OpenLLM-France/wikimedia huggingface dataset:
# LU Decomposition exercises and quiz
## Exercise 1

Problem Given the 3x3 matrix:
    $A={\begin{bmatrix}2&-1&3\\4&2&1\\-6&-1&2\end{bmatrix}}$
Find LU decomposition of A
Solution
The procedure of LU decomposition runs similar to the process of Gaussian Elimination. Firstly A is reduced to upper triangular form, which is U, using just the third elementary row operation, namely: add to one row of matrix a scalar time another row of that same matrix. Those scalar used during this process are co-efficient in the L matrix.
The final result will look like this: $A={\begin{bmatrix}2&-1&3\\4&2&1\\-6&-1&2\end{bmatrix}}={\begin{bmatrix}1&0&0\\l_{21}&1&0\\l_{31}&l_{32}&1\end{bmatrix}}\times {\begin{bmatrix}u_{11}&u_{12}&u_{13}\\0&u_{22}&u_{23}\\0&0&u_{33}\end{bmatrix}}$.
Here are the solution for this problem:
According to Gaussian Elimination, the first number in row 2 must be zero-out by adding the first row of matrix a scalar times second row. This scalar, fortunately, is $-l_{21}$.
Therefore:

## Exercise 2

Use exercise 1's result to solve the system:
${\begin{bmatrix}2x_{1}-x_{2}+3x_{3}=4\\4x_{1}+2x_{2}+x_{3}=7\\-6x_{1}-x_{2}+2x_{3}=-5\end{bmatrix}}$
Solution:
The idea of using LU decomposition to solve systems of simultaneous linear equations Ax=b is rewriting the systems as L(Ux)=b. To solve x, we first solve the systems Ly=b for y, and then, once y is determined, we solve the systems: Ux=y for x. Both systems are easy to solve, the first by forward substitution and the second by backward substitution.
Here is the solution for this exercise:
This system has the matrix form:
$A={\begin{bmatrix}2&-1&3\\4&2&1\\-6&-1&2\end{bmatrix}}\times {\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}$ = ${\begin{bmatrix}4\\7\\-5\end{bmatrix}}.$
Since $A=LU$, by Exercise 1 we have:
${\begin{bmatrix}1&0&0\\**&1&0\\**&*&1\end{bmatrix}}\times {\begin{bmatrix}2&-1&3\\0&4&-5\\0&0&6\end{bmatrix}}\times {\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}4\\7\\-5\end{bmatrix}}.$
Lets $y=U\times x$, we have:
${\begin{bmatrix}1&0&0\\**&1&0\\**&*&1\end{bmatrix}}\times {\begin{bmatrix}y_{1}\\y_{2}\\y_{3}\end{bmatrix}}={\begin{bmatrix}4\\7\\-5\end{bmatrix}}$.
Use forward substitution we have:
# Numerical Analysis/Lagrange example

We'll find the interpolating polynomial passing through the points $(1,-6)$, $(2,2)$, $(4,12)$, using the Lagrange method.
We first use the formula to write the following:
$p(x)=-6{\frac {(x-2)}{(1-2)}}{\frac {(x-4)}{(1-4)}}+2{\frac {(x-1)}{(2-1)}}{\frac {(x-4)}{(2-4)}}+12{\frac {(x-1)}{(4-1)}}{\frac {(x-2)}{(4-2)}}$
After some simplification, we get:
$p(x)=-2(x-2)(x-4)-1(x-1)(x-4)+2(x-1)(x-2)$
$p(x)=-2(x^{2}-6x+8)-1(x^{2}-5x+4)+2(x^{2}-3x+2)$
And our answer:
$p(x)=-x^{2}+11x-16$.

## Adding a point

Now we'll add a point to our data set, and find a new interpolating polynomial. Let us add the point $(3,-10)$ to our set. Starting over with the Lagrange formula, we write:
$p(x)=-6{\frac {(x-2)}{(1-2)}}{\frac {(x-4)}{(1-4)}}{\frac {(x-3)}{(1-3)}}+2{\frac {(x-1)}{(2-1)}}{\frac {(x-4)}{(2-4)}}{\frac {(x-3)}{(2-3)}}+12{\frac {(x-1)}{(4-1)}}{\frac {(x-2)}{(4-2)}}{\frac {(x-3)}{(4-3)}}-10{\frac {(x-1)}{(3-1)}}{\frac {(x-2)}{(3-2)}}{\frac {(x-4)}{(3-4)}}$
Simplifying, we get:
$p(x)=(x-2)(x-4)(x-3)+(x-1)(x-4)(x-3)+2(x-1)(x-2)(x-3)+5(x-1)(x-2)(x-4)$
$p(x)=(x^{3}-9x^{2}+26x-24)+(x^{3}-8x^{2}+19x-12)+2(x^{3}-6x^{2}+11x-6)+5(x^{3}-7x^{2}+14x-8)$
And our polynomial is:
$p(x)=9x^{3}-64x^{2}+137x-88$.
# Esper/rp/i

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# Polish language

Polish is a Slavic language spoken primarily in Poland. It belongs to the Lechitic subgroup of West Slavic languages. Polish is the official language of Poland, but it is also used throughout the world by Polish minorities in other countries. Its written standard is the Polish alphabet, which has 9 additions to the letters of the basic Latin script (ą, ć, ę, ł, ń, ó, ś, ź, ż).

## Resources

* Alphabet and phonetics
* Być
* Countries and nationalities
* Nouns and adjectives
* Foods and Animals
# Polish language/Być

Before learning this verb, we must first learn the Personal pronouns in the nominative case.

## Personal Pronouns

* Ja - I
* Ty - You (singular)
* On - He
* Ona - She
* Ono - It
* My - We
* Wy - You (plural)
* Oni - They (male or mixed)
* One - They (female only)
When speaking to someone to whom we should show respect, e.g. those older or strangers, we should use the personal pronouns pan (m) and pani (f). When we do so, we use the Wy form of the verb (i.e. the you (plural) version).

## Być - to be

Być is the Polish infinitive form of the English verb "to be". This is the most commonly used verb in the Polish language and should be the first verb learnt in any language. It is an irregular verb and, therefore, provides no pattern from which to learn other verbs. This verb has three tenses: past, present and future.

### Past Tense

The past tense of Polish verbs depends entirely upon the gender of the subject.
| Masculine Ja byłem - I was Ty byłeś - You were On był - He was My byliśmy - We were Wy byliście - You were Oni byli - They were | Feminine Ja byłam - I was Ty byłaś - You were Ona była - She was My byłyśmy - We were Wy byłyście - You were One były - They were | Neuter --------- --------- Ono było - It was --------- --------- --------- |

### Present Tense

The present tense does not depend on the gender of the subject.
* Ja jestem - I am
* Ty jesteś - You are
* On jest - He is
* Ona jest - She is
* Ono jest - It is
* My jesteśmy - We are
* Wy jesteście - You are
* Oni są - They are
* One są - They are

### Future Tense

Just like the present tense, the future tense does not depend on the gender of the subject.
* Ja będę - I will be
* Ty będziesz - You will be
* On będzie - He will be
* Ona będzie - She will be
* Ono będzie - It will be
* My będziemy - We will be
* Wy będziecie - You will be
* Oni będą - They will be
* One będą - They will be

## Summary

The verb "być" has three tenses: past, present and future. The past depends upon the gender of the subject but the present and future tenses do not.
# Real exponential function/Base/Derivative/Fact

The exponential function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto a^{x},$
with base $a>0$,
differentiable with
    ${\left(a^{x}\right)}'={\left(\ln a\right)}a^{x}\,.$
# Real cosine function/One zero between 0 and 2/Fact/Proof

We consider the cosine series
    $\cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}\,.$
For $x=0$, we have $\cos 0=1$. For $x=2$, one can write
    ${\begin{aligned}\cos 2&=1-{\frac {2^{2}}{2!}}+{\frac {2^{4}}{4!}}-{\frac {2^{6}}{6!}}+{\frac {2^{8}}{8!}}-\ldots \\&=1-{\frac {2^{2}}{2!}}{\left(1-{\frac {4}{3\cdot 4}}\right)}-{\frac {2^{6}}{6!}}{\left(1-{\frac {4}{7\cdot 8}}\right)}-\ldots \\&=1-2(2/3)-\ldots \\&\leq -1/3.\end{aligned}}$
Hence, due to the intermediate value theorem, there exists at least one zero in the given interval. To prove uniqueness, we consider the derivative of cosine, which is
    $\cos 'x=-\sin x\,,$
due to fact. Hence, it is enough to show that sine is positive in the interval $]0,2[$, because then cosine is strictly decreasing by fact in the interval and there is only one zero. Now, for $x\in0,2]}$, we have
    ${\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\ldots \\&=x{\left(1-{\frac {x^{2}}{3!}}\right)}+{\frac {x^{5}}{5!}}{\left(1-{\frac {x^{2}}{6\cdot 7}}\right)}+\ldots \\&\geq x{\left(1-{\frac {4}{3!}}\right)}+{\frac {x^{5}}{5!}}{\left(1-{\frac {4}{6\cdot 7}}\right)}+\ldots \\&\geq x/3\\&>0.\end{aligned}}$
# Real estate

| Outline First tier Real estate principles Real estate agency Real estate development Construction fundamentals Real estate finance Second tier Real estate valuation Investment property analysis Property management Real Estate law Real Estate taxation Third tier Income property valuation Real estate market analysis Real estate asset management International real estate Real estate project planning and feasibility Fourth tier Urban development regulations Real estate redevelopment (Adaptive Re-use) Historic preservation Affordable housing Sustainable construction Six tier Current Real Estate Trends | Resources Wikipedia Real estate Real estate pricing Real estate economics Real estate appraisal List of real estate topics Real estate trends Real property Fee simple Land use Zoning Zoning in the United States Variance Form-based code Historic preservation See also School:Urban studies and planning School:Business Real estate analysis exam review External links Open Standards Consortium for Real Estate |
# Real closed interval/Strictly increasing/Inverse function/Continuous/Fact

Let $I\subseteq \mathbb {R}$ be an interval and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a continuous, strictly increasing function.
image
    $J:=f(I)\,$
is also an interval, and the inverse function
    $f^{-1}\colon J\longrightarrow I$
# Real exponential function/Base/Derivative/Fact/Proof

By definition, we have
    $a^{x}=\exp {\left(x\,\ln a\right)}\,.$
The derivative with respect to $x$ equals
    ${\left(a^{x}\right)}'={\left(\exp {\left(x\,\ln a\right)}\right)}'={\left(\ln a\right)}\exp '(x\,\ln a)={\left(\ln a\right)}\exp {\left(x\,\ln a\right)}={\left(\ln a\right)}a^{x}\,,$
due to fact and the chain rule.
# Real cosine function/One zero between 0 and 2/Fact

The cosine function
real interval $[0,2]$, exactly one
# Real estate/Development processes

In this course you will learn about the over all development process of real estate, from Project identification to project completion.
Important aspects of real estate development process.
* Fund management
* Asset managemant
* Accounts and Finance
* Business development
* Marketing
* Customer services
* Planning and Architecture
* Construction
* Operation and Maintainance
Onlyramram 12:16, 9 September 2008 (UTC)
# Progress and Prospects in Parkinson's Research/Symptoms/Palilalia

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Palilalia
Forced repetition of one’s own words
The following references are relevant:-
Ackermann et al (1989) Palilalia as a symptom of levodopa induced hyperkinesia in Parkinson's disease.
Anderson et al (1999) Developmental stuttering and Parkinson's disease: the effects of levodopa treatment
Benke et al (2000) Repetitive speech phenomena in Parkinson's disease
Further Research
References
* ↑ Ackermann, H; Ziegler, W. and Oerte, W. H.J. Neurol. Neurosurg. Psychiatry. 1989 Ju) Palilalia as a symptom of levodopa induced hyperkinesia in Parkinson's disease.http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1032047/
* ↑ Anderson, J.; Hughes, J.; Rothi, L.; Crucian, L. and Crucian, K. (1981) Full Text J. Neurol. Neurosurg. Psychiatry. 66 (6) 776 – 778 Developmental stuttering and Parkinson's disease: the effects of levodopa treatmenthttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC1736378/
* ↑ Benke, T.; Hohenstein, C.; Poewe, W. and Butterworth, B. (2000) Full Text J. Neurol. Neurosurg. Psychiatry. 69 (3) 319 – 324. Repetitive speech phenomena in Parkinson's diseasehttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC1737094/
# Progress and Prospects in Parkinson's Research/Symptoms/Olfactory dysfunction

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Olfactory dysfunction
A reduced or impaired ability to detect odors.
The following references are relevant:-
Hoyles and Sharma (2013) Olfactory loss as a supporting feature in the diagnosis of Parkinson's disease: a pragmatic approach
Takeda (2013) Severe olfactory dysfunction is a predictor of dementia with Parkinson’s disease
Hüttenbrink et al (2013) Olfactory dysfunction: common in later life and early warning of neurodegenerative disease
Further Research
References
* ↑ Hoyles, K. and Shama, (2013) Full Text J.C. J. Neurol. (Feb. 3) Olfactory loss as a supporting feature in the diagnosis of Parkinson's disease: a pragmatic approach. http://www.ncbi.nlm.nih.gov/pubmed/23377435
* ↑ Takeda,A. (2013) Full Text Rinsho Shinkeigaku 53 (2) 91 – 97 Severe olfactory dysfunction is a predictor of dementia with Parkinson’s disease http://www.ncbi.nlm.nih.gov/pubmed/23470887
* ↑ Hüttenbrink. K.B.; Hummel.T.; Berg, D; Gasser, T. and Hähner,A. (2013) Full TextDtsch. Arztebl. Int. 110 (1-2) 1 – 7 Olfactory dysfunction: common in later life and early warning of neurodegenerative disease http://www.ncbi.nlm.nih.gov/pubmed/23450985
# Progress and Prospects in Parkinson's Research/Symptoms/Narcolepsy

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Narcolepsy Sudden urge to sleep.
The following references are relevant:-
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1978248/
http://www.ncbi.nlm.nih.gov/pubmed/22747735
http://www.ncbi.nlm.nih.gov/pubmed/22939103
Further Research
References
# Progress and Prospects in Parkinson's Research/Symptoms/Oculogyric crises

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## Oculogyric crises

Attacks of forced deviation of gaze;
The following references are relevant:-
Clough et al (1983) Oculogyric crises and parkinsonism. A case of recent onset
Furumoto et al (1989) Oculogyric crisis as an initial symptom of juvenile parkinsonism-like disease
Wilkaniec et al (2013) Toxicity of extracellular secreted alpha-synuclein: Its role in nitrosative stress and neurodegeneration.
# Progress and Prospects in Parkinson's Research/Symptoms/Palipraxia

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## Palipraxia

Forced repetitive movement.
The following references are relevant:-
Ghika (----) Mood and behaviour in disorders of the basal ganglia.
Critchley (1970) Aphasioloy and other aspects of language
Naremore (1981) Language science : recent advances
# Progress and Prospects in Parkinson's Research/Symptoms/Ophthalmoplegia

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Ophthalmoplegia
Paralysis of gaze.
The following references are relevant:-
Guiloff et al (1980)Reversible supranuclear ophthalmoplegia associated with Parkinsonism
Armstrong (2011)Visual signs and symptoms of progressive supranuclear palsy.
Pontieri et al (2012) Sad and happy facial emotion recognition impairment in progressive supranuclear palsy in comparison with Parkinson's disease Further Research
References
* ↑ Guiloff, R.J.; Geirge, R.J. and Marsden, C.D.(1980) Full Text J. Neurol. Neurosurg. Psychiatry. 43 (6) 552 – 554 Reversible supranuclear ophthalmoplegia associated with Parkinsonism http://www.ncbi.nlm.nih.gov/pmc/articles/PMC490600/
* ↑ Armstrong, R.A. (2011) Full TextClin. Exp. Optom. 94 (2) 150 – 160 Visual signs and symptoms of progressive supranuclear palsy. http://www.ncbi.nlm.nih.gov/pubmed/20629667
* ↑ Pontieri, F.E.; Assogna, F.; Stefani, A.; Pierantozzi, M.; Meco, G.; Benincasa, D.; Colosimo, C.; Caltagirone, C. and Spalletta, G.(2012) AbstractParkinsonism Relat. Disord.18 (7) 871 -875 Sad and happy facial emotion recognition impairment in progressive supranuclear palsy in comparison with Parkinson's disease http://www.ncbi.nlm.nih.gov/pubmed/22595619
# Progress and Prospects in Parkinson's Research/Symptoms/Orthostatic hypotension

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Orthostatic hypotension
Drop in blood pressure during changes in body position.
A recent review of Cardiovascular dysautonomia in Parkinson's by Jain and Goldstein (2012) is particularly relevant.
The following references are also relevant:-
Hussain et al (2001) Treatment of erectile dysfunction with sildenafil citrate (Viagra) in parkinsonism due to Parkinson's disease or multiple system atrophy with observations on orthostatic hypotension
Park and Stacy(2011) Dopamine-Induced Nonmotor Symptoms of Parkinson's Disease
Sommer et al (2011) Nondipping in Parkinson's Disease
Further Research
References
* ↑ Jain, Samay, and David S. Goldstein. Cardiovascular Dysautonomia in Parkinson Disease: From Pathophysiology to Pathogenesis. Neurobiology of Disease 46, no. 3 (June 2012): 572–580. Full Text PubMed Central http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3299874/
* ↑ Husain, I.; Brady, C.; Swinn, M.; Mathias, C. and Fowler, C. (2001) Full TextJ. Neurol. Neurosurg. Psychiatry. 71 (3) 371 – 374 Treatment of erectile dysfunction with sildenafil citrate (Viagra) in parkinsonism due to Parkinson's disease or multiple system atrophy with observations on orthostatic hypotension http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1737541/
* ↑ Park, Ariane and Stacy, Mark(2011) Full TextParkinsons Dis. 2011 485063. Dopamine-Induced Nonmotor Symptoms of Parkinson's Diseasehttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC3096061/
* ↑ Sommer, Sita: Aral-Bexhwe, Billur and Jost, Wolfgang (2011) Full TextParkinsons Dis. 2011: 897586. Nondipping in Parkinson's Disease http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3175384/
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Differentiable function/Mean value theorem/Quotient version/Fact/Factreferencenumber

Theorem 15.10
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Endomorphism/Eigenvalue and characteristic polynomial/Fact/Factreferencenumber

Theorem 28.2
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Exponential function/Base/Rational exponent/Compatible/Exercise/Exercisereferencenumber

Exercise 12.17
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Endomorphismus/Finite dimensional/Change of basis/Fact/Factreferencenumber

Corollary 25.9
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Differentiable function/R/Higher derivatives/Positive/Zeroes/Exercise/Exercisereferencenumber

Exercise 15.17
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Elementary row operations/Column rank constant/Exercise/Exercisereferencenumber

Exercise 26.2
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Endomorphism/Characteristic polynomial/Well-defined/Exercise/Exercisereferencenumber

Exercise 28.3
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Endomorphism/Eigenspaces are linear subspaces/Zero/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 27.14
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Endomorphism/Eigenvectors/Linearly independent/Fact/Factreferencenumber

Lemma 27.14
# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Even number/Last digit/Exercise/Exercisereferencenumber

Exercise 1.21