OpenLLM-France/wikimedia huggingface dataset:
# Taylor series/R/One variable/Introduction/Section
## Definition

Let $I\subseteq \mathbb {R}$ denote an interval,
    $f\colon I\longrightarrow \mathbb {R}$
an infinitely often differentiable function, and $a\in I$. Then
    $\sum _{k=0}^{\infty }{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}$
$f$
$a$

## Theorem

Let $\sum _{n=0}^{\infty }c_{n}x^{n}$ denote a power series which converges on the interval $]-r,r[$, and let
    $f\colon ]-r,r[\longrightarrow \mathbb {R}$
denote the function defined via fact. Then $f$ is infinitely often differentiable, and the Taylor series
$f$
$0$

### Proof

That $f$ is infinitely often differentiable, follows directly from fact by induction. Therefore, the Taylor series exists in particular in the point $0$. Hence, we only have to show that the $n$-th derivative has $c_{n}n!$ as its value. But this follows also from fact.
$\Box$

## Example

We consider the function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x),$
given by
    $f(x):={\begin{cases}0,\,{\text{ if }}x\leq 0\,,\\e^{-{\frac {1}{x}}},\,{\text{ if }}x>0\,.\end{cases}}\,$
We claim that this function is infinitely often differentiable, which is only in $0$ not directly clear. We first show, by induction, that all derivatives of $e^{-{\frac {1}{x}}}$ have the form $p{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}$ with certain polynomials $p\in \mathbb {R} [Z]$, and that therefore the limit for $x\rightarrow 0,\,x>0$ equals $0$ (see exercise and exercise). Therefore, the limit exists for all derivatives and is $0$. So all derivatives in $0$ have value $0$, and therefore the Taylor series in $0$ is just the zero series. However, the Function $f$ is in no neighborhood of $0$ the zero function, since $e^{-{\frac {1}{x}}}>0$.
# PSI Lectures/Statistical Mechanics

* 2009 CORE Statistical Mechanics - Leo Kadanoff
* 2010 CORE Statistical Mechanics - Leo Kadanoff
* 2011 CORE Statistical Mechanics - Leo Kadanoff
* 2012 CORE Statistical Mechanics - John Berlinsky
* 2013 CORE Statistical Mechanics - Anton Burkov
* 2014 CORE Statistical Mechanics - Anton Burkov
* 2015 CORE Statistical Mechanics - Ganapathy Baskaran
* 2016 CORE Statistical Mechanics - David Kubiznak
* 2017 CORE Statistical Mechanics - David Kubiznak
* 2018 CORE Statistical Mechanics - Pedro Vieira
* 2019 CORE Statistical Mechanics -Pedro Vieira
* 2019 ??? Statistical Physics - David Kubiznak ??? SECOND COURSE ???
# PSI Lectures/Variational Calculus and Gaussian Integrals

* 2011 FRONT Evaluation of Integrals and Calculous of Variations -Denis Dalidovich
* 2012 FRONT Calculus of Variation and Gaussian Integrals - Lilia Anguelova
* 2013 FRONT Variational Calculus Gaussian Integrals - Denis Dalidovich
# PSI Lectures/Wolfram Mathematica

* 2010 FRONT Maths and Mathematica - Pedro Vieira
* 2011 FRONT Mathematica - Pedro Vieira
* 2012 FRONT Mathematica - Pedro Vieira
* 2013 FRONT Mathematica and Computational Methods in Physics - Erik Schnetter
* 2015 FRONT Mathematica Introduction - Pedro Vieira
* 2016 FRONT Mathematica - Erik Schnetter
* 2017 FRONT Mathematica - Erik Schnetter
* 2018 FRONT Introduction to Mathematica - Gang Xu (originally called "Numerical Methods")
# Package insert uniformity

This article deals around making the w:Package inserts of both chemical and herbal medicines uniform. Although most pharmaceutical companies (less so with the companies making herbal medicines falling under the "food additives section") already make package inserts bearing similar information, no true uniformity already exists, and this should definitively be fixed.
The following headlines should be made in each package insert, and information should be introduced under the headline:
* A description of the product
* The characteristics of the product
* The ingredients of the product
* The amount of material of the product ingredient per capsule, 100 grams, ...
* The daily dose
* The duration of intake
* The method of administration
* Contraindications
* The potential adverse effects
* The license under which the product is distributed
* The various names under which the product is on the market in several countries
* The best method of preservation
* Expiration date
* Impact on driving and using machinery
* Measures when using large quantities
# Paideia High School

Paideia High School is a set of high school curriculum based on the Paideia Proposal by Mortimer Adler.

## Resources

* Curriculum Plan
* Year One Curriculum
* Year Two Curriculum
* Year Three Curriculum
* Year Four Curriculum
* Glossary

## Subpages

Special:Prefixindex/Paideia High School/
# Paideia High School/Boeke, Kees

Kees Boeke
Cosmic View, a work covered in Paideia High School's Year One Curriculum
# Sound and psychology

Sound may be defined as the sensation felt/detected by our ears due to vibration (compression and expansion) in the air surrounding the ears. Sounds have been classified as forms of energy. Sounds put various effects on our psychology which are, if within ranges or limits, interpreted by our minds into useful information.
# Sound powered phones/Howard Community College

    Fall 2011
# Sound powered phones/Howard Community College/Fall 2011

       550_EPC
       550_EPC Project 2
# Sound synthesis

Sound synthesis is the process of using specialised audio synthesising devices to create sounds. Sound synthesis was originally intended to provide a means of recreating acoustic instrumental tones, but the resulting attempts were generally unrealistic and inadequate as a substitute for the original instrument. What resulted was the creation of new sounds, never before heard nor conceived.
There are many different types of synthesis in common use today:
* Subtractive Synthesis (otherwise knows as Analogue Synthesis)
* Additive Synthesis
* Frequency Modulation (FM) Synthesis
* Phase Distortion Synthesis
* Granular Synthesis
* Wavetable Synthesis
* Physical Modelling Synthesis (a fairly recent development which does enable realistic recreation of acoustic intruments)

## External resources

WikiAudio - Encyclopedia & database for all things audio and sound related
# Somology

This Medicine resource is a "stub". Learn how you can help Wikiversity to develop it.
Somology refers to the study of the caring of the body, e.g., in nursing.

## Notation

Notation: let the symbol Def. indicate that a definition is following.
Notation: let the symbols between [ and ] be replacement for that portion of a quoted text.
Notation: let the symbol ... indicate unneeded portion of a quoted text.
Sometimes these are combined as [...] to indicate that text has been replaced by ....

## Theoretical somology

Def. "understanding the body as an integration of the object body (the thing) into experience so that it is simultaneously an object, a means of experience, a means of expression, a manner of presence among other people, and a part of one's personal identity" is called somology.
# Sound powered phones

    Indented line

## Idea

Sound powered phones require no electricity. They are found on air craft carriers, submarines and other navy ships. They are also found on ski lifts, power plants.

## SubSystems

    Speakers
    Microphones
    Surplus equipment

## College/Universities

    Howard Community College
# UTPA STEM/CBI Courses/Computer Numerical Controls

Intro to Computer Numerical Controls
# UTPA STEM/CBI Courses/Computer Science

Topics
Algorithm Design
Arrays
Linked Lists
Operating Systems
Repetition
Selection
# UTPA STEM/CBI Courses/Computer Networks

Topics
Socket Programming
# UTPA STEM/CBI Courses/Computer Science/Algorithm Design

Course Title: Computer Science
Lecture Topic: Algorithm Desigm
Instructor: Bin Fu
Institution: University of Texas - Pan American

## Backwards Design

Course Objectives
* Primary Objectives- By the next class period students will be able to:
** Understand the basic dynamic programming method
** Find the recursion for dynamic programming
** Find the right order to build up the table for dynamic programming
** Solve a knapsack problem with dynamic programming
* Sub Objectives- The objectives will require that students be able to:
** Input of data
** Implementation
** Performance
** Complexity analysis
* Difficulties- Students may have difficulty:
** Identifying the recursion for dynamic programming, and the hardship to build up the table.
* Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
** A knapsack problem has many interesting applications. It is one of the classical NP-hard problems.
Model of Knowledge
* Concept Map
** Recursion
** Dynamic Programming
** Greedy Method
** Approximation
** NP-completeness
* Content Priorities
** Enduring Understanding
*** Recursion
*** Dynamic Programming
*** Greedy Method
*** Approximation
** Important to Do and Know
*** Find the size of sub-problems
*** Determine the size of the table of dynamic programming
*** Finding an accurate approximation
** Worth Being Familiar with
*** Memory allocation
*** Complexity analysis
*** Approximation
Assessment of Learning
* Formative Assessment
** In Class (groups)
*** Give an example to the students and ask them solve it with dynamic programming method.
** Homework (individual)
*** Ask them to solve the Knapsack problem with dynamic programming method.
* Summative Assessment
** Try to solve more problems with dynamic programming method.

## Legacy Cycle

OBJECTIVE
By the next class period, students will be able to:
* Be efficient in finding a recursion for solving a computational problem.
The objectives will require that students be able to:
* Effectively shrink a computational problem.
THE CHALLENGE
How to handle the exponential number of cases in a knapsack problem.
GENERATE IDEAS
When the size of a knapsack is not too large, using a table to save the sum of subsets is better than trying all subsets.
MULTIPLE PERSPECTIVES
NP-hardness.
The methods of algorithm design depend on problems.
Each method can be applied to a specific class of problems.
RESEARCH & REVISE
Find new method to attack the problem with large Knapsack.
TEST YOUR METTLE
Implement the method with dynamic programming.
GO PUBLIC
Post software implementation over internet.

## Pre-Lesson Quiz

* What is recursion?
* What is the limitation of divide and conquer in solving a knapsack problem?
* What is computational time for brute force method in solving a knapsack problem?

## Test Your Mettle Quiz

* What is the computational complexity of dynamic programming method for solving a knapsack problem?
* Apply the method to the longest common sequence problem.
# Measure Theory/Bounded Variation
## Bounded Variation

In Lesson 0 of this section, we already introduced functions of bounded variation as an extension of monotone functions. We also showed that every function of bounded variation has an up-down decomposition.
This was all in the hope of proving the integral of the derivative equation, under nice conditions for a given function. We can now state that, at least one nice condition, is for a function to be of bounded variation.
* Exercise 1. BV Differentiable A.E.
* Show that any function, $f:[a,b]\to {\mathbb {R}}$, of bounded variation is differentiable a.e.
* This should be a quick proof from the observation that f has an up-down decomposition, each component function being monotone and therefore differentiable a.e. Therefore the set of points at which either is not differentiable has measure zero, and so on.
This ensures that the f' exists in the equation
    $\int _{(a,b)}f'=f(b)-f(a)$
We'll also wish for a guarantee that f' is integrable, so that we may be assured $\int _{(a,b)}f'$ exists. This will occupy our concerns in this lesson.

### BV Integrable f'

* Exercise 2. BV Integrable f'
* Let $f:[a,b]\to {\mathbb {R}}$ be a function of bounded variation, and prove that f' is integrable on [a,b], and
* $\int _{(a,b)}f'\leq f(b)-f(a)$
* Hint: First show this for any monotonically increasing function.
* Consider the set of points at which the derivative exists, and "fill the gaps" by defining g to be equal to the one of the Dini derivatives of f. Therefore $f=g$ a.e. and g has a finite Dini derivative everywhere on [a,b]. You can then show that $\int _{(a,b)}g'$ exists and infer that $\int _{(a,b)}g'=\int _{(a,b)}f'$ at the end.
* Now to show that $\int _{(a,b)}g'$ exists, use the Dini derivative which you know exists to infer that the sequence of functions,
* $g_{k}(x)={\frac {g(x+1/k)-g(x)}{1/k}}$
* converges pointwise to $g'(x)$ and $0\leq g'$.
* Then apply Fatou's to infer the desired inequality.
Note that in fact, we not only have the existence of $\int _{(a,b)}f'$, but also half of the equality that we're seeking!
Can we prove the reverse inequality? We answer this cliff-hanger in the next lesson.
# Measure Theory/Approximations of Sequences of Functions
## Approximations of Sequences of Functions

The content of this lesson is the final of Littlewood's three principles, which are also known as Egoroff's Theorem and Lusin's Theorem.

### Pointwise Convergence to Nearly Uniform

Here we will show that pointwise convergence can, through an appropriate domain restriction, entail something like uniform convergence.
There are a few theorems that show something like the rough expression above. The first interpretation that we will prove, in a slightly more formal expression, is:
    For $\delta$ as small as you like, you never need to remove more than $100\delta$ percent of the domain, for the convergence to be uniform.
The formal expression of this is: Let
* $E\in {\mathcal {M}}$ has finite measure,
* $\langle f_{n}\rangle$ a sequence of measurable functions defined on E,
* $f_{n}\to f$ pointwise on E,
Then $\forall \delta ,\varepsilon \in {\mathbb {R}}^{+},\quad \exists F\in {\mathcal {M}},\quad \exists N\in {\mathbb {Z}}^{+}$ such that
* $F\subseteq E$
* on $E\smallsetminus F,\quad |f_{n}(x)-f(x)|<\varepsilon \quad \forall n\geq N$
* $\lambda (F)<\delta$
To begin the proof, let $\varepsilon ,\delta \in {\mathbb {R}}^{+}$.
With the assumptions as above, define for each $1\leq n$ the set of points where the sequence is "far apart":
    $G_{n}=\{x\in E:|f_{n}(x)-f(x)|\geq \varepsilon \}$

#### Exercise 1. Where the Sequence Is Far

Definition: sup and limsup
Let $\langle A_{n}\rangle$ be any sequence of sets, each a subset of some set X. Define the supremum of $\langle A_{n}\rangle$ to be their union. Also define the limit superior of $\langle A_{n}\rangle$ to be the intersection over all suprema of tails:
    ${\overline {\lim }}_{k}A_{k}=\bigcap _{N=1}^{\infty }\bigcup _{N\leq k}A_{k}$.
Of course every definition above for suprema has a correlate for infima.
Part H.
Definition: converge a.e.
Let $E\in {\mathcal {M}}$ and $\langle f_{n}\rangle$ be a sequence of measurable functions defined on E. Let $F\subseteq E$ be a measurable subset, such that $\lambda (F)=\lambda (E)$ and $f_{n}(x)\to f(x)$ for all $x\in F$. Then we say that $f_{n}\to f$ pointwise a.e.

### Egoroff's Theorem

Egoroff's theorem states the following.
Let $\langle f_{n}\to \rangle$ pointwise a.e. on a measurable set E of finite measure.
Then $\forall \eta \in {\mathbb {R}}^{+}\quad \exists (F\subseteq E)$ such that
* $\lambda (F)<\eta$
* $f_{n}{\overset {u}{\to }}f$ on $E\smallsetminus F$.

#### Exercise 2. Prove Egoroff's

Prove Egoroff's theorem.
Hint: Use the Part H. of Exercise 1. Where the Sequence Is Far together with the $\eta /2^{n}$ trick.

### Lusin's Theorem

Lusin's theorem states the following.
Let
* $f:[a,b]\to {\mathbb {R}}$ be a measurable real-valued function.
* $\varepsilon \in {\mathbb {R}}^{+}$
Then there exists a continuous $g:[a,b]\to {\mathbb {R}}$ such that $\lambda (\{x\in [a,b]:f(x)\neq g(x)\})<\delta$.

#### Exercise 3. Prove Lusin's

Prove Lusin's theorem.
Hint: Use Egoroff's and the approximation of measurable functions by continuous functions.
# Primary mathematics/Proportions

Ratios are similar to fractions. $x:y={\frac {x}{x+y}}$. Remember, The proportion x:y does not equal the fraction ${\frac {x}{y}}$.
For example, if the ratio of boys to girls in a classroom is 1:2, that does not mean half of the class is boys and the other half is girls. It means that ${\frac {1}{3}}$ of the class is boys and ${\frac {2}{3}}$ of the class is girls. (But that means that the number of boys is half the number of girls.)
But remember that the fraction only applies to the sum of the two sides $\left(x+y\right)$, and can be wrong if there are more than 2 things.
For example, if there is a fruit basket that has 16 fruits, and the ratio of apples to oranges is 3:1, that doesn't always mean that there are 12 apples $\left({\frac {3}{4}}\times 16\right)$ and 4 oranges $\left({\frac {1}{4}}\times 16\right)$. There could be 12 bananas in the fruit basket, which means that there are only 3 apples and 1 orange.
# Primary mathematics/Working with fractions

Mixed numbers and top heavy fractions:
A mixed number is an integer and a fraction, eg 2 and a half, a top heavy fraction is a fraction with a larger numerator than demominator, eg 3/2. For multiplication and division you must not use mixed numbers. They can be used in addition and subtraction (do the integers and fractions seperately and combine the result) if this is not too confusing.
Simplest terms: Multiplying (or dividing) the numerator and denominator by the same number will not change the fraction. A fraction is put into its simplest terms by dividing both by the largest integer that goes into both a whole number of times, this can be done in stages using small integers such as 2 and 3, 5 and 10 repeatedly.
Multiplication:
Where ab=a*b.
Division:
To divide fractions; remember that a fraction is a division. Fractions can be stacked like so:
The rule to remember to simplify is that anything under an odd number of lines goes on the bottom and anything under an even number of lines goes on the top (0 is even for this purpose) thus:
It is easier (perhaps) to remember that the second fraction in the division gets turned upside down.
Addition and subtraction:
For addition the denomintators of the two fractions must be the same - if this is so then simply add (or subtract) the numerators:
If the denominators are not the same then a common denominator must be found, the simplest way to do this is take the product of the denominators allthough if one is a multiple of the other simply multiplying it by the aplicable integer works aswell. Remember to multiply the numerators by the same number. In general:
But for example
Is possible.
Remember that young children will not understand the algebraic formulation here. The art of primary teaching is finding a way to make this make sense to young children however teachers need to know these general rules.
# Prime factorization/Existence/Fact

natural number $n\in \mathbb {N}$, $n\geq 2$,
That means there exists a representation
    $n=p_{1}\cdot p_{2}\cdots p_{r}\,$
with prime numbers $p_{i}$.
# Primary mathematics:Introduction to variables

A variable is any part of an equasion that replaces a number with something that can represent a number. For example, $5+E=14$. In this case, $E$ is the variable. You may have seen this before as $5+$__$=14$. To solve a variable including problem with one step, you have to get rid of the first number. So for example, take $x+36=45$. To get rid of 36, we need to create $-36$. We also need to balance the problem on both sides. Therefore, the problem looks somewhat like this: . Now you solve the problem vertically. So $36+(-36)$ equals 0, so we can get rid of that. Then we take $45-36$ which equals $9$. Therefore, $x=9$.
The same overall process goes for other operations, just remember: Addition is the opposite of subtraction and multiplication is the opposite of division. By doing the opposite of the operation in the equation, you eliminate the variable and come to an answer.
# Prime factorization/Existence/Fact/Proof2

We prove the existence by induction over $n$, and we consider the statement $A(n)$ saying that every natural number $m$ with $2\leq m\leq n$ has a prime factorization. For $n=2$ we have a prime number. So suppose that $n\geq 2$ and assume that, by the induction hypothesis, every number $m\leq n$ has a prime factorization. We have to show that every number $m\leq n+1$ has a prime factorization. The only new number to consider is $n+1$. In the proof of this induction step, another important proof scheme occurs, the proof by cases. Here one argues depending on whether an additional property holds or not, and in both cases one has to prove the result.
Here we consider the cases whether $n+1$ is a prime number or not. If $n+1$ is a prime number, then we have immediately the prime factorization, just take the number itself. In this case we do not even use the induction hypothesis.
So now we consider the case where $n+1$ is not prime. This means that there exists a non-trivial decomposition $n+1=ab$ with smaller numbers $a,b<n+1$. For these numbers $a$ and $b$, there exist, due to the induction hypothesis, factorizations into prime numbers, and we can put these together to gain a prime factorization of $n+1$.
# Primary mathematics/Subtracting numbers

first 5-2=3 second 40-30=10
then add
To get the answer.
When this is grasped move on to harder problems:
There are 2 methods that I would like to illustrate, the first uses carrying:
Start by saying that 5 is bigger than 2 so we take 10 from the 50 and add it to the 2 to get 12
Is the usual way of writing this.
Now subtract 5 from 12 to get 7. This is another 'pair' that needs to be taught - all the numbers up to 10 subtracted from all larger numbers up to 20 are 'required', or it is possible to say that 12-2 is 10 and that 5-2 is 3 and 10-3 is 7, which is how I do it, but is not necessarily the 'easiest' way. Different children will want to do this in different ways and will often develop their own methods for mental subtraction (i.e. when you don't write it down). For instance I do this: 56-38 becomes 50-40+6+2 : note that I have used both the fact that 56=50+6 and that 38=40-2 (this is why I personally feel that every child needs to know the pairs that sum to 10) and that a+b-(c-d) = a-c+b+d although you can 'see' this as the 'big chunk' from 40 to 50 and then the two small end bits added on (6 and 2) without explaining using algebra, again teachers should remember this general truth (that addition can be re-ordered and that minus minus = plus) in order to answer questions and with a view to the child learning it in this form.
To continue:
Then we take 30 from 40 (remember that it is now 40 since we gave the other 10 to the units) to get 10
And finally add these two together
To give the answer.
This is the method I was taught. I was introduced to this second method by being told that a young boy had used it without prompting, proving that young children can (and do) understand negative numbers if told about them:
The same sum:
This time we will take 30 from 50 first (working left to right here - much more 'obvious') to get 20
Then we will take 5 from 2 to get -3
Then we take the 3 from the 20
To finish the sum. Here students need to know what 20-3 is: This is easiest to look at as 10+10-3 and then know that 10-3 is 7 (again, numbers adding to 10).
# Primary mathematics/The Use of Calculators

When it comes to the discussion regarding tools that should be used to facilitate students' learning in the field of mathematics, the calculator and the use the computer/internet is a great part of the debate.
This page will cover some of the following topic as they relate to the teaching of Primary mathematics.
* Calculators support learning
* Calculators are what students will eventually use in the "real world"
* Calculators are not "smart".. students are! For example a calculator might calculate a+b*c as (a+b)*c, while the real answer is a+(b*c).
* So when should students use calculators?
* Ways of teaching students when and when not to use calculators
Back to: Adding numbers, Subtracting numbers, Multiplying numbers, Dividing numbers