OpenLLM-France/wikimedia huggingface dataset:
# Applied linear operators and spectral methods

Welcome to this learning project about Applied linear operators and spectral methods!

## Learning Project Summary

* Project code:
* Suggested Prerequisites:
** Introduction to real analysis
** Introduction to ordinary differential equations
* Time investment: 6 months
* Portal:Mathematics
* School:Applied Mathematics
* Level: Graduate

## Content summary

The course covers the theory of linear operators applied to matrix, differential, and integral equations. We also discuss Fredholdm integral equations and spectral theory.

## Goals

This learning project aims to help applied mathematicians and engineers gain the understanding needed to read and critically assess papers that deal with functional analysis in its various forms.
We will talk about operator theory and linear operators as applied to functional analysis.

## Contents
### Readings and other resources

* Principles of Applied Mathematics, Transformation and Approximation: 2nd Edition, by J. Keener, 2000.
* Green's Functions and Boundary Value Problems, by Ivar Stakgold, Wiley.
* Methods of Modern Mathematical Physics. I - Functional Analysis, by Michael Read and Barry Simon, Elsevier.
* Applied Functional Analysis: Applications to Mathematical Physics, by Eberhard Zeidler, Springer Series on the Applied Mathematical Sciences, vol. 108. This is a more technical book than the others..
* Applied Functional Analysis: Main Principles and their Applications, by Eberhard Zeidler, Springer Series on the Applied Mathematical Sciences, vol. 109. This is a more technical book than the others..

### Lectures

* Lecture 1: Linear vector spaces
* Lecture 2: Normed inner product spaces, orthogonal vectors, linear operators
* Lecture 3: Adjoint operators, spectral theory for matrices
* Lecture 4: More on spectral decompositions
* Differentiation of distributions
* Convergence of distributions
* Differential equations in the sense of distributions
* Differentiation of distributions with severe discontinuities
* Green's functions for linear differential operators

### Assignments

* Assignment 1: ...
# Applied linear operators and spectral methods/Differentiating distributions
## Differentiation of a distribution

If $f(x)$ is a differentiable function in $\mathbb {R}$ whose first derivative is locally integrable, then the derivative $f'(x)$ defines its own distribution
    $\left\langle f',\phi \right\rangle :=\int _{-\infty }^{\infty }f'(x)~\phi (x)~{\text{d}}x$
Integrating by parts and noting that at the boundaries $x=\pm \infty$, $\phi (x)=0$ because of compact support, we have
    $\left\langle f',\phi \right\rangle :=-\int _{-\infty }^{\infty }f(x)~\phi '(x)~{\text{d}}x=:\left\langle f,\phi '\right\rangle$
This suggests defining the derivative of any distribution $f$ via
    ${\text{(1)}}\qquad {\left\langle f',\phi \right\rangle =-\left\langle f,\phi '\right\rangle }$
We need to check that this definition satisfies the properties of a distribution. We observe that if $\phi$ is a test function then so is $-\phi '$. Linearity is obvious. What about continuity? If $\{\phi _{n}\}$ is a zero sequence, so is the sequence $\{-\phi _{n}\}$ and thus $\left\langle f,-\phi '_{n}\right\rangle$ tends to zero as $n\rightarrow \infty$. Hence (1) defines a distribution.
Remark: All distributions are infinitely differentiable is the class of test functions includes only infinitely differentiable functions.
Comment: One can take test functions which are only $C^{n}$ differentiable. Then the distribution will only be $n$ times differentiable. Thus, enlarging the class of test functions reduces the class of distributions.
More generally, if $f^{(n)}$ is the $n$th derivative of the distribution $f$,
    ${\left\langle f^{(n)},\phi \right\rangle =-\left\langle f,\phi ^{(n)}\right\rangle }$
Distributions can be generated by functions which are not differentiable in the ordinary sense. However, we can differentiate them in the distribution sense.
For example, the derivative of the delta distribution gives us the dipole distribution:
    $\left\langle \delta '_{0},\phi \right\rangle =-\left\langle \delta _{0},\phi '\right\rangle =-\phi '(0)$
Also, the derivative of the Heaviside function is given by
    $\left\langle H',\phi \right\rangle =-\left\langle H,\phi '\right\rangle =-\int _{0}^{\infty }\phi '(x)~{\text{d}}x$
Since $\phi (\infty )=0$, from the fundamental theorem of calculus,
    $\left\langle H',\phi \right\rangle =-\phi (0)=\left\langle \delta _{0},\phi \right\rangle$
Therefore the derivative of the Heaviside function is the delta function.
Template:Lectures
# Applied linear operators and spectral methods/Weak convergence
## Convergence of distributions

Definition:
A sequence of distributions $\{t_{n}\}$ is said to converge to the distribution $t$ if their actions converge in $\mathbb {R}$, i.e.,
    $\left\langle t_{n},\phi \right\rangle \rightarrow \left\langle t,\phi \right\rangle \qquad \forall ~\phi \in D\quad {\text{as}}~b\rightarrow \infty$
This is called convergence in the sense of distributions or weak convergence.
For example,
    $t_{n}:=\left\langle \sin(nx),\phi \right\rangle =\int _{-\infty }^{\infty }\sin(nx)~\phi (x)~dx\rightarrow 0\qquad {\text{as}}~~n\rightarrow \infty$
Therefore, $\{\sin(nx)\}$ converges to $\sin(0)$ as $n\rightarrow \infty$, in the weak sense of distributions.
If $\{t_{n}\}\rightarrow t$ if follows that the derivatives $\{t'_{n}\}$ will converge to $t'$ since
    $\left\langle t'_{n},\phi \right\rangle =-\left\langle t_{n},\phi '\right\rangle \rightarrow -\left\langle t,\phi '\right\rangle =\left\langle t',\phi \right\rangle$
For example, $\{t_{n}\}=\left\{{\cfrac {\cos(nx)}{n}}\right\}$ is both a sequence of functions and a sequence of distributions which, as $n\rightarrow \infty$, converge to 0 both as a function (i.e., pointwise or in $L^{2}$) or as a distribution.
Also, $\{t'_{n}\}=\{-\sin(nx)\}$ converges to the zero distribution even though its pointwise limit is not defined.
Template:Lectures
# English-Chinese/Someone bent the metal rod but not yet broken

Someone bent the metal rod but not yet broken.
有人 弄弯了 金属杆 但 还未 折断。
# English-Chinese/Something done, a deed

Something done, a deed.
Going into the house on fire to help a child was a brave act.
做的事，行为。
走进着火房子救孩子是一个勇敢的行为。
# English-Chinese/Source text

A source text is a text (sometimes oral) from which information or ideas are derived. In translation, a source text is the original text that is to be translated into another language.
源文本是从中获取信息或想法的文本（有时是口头的）。在翻译中，源文本是要翻译成另一种语言的原始文本。
# English-Chinese/Some subjects

Some subjects, such as history, are sometimes social sciences and sometimes humanities.
一些主题，如历史，有时是社会科学，有时是人文科学。
# English-Chinese/Some symptoms of heart disease are

Some symptoms of heart disease are:
* Pain in the chest--the heart muscle is not getting enough flow to keep it going.
* Trouble breathing--blood may back up into the lungs.
* Palpitations (a feeling that the heart is beating too fast, too hard, or not regularly).
* Swelling of feet or legs--blood is backing up from the heart into the lower body.
* Feeling weak because the body and brain are not getting enough blood to supply them with oxygen.
* Cyanosis (skin turning a blue colour) means that too little oxygen is in the bloodstream to supply the cells in the body.
心脏病的一些症状是：
* 胸部疼痛 - 心肌没有足够的血液来保持活力。
* 呼吸困难 - 血液可能堵塞于肺部。
* 心悸（心脏跳得太快，太硬或不规律的感觉）。
* 脚或腿肿胀 - 血液从心脏堵塞于下半身。
* 感觉虚弱，因为身体和大脑没有足够的血液为他们提供氧气。
* 发绀（皮肤变成蓝色）意味着血液中的氧气太少，无法供给体内的细胞。
# English-Chinese/Some things that might be stored in memory might be

Some things that might be stored in memory might be:
* Photos from a Camera (A camera is an input)
* A Program
* A Text Document
可能存储在内存中的一些东西可能是：
* 来自相机的照片（相机是输入）
* 一个程序
* 文本文件
# English-Chinese/Some stocks migrate in large schools along the coast to suitable spawning grounds, where they spawn in fairly shallow waters

Some stocks migrate in large schools along the coast to suitable spawning grounds, where they spawn in fairly shallow waters.
一些在沿海的大型鱼群迁移到合适的产卵场，在那里它们在相当浅的水域产卵。
# English-Chinese/South America is located between the Caribbean Sea, the South Pacific and the South Atlantic

South America is located between the Caribbean Sea, the South Pacific and the South Atlantic. It is the wilder part of the Americas.
南美洲位于加勒比海，南太平洋和南大西洋之间。 它是美洲较为荒凉的地区。
# English-Chinese/Some spices, such as allspice and chili pepper, are fruits, botanically speaking

Some spices, such as allspice and chili pepper, are fruits, botanically speaking.
从植物学角度讲，一些香料，如多香果和辣椒，都是果实。
# English-Chinese/South America

South America is the continent to the south of North America. These two continents are separated by the Panama Canal.
南美洲 是 北美洲 南部的 大洲。这 两个 大洲 被 Panama运河 隔开。
# Bloom Clock/Keys/Global/Temperate/Early Summer/Purple Flowers/DPL

This is a DPL-generated list, used to create the key for Purple-flowered plants in Early Summer
* BCP/Leonurus cardiaca
* BCP/Scabiosa graminifolia
* BCP/Angelonia angustifolia
* BCP/Stachys byzantina
* BCP/Geranium maculatum
* BCP/Clematis
* BCP/Cleome hassleriana
* BCP/Salvia lyrata
* BCP/Echinacea purpurea
* BCP/Callicarpa
* BCP/Nepeta x faassenii
* BCP/Salvia nemorosa 'Caradonna'
* BCP/Petunia x hybrida
* BCP/Thymus 'Spicy Orange'
* BCP/Torenia fournieri
* BCP/Allium schoenoprasum
* BCP/Phlox paniculata
* BCP/Cirsium arvense
* BCP/Allium canadense
* BCP/Iris sibirica
* BCP/Allium vineale
* BCP/Geranium
* BCP/Allium sativum
* BCP/Vinca sardoa
* BCP/Allium cernuum
* BCP/Wisteria frutescens
* BCP/Lythrum salicaria
* BCP/Lavandula angustifolia
* BCP/Platycodon grandiflorus
* BCP/Salvia nemorosa
* BCP/Geranium phaeum
* BCP/Antirrhinum majus
* BCP/Dianthus plumarius
* BCP/Solanum dulcamara
* BCP/Iris
* BCP/Ageratum houstonianum
* BCP/Rhododendron catawbiense
* BCP/Glechoma hederacea
* BCP/Hesperis matronalis
* BCP/Aquilegia x hybrida
* BCP/Trifolium arvense
* BCP/Hosta
* BCP/Prunella vulgaris
* BCP/Trifolium pratense
* BCP/Viola tricolor
* BCP/Securigera varia
* BCP/Tradescantia
* BCP/Hemerocallis
* BCP/Tradescantia virginiana
* BCP/Salvia guaranitica
* BCP/Salvia verticillata
* BCP/Rhododendron
* BCP/Rosa
* BCP/Digitalis purpurea
* BCP/Allium
* BCP/Viola
* BCP/Dianthus
* BCP/Mentha requienii
* BCP/Astilbe
* BCP/Cuphea hyssopifolia
* BCP/Nepeta subsessilis
* BCP/Leonurus cardiaca
* BCP/Scabiosa graminifolia
* BCP/Angelonia angustifolia
* BCP/Stachys byzantina
* BCP/Geranium maculatum
* BCP/Clematis
* BCP/Cleome hassleriana
* BCP/Salvia lyrata
* BCP/Echinacea purpurea
* BCP/Callicarpa
* BCP/Nepeta x faassenii
* BCP/Salvia nemorosa 'Caradonna'
* BCP/Petunia x hybrida
* BCP/Thymus 'Spicy Orange'
* BCP/Torenia fournieri
* BCP/Allium schoenoprasum
* BCP/Phlox paniculata
* BCP/Cirsium arvense
* BCP/Allium canadense
* BCP/Iris sibirica
* BCP/Allium vineale
* BCP/Geranium
* BCP/Allium sativum
* BCP/Vinca sardoa
* BCP/Allium cernuum
* BCP/Wisteria frutescens
* BCP/Lythrum salicaria
* BCP/Lavandula angustifolia
* BCP/Platycodon grandiflorus
* BCP/Salvia nemorosa
* BCP/Geranium phaeum
* BCP/Antirrhinum majus
* BCP/Dianthus plumarius
* BCP/Solanum dulcamara
* BCP/Iris
* BCP/Ageratum houstonianum
* BCP/Rhododendron catawbiense
* BCP/Glechoma hederacea
* BCP/Hesperis matronalis
* BCP/Aquilegia x hybrida
* BCP/Trifolium arvense
* BCP/Hosta
* BCP/Prunella vulgaris
* BCP/Trifolium pratense
* BCP/Viola tricolor
* BCP/Securigera varia
* BCP/Tradescantia
* BCP/Hemerocallis
* BCP/Tradescantia virginiana
* BCP/Salvia guaranitica
* BCP/Salvia verticillata
* BCP/Rhododendron
* BCP/Rosa
* BCP/Digitalis purpurea
* BCP/Allium
* BCP/Viola
* BCP/Dianthus
* BCP/Mentha requienii
* BCP/Astilbe
* BCP/Cuphea hyssopifolia
* BCP/Nepeta subsessilis
# Bloom Clock/Keys/Global/Temperate/Early Summer/Red Flowers/DPL

This is a DPL-generated list, used to create the key for Red-flowered plants in Early Summer
* BCP/Opuntia
* BCP/Paeonia
* BCP/Rosa 'Knockout'
* BCP/Clematis
* BCP/Lonicera x heckrottii
* BCP/Petunia x hybrida
* BCP/Antirrhinum majus
* BCP/Begonia
* BCP/Geranium phaeum
* BCP/Kniphofia uvaria
* BCP/Dianthus plumarius
* BCP/Monarda didyma
* BCP/Weigela
* BCP/Geranium
* BCP/Salvia splendens
* BCP/Phlox paniculata
* BCP/Kalmia latifolia
* BCP/Lonicera sempervirens
* BCP/Iris
* BCP/Hesperis matronalis
* BCP/Aquilegia x hybrida
* BCP/Calycanthus floridus
* BCP/Aquilegia canadensis
* BCP/Viola tricolor
* BCP/Hemerocallis
* BCP/Rhododendron
* BCP/Rosa
* BCP/Viola
* BCP/Dianthus
* BCP/Rumex acetosella
* BCP/Spigelia marilandica
* BCP/Rhus typhina
* BCP/Astilbe
* BCP/Opuntia
* BCP/Paeonia
* BCP/Rosa 'Knockout'
* BCP/Clematis
* BCP/Lonicera x heckrottii
* BCP/Petunia x hybrida
* BCP/Antirrhinum majus
* BCP/Begonia
* BCP/Geranium phaeum
* BCP/Kniphofia uvaria
* BCP/Dianthus plumarius
* BCP/Monarda didyma
* BCP/Weigela
* BCP/Geranium
* BCP/Salvia splendens
* BCP/Phlox paniculata
* BCP/Kalmia latifolia
* BCP/Lonicera sempervirens
* BCP/Iris
* BCP/Hesperis matronalis
* BCP/Aquilegia x hybrida
* BCP/Calycanthus floridus
* BCP/Aquilegia canadensis
* BCP/Viola tricolor
* BCP/Hemerocallis
* BCP/Rhododendron
* BCP/Rosa
* BCP/Viola
* BCP/Dianthus
* BCP/Rumex acetosella
* BCP/Spigelia marilandica
* BCP/Rhus typhina
* BCP/Astilbe
# Czech Language/Vocabulary Resources
## Dictionaries

* slovnik.cz - a bi-directional, multilingual dictionary, where diacritics are optional.
* seznam.cz - another simple multilingual dictionary.

## General Pages

* SlovoDne.cz - Czech word-of-the-day.
* Czech Pronunciator
* Local Lingo.
* Owen Leonard's Little Czech Primer, with pictures!
* Omniglot.
* James Naughton's Pages. Dr James Naughton is a lecturer in Czech and Slovak at Oxford.
* Unilang wiki.
# DADA 99th anniversary celebration

DADA 99 th anniversary celebration of the program of events in Alytus
Any international seminar-conference of art schools, teachers and communities Daily Education and Dada - 2015. 8-9 October.
Lithuanian art teachers art exhibition - in 2015. October 9-23 d.
Location: Alytus Ethnographic Museum, Volunteer g. 6, Alytus.

## Practical sessions

* /Constructivism_Session
* /Metagraphy session

## External Links

Radio 99 interviews, images and audio recordings from the conference: http://www.fm99.lt/prie-puodelio-arbatos-pakalbekim-dada-po-99-metu-vis-dar-antimenas/

## Internal links

* Dada
* Art practices
# Boundary Value Problems/Review: ODEs

Return to Boundary Value Problems
For more of an introduction to ODEs you are referred to Boundary Value Problems/BVP-Ordinary-Differential-Equations and for examples please look at Examples of ordinary differential equations

## First order differential equation:

Let $t\in [a,b]$ the equation
${\frac {dy}{dt}}=k(t)y+f(t)$
is a linear ordinary (single independent variable) differential equation of first order (first derivative). If $f(t)=0$ on $[a,b]$ the equation is called "homogeneous" differential equation. Otherwise it is called nonhomogeneous otherwise.
Examples of homogeneous and non-homogeneous:
$(Homogeneous){\frac {d}{dt}}y=10y$ where $k(t)=10$ is a constant coefficient.
$(Homogeneous){\frac {d}{dt}}y=ty$ where $k(t)=t$ is a coefficient that increases as $t$ increases.
$(Nonhomgeneous){\frac {d}{dt}}y=10y+sin(t)$ where $f(t)=sin(t)$.

## Solving homogeneous ODE with constant coefficients: Separable case

Solving ${\frac {d}{dt}}y=10y$
${\frac {1}{y}}{\frac {d}{dt}}y=10$
$\int {\frac {1}{y}}{\frac {dy}{dt}}dt=\int 10dt$
$ln(|y|)=10t+C_{1}$
where $C_{1}$ is the integration constant.
$e^{ln(|y|)}=e^{10t+C_{1}}$
$y(t)=e^{C_{1}}e^{10t}$
$y(t)=C_{2}e^{10t}$
After integration there is a constant, an unknown. This will happen each time you integrate. To determine this unknown for a particular application you will need another piece of information. Typically it is another equation that requires $y(0)=intialvalue$. This condition with the above differential equation defines an Initial Value Problem. For this problem the value of $C_{2}$is determined by using an intial condition such as $y(0)=5$. Then $y(0)=C_{2}e^{0}=C_{2}=5$ and making the substitution provides the solution $y(t)=5e^{10t}$

## Student work

Solve the following IVPs:
* Differential eq: ${\frac {dx}{dt}}=5x$ where $x(t)$ for all $\infty <t<\infty$. Initial value: $x(0)=-2$
Click here for solution: Boundary Value Problems/BVP-solution-1
* Differential eq: ${\frac {dx}{dt}}=5x+2$ where $x(t)$ for all $\infty <t<\infty$. Initial value: $x(3)=-2$
Click here for solution: Boundary Value Problems/BVP-solution-2

### Pre-class work:

Complete each of the following:
* Read the following:
** Pages 1-10 of Powers
** Lecture notes (web page)
** (View) Associated Lecture Video
* Work the following problems Boundary Value Problems/BVP-Exercise 1-1:Problems and Solutions
# Boundary Value Problems/Solving homogeneous linear ODEs

ODE definition Standard form of an ODE.
Return to Main Page of Boundary Value Problems
# Boundary Value Problems/Review: ODEs/BVP-solution-1

Return Review: ODEs
# Boundary Value Problems/Vector Field and Flow

Return Boundary Value Problems

## Vector Analysis: Basics

We are interested in operations on sets in a multidimensional space. The focus is on physical problems so think about three spatial dimensions, ${x,y,z}$, and the dimension of time $t$. For notation, I will use $x_{1}=x$ ,$x_{2}=y$ and $x_{3}=z$. This is easily scaled to $n$ - dimensions for larger spaces. So $R^{n}$ corresponds to $R^{3}$ for most of the time, or less.

### Two types of fields
#### Scalar Field

A scalar x is a real number, this is denoted as $x\in \mathbf {R}$. A vector in 3-D $v\in \mathbf {R^{3}}$ is a "3-tuple" of real numbers $v=\{v_{1},v_{2},v_{3}\}$ where $v_{i}\in \mathbf {R}$ for $i=1,2,3$. Example: $v=\{1,1,-1\}$
# Bounded interval/Continuous function/Equidistant lower integral/Lower integral/Exercise

Let $I$ be a bounded interval and let $f\colon I\rightarrow \mathbb {R}$ denote a continuous function which is bounded from below. Suppose that the
supremum over all staircase integrals for the equidistant lower staircase functions exists. Show that then the supremum for all staircase integrals for lower staircase functions (that is, the lower integral) exists and coincides with the supremum first mentioned.
# Bounded interval/Continuous image is not/Exercise

Show that the image of a bounded interval under a continuous function is not necessarily bounded.
# Boundary Value Problems/Series Solutions

In your Calculus course you were introduced to Power Series and the representation of certain types of functions using Taylor series.
$\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}\,,$ where n! is the factorial of n and f$^{(n)}$(a) denotes the nth derivative of f evaluated at the point a; the zeroth derivative of f is defined to be f itself and (x − a)$^0$ and 0! are both defined to be 1.
If the Taylor series converges to $f(x)$ we write $f(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}\,,$
# Boundary Value Problems/Solving inhomogeneous linear ODEs

Return Boundary Value Problems
$\displaystyle y''+p(t)y'+q(t)y=f(t)$ is a second order linear inhomogeneous differential equation. $f(t)\neq 0$ for the complete interval $t\in [c,d]$. An example: $\displaystyle 2y''+3y'+y=sin(2t)$ .
General information on Ordinary Differential Equations may be found at [Ordinary Differential Equations http://en.wikipedia.org/wiki/Ordinary_differential_equation]. We are interested in a specific type: One where the coefficients are constants. $\displaystyle ay''+by'+cy=f(t)$.
# Bounded interval/Real Function/Step integral from above/Definition

Let $I$ denote a bounded interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. For a step function from above
    $t\colon I\longrightarrow \mathbb {R}$
of $f$, with respect to the partition $a_{i}$, $i=0,\ldots ,n$, and values $t_{i}$, $i=1,\ldots ,n$, the step integral
    $T:=\sum _{i=1}^{n}t_{i}{\left(a_{i}-a_{i-1}\right)}\,$
is called a step integral from above for $f$ on $I$.
# Bounded interval/Real Function/Step integral from below/Definition

Let $I$ denote a bounded interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. For a step function from below
    $t\colon I\longrightarrow \mathbb {R}$
of $f$, with respect to the partition $a_{i}$, $i=0,\ldots ,n$, and values $s_{i}$, $i=1,\ldots ,n$, the step integral
    $S:=\sum _{i=1}^{n}s_{i}{\left(a_{i}-a_{i-1}\right)}\,$
is called a step integral from below for $f$ on $I$.
# English-Chinese/United Arab Emirates

The United Arab Emirates is a country in Western Asia at the southeast end of the Arabian Peninsula on the Persian Gulf.
阿拉伯联合酋长国是位于波斯湾阿拉伯半岛东南端的西亚国家。
# English-Chinese/University

A university is an institution of higher education and research which awards academic degrees in various academic disciplines. Universities typically provide undergraduate education and postgraduate education.
大学 是 一所 高等教育 和 研究的 机构，授予 各学科的 学位。大学 通常 提供 本科教育 和 研究生教育。
# English-Chinese/Up to 90% of cardiovascular disease may be preventable if established risk factors are avoided

Up to 90% of cardiovascular disease may be preventable if established risk factors are avoided.
如果避免已确定的风险因素，高达90％的心血管疾病可能是可以预防的。
# English-Chinese/Untreated celiac disease can cause the development of many types of cardiovascular diseases, most of which improve or resolve with a gluten-free diet and intestinal healing

Untreated celiac disease can cause the development of many types of cardiovascular diseases, most of which improve or resolve with a gluten-free diet and intestinal healing. However, delays in recognition and diagnosis of celiac disease can cause irreversible heart damage.
未经治疗的乳糜泻可导致许多类型的心血管疾病的发展，其中大多数通过不含麸质饮食和肠道治疗来改善或解决。 然而，延迟识别和诊断乳糜泻可导致不可逆转的心脏损伤。
# English-Chinese/Unicellular organism

Unicellular organisms are organisms that have one cell. They are divided into two quite different types, from different classification kingdoms.
* The prokaryotes, bacteria and archaea, have cells with no nucleus and a simple cell structure.
* Eukaryotes have a nucleus, and a more complex cell structure.
单细胞生物 是 具有 一个 细胞的 生物。它们 分为 两种 完全 不同的 类型，来自 不同的 分类王国。
* 原核生物，细菌 和 古细菌 具有 无细胞核 和 简单 细胞 结构的 细胞。
* 真核生物 具有 细胞核 和 更复杂的 细胞结构。
# English-Chinese/United States is one of the world's 17 megadiverse countries

The extremely diverse geography, climate, and wildlife of the United States make it one of the world's 17 megadiverse countries.
美国极其多样化的地理，气候和野生动植物使其成为世界上17个生物多样性丰富的国家之一。
# English-Chinese/United States

The United States of America (USA), often called the United States (U.S.) or America, is the second largest country in North America. It is made up of 50 states, a federal district, and five territories. It has great influence over world finance, trade, culture, military, politics, and technology.
美国 是 北美洲 第二大的 国家。它 由 50个 州，一个 联邦区 和 五个 地区 组成。它 对 世界 金融，贸易，文化，军事，政治 和 技术 影响 很大。
# English-Chinese/Unlike chicken eggs, turkey eggs are not commonly sold as food due to the high demand for whole turkeys and lower output of eggs as compared with other fowl (not only chickens, but even ducks or quail)

Unlike chicken eggs, turkey eggs are not commonly sold as food due to the high demand for whole turkeys and lower output of eggs as compared with other fowl (not only chickens, but even ducks or quail).
与鸡蛋不同，火鸡蛋通常不作为食品出售，因为与其他家禽（不仅是鸡，甚至是鸭子或鹌鹑）相比，对整只火鸡的需求量很大，火鸡蛋产量较低。
# English-Chinese/Up to one-third of migraine people have an aura

Up to one-third of migraine people have an aura: typically a short period of visual disturbance that signals that the headache will soon occur. Occasionally, there is only a little or even no headache after a migraine aura.
多达三分之一的偏头痛患者有先兆：通常是短暂的视力障碍，表明头痛很快就会发生。 偶尔，偏头痛先兆后只有一点甚至没有头痛。
# Business/Companies/Schiphol/Innovation/theGrounds

theGROUNDS is a Schiphol Group initiative, founded in 2010, to work with other companies and knowledge institutes with the aim of developing innovative applications for a sustainable airport. theGROUNDS is a driving force, a breeding ground and a testing lab.

## Index

* Projects: see http://www.thegrounds.nl/en/projects