OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Gene Nets Physical and Mathemaical Models
### Introduction

Genetic `nets', or networks , $GN$ -- that form a living organism's genome --are mathematical models of functional genes linked through their non-linear, dynamic interactions.
A simple genetic (or gene) network $GN_{s}$ may be thus represented by a directed graph $G_{D}$ -the gene net digraph- whose nodes (or vertices) are the genes $g_{i}$ of a cell or a multicellular organism and whose edges (arcs) are arrows representing the actions of a gene $a_{g}^{i}$ on a linked gene or genes; such a directed graph representing a gene network has a canonically associated biogroupoid ${\mathcal {G}}_{B}$ which is generated or directly computed from the directed graph $G_{D}$.

### Boolean vs. N-state models of genetic networks in LMn- logic algebras

The simplest, Boolean, or two-state models of genomes represented by such directed graphs of gene networks form a proper subcategory of the category of n-state genetic networks, Failed to parse (unknown function "\L"): {\displaystyle '''GN''' _{\LM_n}} that operate on the basis of a \Lukasiewicz-Moisil n-valued logic algebra $LM_{n}$. Then, the category of genetic networks, Failed to parse (unknown function "\L"): {\displaystyle '''GN''' _{\LM_n}} was shown in ref. to form a subcategory of the \htmladdnormallink{algebraic category of \Lukasiewicz algebras}{http://planetphysics.us/encyclopedia/AlgebraicCategoryOfLMnLogicAlgebras.html}, ${\mathcal {LM}}$. There are several published, extensive computer simulations of Boolean two-state models of both genetic and neuronal networks (for a recent summary of such computations see, for example, ref.. Most, but not all, such mathematical models are Bayesian, and therefore involve computations for random networks that may have limited biological relevance as the topology of genomes, defined as their connectivity, is far from being random.
The category of automata (or sequential machines based on Chrysippean or Boolean logic) and the category of $(M,R)$-systems (which can be realized as concrete metabolic-repair biosystems of enzymes, genes, and so on) are subcategories of the category of gene nets Failed to parse (unknown function "\L"): {\displaystyle '''GN''' _{\LM_n}} . The latter corresponds to organismic sets of zero-th order $S_{0}$ in the simpler, Rashevsky's theory of organismic sets.
# PlanetPhysics/Generalized Fourier Transform

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## Fourier-Stieltjes Transform

Given a positive definite, measurable function $f(x)$ on the interval $(-\infty ,\infty )$ there exists a monotone increasing, real-valued bounded function $\alpha (t)$ such that:
   $f(x)=\int _{\mathbb {R} }e^{itx}d(\alpha (t)),$
for all $x\in {\mathbb {R} }$ except a `small' set, that is a finite set which contains only a small number of values. When $f(x)$ is defined as above and if $\alpha (t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of $\alpha (t)$, and it is continuous in addition to being positive definite .
# PlanetPhysics/Generalized Coordinates for Constrained Motion

If the particle is constrained to move on some given surface, any two independent specified functions of its rectangular coordinates $x,y,z$, may be taken as its coordinates $q_{1}$ and $q_{2}$, provided that by the equation of the given surface in rectangular coordinates and the equations formed by writing $q_{1}$ and $q_{2}$ equal to their values in terms of $x,y$, and $z$ the last-named coordinates may be uniquely obtained as explicit functions of $q_{1}$ and $q_{2}$.
If the particle is constrained to move in a given path, any specified function of $x,y,z$ may be taken as its coordinate $q_{1}$, provided that by the two rectangular equations of its path and the equation formed by writing $q_{1}$ equal to its value in terms of $x,y,z$ the last-named coordinates may be uniquely obtained as explicit functions of $q_{1}$.
# PlanetPhysics/Generalized Hurewicz Fundamental Theorem

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The Hurewicz theorem was generalized from connected CW-complexes to arbitrary topological spaces and is stated as follows.
\begin{theorem} If $\pi _{r}(K,L)=0$ for $1\leq r\leq n$ , $(n\geq 2)$, then $h_{\pi }:\pi _{n}^{*}(K,L)\simeq H_{n}(K,L)$ , where $\pi _{n}$ are homotopy groups, $H_{n}$ are homology groups, K and L are arbitrary topological spaces, and `$\simeq$' denotes an \htmladdnormallink{isomorphism {http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}.} \end{theorem}
# PlanetPhysics/Generalized Fourier and Measured Groupoid Transforms

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### Generalized Fourier transforms

Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table (see also [TableOfFourierTransforms Fourier transforms] ) - for the purpose of direct comparison with the latter transform. Unlike the more general Fourier-Stieltjes transform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable over the whole real axis for $t\in {\mathbb {R} }$, or over the entire ${\mathbb {C} }$ domain when ${\check {m}}(t)$ is a complex function.
Fourier-Stieltjes transform .
Given a positive definite, measurable function $f(x)$ on the interval $(-\infty ,\infty )$ there exists a monotone increasing, real-valued bounded function $\alpha (t)$ such that:
   $f(x)=\int _{\mathbb {R} }e^{itx}d(\alpha (t),$
for all $x\in {\mathbb {R} }$ except a small set. When $f(x)$ is defined as above and if $\alpha (t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of $\alpha (t)$, and it is continuous in addition to being positive definite.
\subsubsection*{FT and FT-Generalizations}
| $f(t)$ | ${\mathcal {F}}{f(t)}={\hat {f}}(x)=(2\pi )^{-1}\int {e{(-itx)}dx}$ | Conditions* | Explanation | Description |
| $e^{-t}\theta (t)$ | ${\mathcal {F}}{[f(t)]}(x)=(2\pi )^{-1}\int {\theta (t)e{(it^{2}x)}dx}$ | from $-\infty$ to +$\infty$ | From $Mathematica^{TM**}$ | |
| $c$ | $({\sqrt {2\pi }})^{-1}c$ | | | |
| | | Notice on the next line the overline | bar (${\overline}$) placed above $t(x)$ | |
| $f(t)$ | $\int {\hat {f}}(x){\overline {t(x)}}dx$ | $f(t)\in {L^{1}(G_{l})}$, with $G_{l}$ a | Fourier-Stieltjes transform | ${\hat {f}}(x)\in {C_{0}({\hat {G_{l}}})}$ |
| | | locally compact groupoid; | | |
| | | $\int$ is defined via | | |
| | | a left Haar measure on $G_{l}$ | | |
| ${\hat {m}}(x)$ | ${\check {m}}(t)=\int e^{itx}d{\hat {m}}(x)$ | as above | Inverse Fourier-Stieltjes | ${\check {m}}(t)\in {L^{1}(G_{l})}$, |
| | | | transform | (,). |
| ${\hat {m}}(x)$ | ${\check {m}}(t)=\int e^{itx}d{\hat {m}}(x)$ | When $G_{l}=\mathbb {R}$, and it exists | This is the usual | ${\check {m}}(t)\in {\mathbb {R} }$ |
| | | only when ${\hat {m}}(x)$ is | Inverse Fourier transform | |
| | | Lebesgue integrable on | | |
| | | the entire real axis | | |
* Note the 'slash hat' on ${\hat {f}}(x)$ and ${\hat {G_{l}}}$;
** Calculated numerically using this link to $Mathematica^{TM}$
# Newton’s laws and their applications

In physics Newton's three laws of motion are the basis of kinematics, the subject of this course. All three laws are detailed on this page. See also: Motion - Dynamics

## Newton's Laws of Motion

Newtons law describes the body's motion in terms of physics. It has been widely accepted by the science community for the past 10 centuries, in total there were three laws that affect the laws of motion.

## Newton's First Law

His first law of motion is called the law of inertia, his law is; "Every body continues its state of rest or uniform motion until an external unbalanced force acts on it" ."" If a body is moving ,it will continue to move with the same speed in the same direction unless a force is applied on it" Newton put the above observation in the form of a law which is called the Newton's first law of motion.And the resultant force must be zero.

## Newtons Second Law

His second law is more notably named as the law of acceleration. His law is;
"The rate of change of momentum of an object is directly proportional to the resultant force acting on the body and is in the same direction."
This law means that the force of the object moving will be equal to the opposing force such as air resistance. Mathematically, Newton's second law is stated as:
${\overrightarrow {\mathbf {F} }}={\frac {d{\overrightarrow {\mathbf {p} }}}{dt}}$
where F is the total force, p is the momentum, and t is the time passed. In classical physics, where the object's mass is constant, this equation becomes a more familiar form:
${\overrightarrow {\mathbf {F} }}={\frac {d{\overrightarrow {\mathbf {p} }}}{dt}}={\frac {dm{\overrightarrow {\mathbf {v} }}}{dt}}=m{\frac {d{\overrightarrow {\mathbf {v} }}}{dt}}=m{\overrightarrow {\mathbf {a} }}$
where m is the object's mass and a is the acceleration of the object. Example
A skydiver jumps from a plane and accelerates until he reaches the highest velocity possible, when this happens his acceleration is equal to nothing, this happens when air resistance is equal to the downward force of the skydiver.

## Newton's Third Law of motion

Newton's third and final law is the law of Interaction in which he Doubted;
"All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction." It can be said that "every action has an opposite and equivalent reaction"

### Examples

* When we push a wall, it creates an opposite amount of force, thus, either we move or does the wall move.
This in simple terms is that to every action force there is an equal, but still the same, reaction force. For example if one presses on a wall a equal and same force is pushed back at them.
# Nicene Creed (Roman Catholic Theology)

The Nicene Creed, Niceno-Constantinopolitan Creed or Icon/Symbol of the Faith, is the most widespread Christian statement of faith.
Since its original formulation it continues to be used in the Roman Catholic, Syrian Orthodox (Jacobite) Eastern Orthodox, Oriental Orthodox, Assyrian, Anglican, Lutheran, and most other Protestant Churches.
This page is devoted to the theological, as opposed the historical, significance and interpretation. Note that the Nicene Creed is not the same as the Apostles' Creed.

## History

The Creed was originally created at the First Council of Nicaea in 325. The version created there ended after the words "We believe in the Holy Spirit". The remainder was added at the First Council of Constantinople, in 381. In 431 the Council of Ephesus reaffirmed this version and decreed that "it is unlawful for any man to bring forward, or to write, or to compose a different (ἑτέραν) Faith as a rival to that established by the holy Fathers assembled with the Holy Ghost in Nicæa."
The purpose of a creed is to act as a yardstick of correct belief. A creed is an epitome, not a full definition, of what is required for orthodoxy. It was hoped that by memorizing this summary of the faith, lay people without extensive theological training would still be able to recognize deviations from orthodox doctrines based on the Bible as interpreted in Christian Tradition.

## Text

The Nicene Creed
I believe in one God, the Father, the Almighty, maker of heaven and earth, of all things visible and invisible.
I believe in one Lord, Jesus Christ, the Only Begotten Son of God, born of the Father before all ages. God from God, Light from Light, true God from true God, begotten, not made, consubstantial with the Father; through him all things were made. For us men and for our salvation, he came down from heaven, and by the Holy Spirit was incarnate of the Virgin Mary, and became man. For our sake he was crucified under Pontius Pilate, he suffered death and was buried, and rose again on the third day in accordance with the Scriptures.
He ascended into heaven and is seated at the right hand of the Father. He will come again in glory to judge the living and the dead, and his kingdom will have no end.
I believe in the Holy Spirit, the Lord, the giver of life, who proceeds from the Father and the Son, who with the Father and the Son he is adored and glorified, who has spoken through the Prophets.
I believe in one, holy, catholic, and apostolic Church. I confess one baptism for the forgiveness of sins. I look forward to the resurrection of the dead, and the life of the world to come. Amen.
# Niccolò Machiavelli

Niccolò di Bernardo dei Machiavelli (3 May 1469 – 21 June 1527) was an Italian historian, politician, diplomat, philosopher, humanist, and writer based in Florence during the Renaissance. He was a founder of modern political science, and more specifically political ethics.

## Online Reading Material
### Wikimedia Articles

* Wikipedia: Niccolò Machiavelli

### Biographical Works
## Offline Reading Material
### Primary Texts

The main primary text which will be used in this unit is The Prince by Niccolò Machiavelli. The Prince is a political treatise, and follows the International Relations doctrine of Classical Realism. First published in c. 1516, The Prince has since become one of the foremost historical pieces on politics and statecraft during the Renaissance Period in Italy. The edition used in this unit is the Penguin edition (rev. ed. 2003) translated by George Bull.
# Nicene Creed

The Nicene Creed, Niceno-Constantinopolitan Creed or Icon/Symbol of the Faith, is the most widespread Christian statement of faith.
Since its original formulation it continues to be used in the Roman Catholic, Syrian Orthodox (Jacobite) Eastern Orthodox, Oriental Orthodox, Assyrian, Anglican, Lutheran, and most other Protestant Churches.
This page is devoted to the theological, as opposed the historical, significance and interpretation. Note that the Nicene Creed is not the same as the Apostles' Creed.

## History

The Creed was originally created at the First Council of Nicaea in 325. The version created there ended after the words "We believe in the Holy Spirit". The remainder was added at the First Council of Constantinople, in 381. In 431 the Council of Ephesus reaffirmed this version and decreed that "it is unlawful for any man to bring forward, or to write, or to compose a different (ἑτέραν) Faith as a rival to that established by the holy Fathers assembled with the Holy Ghost in Nicæa."
The purpose of a creed is to act as a yardstick of correct belief. A creed is an epitome, not a full definition, of what is required for orthodoxy. It was hoped that by memorizing this summary of the faith, lay people without extensive theological training would still be able to recognize deviations from orthodox doctrines based on the Bible as interpreted in Christian Tradition.

## Text

The Nicene Creed
We believe in one God, the Father, the Almighty, maker of heaven and earth, of all that is, seen and unseen. We believe in one Lord, Jesus Christ, the only Son of God, eternally begotten of the Father, God from God, Light from Light, true God from true God, begotten, not made, of one Being with the Father. Through him all things were made. For us men and for our salvation, he came down from heaven: by the power of the Holy Spirit he was born of the Virgin Mary, and became man. For our sake he was crucified under Pontius Pilate; he suffered died and was buried. On the third day he rose again in fulfillment of the Scriptures; he ascended into heaven and is seated at the right hand of the Father. He will come again in glory to judge the living and the dead, and his kingdom will have no end.
We believe in the Holy Spirit, the Lord, the giver of life, who proceeds from the Father and the Son. With the Father and the Son he is worshipped and glorified. He has spoken through the Prophets. We believe in one holy catholic and apostolic Church. We acknowledge one baptism for the forgiveness of sins. We look for the resurrection of the dead, and the life of the world to come. Amen.
# Newton-Leibniz theorem

Let $F(x)$ be such function that the (continuous) function $f(x)$ is its derivative i.e $f(x)=dF(x)/dx$ or $F(x)$ is the primitive function of $f$ then the definite integral $\int \limits _{a}^{b}f(x)\mathrm {d} x$ is the area under the curve drawn by (positive) $f$ and
$\int \limits _{a}^{b}f(x)\mathrm {d} x=F(b)-F(a)$.

#### Proof

We will limit ourselves to the positive function in the interval $\scriptstyle [a,b]$.
Let us estimate the area under the graph of the function $f$ by dividing densely the interval $\scriptstyle [a,b]$ into sub-intervals with the ending points $x_{i}$ and with the length $dx$ and such that $x_{0}=a$ and $x_{n}=b$. If the $dx$ is small then between the two consecutive nodes $x_{i}$ and $x_{i+1}$ , we can assume that $f(x_{i})$ approximate $f(x_{i+1})$, and the function $f$ is approximately linear and therefore the area under it can be approximated by the rectangle area with the width $dx$ and height $f(x_{i})$
    $P_{i}=f(x_{i}){dx}$
Then the total area under the graph of the function will be estimated as
    $P=\sum P_{i}$
Let $F$ be the primitive function of the function $f$. Then we can also estimate its derivative as
    $f(x_{i})=[F(x_{i+1})-F(x_{i})]/dx$
Now in the formula for $P$ we can re-express each value of $f(x_{i})$ by values of $F(x_{i})$ and $F(x_{i+1})$ (the same $dx$ in nominator and denominator cancels out)
    $P=F(x_{1})-F(x_{0})+F(x_{2})-F(x_{1})+F(x_{3})-F(x_{2})+...+F(x_{n})-F(x_{n-1})+\Theta (dx)$
Because the majority of the sum contributions show up it the sum twice but with the opposite sign and therefore cancels out we get the expression for the total area with the accuracy up the the small term in $dx$, $\Theta (dx)$ which contains the endpoints terms, as
    $P=F(x_{n})-F(x_{0})+\Theta (dx)=F(b)-F(a)+\Theta (dx)$.
This proves the theorem. Note that the prove readily extends to the function $f$ that is nether continuous or positive everywhere when the above arguing is used piecewise in intervals between the sign change (when the function $f$ is negative the area between the x-axis and the function graph is than taken negative or with the negative sign) or between the discontinuity (the primitive function $F$ stays continuous but its curve may have sharp edges and $f$ may have a sudden jump in value).
# Numerical Analysis/Inverse iteration exercises

Consider the matrix, $A=\left[{\begin{array}{c c c}6&2&-1\\2&5&1\\-1&1&4\end{array}}\right]$, and the vector, ${\textbf {x}}^{(0)}=\left[{\begin{array}{c}1\\1\\1\\\end{array}}\right].$
The LU decomposition of A is $L=\left[{\begin{array}{c c c}1&0&0\\0.3333&1&0\\-0.167&0.3077&1\end{array}}\right],~~U=\left[{\begin{array}{c c c}6&2&-1\\0&4.333&1.333\\0&0&3.423\end{array}}\right]$.
By hand, use the LU decomposition to do two iterations of the inverse power method (without shift) starting with ${\textbf {x}}^{(0)}.$
Solution, first iteration:
$LU{\textbf {y}}^{(1)}={\textbf {x}}^{(0)}\Rightarrow {\textbf {y}}^{(1)}=\left[{\begin{array}{c}0.1910\\0.0674\\0.2809\end{array}}\right]$. The estimated eigenvalue is $\mu _{1}={\frac {1}{0.2809}}=3.56$ and the estimated eigenvector is ${\textbf {x}}^{(1)}={\frac {{\textbf {y}}^{(1)}}{0.2809}}=\left[{\begin{array}{c}0.6800\\0.2400\\1.0000\end{array}}\right]$.
Solution, second iteration:
$LU{\textbf {y}}^{(2)}={\textbf {x}}^{(1)}\Rightarrow {\textbf {y}}^{(2)}=\left[{\begin{array}{c}0.1996\\-0.0966\\0.3240\end{array}}\right]$. The estimated eigenvalue is $\mu _{2}={\frac {1}{0.3240}}=3.0864$ and the estimated eigenvector is ${\textbf {x}}^{(2)}={\frac {{\textbf {y}}^{(2)}}{0.3240}}=\left[{\begin{array}{c}0.6158\\-0.2982\\1.0000\end{array}}\right]$.
Use a computational software package to do 50 iterations. What was the result?
Solution, 50 iterations:
The estimated eigenvalue after 50 iterations is $\mu _{50}={\frac {1}{0.4375}}=2.2855$ and the estimated eigenvector is ${\textbf {x}}^{(50)}=\left[{\begin{array}{c}0.7750\\-0.9394\\1.0000\end{array}}\right]$.
# Numerical Analysis/LU decomposition concept quiz

Choose the best answer for each question:
# Numerical Analysis/Matlab Sample Code

Matlab code given below can also be used for excuting Romberg Integration method.
Sample Matlab code:
Resource: Numerical Methods using Matlab by Mathews
# Numerical Analysis/Lagrange exercise

Find the interpolating polynomial passing through the points $(1,2)$, $(3,4)$, $(5,6)$, $(7,8)$, using the Lagrange method.
Solution:
By using the Lagrange method, we need to find the lagrange basis polynominals first.Since we know
    ${\begin{aligned}x_{0}&=1&&&&&f(x_{0})&=2\\x_{1}&=3&&&&&f(x_{1})&=4\\x_{2}&=5&&&&&f(x_{2})&=6\\x_{3}&=7&&&&&f(x_{3})&=8.\end{aligned}}$
So we can get the basis polynominals as following:
    $\ell _{0}(x)={x-x_{1} \over x_{0}-x_{1}}\cdot {x-x_{2} \over x_{0}-x_{2}}\cdot {x-x_{3} \over x_{0}-x_{3}}=-{1 \over 48}(x-3)(x-5)(x-7)$
    $\ell _{1}(x)={x-x_{0} \over x_{1}-x_{0}}\cdot {x-x_{2} \over x_{1}-x_{2}}\cdot {x-x_{3} \over x_{1}-x_{3}}={1 \over 16}(x-1)(x-5)(x-7)$
    $\ell _{2}(x)={x-x_{0} \over x_{2}-x_{0}}\cdot {x-x_{1} \over x_{2}-x_{1}}\cdot {x-x_{3} \over x_{2}-x_{3}}=-{1 \over 16}(x-1)(x-3)(x-7)$
    $\ell _{3}(x)={x-x_{0} \over x_{3}-x_{0}}\cdot {x-x_{1} \over x_{3}-x_{1}}\cdot {x-x_{2} \over x_{3}-x_{2}}={1 \over 48}(x-1)(x-3)(x-5).$
Thus the interpolating polynomial then is:
    ${\begin{aligned}L(x)&=f(x_{0})\ell _{0}(x)+f(x_{1})\ell _{1}(x)+f(x_{2})\ell _{2}(x)+f(x_{3})\ell _{3}(x)\\[10pt]&=2\cdot {1 \over 16}(x-1)(x-5)(x-7)+4\cdot {-1 \over 16}(x-1)(x-3)(x-7)+6\cdot {-1 \over 16}(x-1)(x-3)(x-7)+8\cdot {1 \over 48}(x-1)(x-3)(x-5)\\[10pt]&={\frac {5}{12}}x^{3}-{\frac {65}{12}}x^{2}+{\frac {247}{12}}x-{\frac {163}{12}}.\end{aligned}}$
Therefore, we get the Lagrange form interpolating polynomial:
$L(x)={\frac {5}{12}}x^{3}-{\frac {65}{12}}x^{2}+{\frac {247}{12}}x-{\frac {163}{12}}.$
# Numerical Analysis/Learning projects and learning resources

Since there is already a lot of material on this topic freely available, we will try to organize and supplement, rather than duplicate. In particular, here we collect exercises and quizzes, which would be inappropriate for Wikipedia. The topics are organized and ordered as in Wikipedia's w:List of numerical analysis topics. A topic is only listed here if we have something to add, such as a Wikibook link (prefaced with b:), a learning module here, further examples, exercises, quizzes, or links to external learning resources.
# Numerical Analysis/Intro

This topic page is for organising the development of Numerical analysis content on Wikiversity.
If you are knowledgeable in any area Numerical analysis, feel free to improve, we would greatly appreciate your contributions.
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of continuous mathematics (as distinguished from discrete mathematics).
As a topic of study, it has a large overlap with Numerical Algorithms and Scientific Computing.