OpenLLM-France/wikimedia huggingface dataset:
# Introduction

Translation:
We call an expression in the form
X' = A ⋅ X + G(t)
homogeneous if G(t)≡0. Now, in previous methods of differential equations, it turned out that X had an exponential of the transcendental number e in its form, so if a uniqueness theorem is developed, we can define a possible answer with this form, set it in the equation, and determine if this answer works and if so how to obtain the answer and its corresponding exponentials.

# Results

So, because the exponential function appeared many times in simpler differential equations, we will guess that the solution for X is X = u ⋅ $e^{\lambda t}$, where u is a coefficient matrix.
Thus:
$\lambda \mathbf {u} e^{\lambda t}=\mathbf {A} \mathbf {u} e^{\lambda t}$
$\lambda e^{\lambda t}=\mathbf {A} e^{\lambda t}$
$(\mathbf {A} -\lambda \mathbf {I} )\mathbf {u} =0$
There is a lie here, we're also going to make one more assumption: a constant matrix for A; but this is the definition for an eigenvalue-eigenvector pair! Thus with a two-by-two matrix there are two linearly independent solutions, and thus by the principle of superposition the constant matrix multiplication by an augmented matrix of these two solutions makes the fundamental set of solutions of which we are trying to look for.
However, due to the property of these eigenvalues (and that we want real-solutions to help analysis in physical models utilizing these differential techniques), there are different ways of creating the fundamental set of solutions to the three possible cases that the pair of eigenvalues could fall under:
* Real, Distinct Eigenvalues Method
* Imaginary Eigenvalues Method
* Repeated Eigenvalue Method
# Linear Algebra with Differential Equations/Heterogeneous Linear Differential Equations/Laplacian Transforms

Yet AGAIN, very similar to the normal technique. The only nuance is how to take a Laplacian operator of a matrix, however, the Laplacian operator by definition is basically an integral: take the operator of each term inside the matrix. The Laplacian operator then boils the problem to an exercise of linear algebra. and the reverse Laplacian operator works the same way: on each term in the matrix. It's nearly identical to how it worked in normal differential equations.
# Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations/Imaginary Eigenvalues Method

When eigenvalues become complex, mathematically, there isn't much wrong. However, in certain physical applications (like oscillations without damping) there is a problem in understanding what exactly does an imaginary answer mean? Thus there is a concerted effort to try to "mathematically hide" the complex variables in order to achieve a more approachable answer for physicists and engineers. Essentially, we have a solution that in part looks like this:
$(\alpha +\beta \cdot i)\cdot e^{r+i\lambda _{1}\cdot t}$
But by Euler's formula:
$(\alpha +\beta \cdot i)\cdot e^{r}(cos(\lambda _{1}\cdot t)+i\cdot sin(\lambda _{1}\cdot t))$
Now we distribute the terms:
$e^{r}(\alpha \cdot cos(\lambda _{1}\cdot t)-\beta \cdot sin(\lambda _{1}\cdot t))+i\cdot e^{r}(\beta \cdot cos(\lambda _{1}\cdot t)+\alpha \cdot sin(\lambda _{1}\cdot t))$
Since this is a linear combination of two terms, $i$ is a constant (complex, but still a constant), each part is an element of the set of solutions and the general solution can be constructed therein.
# Linear Algebra with Differential Equations/Heterogeneous Linear Differential Equations/Method of Undetermined Coefficients

This is very similar to the Method of Undetermined Coefficients encountered in normal differential equations, with some slight exceptions to the "rules" of guessing. Actually, there's only one rule extra. In the normal method of undetermined coefficients when there was a conflict of the characteristic equation with the particular solution, there was a multiplication by the independent variable. In this case we multiply by $\mathbf {A} t+\mathbf {B}$ to include more possible solutions. That, and when working through the problem thoughts about the signifigance of getting a trivial solution when finding eigenvalues must be kept in mind. Other than that, it's pretty much as it was, and is a very powerful method (although it can be quite tedious).
# Introduction

We now tackle the problem of $\mathbf {G} (t)$ being nonzero, so that we have the following problem:
$\mathbf {X} '=\mathbf {AX} +\mathbf {G} (t)$
There are four reasonable ways to solve this.
* Diagonalization
* Method of Undetermined Coefficients
* Variation of Parameters
* Laplacian Transforms
# Linear Algebra with Differential Equations/Heterogeneous Linear Differential Equations/Diagonalization

First of all (and kind of obvious suggested by the title), $\mathbf {A}$ must be diagonalizable. Second, the eigenvalues and eigenvectors of $\mathbf {A}$ are found, and form the matrix $\mathbf {T}$ which is an augemented matrix of eigenvectors, and $\mathbf {D}$ which is a matrix consisting of the corresponding eigenvalues on the main diagonal in the same column as their corresponding eigenvectors. Then with our central problem:
$\mathbf {X} '=\mathbf {AX} +\mathbf {G} (t)$
We substitute:
$\mathbf {TY} '=\mathbf {ATY} +\mathbf {G} (t)$
Then left multiply by $\mathbf {T} ^{-1}$
$\mathbf {Y} '=\mathbf {T} ^{-1}\mathbf {ATY} +\mathbf {T} ^{-1}\mathbf {G} (t)$
As a consequence of Linear Algebra we take the following identity:
$\mathbf {D} =\mathbf {T} ^{-1}\mathbf {AT}$
Thus:
$\mathbf {Y} '=\mathbf {DY} +\mathbf {T} ^{-1}\mathbf {G} (t)$
And because of the nature of the diagonal the problem is a series of one-dimensional normal differential equations which can be solved for $\mathbf {Y}$ and used to find out $\mathbf {X}$.
# Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations/Real, Distinct Eigenvalues Method

If the eigenvalues for the characteristic equation are real and distinct, mathematically, nothing is really wrong. Thus, by our guess and the existence and uniqueness theorem, for an n-size square matrix, the solution set is determined by:
$\mathbf {X} =\{e^{\lambda _{1}\cdot t};e^{\lambda _{2}\cdot t};...;e^{\lambda _{n-1}\cdot t};e^{\lambda _{n}\cdot t}\}$
Then since the linear combination of two solutions is also a solution (which can be verified directly from the structure of the problem), we can form the general solution as such:
$\mathbf {X} =c_{1}e^{\lambda _{1}\cdot t}+c_{2}e^{\lambda _{2}\cdot t}+...+c_{n-1}e^{\lambda _{n-1}\cdot t}+c_{n}e^{\lambda _{n}\cdot t}$
What's interesting is when the eigenvalues are not so simple.
# Some Graphical Analysis

So far we've dealt with $\mathbf {A}$ being a constant matrix, and other niceties; but when it is otherwise, and thus a non-linear differential equation, the best way to find a solution is by graphical means. By taking the independent variables on the axis of a graph, we can note several types of behavior that suggest the form of a solution.
So without adue, here are the main types of behaviors, and their suggested causes:
A nodal source (the graph tends away from a point): real, distinct positive eigenvalues.
A nodal sink (the graph approaches in towards a point): real, distinct negative eigenvalues.
A saddle point (the graph approaches from one end and deviates away at another): real, distinct, opposite eigenvalues.
A spiral point (spirals in or away from a point): a complex eigenvalues.
A series of ellipses around a point: a purely imaginary eigenvalue.
A star point (straight lines deviating or coming towards a point): repeated eigenvalues.
# Neuroimaging Data Processing/Processing/Steps

This is a collection of possible preprocessing steps. Which steps you include and also their order depends on your research question and to an extent also on the software you use
# Neuroimaging Data Processing/Processing/Steps/Skull Stripping
## Concept

Removal of the skull and other non-brain tissue like dura and eyes from anatomical images, which could otherwise complicate e.g. coregistration and normalization steps. Skull stripping can be part of the tissue segmentation (e.g. in SPM) but is mostly done by specialized algorithms that delineate the brain boundary. See for a comparison of some brain extraction algorithms (BSE, BET, SPM, and McStrip), which suggests that all algorithms perform well in general but results highly depend on the particular dataset.

## Implementation
#### SPM

Skull stripping in SPM can be achieved by segmenting the anatomical scan, and using a thresholded version of the sum of grey and white matter probability maps to mask out the bias corrected structural scan.
This masking can be done using SPM --> Util --> Image Calculator (ImCalc button) directly in the batch editor or by pressing the ImCalc button on the SPM GUI. The image calculator allows to perform algebraic manipulations on a set of structural images.
Input Images = segmentation generated probability maps of grey matter(i1) and white matter(i2) and the original anatomical scan (i3) Expression = defining the threshold to mask out the skull based on the selected tissue probability maps e.g ·[i3.*((i1 +i2) > 0.2) ] --> if you set the threshold higher then your skull strip is more stringent For older versions of SPM (or OldSeg), you can use the following expression: (i3.*(i1+i2))

#### FSL

BET and BET2

#### AFNI

3dSkullStrip is a program in AFNI to extract the brain from surrounding tissue from MRI T1-weighted images. The simplest command would be:
The process by default includes spatial normalization, some intensity normalization and repositioning of the brain, but to a certain extent those can be switched off. The actual skull stripping is a modified version of the BET algorithm, expanding a spherical surface iteratively until it envelopes the brain. The output can be a skull stripped (masked) brain, the mask itself or different surface formats.
3dSkullStrip is called by a set of other afni functions like align_epi_anat.py, @auto_tlrc and 3dSeg. Therefore in afni_proc.py there is no separate block for skull-stripping but options can be adjusted as parts of the respective blocks.

## References

* ↑ Kristi Boesen, Kelly Rehm, Kirt Schaper, Sarah Stoltzner, Roger Woods, Eileen Lüders, David Rottenberg, Quantitative comparison of four brain extraction algorithms, NeuroImage, Volume 22, Issue 3, July 2004, Pages 1255-1261, ISSN 1053-8119, http://dx.doi.org/10.1016/j.neuroimage.2004.03.010.
* ↑ http://afni.nimh.nih.gov/pub/dist/doc/program_help/3dSkullStrip.html
* ↑ Smith, Stephen M., Fast robust automated brain extraction, Human Brain Mapping 2002, Volume 17 Issue 3, page 143-155, http://dx.doi.org/10.1002/hbm.10062
# Neuroimaging Data Processing/Processing/Steps/Discarding First Volumes
## Discarding first volumes

The signal in first volume of an EPI sequence can be off-scale due to longitudinal magnetization not having reached steady-state yet. Therefore, they are often discarded and not used for analysis. A visual inspection of voxel time-series can give a good idea of which volumes are affected.
Keep in mind that discarding the first volumes will have an effect on the number of volumes in your scan as well as on the timing information in stimulus onset files and physiological parameter files.

#### AFNI

3dTcat is AFNIs function to discard the first volumes. It basically copies the original data, leaving out specified volumes. A simple command could look similar to this:
which in this case writes all but the first two volumes (0 and 1) of the original data (inputfile) into the outputfile. This step is typically used to copy the (interesting part) of the raw data into the results directory for further preprocessing, so that the original data remains untouched. Note that volume number and timing information might have to be adjusted accordingly.
3dTcat can also be used for linear detrending using the -rlt option. Check manual page for more details and options.
In afni_proc.py this is a default step, however with the number of TRs to remove set to 0. The respective option to remove the first n TRs is

## References

* ↑ AFNI program: 3dTcat: Output of -help
# Old School RuneScape/Quests/Shilo Village

The inhabitants of Shilo Village are under siege by the undead, give them a hand and beat the ruler of the undead while you're at it would you? This quest gives the player access to Shilo Village which is an excellent place for fly fishing as well as containing a bank very close to the Kharazi Jungle during the Legends Quest. This town also has 7 rocks which can be mined for all gems up to diamond, Duradel (the highest level slayer master), and the start of the One Small Favour quest.
# Old School RuneScape/Quests/Sheep Shearer

Description: Farmer Fred's sheep are getting mighty woolly. He will pay you to shear them.
Starting Point: Northwest of Lumbridge, there is a person named Farmer Fred inside the house surrounded with chickens. Go inside and talk to him.
Difficulty: Easy
Length: Short
Requirements: None
Items Needed: Shears
First of all, buy some shears from the General Store. Next, head to Farmer Fred and talk to him. Tell him you're looking for a quest. He will give you one, which will be to get him 20 balls of wool. Leave his house and go south a a bit and enter the sheep pen. In here, there will be sheep roaming around. You have to use your shears with the sheep. Sheep with wool will look puffy, and sheep without wool will look skinny. If a sheep has no wool, wait a few minutes and wool will appear on them. They will run away sometimes, but keep trying and you should get wool from them.
Once you have 20 pieces of wool, you must now spin them into balls of wool. Go to Lumbridge Castle to the second floor. In one of the rooms is a spinning wheel. Click on it and spin the 20 pieces of wool you have. When they are all balls of wool, go back to Farmer Fred to finish the quest.
Reward: 1 Quest Point, 60 gp, and 150 crafting experience.
# Old School RuneScape/Quests/Spirits of the Elid

This quest is helpful as it allows access to a fountain in the middle of the desert and a statuette that will temporarily restore your hitpoints to above your actual level and still restores prayer. Nardah is a very useful city for banking when training at the Agility Pyramid and when this quest is completed it becomes even more handy. Players training agility can use a bowl on the water fountain and then use it on their empty waterskins to fill them back up. Players who have not completed the quest will notice that the fountain is dry for them.
# Old School RuneScape/Quests/Rune Mysteries Quest
## Rune Mysteries Quest

Description: Recent research at the Wizards Tower has found a way to create Runes for the first time in centuries. Assist the head wizard Sedridor in his research and he too may teach you these secrets!
Starting Point: Lumbridge Castle. Go to the second floor and talk to the Duke of Lumbridge.
Difficulty: Easy
Length: Medium (lots of walking)
Requirements: 37 magic (if you don't want to walk)
Items Needed: None
The Duke of Lumbridge will give you an Air Talisman and tell you to bring it to the head wizard in the Wizards' Tower. Go to the Wizards' Tower, which is south of Draynor, and go downstairs to the basement.
Wizard Sedridor is the head wizard. Look for him in the basement and then talk to him. He will take your Air Talisman and give you a package that you have to deliver to the Aubury in Varrock. The Aubury is south of Varrock's west bank, and is inside the runes shop. So go to Varrock and talk to the Aubury.
Talk to him twice and he'll want you to deliver some notes to Wizard Sedridor. So go back to the Wizards' Tower and go down into the basement and talk to Wizard Sedridor.
The wizard will give you back your Air Talisman and will explain to you the rune crafting skill. When he is done talking, you will be done the quest.
Reward: 1 Quest Point and the ability to do the Runecrafting skill.
# Old School RuneScape/Quests/Tears of Guthix

This is a relatively short quest for the awesome reward it gives (in my opinion at least). The only hitch is that players need 49 Firemaking and Crafting to complete it. For those of us who had trouble finding the start point of the quest and spend about half an hour searching around for it (like me), it is located in the Lumbridge slayer caves below the swamp (see the Slayer guide for details on the caves). Bring a bullseye lantern (you'll need it for the quest anyway) and a tinderbox. It really helps to have started the The Lost Tribe quest because this allows you to avoid wall beasts (if you haven't started the quest bring a spiny helmet and wear it in the caves).
Once you're in the right cave, follow the path to the climbing rocks and go down. Talk to Juna (NPC snake, not dangerous) to start the quest. Now to make the special bowl: get a pickaxe, a chisel, a sapphire, and a bullseye lantern (unless you were smart and already have one). Go up the rockslide and extinguish your lantern. Now use your sapphire with it and light it again. Use it on one of the light creatures for a (slightly terrifying, if you're afraid of heights close your eyes) trip across the chasm. Once across, mine the rocks and use a chisel on the rock you get to make the bowl. Use your sapphire lantern on a light creature again to get back across. Now extinguish your lantern, use your lens on it and light it again. Go back to Juna to access the Tears of Guthix cave.
Juna will also tell you some interesting stories about how Geilinor (RuneScape) was formed such as the God Wars, Cave Goblins, and some of the ancient civilizations that existed before Zamorak invaded.
The minigame: Once every week you may access the Tears of Guthix minigame (if you're there an hour early too bad for you, you have to wait). This will let you collect tears from the blue walls in the cave, after your time in the cave is up you go outside and drink the tears. Depending on how many tears you got, you get experience in your lowest skill (this is how I got 42 farming). There are only three problems. First is that the green walls will take away tears. The second is that the walls will move around the cave. Figure out your own strategy for dealing with this. By the way, no need to keep clicking on a wall, when you start draining tears from it your character keeps going until you say differently (they also don't stop if the wall goes empty or turns green). The third problem is that every week, you must have either 100k exp (can be spread over all your skills) or 1 quest point more than you had the week before or you won't be allowed into the cave. I prefer to start preparing for any quest I choose on Mondays, go to the cave on Wednesdays, keep training and do a quest over the weekend. This system has worked pretty well for me but you might like something different. I do quests because the more quest points you have, the longer you can stay in the cave.
# Old School RuneScape/Quests/Roving Elves

Part of the ongoing West Ardougne storyline, at the end players receive either a crystal bow or shield (their choice) and they can buy more of either if they want.
# Old School RuneScape/Quests/The Fremennik Trials

Become a member of the local Viking tribe by going through a series of trials to gain enough votes on the village council. You will have to fight twice, use your smarts several times, and go through a series of NPCs very like in the One Small Favour quest to obtain one tiny item. The trials are the reason you frequently see people in Seer's Village buying a raw shark so you might want to think about getting one to prevent yourself suffering stress during the quest because of a lack of sellers. This quest provides experience in several skills, a teleport to the Fremennik village, the ability to communicate with the Fremennik tribe (buy from stores, steal from stalls, talk without getting snubbed, etc), a free ride to Waterbirth Island (otherwise you'd have to pay 1k), and access to the islands of Miscellania and Etceteria.
# Old School RuneScape/Quests/The Giant Dwarf

To start this quest, go to the Keldagrim entrance in northeast Relekka. Go into the first cave and then into the second, talk to the dwarven boatman. He will take you to Keldagrim, on the way a cutscene happens and the quest is started. The quest does not have to be completed to access the city again but it is very helpful as completing it allows you to start the Between a Rock... quest. Completing the Giant Dwarf also grants you access to the mine cart transportation system. This connects Keldagrim to the tunnel below White Wolf Mountain (see Fishing Contest) and the dwarven mines below Ice Mountain. Every time the carts are used a fee has to be paid, with the Ring of Charos (U) (see Garden of Tranquility) the ticket price is halved. The city has some useful features such as the Blast Furnace minigame, a few stores, some stalls that are useful for Thieving, and most importantly a bank that can be very helpful when killing Slayer monsters in the Relekka slayer dungeon.
# Rebol Programming/off

OFF is a logic of value: false
# DESCRIPTION:

Returns the offset between two series positions.
OFFSET? is a function value.

# ARGUMENTS:

* series1 -- (Type: series)
* series2 -- (Type: series)
# Rebol Programming/olive

OLIVE is a tuple of value: 128.128.0
# Rebol Programming/on

ON is a logic of value: true
# DESCRIPTION:

Opens a new port connection.
OPEN is a native value.

# ARGUMENTS:

* spec -- (Type: file url port object block)

# REFINEMENTS:

* /binary -- Preserves contents exactly.
* /string -- Translates all line terminators.
* /direct -- Opens the port without buffering.
* /seek -- Opens port in seek mode without buffering.
* /new -- Creates a file. (Need not already exist.)
* /read -- Read only. Disables write operations.
* /write -- Write only. Disables read operations.
* /no-wait -- Returns immediately without waiting if no data.
* /lines -- Handles data as lines.
* /with -- Specifies alternate line termination.
** end-of-line -- (Type: char string)
* /allow -- Specifies the protection attributes when created.
** access -- (Type: block)
* /mode -- Block of above refinements.
** args -- (Type: block)
* /custom -- Allows special refinements.
** params -- (Type: block)
* /skip -- Skips a number of bytes.
** length -- (Type: number)
# Rebol Programming/oldrab

OLDRAB is a tuple of value: 72.72.16
# DESCRIPTION:

Returns the first value ORed with the second.
OR is an op value.

# ARGUMENTS:

* value1 -- (Type: logic number char tuple binary string image)
* value2 -- (Type: logic number char tuple binary string image)
# DESCRIPTION:

Returns TRUE for op values.
OP? is an action value.

# ARGUMENTS:

* value -- (Type: any-type)
# DESCRIPTION:

Returns the offset in the face's text corresponding to the offset pair.
OFFSET-TO-CARET is a native value.

# ARGUMENTS:

* face -- The face containing the text. (Type: object)
* offset -- The XY offset relative to the face. (Type: pair)
# Pinyin/Flat

See also
# Pinyin/Flood

Search flood
# Pinyin/Flavor

A flavor is the particular taste and smell that a food or drink has.
weihdaoh shih shirwuh huoh yiinpiin tehyoou d koouweih her qihweih.
# Pinyin/Flight

Search Pinyin flight
# Pinyin/Flooding

Search flood
# Pinyin/Flow

A flow is a continuous movement of something in one direction.
Liúdòng shì mǒushìwù zài yīgè fāngxiàng de liánxù yùndòng.
# Pinyin/Five

See also
# Pinyin/Flesh

See also

## Noun

* flesh: the soft part between the skin and bones of people’s or animals’ bodies
** rouh: rern huoh dohngwuh shentii pirfu her guutour zhijian d rourruaan buhfehn
* flesh: the skin of the human body
** pirfu: rerntii d pirfu
* flesh: the soft part of a fruit that is under the skin
** guoorouh: shuiiguoo pirxiah d rourruaan buhfehn
* flesh: the soft part of a vegetable that is under the skin
** shucaihrouh: shucaih pirxiah d rourruaan buhfehn