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hide-koba · 1 year
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. . . 新聞を取りに出た時の嬉しい景色 . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CqYsMSTPDvO/?igshid=NGJjMDIxMWI=
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suzuk1yas · 5 years
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#higashimuroran #morningsun #landscape #streetphotography #pentaxmx1 #mx_1 https://www.instagram.com/p/BuUiOymnDvc/?utm_source=ig_tumblr_share&igshid=bmo0ovz6ibsj
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crazymofo131369 · 4 years
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Planning it
PRESCRIPTION
Valium 10mg 1 c/8h 60 días  
Fenodid solucion inyectable 0,5 mg/10 ml 1 c/24h por 18 dias  (fentanilo)
Omeprazol 2mg 1 c/12h 15 dias 
ALSO
1 bottle of alcohol (vodka)
OPIOD  AND BDZ OVERDOSE
Opioids affect the part of the brain that controls breathing. Taking high doses of opioids can lead to an overdose and a slowdown or interruption of breathing, which can lead to death.
Written is in pharmacology texts and in vademecums that fentanyl, a powerful opioid (100 times more than morphine)
Mix an opioid with other medications, illegal drugs or alcohol. An overdose can be fatal when mixing an opioid and certain medications for the treatment of anxiety, such as Xanax or Valium.
SOURCES
https://sci-hub.tw/https://ascpt.onlinelibrary.wiley.com/doi/abs/10.1002/cpt1977214497
https://www.dea.gov/factsheets/fentanyl
https://www.drugabuse.gov/es/publicaciones/drugfacts/el-fentanilo
https://medlineplus.gov/spanish/opioidoverdose.html
https://www.vademecum.es/equivalencia-lista-fenodid+solucion+inyectable+0%2C5+mg%2F10+ml-mexico-n01ah01-mx_1
https://www.elsevier.es/es-revista-farmacia-profesional-3-articulo-analgesicos-opiaceos-X0213932412941155
https://www.cancer.org/es/tratamiento/tratamientos-y-efectos-secundarios/efectos-secundarios-fisicos/dolor/medicamentos-opioides-para-aliviar-el-dolor-causado-por-el-cancer.html
https://cima.aemps.es/cima/dochtml/ft/39905/FichaTecnica_39905.html
GOODBYE.
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sangklp · 5 years
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一センチマクロできるらしい #PENTAX #MX_1 https://t.co/pebCFlgvyD https://www.youtube.com/c/lifesang
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mathematicianadda · 4 years
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Integrating the product of lines.
The Question.
Suppose we have $n$ linear functions $f_k$ defined on $[x_1,x_2]$. Let $f_k(x_1)=y_k$ and $f_k(x_2)=z_k$ denote the function values at the endpoints of the interval. We would like to calculate
$$\mathfrak{P}(n)=\int_{x_1}^{x_2} \prod_{k=1}^n f_k(x) \, dx$$
in terms of $x_1,x_2,$ and $y_k, z_k$ for $k\in\{1,\ldots,n\}$.
The motivation for this comes from mathematical programming-- specifically, an algorithm I'm writing that requires integrating the product of a large number of piecewise linear functions. What I would like is to find a closed form for $\mathfrak{P}(n)$.
Let's work through a few examples and see if a formula jumps out at us.
Example: $n=2$
The first thing to do is write the $f_k$ in terms of the endpoint values. Solving $f_k(x_1)=mx_1+b=y_k$ and $f_k(x_2)=mx_2+b=z_k$, we get
$$f_k(x)=\frac{z_k-y_k}{x_2-x_1}x+\frac{y_k x_2-z_kx_1}{x_2-x_1}$$
So, then, we are calculating
$$ \mathfrak{P}(2)=\int_{x_1}^{x_2} \left(\frac{z_1-y_1}{x_2-x_1}x+\frac{y_1 x_2-z_1x_1}{x_2-x_1}\right)\left(\frac{z_2-y_2}{x_2-x_1}x+\frac{y_2 x_2-z_2x_1}{x_2-x_1}\right) \, dx $$
which, God help us, is
$$ \left.\frac{x \left(3 x (y_1 z_2 (x_1+x_2)+y_2 z_1 (x_1+x_2)-2 x_1 z_1 z_2-2 x_2 y_1 y_2)+6 (x_1 z_1-x_2 y_1) (x_1 z_2-x_2 y_2)+2 x^2 (y_1-z_1) (y_2-z_2)\right)}{6 (x_1-x_2){}^2}\right|_{x_1}^{x_2} $$ Luckily, this simplifies quite a bit down to $$ \frac{1}{6} (x_2-x_1)(y_1 (2 y_2+z_2)+z_1 (y_2+2 z_2)). $$
Alright! It didn't turn out that bad.
Example: $n=3,\ldots, 6$
Let's take a look at some other values of $\mathfrak{P}$. I'll spare you the intermediate calculations.
$$\begin{eqnarray*}\mathfrak{P}(3)&=&\frac{1}{12} (x_2-x_1) (y_1 y_3 (3 y_2+z_2)+y_1 z_3 (y_2+z_2)+y_3 z_1 (y_2+z_2)+z_1 z_3 (y_2+3 z_2))\\ \mathfrak{P}(4)&=&\frac{1}{60} (x_2-x_1) (y_1 y_4 (3 y_2 (4 y_3+z_3)+z_2 (3 y_3+2 z_3))+y_1 z_4 (3 y_2 y_3+2 y_2 z_3+2 y_3 z_2+3 z_2 z_3)+y_4 z_1 (3 y_2 y_3+2 y_2 z_3+2 y_3 z_2+3 z_2 z_3)+z_1 z_4 (y_2 (2 y_3+3 z_3)+3 z_2 (y_3+4 z_3)))\\ \mathfrak{P}(5)&=&\frac{1}{60}(x_2-x_1) (y_1 y_5 (2 y_2 y_4 (5 y_3+z_3)+y_2 z_4 (2 y_3+z_3)+y_4 z_2 (2 y_3+z_3)+z_2 z_4 (y_3+z_3))+y_1 z_5 (y_2 y_4 (2 y_3+z_3)+y_2 z_4 (y_3+z_3)+y_4 z_2 (y_3+z_3)+z_2 z_4 (y_3+2 z_3))+y_5 z_1 (y_2 y_4 (2 y_3+z_3)+y_2 z_4 (y_3+z_3)+y_4 z_2 (y_3+z_3)+z_2 z_4 (y_3+2 z_3))+z_1 z_5 (y_2 y_4 (y_3+z_3)+y_2 z_4 (y_3+2 z_3)+y_4 z_2 (y_3+2 z_3)+2 z_2 z_4 (y_3+5 z_3)))\\ \mathfrak{P}(6)&=&\frac{1}{420} (x_2-x_1) (y_1 y_2 (2 y_3 y_6 (5 y_4 (6 y_5+z_5)+z_4 (5 y_5+2 z_5))+y_3 z_6 (10 y_4 y_5+4 y_4 z_5+4 y_5 z_4+3 z_4 z_5)+y_6 z_3 (10 y_4 y_5+4 y_4 z_5+4 y_5 z_4+3 z_4 z_5)+z_3 z_6 (4 y_4 y_5+3 y_4 z_5+3 y_5 z_4+4 z_4 z_5))+y_1 z_2 (y_3 y_6 (10 y_4 y_5+4 y_4 z_5+4 y_5 z_4+3 z_4 z_5)+y_3 z_6 (4 y_4 y_5+3 y_4 z_5+3 y_5 z_4+4 z_4 z_5)+y_6 z_3 (4 y_4 y_5+3 y_4 z_5+3 y_5 z_4+4 z_4 z_5)+z_3 z_6 (3 y_4 y_5+4 y_4 z_5+4 y_5 z_4+10 z_4 z_5))+y_2 z_1 (y_3 y_6 (10 y_4 y_5+4 y_4 z_5+4 y_5 z_4+3 z_4 z_5)+y_3 z_6 (4 y_4 y_5+3 y_4 z_5+3 y_5 z_4+4 z_4 z_5)+y_6 z_3 (4 y_4 y_5+3 y_4 z_5+3 y_5 z_4+4 z_4 z_5)+z_3 z_6 (3 y_4 y_5+4 y_4 z_5+4 y_5 z_4+10 z_4 z_5))+z_1 z_2 (y_3 y_6 (4 y_4 y_5+3 y_4 z_5+3 y_5 z_4+4 z_4 z_5)+y_3 z_6 (3 y_4 y_5+4 y_4 z_5+4 y_5 z_4+10 z_4 z_5)+y_6 z_3 (3 y_4 y_5+4 y_4 z_5+4 y_5 z_4+10 z_4 z_5)+2 z_3 z_6 (y_4 (2 y_5+5 z_5)+5 z_4 (y_5+6 z_5))))\end{eqnarray*}$$
There certainly seems to be some pattern here.
The coefficient has a denominator of $\operatorname{lcm}\{1,\ldots,n+1\}$ and we're always multiplying by $x_2-x_1$, so let's just get rid of that first term by looking at $\frac{\operatorname{lcm}\{1,\ldots,n+1\}}{x_2-x_1}\mathfrak{P}_n$. Maybe it will help us see the pattern if we expand everything out.
$$ \begin{eqnarray*}\frac{\operatorname{lcm}\{1,\ldots,4\}}{x_2-x_1}\mathfrak{P}_3&=&3 y_1 y_2 y_3+y_1 y_2 z_3+y_1 y_3 z_2+y_1 z_2 z_3+y_2 y_3 z_1+y_2 z_1 z_3+y_3 z_1 z_2+3 z_1 z_2 z_3\\ \frac{\operatorname{lcm}\{1,\ldots,5\}}{x_2-x_1}\mathfrak{P}_4&=&12 y_1 y_2 y_3 y_4+3 y_1 y_2 y_3 z_4+3 y_1 y_2 y_4 z_3+2 y_1 y_2 z_3 z_4+3 y_1 y_3 y_4 z_2+2 y_1 y_3 z_2 z_4+2 y_1 y_4 z_2 z_3+\cdots\\ &\cdots&3 y_1 z_2 z_3 z_4+3 y_2 y_3 y_4 z_1+2 y_2 y_3 z_1 z_4+2 y_2 y_4 z_1 z_3+3 y_2 z_1 z_3 z_4+2 y_3 y_4 z_1 z_2+3 y_3 z_1 z_2 z_4+\cdots\\&\cdots&3 y_4 z_1 z_2 z_3+12 z_1 z_2 z_3 z_4\\ \frac{\operatorname{lcm}\{1,\ldots,6\}}{x_2-x_1}\mathfrak{P}_5&=&10 y_1 y_2 y_3 y_4 y_5+2 y_1 y_2 y_3 y_4 z_5+2 y_1 y_2 y_3 y_5 z_4+y_1 y_2 y_3 z_4 z_5+2 y_1 y_2 y_4 y_5 z_3+y_1 y_2 y_4 z_3 z_5+\cdots\\&\cdots&y_1 y_2 y_5 z_3 z_4+y_1 y_2 z_3 z_4 z_5+2 y_1 y_3 y_4 y_5 z_2+y_1 y_3 y_4 z_2 z_5+y_1 y_3 y_5 z_2 z_4+y_1 y_3 z_2 z_4 z_5+\cdots\\&\cdots&y_1 y_4 y_5 z_2 z_3+y_1 y_4 z_2 z_3 z_5+y_1 y_5 z_2 z_3 z_4+2 y_1 z_2 z_3 z_4 z_5+2 y_2 y_3 y_4 y_5 z_1+y_2 y_3 y_4 z_1 z_5+\cdots\\&\cdots&y_2 y_3 y_5 z_1 z_4+y_2 y_3 z_1 z_4 z_5+y_2 y_4 y_5 z_1 z_3+y_2 y_4 z_1 z_3 z_5+y_2 y_5 z_1 z_3 z_4+2 y_2 z_1 z_3 z_4 z_5+\cdots\\&\cdots&y_3 y_4 y_5 z_1 z_2+y_3 y_4 z_1 z_2 z_5+y_3 y_5 z_1 z_2 z_4+2 y_3 z_1 z_2 z_4 z_5+y_4 y_5 z_1 z_2 z_3+2 y_4 z_1 z_2 z_3 z_5+\cdots\\&\cdots&2 y_5 z_1 z_2 z_3 z_4+10 z_1 z_2 z_3 z_4 z_5\\ \frac{\operatorname{lcm}\{1,\ldots,7\}}{x_2-x_1}\mathfrak{P}_6&=& 60 y_1 y_2 y_3 y_4 y_5 y_6+10 y_1 y_2 y_3 y_4 y_5 z_6+10 y_1 y_2 y_3 y_4 y_6 z_5+4 y_1 y_2 y_3 y_4 z_5 z_6+10 y_1 y_2 y_3 y_5 y_6 z_4+\cdots\\&\cdots& 4 y_1 y_2 y_3 y_5 z_4 z_6+4 y_1 y_2 y_3 y_6 z_4 z_5+3 y_1 y_2 y_3 z_4 z_5 z_6+10 y_1 y_2 y_4 y_5 y_6 z_3+4 y_1 y_2 y_4 y_5 z_3 z_6+\cdots\\&\cdots& 4 y_1 y_2 y_4 y_6 z_3 z_5+3 y_1 y_2 y_4 z_3 z_5 z_6+4 y_1 y_2 y_5 y_6 z_3 z_4+3 y_1 y_2 y_5 z_3 z_4 z_6+3 y_1 y_2 y_6 z_3 z_4 z_5+\cdots\\&\cdots& 4 y_1 y_2 z_3 z_4 z_5 z_6+10 y_1 y_3 y_4 y_5 y_6 z_2+4 y_1 y_3 y_4 y_5 z_2 z_6+4 y_1 y_3 y_4 y_6 z_2 z_5+3 y_1 y_3 y_4 z_2 z_5 z_6+\cdots\\&\cdots& 4 y_1 y_3 y_5 y_6 z_2 z_4+3 y_1 y_3 y_5 z_2 z_4 z_6+3 y_1 y_3 y_6 z_2 z_4 z_5+4 y_1 y_3 z_2 z_4 z_5 z_6+4 y_1 y_4 y_5 y_6 z_2 z_3+\cdots\\&\cdots& 3 y_1 y_4 y_5 z_2 z_3 z_6+3 y_1 y_4 y_6 z_2 z_3 z_5+4 y_1 y_4 z_2 z_3 z_5 z_6+3 y_1 y_5 y_6 z_2 z_3 z_4+4 y_1 y_5 z_2 z_3 z_4 z_6+\cdots\\&\cdots& 4 y_1 y_6 z_2 z_3 z_4 z_5+10 y_1 z_2 z_3 z_4 z_5 z_6+10 y_2 y_3 y_4 y_5 y_6 z_1+4 y_2 y_3 y_4 y_5 z_1 z_6+4 y_2 y_3 y_4 y_6 z_1 z_5+\cdots\\&\cdots& 3 y_2 y_3 y_4 z_1 z_5 z_6+4 y_2 y_3 y_5 y_6 z_1 z_4+3 y_2 y_3 y_5 z_1 z_4 z_6+3 y_2 y_3 y_6 z_1 z_4 z_5+4 y_2 y_3 z_1 z_4 z_5 z_6+\cdots\\&\cdots& 4 y_2 y_4 y_5 y_6 z_1 z_3+3 y_2 y_4 y_5 z_1 z_3 z_6+3 y_2 y_4 y_6 z_1 z_3 z_5+4 y_2 y_4 z_1 z_3 z_5 z_6+3 y_2 y_5 y_6 z_1 z_3 z_4+\cdots\\&\cdots& 4 y_2 y_5 z_1 z_3 z_4 z_6+4 y_2 y_6 z_1 z_3 z_4 z_5+10 y_2 z_1 z_3 z_4 z_5 z_6+4 y_3 y_4 y_5 y_6 z_1 z_2+3 y_3 y_4 y_5 z_1 z_2 z_6+\cdots\\&\cdots& 3 y_3 y_4 y_6 z_1 z_2 z_5+4 y_3 y_4 z_1 z_2 z_5 z_6+3 y_3 y_5 y_6 z_1 z_2 z_4+4 y_3 y_5 z_1 z_2 z_4 z_6+4 y_3 y_6 z_1 z_2 z_4 z_5+\cdots\\&\cdots& 10 y_3 z_1 z_2 z_4 z_5 z_6+3 y_4 y_5 y_6 z_1 z_2 z_3+4 y_4 y_5 z_1 z_2 z_3 z_6+4 y_4 y_6 z_1 z_2 z_3 z_5+10 y_4 z_1 z_2 z_3 z_5 z_6+\cdots\\&\cdots& 4 y_5 y_6 z_1 z_2 z_3 z_4+10 y_5 z_1 z_2 z_3 z_4 z_6+10 y_6 z_1 z_2 z_3 z_4 z_5+60 z_1 z_2 z_3 z_4 z_5 z_6 \end{eqnarray*} $$
The pattern in the variables is easy to see-- there are $2^n$ terms, each of which has terms $1$ through $n$ of either the $y$ or the $z$. (What I mean is, the terms are in 1-to-1 correspondence with $\{y_1,z_1\}\times \cdots \times \{y_n,z_n\}$.)
But, what are the coefficients?
from Hot Weekly Questions - Mathematics Stack Exchange from Blogger https://ift.tt/3divoEa
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hide-koba · 1 year
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. . . 桃の節句 .  気づいたら地元のひな飾り展にも行っていない。明日が最終日とのこと。 . . .  心が亡くならないようにしなくてはと...。 . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CpTJmnGSDgJ/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . 橋の上での渋滞時。3.11の後はこの状況に恐怖を感じていたことを思い出した。 . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CodKMbGyWkX/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . 夕暮れ方 . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/Cn7cm3ivaPo/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . 芸備線の朱 . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CoQb0fISysI/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . すっかりオブジェ化したこの楽器を知ったのは,キャプテンハーロックだったなと。 . #松本零士先生 #キャプテンハーロック #オカリナ #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CpI3IQUyszi/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . 見上げて . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CnKsJ88SUc6/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . 目にすると毎回「富士山だ〜!」と心の中で声を出している。 . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CnC75Jiyl4N/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . 濃厚接触→陽性3日目 発熱に加えて悪寒,頭痛,咳,喉の痛みにめまいかな。めちゃくちゃしんどい...。 . #アラジンストーブ #ブルーフレームヒーター #Aladdinstove #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/CmTDDgLPRxY/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . 青い炎 . #アラジンストーブ #ブルーフレームヒーター #Aladdinstove #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/Clb_UFZSVyr/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . おしまい .  寂しかった話ではなく,テイクアウトをお願いておいて良かっと。  受け取り時に撮影。 . #茨城県 #坂東市 #ハリオ商店 #PENTAX #PENTAXIAN #MX1 #MX_1 📷 2023/01/08 https://www.instagram.com/p/CnP1cqnS3JN/?igshid=NGJjMDIxMWI=
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hide-koba · 1 year
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. . . MX-1は改めて良いな,と。 . #PENTAX #PENTAXIAN #MX1 #MX_1 https://www.instagram.com/p/ClGaFn3vgL_/?igshid=NGJjMDIxMWI=
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