OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Stability Analysis/Continuous Time/Stability of Structured, Norm-Bounded Uncertainty

LMIs in Control/Stability Analysis/Continuous Time/Stability of Structured, Norm-Bounded Uncertainty
Given a system with matrices A,M,N,Q with structured, norm-bounded uncertainty, the stability of the system can be found using the following LMI. The LMI takes variables P and $\Theta$ and checks for a feasible solution.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in {\bf {{\Delta }\;,||\Delta ||\leq 1}}\\\end{aligned}}$

## The Data

The matrices $A,M,N,Q$.

## The LMI:

       ${\begin{aligned}{\text{Find}}\;&P>0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+{\begin{bmatrix}M\Theta M^{T}&M\Theta Q^{T}\\Q\Theta M^{T}&Q\Theta Q^{T}-\Theta \end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

${\begin{aligned}{\text{The system above is quadratically stable if and only if there exists some }}\Theta \in P\Theta {\text{ and }}P>0{\text{ such that the LMI is feasible.}}\end{aligned}}$

## Implementation

https://github.com/mcavorsi/LMI

## Related LMIs

Parametric, Norm-Bounded Uncertain System Quadratic Stability
Stability under Arbitrary Switching
Quadratic Stability Margins

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
# LMIs in Control/Stability Analysis/Continuous Time/Strong Stabilizability
## The System

Consider the continous-time LTI system, $G:L_{2e}\rightarrow L_{2e}$ with state-space realization (A,B,C,0)
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t),\end{aligned}}$
where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable, and the transfer matrix $G(s)=C(s1-A^{-1})B$ has no poles on the imaginary axis.

## The Data

The matrices $A,B,C$.

## The Optimization Problem

The system G is strongly stabilizable if there exist $P\in \mathbb {S} ^{n}$, $Z\in \mathbb {R} ^{n\times p}$, and $\gamma \in \mathbb {R} _{>0}$, where $P>0$, such that
    ${\begin{aligned}PA+A^{T}+ZC+C^{T}Z^{T}<0\\{\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^{T}P+ZC+C^{T}Z^{T}&&-Z&&-XB\\*&&-\gamma I&&0\\*&&*&&-\gamma I\end{bmatrix}}<0\end{aligned}}}\\\end{aligned}}$

## Conclusion:

where $F=-B^{T}X$ and $X\in S_{n}$ , $X\geq 0$ is the solution to the Lyapunov equation given by
    ${\begin{aligned}XA+A^{T}X-XBB^{T}X=0\end{aligned}}$
Moreover, a controller that strongly stabilizes G is given by the state-space realization
    ${\begin{aligned}{\dot {x}}_{c}=(A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\y_{C}=-B^{T}Xx(t)\end{aligned}}$

## Implementation

* Example Code

## Related LMIs

* https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability

## External Links

* http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.
* https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.
* https://web.stanford.edu/~boyd/lmibook/ LMIs in Systems and Control Theory- A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Stability Analysis/Continuous Time/Time-Delay Systems

      * Delay Dependent Time-Delay Stabilization
      * Delay Independent Time-Delay Stabilization
# LMIs in Control/Stability Analysis/Continuous Time/Quadratic Stability Margins

LMIs in Control/Stability Analysis/Continuous Time/Quadratic Stability Margins
${\begin{aligned}{\text{The quadratic stability margin of the system is defined as the largest }}\alpha \geq 0{\text{ for which the system is quadratically stable.}}{\text{This LMI applies for systems with norm-bounded uncertainty.}}\end{aligned}}$

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t),&&p^{T}p\leq \alpha ^{2}x^{T}C_{q}^{T}C_{q}x\\\end{aligned}}$

## The Data

The matrices $A,B_{p},C_{q}$.

## The Optimization Problem

${\text{Maximize }}\beta =\alpha ^{2}{\text{ subject to the LMI constraint.}}$

## The LMI:

       ${\begin{aligned}{\text{Find}}\;&P,\lambda ,{\text{ and }}\beta =\alpha ^{2}:\\{\begin{bmatrix}A^{T}P+PA+\beta \lambda C_{q}^{T}C_{q}&PB_{p}\\B_{p}^{T}P&-\lambda I\end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

If there exists an $\alpha \geq 0$ then the system is quadratically stable, and the stability margin is the largest such $\alpha$.

## Implementation

https://github.com/mcavorsi/LMI

## Related LMIs

Parametric, Norm-Bounded Uncertain System Quadratic Stability
Stability of Structured, Norm-Bounded Uncertainty
Stability under Arbitrary Switching

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Stability Analysis/Continuous Time/Parametric, Norm-Bounded Uncertain System Quadratic Stability

LMIs in Control/Stability Analysis/Continuous Time/Parametric, Norm-Bounded Uncertain System Quadratic Stability
Given a system with matrices A,M,N,Q the quadratic stability of the system with parametric, norm-bounded uncertainty can be determined by the following LMI. The feasibility of the LMI tells if the system is quadratically stable or not.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in \mathbf {\Delta } \;:=\{\Delta \in \mathbb {R} ^{n\times n}:\|\Delta \|\leq 1\}\\\end{aligned}}$

## The Data

The matrices $A,M,N,Q$.

## The LMI:

       ${\begin{aligned}{\text{Find}}\;&P>0,\mu \geq 0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+\mu {\begin{bmatrix}MM^{T}&MQ^{T}\\QM^{T}&QQ^{T}-I\end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.

## Implementation

https://github.com/mcavorsi/LMI

## Related LMIs

Stability of Structured, Norm-Bounded Uncertainty
Stability under Arbitrary Switching
Quadratic Stability Margins

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
# LMIs in Control/Stability Analysis/Continuous Time/Quadratic Hurwitz Stabilization for Polytopic Systems

This section studies the Quadratic Hurwitz stabilization for polytopic systems.

## The System

Given a state-space representation of a linear system
    ${\begin{aligned}\ {\dot {x}}(t)=A(\delta (t)x(t)+B\delta (t)u(t)\\A(\delta (t))=A_{0}+\delta _{1}(t)A_{1}+\delta _{2}(t)A_{2}\\B(\delta (t))=B_{0}+\delta _{1}(t)B_{1}+\delta _{2}(t)B_{2}\end{aligned}}$         (1)

## LMI Condition

With $\Delta =\Delta _{p}$ , the quadratic Hurwitz Stabilization problem has a solution if and only if there exists a symmetric positive definite matrix $P$ and a matrix $W$ satisfying the below LMI :
    ${\begin{aligned}(A_{0}+A_{i})P+P(A_{0}+A_{i})^{T}+(B_{0}+B_{i})W+W(B_{0}+B_{i})^{T}<0,i&=1,2,....k\end{aligned}}$         (2)
In this case, a solution to the problem is given by
    ${\begin{aligned}K=WP^{-1}\\\end{aligned}}$         (3)

## Conclusion

Stability of a system does not guarantee quadratic stability. Since quadratic stability can represent infinite LMI constraints, it is not tractable. Therefore, to make it feasible and tractable, polytopic sets are helpful.

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
# LMIs in Control/Stability Analysis/Continuous Time/Stability under Arbitrary Switching

LMIs in Control/Stability Analysis/Continuous Time/Stability under Arbitrary Switching
Using the LMI below, find a P matrix that fits the constraints. If there exists one, then the system can switch between subsystems $A_{1}$ and $A_{2}$ arbitrarily and remain stable.

## The System

    ${\begin{aligned}{\dot {x}}(t)\in \{A_{1}x(t),A_{2}x(t)\}\end{aligned}}$

## The Data

The matrices $A_{1}\in R^{n\times n},A_{2}\in R^{n\times n}$.

## The LMI

       ${\begin{aligned}{\text{Find}}\;&P>0:\\A_{1}^{T}P+PA_{1}<0{\text{ and }}\\A_{2}^{T}P+PA_{2}<0\end{aligned}}$

## Conclusion

The switched system is stable under arbitrary switching if there exists some P > 0 such that the LMIs hold.

## Implementation

https://github.com/mcavorsi/LMI

## Related LMIs

Parametric, Norm-Bounded Uncertain System Quadratic Stability
Stability of Structured, Norm-Bounded Uncertainty
Quadratic Stability Margins

## External links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
# LMIs in Control/Stability Analysis/Continuous Time/Transient Output Bound for Autonomous LTI Systems
## The System

Consider a continuous-time LTI system with a state-space representation of:
${\dot {x}}=Ax,$
$y=Cx,$
where $A\in \mathbb {R} ^{n\times n}$, $C\in \mathbb {R} ^{p\times n}$ and $x(0)=x_{0}$,
the transient bound on the output can be evaluated with the following LMI.

## The Data

$A\in \mathbb {R} ^{n\times n}$, $C\in \mathbb {R} ^{p\times n}$ and $x(0)=x_{0}$.

## The LMI:

The Euclidean norm of the output satisfies
$\lVert y(T)\rVert _{2}\leq \gamma \lVert x_{0}\rVert _{2},\forall T\in \mathbb {R} _{+},$
if there exists $P\in \mathbb {S} ^{p}$ and $\gamma \in \mathbb {R} _{++}$, where $P>0$, such that
* $P-\gamma I\leq 0,$
* ${\begin{bmatrix}P&C^{T}\\*&\gamma I\end{bmatrix}}\geq 0,$
* $PA+A^{T}P\leq 0.$

## Conclusion

By using this LMI the transient output bound can be found for a given autonomous LTI system.

## Implementation

The LMI given above can be implemented and solved using a parser such as YALMIP, along with an LMI solver such as SeDuMi, or MOSEK. (For a detailed list of solvers, please visit this page on Relevant Solvers).

## References

Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.
# General Biology/Classification of Living Things/Eukaryotes/Protists/Answers

1. f)
2.
    a) G. lamblia
    b) parasitic
3. Brown Algae (Phaeophytes) & Red Algae (Rhodophytes)
4. c); these Euglenoids cannot photosynthesis to produce energy, but they can still produce carbonic compounds by themselves.
5. Yes; zooxanthellae species photosynthesise and pass carbohydrates onto coral reefs in these nutrient-poor regions.
# General Biology/Classification of Living Things/Prokaryotes

For Eubacteria, please visit General Biology/Classification of Living Things/Eubacteria.

## Archaea

* Proposed as separate group from (eu)bacteria by Carl Woese
** based on structure and metabolic pathways
** inhabit extreme environments
** unique branched lipids in membrane
* Share traits with both eukaryotes and eubacteria, e.g., RNA polymerase, introns
* Biochemically diverse
* Economically important
** Taq polymerase used in PCR

### Types

* Methanogens
* Halophiles
* Thermophiles

#### Underground bacteria

* Metabolism
** built around inorganic energy sources
* e.g., basalt reacts with H$_2$O to release hydrogen which is catalytically combined with CO$_2$ to form carbohydrate (akin to photosynthesis)
* may result in deposit of minerals
* Unresolved problems
** Did bacteria move downward from surface or did they first evolve there, protected from harsh surface conditions?
** Could bacteria be ejected into space in rocks?

## Prokaryote evolution

* Tentative, subject to change
* Derived largely from molecular systematics (rRNA sequences)
* Note: most bacteria can’t be cultured, thus hard to study! (Studied by PCR of water/soil samples)

## Domains of life: characteristics

This text is based on notes very generously donated by Paul Doerder, Ph.D., of the Cleveland State University.
# General Biology/Classification of Living Things/Eukaryotes

* Protists
* Plants
* Fungi
* Animals
** Phyla
** Chordates
# General Biology/Classification of Living Things/Eubacteria/Answers

1. Answers may vary. Some possibilities are:
    1) Binary fission produces two identical daughter cells.
    2) Daughter cells have 2n chromosomes; the daughter cells have the same number of chromosomes as the parent cell.
    3) Binary fission commences with a duplicate of the original chromosome.
2. Answers may vary. Some possibilities:
    1) Binary fission produces only two identical daughter cells.
    2) Daughter bacterial cells have 2n chromosomes, unlike n chromosomes for gametes.
    3) Binary fission transpires only in bacteria, and not in gametes.
    4) Binary fission does not involve any spindle fibres or asters.
3.
    a) photoautotrophic
    b) chemoheterophic
    c) chemoheterophic
    d) photoautotrophic
    e) chemoautotrophic
4. Cocci; least surface area.
5.
    a) Plasmids
    b) Conjugation.
6. Yes, because endospores are quite robust.
# High School Mathematics Extensions/Supplementary/Basic Counting/Solutions
## Solutions to exercises
### Exercise 1

There would be 4! ways if the letters were all distinct. Since "O" is repeated twice (2! permutations), every different arrangement is counted 2! times; therefore there are
    $4!/2!=12$
ways.

### Exercise 2

There are thirteen diamonds in the deck; hence, there are
    ${13 \choose 5}={\frac {13!}{8!\times 5!}}={\frac {13\times 12\times 11\times 10\times 9}{120}}=1287$
ways.

### Exercise 3

Since the lettuce is always on top, we just consider all the possible arrangements of ham - cheese - tomato - salami such that the ham is never next to the salami. We count the permutations of the four elements,
    $4!=24$
and then deduct the permutations in which the ham and the salami are together. That is, the permutations of (ham & salami) - cheese - tomato and (salami & ham) - cheese - tomato:
    $2\times 3!=2\times 6=12$
Joey can arrange the ingredients in
    $24-12=12$
different ways.

### Exercise 4

We count the number of ways of choosing three flavours out of ten and discount the number of ways of choosing one flavour, chocolate, and vanilla, that is the number of ways of choosing one flavour out of the eight left:
    ${10 \choose 3}-{8 \choose 1}={\frac {10!}{7!\times 3!}}-{\frac {8!}{7!\times 1!}}=120-8=112$
Rachel can order 112 different ice cream cups.

### Exercise 5

We count the number of combinations of one, two, three, four, and five ingredients out of five available. That is the sum of five terms 5 choose k, where k equals 1, 2, 3, 4, 5:
    ${5 \choose 1}+{5 \choose 2}+{5 \choose 3}+{5 \choose 4}+{5 \choose 5}$
Then we subtract to each term the number of combinations of ham, salami and k - 2 ingredients out of the 3 left (note that 5 choose 1 does not need any deduction):
* ${5 \choose 1}-0=5-0=5$
* ${5 \choose 2}-{3 \choose 0}=10-1=9$
* ${5 \choose 3}-{3 \choose 1}=10-3=7$
* ${5 \choose 4}-{3 \choose 2}=5-3=2$
* ${5 \choose 5}-{3 \choose 3}=1-1=0$
Joey can make
    $5+9+7+2+0=23$
different sandwiches.
# High School Mathematics Extensions/Supplementary

* Supplementary Chapters
* Content
* Basic Counting
* Polynomial Division
* Partial Fractions
* Summation Sign
* Complex Numbers
* Differentiation
* Problems & Projects
* Problem Set
* Solutions
* Exercise Solutions
* Problem Set Solutions
About supplementary chapters
All supplementary chapters contain materials that are part of the standard high school mathematics curriculum, therefore the material is only provided for completeness and should mostly serve as revision.
For more detailed explanations consider checking other Wikibooks such as
      * Basic Algebra
      * Algebra
      * Geometry
      * High School Trigonometry
# Magic: The Gathering/5-Color Format

5-Color Format exists in slightly different variations in the online and offline Magic communities. 5-Color was initially a fan-created format, not supported by the DCI. The basic rules could be summarized as follows:
* Decks must contain a minimum of 250 cards
* Decks must have no less than 20 cards of each color
* No more than four of each non basic land card
* The game is played for ante, and ante cards, including Contract from Below, are legal
A full list of rules, including banned and restricted cards, is maintained at: http://www.5-color.com/
Due to the popularity of this non-sanctioned format, WotC created a similar format in the Magic online game.
# Magic: The Gathering/Extended Format

As of October 7, 2013, not supported by the DCI. http://www.wizards.com/default.asp?x=judge/resources/sfrextended
# Magic: The Gathering/Legacy Format

Legacy (along with Vintage) is considered an eternal format because the card pool never rotates. This means that all the sets that are currently legal will continue to be legal and any new sets will automatically be included in the legal card pool. (Except of course for cards on the banned and restricted list)
It evolved from Type 1.5, a format defined by a banned list that merely consisted of all banned and restricted cards in the old Type 1. In 2004, the format was revitalized by separating the banned list from the rechristened Vintage and banning many old, powerful, and expensive cards such as Mishra's Workshop, Mana Drain, and Bazaar of Baghdad. The result is that Legacy has a lower power level than Vintage, which makes for longer games, and is considerably more affordable. The DCI has attempted to promote the format with the addition of a Legacy Grand Prix circuit. In 2007, the company announced that this format had selected for the final individual portion of the World Championships prior to the fixing a problem that they had made after moving the championships from summer/fall to fall/winter.
# Magic: The Gathering/Assassin Format

Assassin Format is typically played in groups of five or more. The name of each player is put into a hat (or whatever) and each player draws a name, to be kept secret. The rules are thus: A player may only attack the player whose name they drew, or any player that has attacked them at any point in the game (universal effect cards can be played, just as long as your target could be hit with it, e.g. Skullcage, Radiance). If they drew their own name they may attack any or all other players (strategy here, simply build up your forces until you can swamp a player, since no one can target you until you attack them). All types of spells, with the exception of those with universal effects, may only target the player drawn or players who have since attacked you (counter spells, creatures, instants, sorceries...). Any method of attack goes. Speed rules are usually enforced, since this format tends to take over an hour to play.
# Magic: The Gathering/Commander Format

Originally and alternatively known as Elder Dragon Highlander, or EDH, Commander is a multiplayer Constructed format of the Magic: the Gathering trading card game. It is characterized by the use of a Commander, or General, and the 100-card singleton deck.

# Deck Construction

There are four rules for building a commander deck:
* Commander/General: The deck must include a Commander, also known as a general, which must be a Legendary Creature. "Commanderness" is an inherent quality of the Commander: it can neither be removed nor copied, and a Commander is still a Commander even if that Commander is being controlled by another player.
* Color Identity: The deck cannot include any cards that feature mana symbols not found on the Commander, e.g. a deck whose Commander is Alesha, Who Smiles at Death may include cards that contain Mountain and/or Swamp and/or Plains symbol but may not contain cards that include Forest and/or Island symbols; the exception to this rule is reminder text, e.g. a deck whose commander is Gonti, Lord of Luxury may contain a Crypt Ghast, even though the reminder text includes a Plains symbol.
* Card Copies: Except for basic lands and cards that mention otherwise, such as Relentless Rats or Shadowborn Apostle, a deck may not contain more than one copy of any card.
* Size: A Commander deck must be exactly 100 cards, including the Commander. In other words, an EDH deck consists of one Commander (or two if both have Partner) and 99 (or 98) other cards.
There is also an official banlist for Commander, and a deck should not include any cards on the banlist without prior consent of other players of a particular instance of a Commander game.

# Gameplay

The game starts as other Magic formats with players shuffling their decks, cutting the decks of their opponents, and drawing an initial hand of seven cards. Given that Commander is traditionally a multiplayer format, all players draw a card during their first turn. The only differences between EDH gameplay and traditional gameplay concern the Commander:
The Commander begins the game in the Command Zone, and may be cast at any legal time that its controller has sufficient mana. If at any point the Commander would be removed from the battlefield and placed into a different zone (e.g. graveyard, hand, library, or exile), its controller may instead redirect the Commander to the Command Zone. When casting a Commander from the Command Zone, a Commander's mana cost is equal to its printed mana cost plus two colorless mana for each time it has been previously cast from the Command Zone; this is colloquially known as the "command tax". However, when casting a Commander from anywhere other than the Command Zone, its cost is equal to its printed cost. Finally, if any player takes 21 or more combat damage from a single Commander over the course of a game, that player loses in spite of that player's current life total.
# Magic: The Gathering/Highlander Format

Highland Format takes its name from a memorable line from the movie 'Highlander': There can be only one!. Only one copy of any Magic card (other than basic land) is allowed in a highlander deck. Any magic format can be converted into a highlander-type format if this rule is followed.
# Magic: The Gathering/Emperor Format

In Emperor (a six player format), two players are the emperors, and the remaining four players are the guards (two for each emperor). Players can only attack, target, and affect the players sitting immediately next to them. If an emperor dies, his or her team loses.
The main strategy for the emperor player is to use cards that give everybody bonuses (card draws, mana production, creature power, life gain, etc.) This enables the emperor's guards to play larger threats more quickly and ultimately should cause the opponents to lose due to being overwhelmed by powerful threats and answers.
Some Emperor formats ban alternate win cards (Barren Glory, Mortal Combat, etc.) to prevent abuse of the amount of time it would take to get past the emperor's guards and stop that condition from happening. Generally by suspending a Greater Gargadon with Barren Glory to sacrifice all of his other permanents or use Cephalid Illusionist and Shuko to empty one's deck into their graveyard with Mortal Combat.
# Magic: The Gathering/Block Format

http://www.wizards.com/default.asp?x=judge/resources/sfrblock
# MATLAB Programming/Psychtoolbox/Screen Commands/SelectMATLAB

Usage:
Bring forward n-th window in MATLAB Window menu. If not specified, n defaults to 0 (the command window).
# MATLAB Programming/Psychtoolbox/Screen Commands/SetDrawingRegion

Usage:
Make subsequent drawing to the window visible only if it is within the specified region. This does not apply to PutImage or CopyWindow operations. Call with rect equal to screenRect to reestablish the entire screen. Drawing outside the region is suppressed, but still takes the normal amount of time to complete.
* If shapeFlag == 0 [default], shapeSpec is a rect, region is rectangular.
* If shapeFlag == 1, shapeSpec is a rect, region is oval.
* If shapeFlag == 2, shapeSpec is a point list, region is polygonal.
# MATLAB Programming/Psychtoolbox/Screen Commands/SetGamma

Usage:
Get/set the gamma table. The gamma table will usually be 256x1 or 256x3. Most users won't need this function.