OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Integral Equation

An integral equation involves an unknown function under the integral sign.\, Most common of them is a linear integral equation
   ${\begin{matrix}\alpha (t)\,y(t)+\!\int _{a}^{b}k(t,\,x)\,y(x)\,dx=f(t),\end{matrix}}$
where $\alpha ,\,k,\,f$ are given functions.\, The function\, $t\mapsto y(t)$\, is to be solved.
Any linear integral equation is equivalent to a linear differential equation; e.g. the equation\, $y(t)\!+\!\int _{0}^{t}(2t-2x-3)\,y(x)\,dx=1+t-4\sin {t}$\, to the equation\, $y''(t)-3y'(t)+2y(t)=4\sin {t}$\, with the initial conditions \,$y(0)=1$\, and\, $y'(0)=0$.\\
The equation (1) is of
* 1st kind if\, $\alpha (t)\equiv 0$,
* 2nd kind if $\alpha (t)$ is a nonzero constant,
* 3rd kind else.
If both limits of integration in (1) are constant, (1) is a Fredholm equation , if one limit is variable, one has a Volterra equation .\, In the case that\, $f(t)\equiv 0$,\, the linear integral equation is homogeneous .\\
Example. \, Solve the Volterra equation\, $y(t)\!+\!\int _{0}^{t}(t\!-\!x)\,y(x)\,dx=1$\, by using Laplace transform.
Using the convolution, the equation may be written\, $y(t)+t*y(t)=1$.\, Applying to this the Laplace transform, one obtains\, $Y(s)+{\frac {1}{s^{2}}}Y(s)={\frac {1}{s}}$,\, whence\, $Y(s)={\frac {s}{s^{2}+1}}$.\, This corresponds the function \,$y(t)=\cos {t}$,\, which is the solution.\\
Solutions on some integral equations in EqWorld.
# PlanetPhysics/Index of Categories
### Index of categories

The following is a contributed listing, or Index of Categories:
* Dual category #Double dual category, $V^{**}$
* Category of sets, $Set$ or $Ens$
* Cartesian closed category, or $Cc$-category
* category of molecular sets: a $Cc$-category
* Category of $(M,R)$--systems: a $Cc$-category
* Category of dynamical systems
* abelian categories: Grothendieck category, category of Abelian groups, category of sheaves of abelian groups, category of reversible automata ${\mathcal {A}}_{R}$
* Complete category
* Comma category
* $C_{1}$-category
* $C_{2}$-category
* $C_{3}$-category
* $Ab5$-category
* category of additive fractions #Category of fractions
* quotient category #Grassmann category
* Category of groups #Category of crossed modules of groups
* Category of crossed modules of algebroids #Category of matrices #Category of Abelian (or commutative) groups
* Category of Polish groups
* Category of Lie 2-groups, (also an example of internal category)
* Categorical groups
* Internal categories
* Monoidal category: a. symmetric , b. braided , c. closed
* Category of semigroups #Category of rings
* Category of modules #Category of crossed modules
* Category of crossed complexes #groupoid category #Category of topological spaces
* homotopy category #Category of sheaves of abelian groups
* category of Riemannian manifolds #Borel category
* Category of graphs: category of reflexive (or directed) graphs
* Category of paths on a graph
* Category of hypergraphs #category of groupoids, Groupoid category
* Category of topological groupoids
* category of Borel groupoids #Cohomology of small categories
* category of C*-algebras #\htmladdnormallink{$2-C^{*}$ -category,${{\mathcal {C}}^{*}}_{2}$}{http://planetphysics.us/encyclopedia/2CCategory.html}
* Category of H $*$ -algebras
* category of Hilbert spaces #category of automata #category of quantum automata #algebraic categories
* category of logic algebras #Category of lattices
* Category of Boolean algebras
* Algebraic Category of \Lukasiewicz-Moisil, $LM_{n}$--Algebras
* Category of Heyting algebras
* Category of topoi
* Category of genetic nets
* Category of quantum logic algebras
* R-category
* Category of algebroids
* Category of double algebroids
* Grassmann-Hopf algebroid category
* Topological category
* Galois category
* Subcategory
* Internal category
* Groupoid groups
* Groupoid category
* Fibered categories
* Non-Abelian categories
* Category of diagrams
* Functor category, or category of functors and functorial morphisms (natural transformations)
* Category of small categories
* Category of categories, super-category
* Supercategory
* Meta-category
* Category of functors, (or Functor category)
* 2-category; specific examples are: a Lie 2-group, Abelian 2-category,
2-category of commutative (or Abelian) groupoids.
* Double category
* n-category
* Double groupoid category
# PlanetPhysics/Index of Quantum Theories

This is a contributed entry in progress on quantum concepts and quantum theories.

## Index of Quantum Theories
### Fundamental Concepts

observable, operators, Quantum States, Entanglement, Schr\"odinger and Heisenberg equations, quantum state space, Hilbert space, von Neumann algebra, C*-algebra, Hamiltonian operator, Hermitian operators, unitary operators, representations, quantum groups, quantum groupoids, quantum algebroids, Hamiltonian algebroids.
# PlanetPhysics/Inflexion Point

In examining the graphs of differentiable real functions, it may be useful to state the intervals where the function is convex and the ones where it is concave.
* A function $f$ is said to be convex on an interval if the restriction of $f$ to this interval is a (strictly) convex function; this may be characterized more illustratively by saying that the graph of $f$ is concave upwards or concave up . On such an interval, the tangent line of the graph is constantly turning counterclockwise, i.e., the derivative $f'$ is increasing and thus the second derivative $f''$ is positive. In the picture below, the sine curve is concave up on the interval\, $(-\pi ,\,0)$.
* The concavity of the function $f$ on an interval correspondingly: On such an interval, the graph of $f$ is concave downwards or concave down , the tangent line turns clockwise, $f'$ decreases, and $f''$ is negative. In the picture below, the sine curve is concave down on the interval\, $(0,\,\pi )$.
* The points in which a function changes from concave to convex or vice versa are the inflexion points (or inflection points ) of the graph of the function. At an inflexion point, the tangent line crosses the curve, the second derivative vanishes and changes its sign when one passes through the point.
   \begin{pspicture}(-5,-2.5)(5,2) \psaxes[Dx=9,Dy=1]{->}(0,0)(-4.5,-1.5)(5,2) \rput(5,-0.2){$x$} \rput(0.2,2){$y$} \rput(3,-0.2){$\pi$} \rput(-3.1,-0.2){$-\pi$} \psplot[linecolor=blue]{-4}{4}{x 60 mul sin} \psdot[linecolor=red](0,0) \rput(0.2,-2.3){The origin is an inflexion point of the sinusoid \,$y=\sin {x}$.} \end{pspicture}
Since the sine function is $2\pi$-periodic, the sinusoid possesses infinitely many inflexion points. Indeed,\, $f(x)=\sin x$;\, $f''(x)=-\sin x=0$\, for\, $x=0,\,\pm \pi ,\,\pm 2\pi ,\,\dots$;\, $f'''(x)=-\cos x$, $f'''(n\pi )=-\cos n\pi =(-1)^{n+1}\neq 0$. Non-nullity of the third derivative at these critical points assures us the existence of those inflexion points.
Remarks
1. For finding the inflexion points of the graph of $f$ it does not suffice to find the roots of the equation\, $f''(x)=0$, since the sign of $f''$ does not necessarily change as one passes such a root. If the second derivative maintains its sign when one of its zeros is passed, we can speak of a plain point (?) of the graph. E.g. the origin is a plain point of the graph of\, $x\mapsto x^{4}$.
2. Recalling that the curvature $\kappa$ for a plane curve \,$y=f(x)$\, is given by $\kappa (x)={\frac {f''(x)}{[1+f'(x)^{2}]^{3/2}}},$ we can say that the inflexion points are the points of the curve where the curvature changes its sign and where the curvature equals zero.
3. If an inflexion point\, $x=\xi$\, satisfies the additional condition \,$f'(\xi )=0$,\, the point is said to be a stationary inflexion point or a saddle-point , while in the case\, $f'(\xi )\neq 0$\, it is a non-stationary inflexion point .
# PlanetPhysics/Inkscape Graphics on PlanetPhysics

Test image using Inkscape
\begin{figure}[h] \centering \includegraphics{inkscapeTest}
\centerline{Figure 1: Latex and eps images} (1) \end{figure}
Save Inkscape drawing to eps with option export area is drawing, then include the package pdftex graphicx and psfrag (see preamble)
Even better, the newest version 0.48 has option to save as pdf and then choose option to use Latex. (not working on PP yet)
# PlanetPhysics/Index of Differential Geometry

This is a contributed entry in progress on Differential Geometry.

## Index of Differential Geometry

* 1.0 Euclidean and analytical geometry
* 1.1 Riemannian geometry
* 1.2 Pseudo-Riemannian geometry
* 1.3 Cauchy-Riemannian geometry
* 1.4 Finsler geometry
* 1.5 Symplectic geometry
* 1.6 Contact geometry
* 1.7 Complex and K\"ahler geometry
* 1.8 Affine differential geometry
* 1.9 Projective differential geometry
* 2.0 noncommutative geometry
* 2.1 Synthetic differential geometry
* 2.2 Abstract differential geometry
* 2.3 Discrete differential geometry

### Fundamental concepts of differential geometry

* Bundles
* Connections (connexions)
* manifolds
* Submanifolds
* Differentiable manifolds
* Cross manifolds
* Hypersurfaces
* tensors and Forms
* operators on forms and Integration
* Rigidity
* Involutive distributions
* Jacobi fields and conjugate points
* First and second variations
* geodesics
* Lie groups in differential geometry
* Lie derivatives
* xyz
* xwz
* tangent spaces

### Differential geometry (DG) applications

* In physics both Electromagnetism and general relativity (GR) employ extensively DG concepts and tools; our Universe was represented in Einstein's GR as a smooth manifold equipped with a pseudo-Riemannian metric, that describes the curvature of space-time; however, refined GR models of space-times in the inflationary universe regard the space-times as a directed sequence of space-times or a limit .
* Symplectic manifolds are especially useful to the study of Hamiltonian systems.
* In engineering there are numerous applications of differential geometry in digital signal processing, architectural design, image enhancement, and so on.
* The Fisher information metric is used in information theory and advanced statistics.
* computer graphics and CAD computer-aided design are based on differential geometry.
* In Geophysics, differential geometry is routinely used to analyze and describe geologic structures, as well as several other applications
Often differential geometry is considered to be one of the more applied areas of mathematics.
# PlanetPhysics/Index of Experimental Physics

This is a contributed entry in progress- an index of experimental physics.

## Index of Experimental Physics

* Classical dynamics
* Molecular Physics and Thermodynamics
* Experimental Biophysics
* Nuclear Medicine and Medical physics
* Polymer physics
* Computational Physics
* Relativistic experiments
* Quantum experiments
* Electromagnetism
* Coherent Optics/Lasers
* Light scattering
* Magnetic resonance (NMR, ESR/, ENDOR)
* Spectroscopy
* Atomic Physics
* nuclear physics
* superconductivity
* Low temperature physics
* condensed matter Physics
* Complex system analysis
* solid state physics
* Physical Crystallography
* neutron scattering
* High energy physics experiments
* Accelerator physics and accelerator systems
* Fission Reactor physics
* Nuclear Fusion experiments
* Gravitational physics
* Experimental Astrophysics
* Experimental Geophysics
* Solar physics
* Alternative energy physics
* Experimental Physical Chemistry
# Stars/Oranges/Quiz

Orange stars is a lecture from the radiation astronomy department for possible inclusion in the course on the principles of radiation astronomy and is already in the star courses.
You are free to take this quiz based on orange stars at any time.
To improve your score, read and study the lecture, the links contained within, listed under See also, External links, and in the {{stars resources}} template. This should give you adequate background to get 100 %.
As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score.
Suggestion: Have the lecture available in a separate window.
To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association.
Enjoy learning by doing!

## Hypotheses

* Orange stars reveal details otherwise not available at other wavelengths.
# Stars/Reds/Quiz

Red stars is a stand-alone radiation astronomy lecture studying astronomical radiating sources, or natural radiating sources called red stars.
You are free to take this quiz based on red stars at any time.
To improve your score, read and study the lecture, the links contained within, listed under See also, External links, and in the {{stars resources}} template.
This should give you adequate background to get 100 %.
As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score.
Suggestion: Have the lecture available in a separate window.
To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association.
Enjoy learning by doing!

## Hypotheses

* Nebulas are composed of red dust and rock.
# Stars/Quasars/Quiz

Quasars is a lecture about a specific theory from astrophysics for the origin and accumulated changes of star systems. It is an offering from the radiation astronomy department.
You are free to take this quiz based on quasars at any time.
To improve your score, read and study the lecture, the links contained within, listed under See also, External links, and in the {{stars resources}} template. This should give you adequate background to get 100 %.
As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score.
Suggestion: Have the lecture available in a separate window.
To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association.
Enjoy learning by doing!
# Stars/Quiz

Stars is a lecture about an entity, source, and/or object observed using radiation astronomy.
You are free to take this quiz based on stars. Once you've read and studied it, the links contained within, listed under See also, External links and the {{stars resources}} template, you should have adequate background to score 100 %.
Enjoy learning by doing!

## Hypotheses

* Stars that emit light do so because they are receiving electron irradiation.
# Stars/Os/Quiz

O stars is a lecture about an entity, source, and/or object observed using radiation astronomy that is classified as an O-type star.
You are free to take this quiz based on O stars. Once you've read and studied it, the links contained within, listed under See also, External links and the {{stars resources}} template, you should have adequate background to score 100 %.
Enjoy learning by doing!

## Hypotheses

* Stars that emit light do so because they are receiving electron irradiation.
# Stars/Radiative dynamo/Quiz

Radiative dynamo is a lecture and an article about a specific aspect of astrophysics applied to plasma, gaseous, liquid, and rocky objects. It is from the Radiation astronomy department initially for the course on the principles of radiation astronomy.
You are free to take this quiz based on radiative dynamo at any time.
To improve your score, read and study the lecture, the links contained within, listed under See also, External links and in the {{radiation astronomy resources}} and the {{principles of radiation astronomy}} templates. This should give you adequate background to get 100 %.
As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score.
Suggestion: Have the lecture available in a separate window.
To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association.
Enjoy learning by doing!

## Hypotheses

* The dynamo lectures may be better as a part of a magnetohydrodynamics course.
# Polynomial/C/Linear factors/Exercise

Let $F\in \mathbb {C} [X]$ be a non-constant polynomial. Prove that $F$ can be decomposed as a product of linear factors.
# Polynomial/Continuity/Check delta/1/Exercise

Let
    $f(x)=2x^{3}-4x+5\,.$
Show that for all $x\in \mathbb {R}$, the following relation holds: If
    $\vert {x-3}\vert \leq {\frac {1}{800}}\,,$
then
    $\vert {f(x)-f(3)}\vert \leq {\frac {1}{10}}\,.$
# Polymers/Introduction
## Concept Of Polymers

A polymer is a large macromolecule of high to very high molecular weight which consists of many repeating units called monomers which are covalent bond to one another. In many cases, monomers are linked together in a row, like links in a chain. However branching and cross-linking between chains does occur. Monomers can even be bonded together in a two- or three-dimensional polymer network.
Polymers are very abundant on the surface of earth; either in nature or artificially prepared. We come across polymers each and every minute we live. Artificially produced polymers include plastics, elastomers, fibers, and materials of intermediate characteristics. When you get up in the morning and brush your teeth, the toothbrush you have is made of plastic-that is a polymer. Its bristles are fibers-again a polymer.
We use plastic bags, cards, paper, wood, clothing, paints, and endless items that have the basic chemical structure of a polymer. Human and animal bodies are mainly made up of protein. Proteins are the building blocks of life. They are polymers too, with α-amino acids as repeating units linked together with peptide bonds.
A polymer can be classified into a homopolymer, which consists of only one type of monomer, or a copolymer (sometimes called a heteropolymer), which consists of two or more types of monomers.
To create polymers from monomers (single units), a chemical reaction called polymerization (linking of monomers) occurs.
There are two types of polymerizations, that of addition polymerization and of condensation polymerization.
The new broad terms for addition and condensation polymerizations are step-growth polymerization and chain-growth polymerization. This is the case because nowadays polymerization is also carried out by ring-opening or by complex mechanisms other than addition or condensation.

## See Also

* polymers on Wikipedia
# Polymers/Properties

Tacticity (stereochemistry) can change the properties of the polymer.
| Isotactic | Syndiotactic | Atactic |
Molecular mass
Number average (M$_n$) and weight average (M$_w$).
$M_{n}={\frac {\sum M_{i}N_{i}}{\sum N_{i}}},\quad M_{w}={\frac {\sum M_{i}^{2}N_{i}}{\sum M_{i}N_{i}}},\quad M.W.D.orPDi={\frac {M_{w}}{M_{n}}}$
Glass transition (T$_g$) and melting temperature (T$_m$)
* T$_m$: crystalline regions, first order transition
* T$_g$: amorphous regions, second order transition
* hydrogen bonding increases T$_g$
* longer side chains lowers T$_g$
* at or below T$_g$, the ratio of free volume to total volume of polymer is constant
# Polymers

Polymer is a term used to describe large molecules consisting of repeating structural units, or monomers, connected by covalent chemical bonds. The term is derived from the Greek words: polys meaning many, and meros meaning parts [1]. A key feature that distinguishes polymers from other molecules is the repetition of many identical, similar, or complementary molecular subunits in these chains. These subunits, the monomers, are small molecules of low to moderate molecular weight, and are linked to each other during a chemical reaction called polymerization.

## Resources

* Introduction
* Properties
* Electron beam

## See Also

* Portal: Polymer chemistry
* Wikipedia: Polymer
* Wikipedia: Polymer science
# Polynomial/Composition/Polynomial/Exercise

Show that the composition (the inserting of a polynomial into another one) of two polynomials is again a polynomial.
# Polynomial/Degree/Derivation as operator/Eigenspaces/Exercise

Let $V=\mathbb {R} [X]_{\leq d}$ denote the set of all real polynomials of degree $\leq d$. Determine the eigenvalues, eigenvectors and eigenspaces of the derivation operator
    $V\longrightarrow V,P\longmapsto P'.$
# Polycystic Ovarian Syndrome

The polycystic ovarian syndrome (PCOS) is a common dysfunction that appears in women at reproductive age. The etiology of this syndrome is still unknown. The PCOS reproductive manifestations include hyperandrogenism, impaired ovulation, amenorrhea, polycystic ovaries and are generally associated with metabolic alterations like insulin resistance, obesity, dyslipemia and a high risk of cardiovascular disease. Therefore, consensus about the main clinic manifestations is needed to diagnose the PCOS, since this syndrome has many associated phenotypes. A recent scientific meeting suggested that the PCOS diagnosis should exclude other hyperandrogenic disorders, thyroid disease or another related disorder. Furthermore, PCOS diagnosis should have at least two of the following three criteria: oligo/anovulation, hyperandrogenism (clinical or biochemical) or polycystic ovary on ultrasonography. There are many phenotypes of this syndrome and the discovery of the DNA genome has allowed the association between the clinical and the genetic basis of the PCOS. This syndrome is associated with genes involved in the biosynthetic pathway of androgens in adrenal and ovarian glands, in the androgen receptor expression and in LH and FSH regulation. The metabolic characteristics of this syndrome may be associated with the expression of metabolic genes, for example, the insulin gene. Nevertheless, the evidence is not enough for a monogenic association; it suggests that PCOS is a polygenic syndrome. The basic research of the PCOS is based on animal models. The neonatal or fetal administration of steroids can affect the metabolic and reproductive axis of the rodents, and scientist can study the basis of the syndrome. Furthermore, for a better approximation at human physiology, there are other models in rhesus monkeys. These researches can study the characteristics of women with PCOS during adulthood with an early modification in hormone levels; it allows us to study the target tissue of hyperandrogenism. Finally, PCOS etiology may include a polygenetic basis that modifies the reproductive and metabolic physiology when it interacts with a specific environment such as fetal hyperandrogenism or insulin resistance.

## Bibliography:

“Review: Fetal Programming of PCOS by androgen Excess: Evidence from experimental, clinical and genetics association studies” Xita, 2006, J. of Clinical Endocrinology and Metabolism.
“Insulin resistance, dyslipemia, and metabolic syndrome in women with PCOS” El-Mazny, 2010, International Journal of Gynecology and Obstetrics