OpenLLM-France/wikimedia huggingface dataset:
# Photography Equipment/DSLR Systems

Unless one purchases and uses only a single body and the kit lens, one's equipment is not a DSLR camera, but a DSLR system, meaning a compatible set of bodies, lenses, and accessories. Indeed, SLR and DSLR bodies in the same system can be interchanged (notably, the same lenses can be used). However, bodies from one system (e.g. Canon EOS) cannot generally be used with lenses from another system (e.g. Nikon F-mount), which becomes a major concern if one has a significant investment in lenses.
Because components are not interchangeable between systems, photographers typically choose one system and remain with it for as long as possible. Few consider owning multiple systems to be economical and switching between systems often entails considerable expense.
The key consideration in selecting a system is the lenses and bodies that one expects to use in future, typically over a 10-30 year time frame: one can expect to acquire additional lenses, and periodically upgrade the body.

## Canon and Nikon

The two leading DSLR systems are Canon and Nikon, or more properly the Canon EOS system with Canon EF-mount and the Nikon F-mount system, referring to the lens systems. There are other brands, detailed below, but Canon and Nikon are the preeminent systems, with combined market share of about 89%. While there are significant differences, discussed at Canon and Nikon, they are both widely respected and widely used.
Canon and Nikon have broad ranges of lenses and accessories, and extensive support, so one can easily:
* rent lenses for them,
* purchase components and get repairs, and
* resell equipment in a large market.

## Other brands

Other brands of DSLRs include:
* Fujifilm
* Kodak
* Leica
* Olympus
* Panasonic
* Pentax : Pentax K mount (photo.net) (dpreview.com)
* Sigma (notable for using the Foveon X3 sensor)
* Sony (was Konica Minolta) : Sony Alpha mount (photo.net) (dpreview.com)
They use various lens mounts, and have much smaller selections of lenses.

### Why use another brand?

If you are building a new system, any of the major manufacturers can be considered a good choice. Factors to consider would be:

#### Legacy glass

    If you have a large collection of older lenses, a DSLR from the same manufacturer may allow you to continue to use them.

#### Features

    Different manufacturers take different approaches to the same issues, and you may find one suits you better than the others. An example would be image stabilisation – some makers have chosen to include it as an in-body option, which thus applies to all lenses, while others build stabilising mechanics into the lenses themselves.

#### Price

    Naturally, the perceived price to performance ratio will be a major consideration for most buyers.

## References

* ↑ http://www.kenrockwell.com/olympus/lenses.htm
# Photography Equipment/Lenses/Canon standard

A standard lens is a general-purpose lens, which captures the middle of the focal length range. These can be a normal lens (standard prime), or a standard zoom, often the kit lens.
Some common recommendations follow.

## APS-C

Canon's small sensor cameras mostly use a 1.6x crop factor (except for the 1D series, which is 1.3), so a normal lens would be 50mm/1.6 = 31.25mm, and focal lengths for standard zooms are correspondingly shorter than on full frame standard zooms.

### Standard Zoom

Kit lenses
* Canon EF-S 18-55mm: kit on Rebel series, considered very soft
    FF FOV equivalent of 29-88mm, EF-S analog of Canon EF 28-90mm
* Canon EF-S 17-85mm IS USM: kit on 30D, 40D
    FF FOV equivalent of 27-136mm, EF-S analog of Canon EF 28-135mm
    (The-Digital-Picture review) (Bob Atkins review)
Non-kit lenses
* Canon EF-S 17-55mm ($1,300) excellent standard lens
    FF FOV equivalent of 27-88mm
    Notable features: high-quality glass (similar to L series, though lower build quality), and at f/2.8, quite fast for a zoom.
    (Philip Greenspun Review at photo.net) (Fred Miranda Review) (The-Digital-Picture.com Review) (dpreview.com Review)

### Standard Prime

A normal lens on a 1.6x crop factor camera would be about 31mm. Canon doesn't make a prime lens at this point, but does make some EF wide-angle lenses that can work. Others instead suggest the Sigma 30mm f/1.4 as the best prime normal lens for Canon APS-C cameras.
* Canon EF 28mm: f/2.8, f/1.8
* Canon EF 35mm: f/2 ($200); f/1.4 L USM ($1,100)
* Sigma 30mm f/1.4 for Canon ($400)
    Recommended by Philip Greenspun at photo.net.

## Full Frame
### Standard Zoom

* Canon EF 28-135mm

### Standard Prime

* a Canon EF 50mm: (Comparison of Canon 50mm lenses, at The-Digital-Picture) (Other comparison, at photo.net)
** 50/1.8 II is high quality and excellent value ($75),
** 50/1.4 is more expensive ($300) but widely praised,
** and 50/1.2 is very expensive ($1,300) but very fast.
# Ordinary Differential Equations/Applications to Linear Equations
## Existence of Solutions

Just like with separable equations, not all initial value problems for linear equations have a solution.
    Theorem 1: If P(x) and Q(x) are continuous on an interval I containing the point $x_{0}$, then the initial value problem has a single unique solution.
This is different from separable equations where the conditions for uniqueness and existence are different - with linear equations, if it exists, it will be unique.
Proof We will use the method of successive approximations just as we did before.
# Order Theory/Series-parallel order

Definition (parallel order):
Let $(S_{\alpha },\leq _{\alpha })_{\alpha \in A}$ be a family of ordered sets, and define $S:=\bigsqcup _{\alpha \in A}S_{\alpha }$. Then the parallel order on $S$ is defined to be the order
    $s\leq t:\Leftrightarrow \left(\exists \alpha \in A:s\in S_{\alpha }\wedge t\in S_{\alpha }\right)\wedge s\leq _{\alpha }t$.
Definition (parallel composition):
Let $(S_{\alpha },\leq _{\alpha })_{\alpha \in A}$ be a family of ordered sets, and define $S:=\bigsqcup _{\alpha \in A}S_{\alpha }$. Then the pair $(S,\leq )$, where $\leq$ is the parallel order on $S$, is called the parallel composition of the sets $S_{\alpha }$.
Definition (series order):
Let $(A,\leq _{A})$ be a preordered set, and let $(S_{\alpha },\leq _{\alpha })_{\alpha \in A}$ be a family of ordered sets over $A$. The series order on $S:=\bigsqcup _{\alpha \in A}S_{\alpha }$ induced by $A$ is the order $\leq$ on $S$ given by
    $s\leq t:\Leftrightarrow \left(\left(\exists \alpha \in A:s\in S_{\alpha }\wedge t\in S_{\alpha }\right)\wedge s\leq _{\alpha }t\right)\vee \exists \alpha ,\beta \in A:\alpha \leq \beta \wedge s\in S_{\alpha }\wedge t\in S_{\beta }$.
Definition (series composition):
Let $(A,\leq _{A})$ be a preordered set, and let $(S_{\alpha },\leq _{\alpha })_{\alpha \in A}$ be a family of ordered sets, and define $S:=\bigsqcup _{\alpha \in A}S_{\alpha }$. Then the pair $(S,\leq )$, where $\leq$ is the series order on $S$, is called the series composition of the sets $S_{\alpha }$ over $(A,\leq _{A})$.
Definition (series-parallel order):
A series-parallel order is the order of an ordered set that arises from a family of singleton ordered sets $(\{x_{\alpha }\},\leq _{\alpha })_{\alpha \in A}$ (the $\leq _{\alpha }$ order being the order that turns $(\{x_{\alpha }\},\leq _{\alpha })$ into a poset) by applying parallel composition and series composition over $A=(\{1,2\},\{(1,2)\})$ a finite number of times.
# Ordinary Differential Equations

* Definition, reduction of explicit equations to first order
* Existence and uniqueness of solutions
* Differential inequalities
* Solutions to specific equations
** One-dimensional time-independent equations of first or second order

## OLD TOC

This book aims to lead the reader through the topic of differential equations, a vital area of modern mathematics and science. This book provides information about the whole area of differential equations, concentrating first on the simpler equations.

## Table of contents

* Introduction
* Preliminaries from calculus
* Form and Solutions of Differential Equations
* First-Order Differential Equations
** Separation of Variables
** Linear Differential Equations
** Exact Differential Equations
** Substitution Methods
** Bernoulli Equations
** Ricatti Equations
** Orthogonal and Oblique Trajectories
** Equations of higher degrees
** Equations without x or y
** Equations that are homogeneous in x and y
** d'Alembert's Equation
** Clairaut Equations
** Legendre Transformations
** Graphing Differential Equations
* Second-Order Differential Equations
** Constant Coefficients
** Series Solutions
** Hypergeometric Equation
** Frobenius Solution to the Hypergeometric Equation
** Legendre Equation
** Bessel Equation
** Mathieu Equation
** Continued Fraction Solutions
** Applications of Second-Order Differential Equations
* Higher Order Differential Equations
** Linear equations
*** General Linear Equations
*** Infinite Series Solutions to Linear Equations
** Integration methods
*** Laplace Transform
*** Bessel Function
*** Euler Transform
*** Mellin Transform
*** Hypergeometric Series
*** Double Integration
* Sturm-Liouville theory
* Systems of linear differential equations
* Nonlinear Systems
* Green's Functions
* Existence and Uniqueness of Solutions
** The Picard–Lindelöf theorem
** Peano's theorem
** Blow-ups and moving to boundary
** Global uniqueness of solution over interval
** Maximum domain of solution
** The Successive Approximations Method of Proof
** Applications to Linear Equations
** The Cauchy-Lipschitz Method of Proof
** Existence Theorems for Complex Numbers
* Continuous Transformation Groups
** Infinitesimal Transformations
** Invariant Functions
* Glossary
* List of Some Equations
* Help Needed
* Roadmap

## Sources

Differential Equations and Boundary Value Problems- C.H. Edwards Jr and David E. Penny MIT Open Courseware- http://ocw.mit.edu/index.html
* Kong, Qingkai (0000). A Short Course in Ordinary Differential Equations. Universe: Publisher.
* Walter, Wolfgang (1998). Ordinary Differential Equations. New York: Springer.
# Ordinary Differential Equations/Bernoulli

An equation of the form
${dy \over dx}+f(x)y=g(x)y^{n}$
can be made linear by the substitution $z=y^{1-n}$
Its derivative is
${dz \over dx}=(1-n)y^{-n}{dy \over dx}$
So that multiplying it by $y^{-n}$
The equation can be turned into
${dy \over dx}y^{-n}+f(x)y^{1-n}=g(x)$
Or
${dz \over dx}+(1-n)f(x)z=(1-n)g(x)$
Which is linear.

### Jacobi Equation

The Jacobi equation
$(a_{1}+b_{1}x+c_{1}y)(xdy-ydx)-(a_{2}+b_{2}x+c_{2}y)dy+(a_{3}+b_{3}x+c_{3}y)dx=0$
can be turned into the Bernoulli equation with the appropriate substitutions.
# Order Theory/Preordered classes and poclasses

Definition (preordered class):
A preordered class is a set $S$ together with a binary relation $\leq \subset S\times S$ satisfying the following axioms:
* $\forall s\in S:s\leq s$ (reflexivity)
* $\forall r,s,t\in S:(s\leq r\wedge r\leq t\Rightarrow s\leq t)$ (transitivity)
Definition (poclass):
A poclass (shorthand for partially ordered class) is a preordered class $(S,\leq )$ such that the following additional axiom is satisfied:
    3. $\forall s,t\in S:s\leq t\wedge t\leq s\Rightarrow s=t$ (antisymmetry)
Example (subsets of the power set are ordered by inclusion):
Let $S$ be any set, and let $\sigma \subset {\mathcal {P}}(X)$. Then the relation on $\sigma$ defined by
    $S\leq T:\Leftrightarrow S\subseteq T$
is an order on $\sigma$.
Definition (order homomorphism):
Let $(S,\leq )$ and $(T,\preceq )$ be preordered classes. An order homomorphism from $(S,\leq )$ to $(T,\preceq )$ is a class function $f:S\to T$ so that $x\leq y\Rightarrow f(x)\preceq f(y)$ for all $x,y\in S$.
Definition (isotonic class function):
Let $S,T$ be sets, and let $\leq _{S}$ be a preorder on $S$, and $\leq _{T}$ a preorder on $T$. A class function $f:S\to T$ is said to be isotonic with respect to $\leq _{S}$ and $\leq _{T}$ iff $f$ is an order homomorphism from $(S,\leq _{S})$ to $(T,\leq _{T})$.
Definition (antitonic class function):
Let $S,T$ be sets with preorders $\leq _{S},\leq _{T}$ respectively. Then an antitonic class function from $S$ to $T$ with respect to the partial orders $\leq _{S}$ and $\leq _{T}$ is a class function $f:S\to T$ such that
    $x\leq _{S}y\Rightarrow f(y)\leq _{T}f(x)$.
Definition (product order):
Let $(S_{\alpha },\leq _{\alpha })_{\alpha \in A}$ be a family of preordered classes. The product order on the cartesian product $\prod _{\alpha \in A}S_{\alpha }$ is the order given by
    $(s_{\alpha })_{\alpha \in A}\leq (t_{\alpha })_{\alpha \in A}:\Leftrightarrow \forall \alpha \in A:s_{\alpha }\leq t_{\alpha }$.
# Order Theory/Total orders

Definition (total order):
An order $\leq$ on a set $S$ is said to be total if and only if for each $x,y\in S$, exactly one of the possibilities $x<y$, $x=y$ or $x>y$ occurs.
{{proposition|series order induced by total orders is total|Whenever $A$ is a totally ordered set
Proposition (lexicographic order induced by total orders is total):
Whenever $A$ is a well-ordered set and $(S_{\alpha },\leq _{\alpha })_{\alpha \in A}$ are totally ordered sets, the lexicographic order on $\prod _{\alpha \in A}S_{\alpha }$ is total.
Proof: Let any two elements $(s_{\alpha })_{\alpha \in A}$ and $(t_{\alpha })_{\alpha \in A}$ of $\prod _{\alpha \in A}S_{\alpha }$ be given. Then either $(s_{\alpha })_{\alpha \in A}=(t_{\alpha })_{\alpha \in A}$, or there exists a smallest $\beta \in A$ so that $s_{\beta }\neq t_{\beta }$. Since $\leq _{\beta }$ is total, either $s_{\beta }<t_{\beta }$ or $s_{\beta }>t_{\beta }$, and thus either $(s_{\alpha })_{\alpha \in A}<(t_{\alpha })_{\alpha \in A}$ or $(s_{\alpha })_{\alpha \in A}>(t_{\alpha })_{\alpha \in A}$. $\Box$
# Order Theory/The Grothendieck axioms

AB3, AB3*, AB5 and AB5* are actually due to von Neumann, but Grothendieck's name stuck.
Definition (AB3):
A preordered class $(S,\leq )$ is said to satisfy the third Grothedieck axiom AB3 iff for all families $(x_{\alpha })_{\alpha \in A}$ indexed over a set $A$, a least upper bound $\bigvee _{\alpha \in A}x_{\alpha }$ exists.
Definition (AB3*):
A preordered class $(S,\leq )$ is said to satisfy the dual third Grothedieck axiom AB3* iff for all families $(x_{\alpha })_{\alpha \in A}$ indexed over a set $A$, a greatest lower bound $\bigwedge _{\alpha \in A}x_{\alpha }$ exists.
Definition (AB5):
A class ${\mathcal {B}}$ of preordered sets is said to satisfy the fifth Grothendieck axiom AB5 if and only if for all $(S,\leq )\in {\mathcal {B}}$ and all families of elements $(x_{\alpha })_{\alpha \in A}$ of elements of $S$ and each element $y\in S$, we have
    $\left(\bigvee _{\alpha \in A}x_{\alpha }\right)\wedge y=\bigvee _{\alpha \in A}(x_{\alpha }\wedge y)$
Definition (AB5*):
A class ${\mathcal {B}}$ of preordered sets is said to satisfy the dual fifth Grothendieck axiom AB5* if and only if for all $(S,\leq )\in {\mathcal {B}}$ and all families of elements $(x_{\alpha })_{\alpha \in A}$ of elements of $S$ and each element $y\in S$, we have
    $\left(\bigwedge _{\alpha \in A}x_{\alpha }\right)\vee y=\bigwedge _{\alpha \in A}(x_{\alpha }\vee y)$
Definition (AB6):
A class ${\mathcal {B}}$ of preordered sets is said to satisfy the sixth Grothendieck axiom AB6 if and only if for all $(S,\leq )\in {\mathcal {B}}$
# Ordinary Differential Equations/Applications of Second Order DEQs

There are several uses for second-order differential equations. In this chapter, I will cover the use of second-order differential equations to describe the motion of a mass at the end of a spring.
The chapter is broken up into three sections:
* Simple Harmonic Motion
* Motion with a Damping Force
* Motion with an Outside Force

## Chapter Notation

The formulae in this chapter are written with the following notation in mind. If you've learned a different manner of notation, please take note of the differences. I made every attempt to use a standard set of notation.

#### Important Terms

Terms that I feel deserve your undivided attention will appear like This. You will see this term referred to often in the text that follows, so it's recommended that you fully understand what it means.

#### Derivatives with respect to time

If a derivation is taken with respect to time (t), then an equivalent symbol is used and is pronounced x double-dot, x triple-dot, etc.
* Example:${\frac {d^{2}x}{dt^{2}}}\equiv {\ddot {x}}$

## Rendering MATH PNG Images

This chapter was written using the built-in TeX markup language present in MediaWiki. It's recommended that you view the chapter with your preferences set to render all Math in PNG. Check your preferences for this setting.
# Ordinary Differential Equations/Constant Coefficients
## Constant coefficients
### Lesson 2: Homogenous Equations

    Explanation
    Examples
    Probems
    Answers

### Lesson 3: Nonhomogenous Equations

    Explanation
    Examples
    Probems
    Answers
# Linear Algebra/Orthogonal Sets
## Orthogonal Sets

Given a set $A=(a_{1},a_{2},\ldots ,a_{n})$, where $a_{1}$ through $a_{n}$ are nonzero vectors of the same dimension, is an orthogonal set if
$a_{i}\cdot a_{j}=0$
where $i\neq j$.
So, for example, if one has a set of 3 vectors with the same dimension (for example $4\times 1$) and taking the dot product of each vector with each other vector all equal zero, it is an orthogonal set. This is illustrated below.

##### Example of Orthogonal Set

${\boldsymbol {\Omega }}=(\omega _{1},\omega _{2},\omega _{3})$
$\omega _{1}={\begin{bmatrix}1\\0\\2\\1\\\end{bmatrix}},\omega _{2}={\begin{bmatrix}2\\3\\-2\\2\\\end{bmatrix}},\omega _{3}={\begin{bmatrix}1\\0\\0\\-1\\\end{bmatrix}}$
We see that
$\omega _{1}\cdot \omega _{2}=0$
$\omega _{1}\cdot \omega _{3}=0$
$\omega _{2}\cdot \omega _{3}=0$
Thus, ${\boldsymbol {\Omega }}$ is an orthogonal set of vectors.
# Linear Algebra/Orthogonality

< Linear algebra

## Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality states that the magnitude of the inner product of two vectors is less than or equal to the product of the vector norms, or: $|\langle x,y\rangle |\leq \|x\|\|y\|$.

## Definition

For any vectors $x$ and $y$ in an inner product space $V$, we say $x$ is orthogonal to $y$, and denote it by $x\bot y$, if $\langle x,y\rangle =0$.

## Applications
### Linear least squares
### How to orthogonalize a basis

Suppose to be on a vector space V with a scalar product (not necessarily positive-definite),
Problem: Construct an orthonormal basis of V starting by a random basis { v$_1$, ... }.
Solution: Gram-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.
# Linear Algebra/Partitioned Matrices

A partitioned matrix also called a block matrix is a partition of a matrix into rectangular smaller matrices called blocks.
The matrix $P$ can be partitioned into 4 2×2 blocks
    $P={\begin{bmatrix}1&1&2&2\\1&1&2&2\\3&3&4&4\\3&3&4&4\end{bmatrix}}$
    $P_{11}={\begin{bmatrix}1&1\\1&1\end{bmatrix}},P_{12}={\begin{bmatrix}2&2\\2&2\end{bmatrix}},P_{21}={\begin{bmatrix}3&3\\3&3\end{bmatrix}},P_{22}={\begin{bmatrix}4&4\\4&4\end{bmatrix}}$
Then we can write the partitioned matrix like this
    $P_{\text{partitioned}}={\begin{bmatrix}P_{11}&P_{12}\\P_{21}&P_{22}\end{bmatrix}}$
# Linear Algebra/Other Formulas for Determinants

(This section is optional. Later sections do not depend on this material.)
Determinants are a fount of interesting and amusing formulas. Here is one that is often seen in calculus classes and used to compute determinants by hand.
# Pinyin/Carrot

See also
Carrots are a type of plant. Many different types exist. The Latin name of the plant is Daucus carota. Many people use it as a vegetable. The plant has an edible, orange root, and usually white flowers. Wild carrots grow naturally in Eurasia. Domesticated carrots are grown for food in many parts of the world. Carrots are rich in vitamins A and C.
Hóngluóbo shì yīzhòng zhíwù. Yǒu xǔduō bùtóng zhǒnglèi. Lādīng míngchēng Daucus carota. Hěnduō rén bǎ tā zuòwéi yīzhǒng shūcài. Chéngsè de gēnbù kěshíyòng. Tōngcháng kāi báisè de huāduǒ. Yěshēng hóngluóbo zài Ōuyàdàlù zìrán shēngzhǎng. Shìjièshàng xǔduō dìfāng zhòngzhí hóngluóbo zuò shíwù. Hóngluóbo fùhán vitamin A hé C.
# Pinyin/Carry

See also
# Pinyin/Caribbean Sea

Search Pinyin Caribbean
Search Pinyin sea
# Pinyin/Carotene

See also
# Pinyin/Cat

See also
# Pinyin/Care

See also
# Pinyin/Casino

See also
# Pinyin/Carditis

See also
# MagicJack/Sentiment/Pros

MagicJack is an incredibly inexpensive telephone service, costing as low as $1.39 per month for free calls to the US and Canada.

## Features

MagicJack includes:
* MagicJack PLUS is now available that runs in a stand alone mode without the need of a computer.
* A US phone number in any available area code. Many are available.
* Customizable voice mail. The voice mail system will mail new voice msgs to your email as a .wav file. The customer does not have to run MagicJack for the voice mail system to answer calls.
* Call waiting.
* Caller ID.
* Number porting is now supported (the ability to bring your number with you to MagicJack, and leave MagicJack with your number).
* Free long distance to the US, Canada and Puerto Rico. (For most people, this will appear to be unlimited. However, it appears MagicJack is developing limits.).
* International numbers can be called after purchasing minutes.
* Enhanced 911 (E9-1-1) service.

## Traveling

MagicJack is especially useful for traveling. It can be taken to anywhere broadband internet access exists and used transparently as if it were at the US location associated with the chosen area code. People can call the customer's phone number, and the customer will receive the call wherever they are. And, the customer can place calls while outside the US as if they were inside the US.

## Landline Replacement

Many Magicjack users have successfully used MagicJack to replace their landline saving $200 to $500 a year. Considerations such as 911 service, and the need for reliable telephone access should always be taken into consideration. However, for many, they have been able to do this using a cell phone and/or other pay-as-you-go VOIP services as a backup service.

## International Use

International calls can be made after minutes are purchased (through your account portal). For current rates, see MagicJack's Knowledgebase page.
MagicJack can be taken to foreign countries (that don't block VOIP) and used in the same way as in the United States or Canada. The fact that the user is outside the US is transparent to callers, and anyone the user calls.
* It is common for individuals with friends and relatives outside the US to purchase a MagicJack, register it, and mail it to the individual outside the US. Essentially giving their foreign acquaintance a US phone number which can be called toll-free. Also, the ability of their foreign acquaintance to call the US toll-free.
* ↑ See Cost.
* ↑ vbuzzer, Skype, voipvoip