OpenLLM-France/wikimedia huggingface dataset:
# Real numbers/Convergent sequences/Rules/3/Fact/Proof/Exercise

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real convergent sequence and $c\in \mathbb {R}$. Show that the sequence ${\left(c\cdot x_{n}\right)}_{n\in \mathbb {N} }$ is also convergent and
    $\lim _{n\rightarrow \infty }{\left(c\cdot x_{n}\right)}=c\cdot {\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,$
holds.
# Real numbers/Convergent sequence/Bounded/Fact/Proof

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be the convergent sequence with $x\in \mathbb {R}$ as its limit. Choose some $\epsilon >0$. Due to convergence there exists some $n_{0}$ such that
    $\vert {x_{n}-x}\vert \leq \epsilon {\text{ for all }}n\geq n_{0}.$
So in particular
    $\vert {x_{n}}\vert \leq \vert {x}\vert +\vert {x-x_{n}}\vert \leq \vert {x}\vert +\epsilon {\text{ for all }}n\geq n_{0}.$
Below $n_{0}$ there are ony finitely many members, hence the maximum
    $B:=\max _{n<n_{0}}\{\vert {x_{n}}\vert ,\,\vert {x}\vert +\epsilon \}\,$
is welldefined. Therefore $B$ is an upper bound and $-B$ is a lower bound for ${\left\{x_{n}\mid n\in \mathbb {N} \right\}}$.
# Real numbers/Convergent sequence/Cauchy sequence/Fact

Every convergent sequence
# Real numbers/Convergent sequences/Compare/Fact

Suppose that ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(y_{n}\right)}_{n\in \mathbb {N} }$ are convergent sequences such that $x_{n}\geq y_{n}$ for all $n\in \mathbb {N}$.
$\lim _{n\rightarrow \infty }x_{n}\geq \lim _{n\rightarrow \infty }y_{n}$
# Real numbers/Convergent sequence/Cauchy sequence/Fact/Proof

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a convergent sequence with limit $x$. Let $\epsilon >0$ be given. We apply the convergence property for $\epsilon /2$. Therefore there exists an $n_{0}$ with
    $\vert {x_{n}-x}\vert \leq \epsilon /2{\text{ for all }}n\geq n_{0}.$
For arbitrary $n,m\geq n_{0}$ we then have due to the triangle inequality
    $\vert {x_{n}-x_{m}}\vert \leq \vert {x_{n}-x}\vert +\vert {x-x_{m}}\vert \leq \epsilon /2+\epsilon /2=\epsilon \,.$
# Real numbers/Convergent sequences/Compare/Fact/Proof/Exercise

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(y_{n}\right)}_{n\in \mathbb {N} }$ be convergent real sequences with $x_{n}\geq y_{n}$ for all $n\in \mathbb {N}$. Prove that $\lim _{n\rightarrow \infty }x_{n}\geq \lim _{n\rightarrow \infty }y_{n}$ holds.
# Real numbers/Convergent sequence/Average sequence/Exercise

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a convergent real sequence with limit $x$. Show that the sequence defined by
    $y_{n}:={\frac {x_{0}+x_{1}+\cdots +x_{n}}{n+1}}\,$
also converges to $x$.
# Real numbers/Convergent sequence/Bounded/Fact

A convergent real sequence
# Real numbers/Convergent sequence/Modulus/Exercise

${\left(x_{n}\right)}_{n\in \mathbb {N} }$
$x$
    ${\left(\vert {x_{n}}\vert \right)}_{n\in \mathbb {N} }$
$\vert {x}\vert$
# PlanetPhysics/Quantum Fundamental Groupoid 4

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### Fundamental groupoid functors and functor categories

The natural setting for the definition of a quantum fundamental groupoid $F_{\mathbb {Q} }$ is in one of the functor categories-- that of fundamental groupoid functors, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}} , and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathbb {Q} }$ represented by Hilbert space bundles or rigged Hilbert (also called Frech\'et) spaces ${\mathbb {H} }_{B}$.
Other related functor categories are those specified with the general definition of the fundamental groupoid functor, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}: '''Top''' \to \grp_2} , where Top is the category of topological spaces and Failed to parse (unknown function "\grp"): {\displaystyle \grp_2} is the groupoid category.
A specific example of a quantum fundamental groupoid can be given for spin foams of spin networks, with a spin foam defined as a functor between spin network categories. Thus, because spin networks or graphs are specialized one-dimensional CW-complexes whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a functor representation of CW-complexes on rigged Hilbert spaces (also called Frech\'et nuclear spaces).
# PlanetPhysics/Quantum Categories

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }
The category of quantum groupoids, Failed to parse (unknown function "\grp"): {\displaystyle [Q_{\grp}]_i} , is trivially a subcategory of the groupoid category, that can also be regarded as a functor category, or $2$-category, if Failed to parse (unknown function "\grp"): {\displaystyle \grp} is small, that is, if $G^{0}$ is a set rather than a class.
A physical mathematics definition of quantum category has also been reported as a rigid monoidal category, or its equivalents.
# PlanetPhysics/Quantum Fundamental Groupoid

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

### Fundamental Groupoid Functors in Quantum Theories

The natural setting for the definition of a quantum fundamental groupoid $F_{\mathbb {Q} }$ is in one of the functor categories-- that of fundamental groupoid functors, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}} , and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathbb {Q} }$ represented by Hilbert space bundles or `rigged' Hilbert (or Frech\'et) spaces ${\mathbb {H} }_{B}$.
Let us briefly recall the description of quantum fundamental groupoids in a quantum functor category, ${\mathbb {Q} }_{F}$: The quantum fundamental groupoid , QFG is defined by a functor $F_{\mathbb {Q} }:\mathbb {H} _{B}\to {\mathbb {Q} }_{G}$, where ${\mathbb {Q} }_{G}$ is the category of quantum groupoids and their homomorphisms.

#### Fundamental Groupoid Functors

Other related functor categories are those specified with the general definition of the fundamental groupoid functor , Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}: '''Top''' \to \grp_2} , where Top is the category of topological spaces and Failed to parse (unknown function "\grp"): {\displaystyle \grp_2} is the groupoid category.

#### Specific Example of QFG

One can provide a physically relevant example of QFG as spin foams, or functors of spin networks; more precise the spin foams were defined as functors between spin network categories that realize dynamic transformations on the spin space. Thus, because spin networks (or graphs) are specialized one-dimensional CW-complexes whose cells are linked quantum spin states their quantum fundamental groupoid is defined as a representation of CW-complexes on `rigged' Hilbert spaces, that are called Frech\'et nuclear spaces .
# PlanetPhysics/Quantum Chromodynamics

QCD or Quantum chromodynamics is the advanced, standard mathematical and quantum physics treatment, or theory, of strong force or nuclear interactions such as those among quarks and gluons, partons, `Yukawa' mesons, and so on, with an intrinsic threefold symmetry for RGB quarks (or `eightfold-way' diagrams resulting from representations of the quantum group $SU(3)$ first reported by the US Nobel Laureate Gell-Mann and others. This is not only a rather `colorful' theory (as its name suggests) but also a very highly formalized, mathematical one that affords major simplifications by postulating intrinsic symmetries of magnetic--like `color', `flavor', `strangeness' and top/down quark (and anti-quark) intrinsic properties, each time involving three color charges. Single quarks , such as the $u$ or $d$ ones have however never been observed, with the proton and neutron `consisting of' three such quarks with a resulting `white' color, or charge colorless proton and neutron, as well as stable `white' nuclei `consisting of' the latter two quantum particles, dynamically confined by the very short range, nuclear strong interactions. The quark interactions are mediated by gluons --as well as their exchange-- and the latter also carry charge color--but unlike the photons that mediate the electromagnetic interactions in QED-- gluons have multiple interactions with each other leading to major computational difficulties in QCD, that are not encountered in QED. Major obstacles in QCD computations of observable nuclear (quantum) eigenvalues are therefore encountered in attempting approximate, perturbative approaches that work extremely well for electromagentic interactions (governed by the charge $U(1)$ symmetry group), for example with Richard Feynman's approach in QED. Electro-weak (QEW) interactions were successfully approached in QED--like fashion but with quantum field carriers that are--unlike the photon--massive, and therefore the electro-weak interactions have limited range, unlike the photons of zero mass at rest. Thus, QCD and QED are more than just `one pole apart', as $U(1)$ and $SU(3)$ are very different group symmetries. This makes obvious the need for more fundamental, or extended quantum symmetries, such as those afforded by either several larger groups such as $U(1)\times SU(2)\times SU(3)$, or by spontaneously broken, multiple (or localized) symmetries of a less restrictive kind present in quantum groupoids, such as for example in weak Hopf algebra representations, locally compact groupoid Failed to parse (syntax error): {\displaystyle G_{lc } }, unitary representations, and so on, to the higher dimensional (quantum) symmetries of quantum double groupoids, quantum double algebroids, quantum categories/ supercategories and/or quantum supersymmetry superalgebras (or graded `Lie' algebras, see- for example- the QFT books by Weinberg (1995, 2003) superalgebroids in quantum gravity, or in QCD of the extremely hot, very early, physical Universe, extremely close to the time of the `Big Bang'.
# PlanetPhysics/Quantum Geometry 3

This is a contributed topic on spacetime quantization and loop quantum gravity often descibed as quantum geometry.

## Quantum Geometry

In 4 dimensions, one of the attractive programs of spacetime quantization is "quantum geometry", often represented as "loop quantum gravity" . Loop quantum gravity starts with a Hamiltonian formulation of the first order formalism, with constraints, written in analogy to the (3+1)-dimensional case that take the form:
$D_{i}E^{ia}=0,$ $E_{a}^{i}R_{ij}^{a}=0,$ and $\epsilon _{abc}E^{ib}E^{jc}R_{ij}^{a}=0,$
where the indices $i,j$ and $k$ are the spatial indices on a surface of constant time, $E^{ia}=\epsilon ^{ij}e_{j}^{a},$ $D_{i}$ is the $SO(2,1)$ gauge-covariant derivative for the connection $\omega$, and the $R_{ij}^{a}$ are the spatial components of the curvature two-form.

### Lattice methods and Spin Foams

Ponzano-Regge and Turaev-Viro models are examples of "spin foam" models that is, they are models based on simplicial complexes with faces, edges, and vertices labeled by group representations and intertwiners. spin foam models are based on a fixed triangulation of spacetime, with edge lengths serving as the basic gravitational variables. An alternative scheme is "dynamical triangulation", in which edge lengths are fixed and the path integral is represented as a sum over triangulations.
Dynamical triangulation is a useful alternative to spin foams that has been shown to provide a useful method in two-dimensional gravity.

### Discussion

Several quantum observables whose expectation values generally give topological information about the nature of quantized spacetime have been already considered but-- with very few exceptions-- the results in this area has remain largely mathematical in nature; thus, surprisingly little is understood about the physics of such observables, although some are most likely to be related to length and perhaps volumes, whereas other observables are connected to scattering amplitudes for quantum paricles.
"Perhaps the most important lesson of (2+1)-dimensional quantum gravity is that general relativity can, in fact, be quantized." (download here a concise online review)
# PlanetPhysics/Quantum Gravity Programs 2
### Quantum Gravity Programs

There are several distinct programs aimed at developing a quantum gravity theory. These include--but are not limited to-- the following.
* $\bullet$ The Penrose, twistors programme applied to an open curved space-time (ref.), (which is presumably a globally hyperbolic, relativistic space-time). This may also include the idea of developing a `sheaf cohomology' for twistors (ref. \cite {Hawking and Penrose}) but still needs to justify the assumption in this approach of a charged, fundamental fermion of spin-3/2 of undefined mass and unitary `homogeneity' (which has not been observed so far);
* $\bullet$ The Weinberg, supergravity theory, which is consistent with supersymmetry and superalgebra, and utilizes graded Lie algebras and matter-coupled superfields in the presence of weak gravitational fields;
* $\bullet$ The programs of Hawking and Penrose) in quantum cosmology, concerned with singularities, such as black and `white' holes; S. W. Hawking combines, joins, or `glues' an initially flat Euclidean metric with convex Lorentzian metrics in the expanding, and then contracting, space-times with a very small value of Einstein's cosmological `constant'. Such `Hawking', double-pear shaped, space-times also have an initial Weyl tensor value close to zero and, ultimately, a largely fluctuating Weyl tensor during the `final crunch' of our Universe, presumed to determine the irreversible arrow of time; furthermore, an observer will always be able to access through measurements only a limited part of the global space-times in our universe;
* $\bullet$ The TQFT/ approach that aims at finding the `topological' invariants of a manifold embedded in an abstract vector space related to the statistical mechanics problem of defining extensions of the partition function for many-particle quantum systems;
* $\bullet$ The string and superstring theories/M-theory that `live' in higher dimensional spaces (e.g., $n\geq 6$, preferred $n-dim=11$), and can be considered to be topological representations of physical entities that vibrate, are quantized, interact, and that might also be able to 'predict' fundamental masses relevant to quantum 'particles';
* $\bullet$ The Baez `categorification' programme (,) that aims to deal with quantum field and QG problems at the abstract level of categories and functors in what seems to be mostly a global approach;
* $\bullet$ The `monoidal category' and valuation approach initiated by Isham (ref.) to the quantum measurement problem and its possible solution through local-to-global, finite constructions in small categories.
Most of the quantum gravity programs are consistent with the Big-Bang theory, or the theory of a rapidly expanding universe, although none `prove' the necessity of its existence. Several competing and conflicting theories were reported that deal with singularities in spacetime, such as black holes `without hair', evaporating black holes and naked singularities.
# PlanetPhysics/Quantum Circuit
## Quantum circuit simulation

The following is an imaginary experiment or simulation of a quantum circuit avalaible as a free download from \htmladdnormallink{$Mathematica^{(TM)}$ website}{http://demonstrations.wolfram.com/GeneratingEntangledQubits/}. The link shows how a `quantum circuit' of the selected design might perform a Hadamard operation followed by a simple CNOT operation with two qubits thus yielding an entangled qubit (or E.P.R) pair which is one the four Bell basis states (cf. Brad Rubin, the author of this simulation).
Such quantum circuits represent hypothetical possibilites for building a quantum `computer' that would handle qubits of information instead of the regular bits in the existing, Boolean logic computers.
# Node.js

Node.js is an open-source, cross-platform, back-end JavaScript runtime environment that runs on the V8 engine and executes JavaScript code outside a web browser. Node.js lets developers use JavaScript to write command line tools and for server-side scripting—running scripts server-side to produce dynamic web page content before the page is sent to the user's web browser. Consequently, Node.js represents a "JavaScript everywhere" paradigm, unifying web-application development around a single programming language, rather than different languages for server-side and client-side scripts.

## See Also

* JavaScript Programming
* Server-Side Scripting
# Noli Me Tangere

Noli Me Tángere, Latin for "Touch me not", is an 1887 novel by José Rizal during the colonization of the Philippines by Spain to describe perceived inequities of the Spanish Catholic friars and the ruling government.

## Quizzes

* Chapter 1 Quiz

## See Also

* Wikipedia: Noli Me Tángere (novel)
# Nobel Prizes
## Definitions

Nobel, Alfred Bernhard 1833-1896; Swedish industrialist, philanthropist, and inventor of dynamite: established the Nobel prizes.
(Americanism) Nobelist, Nobellist; noun. A person who has been awarded a Nobel prize.
Nobel prizes; annual international prizes given by the Nobel Foundation for distinction in physics, chemistry, economics, medicine or physiology, and literature, and for promoting peace.
There is scattered information on the Nobel prizes throughout the Wikiversity. I hope to gather most of it together here. I intend to add more information, including dates, recipients, and a summary of their contributions, for the following categories.

## Prizes awarded to More than One Recipient

"A prize may not be shared among more than three people."
Ray Calvin Baker (talk) 18:14, 30 April 2012 (UTC)

## Information Sources

Webster's New World Dictionary, Second College Edition, 1980 and 1979, WILLIAM COLLINS PUBLISHERS, INC., ISBN 60B 0-529-05324-1
Ray Calvin Baker (talk) 20:32, 5 April 2012 (UTC)
# Noli Me Tangere/Chapter 1 Quiz
## Conceptual questions

* If you were Padre Damaso, will you do the same thing like expressing disrespect of your country because of uncontrollable temper? Or will you do the same ill-tempered act as shown by Tenyente Guevara?
* If you are about to decide what will be the role of the lieutenant, how will you tell Padre Damaso that he is speaking in a very offensive way?
* What are the visitors talking about? Is it rightful to talk about this matter inside a Filipino's house? Or in others' house? Why?
# Non-Formal Education

This Education resource is a "stub". Learn how you can help Wikiversity to develop it.
Non-formal Education is the education you get, skills you acquire, training you get in e.g. a youth movement, by taking up responsabilities, using the space you get there, using international exchange programs / trainings they offer or give access to.
# Noetherian ring/Positive characteristic/Existence of Hilbert-Kunz multiplicity/Fact

Let $K$ denote a field of positive characteristic $p$, let $K\subseteq R$ be a noetherian ring and let $I\subseteq R$ be an ideal which is primary to some maximal ideal.
$e_{HK}(I)$
# Noetherian scheme/Affine/Cohomological criterion/Fact

Let $X$ denote a noetherian scheme. Then the following properties are equivalent.
* $X$ is an affine scheme.
* For every quasicoherent sheaf ${\mathcal {F}}$ on $X$ and all $i\geq 1$ we have $H^{i}(X,{\mathcal {F}})=0$.
* For every coherent ideal sheaf ${\mathcal {I}}$ on $X$ we have $H^{1}(X,{\mathcal {I}})=0$.
# Path diagram

Path diagrams are used to visually depict conceptual and statistical relationships between variables.

## Examples

* Example path diagram (conceptual) (.doc MSWord format)
# Pathology

Pathology derives from pathos (disease ) and logos ( study ). Literally meaning "the study of disease". Pathology emphasized four aspects of the disease process. These are the cause (etiology), the mechanism of development (pathogenesis), the alterations of structure and forms (morphology) and functional alterations (pathophysiology). Due to advances and increased complexity of medical sciences, pathology has many branches as clinical medicine. For academic purposes, Pathology has been divided in general and systemic (organ) pathology. Modern pathology has its origin in the cellular studies carried out by the German doctor Rudolf Virchow. After Virchow, cellular adaptations and changes secondary to disease processes have been the main study subject of this medical science.

## Resources

* Open Pathology Project
* w: Pathology
* w: Fibrosis
# Payment Channel

A bidirectional and trustless payment channel is the core building block of the Lightning Network. There are 3 well known methods to construct payment channels.

## Poon Dryja Channels

The Poon Dryja channels are the ones that are currently being implemented in the lightning network. They punish malicious behaviour with a penalty based revocation based system making use of RSMCs and are presented and discussed in the lightning network whitepaper

## Decker Wattenhofer

Based on the idea of invalidation trees and first presented on the ETH Zurich in A Fast and Scalable Payment Network with Bitcoin Duplex Micropayment Channels

## Eltoo

Eltoo tries to reduce the complexity of the penalty based payment channels by dynamically binding settlement transactions to funding transactions. While elegant this method requires a bitcoin softfork. Eltoo was first presented via Blockstream in this publication
# Payment Documentation Requirements/Assessment

| Unit 4.7-Payment Documentation Requirements | Introduction | Required Documents | Summary | Resources | Activities | Assessment Go to Unit 4.1 | Unit 4.2 | Unit 4.3 | Unit 4.4 | Unit 4.5 | Unit 4.6 | Unit 4.7 | Unit 4.8 | Unit 4.9 Return to Trade Finance Module 4: Payment Methods |
# Payment Documentation Requirements

| Unit 4.7-Payment Documentation Requirements | Introduction | Required Documents | Summary | Resources | Activities | Assessment Go to Unit 4.1 | Unit 4.2 | Unit 4.3 | Unit 4.4 | Unit 4.5 | Unit 4.6 | Unit 4.7 | Unit 4.8 | Unit 4.9 Return to Trade Finance Module 4: Payment Methods |

## Unit 4.7-Payment Documentation Requirements

An international manager must be able to identify major documents utilized in international trade. It is also necessary to know when and how these documents may be applied, who is responsible for creating the documents, and who uses the documents.

### Unit Objective

The goal of this material is to introduce US to the documentation required to facilitate international business transactions and the way each relates to receipt of timely payment of the sale of goods and/or services in an international transaction. By the end of this unit you will be able to
* identify the major documents utilized in international trade (commercial invoices, transport documents, etc.).
* identify how and when they are used.
* identify which party of the transaction is responsible for creating each document.

### Unit Outline

* Introduction
* Required Documents and How and When to Apply Documents
* Summary
* Resources
* Activities
* Assessment

### Resources

Correlation: Materials from this unit correlate with NASBITE CGCP's Knowledge Statement 04/04/07: Knowledge of documentation requirements (e.g., commercial invoices, transport documents and documents relating to services, inspection certificate)
# Pauli matrices/Qubit

quantum bit
n. The smallest unit of information in a computer designed to manipulate or store information through effects predicted by quantum physics. Unlike bits in classical systems, a quantum bit has more than two possible states: a state labeled 0, a state labeled 1, and a combination of the two states that obeys the superposition principle.