply the existence of other equilibrium state functions.
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References
This is a derivative work from [1] a Creative Commons Attribution-Noncommercial-Share Alike 3.0 work
[1] MIT OpenCourseWare, 8.333 Statistical Mechanics I: Statistical Mechanics of Particles, Fall 2007
# PlanetPhysics/Thermodynamics an Introduction and Definitions

Thermodynamics is the science of the flow of heat. It applies to macroscopic systems in equilibrium and how to go from one equilibrium state to another. It is entirely empirical and summed up into four laws and basic mathematics.
Zeroth law of thermodynamics: defines temperature $T$
First law of thermodynamics: defines energy $U$
Second law of Thermodynamics: defines entropy $S$
Third Law of Thermodynamics: gives numerical value to entropy $S$
These laws are UNIVERSALLY VALID and cannot be circumvented.
Definitions used in Thermodynamics:
* System : The part of the Universe that we choose to study
* Surroundings : The rest of the Universe
* boundary: The surface dividing the System from the Surroundings
* Homogeneous : A single phase is in the system
* Hetrogeneous : Different phases are in the system
Examples of systems:
* A person
* Hot coffee in a thermos
* glass of ice water
* volume of 4 liters of air in a room
whatever is left over is the surroundings. Between the system and the surroundings is the boundary.
Examples of boundaries:
* Real like the outside of a person's skin
* The inner wall of the thermos
* An imaginary boundary surrounding the 4 liters of air
Systems can be:
* Open : mass and Energy can transfer between the System and the Surroundings
* Closed : Energy can transfer between the System and the Surroundings, but not mass
* Isolated : Neither Mass nor Energy can transfer between the System and the Surroundings
Describing Systems requires:
* A few macroscopic properties: p, T, V, n, m, etc.
* Knowledge if System is Homogeneous or Hetrogeneous
* Knowledge if System is in Equilibrium State
* Knowledge of the number of components
Two classes of Properties:
* Extensive : Depend on the size of the system (n,m,V,...)
* Intensive : Independent of the size of the system (T, p, ${\bar {V}}={\frac {V}{n}}$,...)
A system is in equilibrium if the properties that describe the system, such as $P$, $T$, $V$, etc. do not change in time or space. A gas in a container needs to be the same $P$, $T$, $V$ to be in equilibrium.
References
This is a derivative work from [1] a Creative Commons Attribution-Noncommercial-Share Alike 3.0 work
[1] MIT OpenCourseWare, 5.60 Thermodynamics and Kinetics: Thermodynamics and Kinetics, Spring 2008
# PlanetPhysics/Thin Double Track

Let us consider a square map $u:I^{2}\longrightarrow X$ in a topological space $X$. Then the homotopy class of $u$ relative to the boundary $\partial {I}^{2}$ of $I$ is called a {\it double track}. A double track is {\it thin} if it has a thin representative.
# PlanetPhysics/Thin Square

Let us consider first the concept of a tree that enters in the definition of a thin square. Thus, a simplified notion of thin square is that of {\em a continuous map from the unit square of the real plane into a Hausdorff space $X_{H}$ which factors through a tree}.
A {\it tree}, is defined here as the underlying space $|K|$ of a finite $1$-connected $1$-dimensional simplicial complex $K$ and boundary $\partial {I}^{2}$ of $I^{2}=I\times I$ (that is, a square (interval) defined here as the Cartesian product of the unit interval $I:=[0,1]$ of real numbers).
A square map $u:I^{2}\longrightarrow X$ in a topological space $X$ is thin if there is a factorisation of $u$, $u:I^{2}{\stackrel {\Phi _{u}}{\longrightarrow }}J_{u}{\stackrel {p_{u}}{\longrightarrow }}X,$ where $J_{u}$ is a tree and $\Phi _{u}$ is piecewise linear (PWL) on the boundary $\partial {I}^{2}$ of $I^{2}$.
# Multivariate data analysis
## Learning projects

* How to use R
* How to use SPSS
* Data Analysis using the SAS Language