OpenLLM-France/wikimedia huggingface dataset:
# Phage display

Phage display is a laboratory technique for the study of protein–protein, protein–peptide, and protein–DNA interactions that uses bacteriophages to connect proteins with the genetic information that encodes them. In this technique, a gene encoding a protein of interest is inserted into a phage coat protein gene, causing the phage to display the protein on its outside while containing the gene for the protein on its inside, resulting in a connection between genotype and phenotype. These displaying phages can then be screened against other proteins, peptides or DNA sequences, in order to detect interaction between the displayed protein and those other molecules. In this way, large libraries of proteins can be screened and amplified in a process called in vitro selection, which is analogous to natural selection.
The most common bacteriophages used in phage display are M13 and fd filamentous phage, though T4, T7, and λ phage have also been used.
In the case of M13 filamentous phage display, the DNA encoding the protein or peptide of interest is ligated into the pIII or pVIII gene, encoding either the minor or major coat protein, respectively. Multiple cloning sites are sometimes used to ensure that the fragments are inserted in all three possible reading frames so that the cDNA fragment is translated in the proper frame. The phage gene and insert DNA hybrid is then inserted into E. coli. If a phagemid vector is used (a simplified display construct vector) phage particles will not be released from the E. coli cells until they are infected with helper phage, which enables packaging of the phage DNA and assembly of the mature virions with the relevant protein fragment as part of their outer coat on either the minor (pIII) or major (pVIII) coat protein. By immobilizing a relevant DNA or protein target(s) to the surface of a microtiter plate well, a phage that displays a protein that binds to one of those targets on its surface will remain while others are removed by washing. Those that remain can be eluted, used to produce more phage (by bacterial infection with helper phage) and so produce a phage mixture that is enriched with relevant (i.e. binding) phage. The repeated cycling of these steps is referred to as panning, in reference to the enrichment of a sample of gold by removing undesirable materials. Phage eluted in the final step can be used to infect a suitable bacterial host, from which the phagemids can be collected and the relevant DNA sequence excised and sequenced to identify the relevant, interacting proteins or protein fragments.
# LMIs in Control/Click here to continue/Applications of Linear systems/Hinf Optimal Stabilization of a Class of Uncertain Impulsive Systems

H$\infty$ Optimal Stabilization of a Class of Uncertain Impulsive Systems
Impulsive Systems are a special class of hybrid dynamical system which contains many abrupt changes of states at some certain time instances. Many examples can be found in finance, engineering, and other disciplines. This LMI addressed the problem of designing a feedback control law such that the uncertain closed-loop system is asymptotically stable, and the H$\infty$ norm bound constrains on disturbance attentuation for all admissible uncertainties are satisfied.

## The System

The following system is assumed:
${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Cu(t)\,\,(1)\\\end{aligned}}$
When the system (1) is time-invariant and involves uncertain disturbances and proportional impulsive effects is can be rewritten as seen below in (2).

## The Data

The system can then be described as:
${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bw(t)+Cu(t)\,\,\,&t\neq t_{k}\\\ \Delta x(t)&=d_{k}x(t)&t=t_{k}\\\ z(t)&=Ex(t)\\\ x(t)&=0&t=t_{0}=0\\\end{aligned}}$
Where
$x(t)\in R^{n}$ is the state.
$u(t)\in R^{m}$ is the control input.
$w(t)\in R^{p}$ is the disturbance input with limited energy.
$z(t)\in R^{n}$ is the controlled output.

## The LMI

Using the Lyapunov function, Schur Complement, the time-domain expression of the L2-norm, and Parseval's Theorem.
${\begin{aligned}&\quad \varepsilon >0\,\,\,\&\,\,\,P>0\in R^{nxn}\\&{\begin{bmatrix}A^{T}P+PA+P(\gamma ^{-2}BB^{T}-\varepsilon ^{-2}CC^{T})P&E^{T}\\E&-I\\\end{bmatrix}}<0\end{aligned}}$

## Conclusion:

The goal of this LMI is to provide a state feedback robust H$\infty$ optimal control technique for impulsive dynamical systems with time varying uncertainty. Based on a positive definite solution of a linear matrix inequality, the proposed robust H$\infty$ static state feedback control law guarantees both internally asymptotical stability and robust H$\infty$ optimal performance.
# LMIs in Control/Click here to continue/Applications of Linear systems/H∞-based Synthesis of Quadrotor Guidance and Control System
## System Dynamics

Consider the motion of a quadrotor along a horizontal plane alone. The control law is synthesised for roll motion(rotation wrt. X-axis) coupled with linear motion along Y-axis. The equations of motion governing the 2-D motion are given by(taken from the paper provided below):
${\begin{aligned}&\quad \quad {\begin{cases}{\frac {d^{2}x}{dt^{2}}}=-{\frac {\mu }{m_{Q}}}{\frac {dx}{dt}}-g{\frac {tan\theta }{cos\phi }};\\{\frac {d^{2}y}{dt^{2}}}=-{\frac {\mu }{m_{Q}}}{\frac {dy}{dt}}-gtan\phi ,\\\end{cases}}\end{aligned}}$
$where$
* $x$ and $y$ are the displacements along X & Y axis, respectively.
* $\phi$ and $\theta$ are the angular displacement about X & Y axis, respectively.
* $m_{Q}$ and $\mu$ are constants that depend on the intrinsic properties of the quadrotor.
The state space representation of The $H_{\infty }$ Quadrotor Guidance is given below,
${\begin{aligned}&\quad \quad {\begin{cases}{\frac {dX(t)}{dt}}=AX(t)+B_{u}u(t)+B_{d}(t)d(t);\\Y=CX(t),\\\end{cases}}\end{aligned}}$
$where:$
$X=[X_{1},X_{2},X_{3},X_{4},X_{5}]^{T}=[x,{\frac {dx}{dt}},y,{\frac {dy}{dt}},\Delta \Omega ]^{T},$ (where $\Delta \Omega$ is the increment in motor's rotation rate)
$A={\begin{bmatrix}0&1&0&0&0\\0&-{\frac {\mu }{m_{Q}}}&g&0&0\\0&0&0&1&0\\0&0&0&0&K_{1}\\0&0&0&0&K_{2}\\\end{bmatrix}}$
$B_{u}={\begin{bmatrix}0\\1\\K_{3}\\0\\K_{4}\\\end{bmatrix}}$
$B_{d}={\begin{bmatrix}0\\0\\0\\0\\K_{3}\\\end{bmatrix}}$
$C={\begin{bmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\\end{bmatrix}}$
$C_{2}={\begin{bmatrix}I_{3x3}&0_{3x3}\end{bmatrix}}$
$D_{1}=10^{-3}\times L_{1},D_{2}=10^{-3}\times L_{2}$

## The Optimization Problem

For the above-linearized model of the quadrotor, the linear control would be as below:
    $u=-K*V(t)=-K*C*X(t),$
where V is the voltage input, $K\in R^{1x4}$ and $C\in R^{4x5}.$
The cost function of this controller could be defined as:
    $j=\int _{0}^{\infty }\!||z(t)||^{2}\,\mathrm {d} t=\int _{0}^{\infty }\!||x^{T}Qx+u^{T}Ru||^{2}\,\mathrm {d} t$ ,
where,
$z={\begin{bmatrix}{\sqrt {Q}}&0\\0&{\sqrt {R}}\end{bmatrix}}{\begin{bmatrix}x\\u\end{bmatrix}}.$

## The LMI

${\begin{bmatrix}PA+A^{T}P+Q&PB_{u}&PB_{d}&L^{T}\\B_{u}^{T}P&-R&0&0\\B_{d}^{T}P&0&-\gamma ^{2}I&0\\L&0&0&-R\end{bmatrix}}$ < 0 .
Solving the above LMI gives the value of $\gamma$ for X>0, $\gamma$ >0 and Z.

## Conclusion

This LMI can be used in a problem and can be solved using the solvers like Yalmip,sedumi,gurobi etc,. This can be used to analyze the state feedback control and path tracking of a quadcopter.
# LMIs in Control/Click here to continue/Applications of Linear systems/LMI for Adaptive Barrier Global Sliding Mode

LMI for adaptive barrier global sliding mode control (ABGSMC)
Uncertain systems with faulty actuators can be simplified and modeled via an LMI for adaptive barrier global sliding mode control (ABGSMC). The LMI can be derived by using a global nonlinear sliding surface in order to achieve system stability and the occurrence of sliding when actuator faults are present within the system. This LMI is important for continued research into LMI-based actuator improvements as modern systems are heavily prone to faults (including actuator faults).

## The System

The following system is assumed:
${\begin{aligned}{\dot {x}}(t)&=Ax(t)+BLu(t)+D\\D&=\Delta \ Bu(t)+bf(x)+d\\\end{aligned}}$
where $x(t)=[x1,x2]^{\text{T}}\in R^{2}$ notes the system’s state vector, f(𝑥) is a nonlinear system function, $\Delta f$ and $\Delta g$ are parametric uncertainties and d represents the external disturbances. L is the post-fault coefficient which represents the actuator's effectiveness. We can assume that: ${\begin{aligned}{\begin{bmatrix}A\end{bmatrix}}={\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}}\end{aligned}}$
$\Delta B=[0,\Delta B_{2}]^{\text{T}}$
$B=[0,\Delta B_{2}^{\text{T}}]^{\text{T}}$
and $L=[0,L_{2}]^{\text{T}}$

## The Data

The system can then be described as:
${\begin{aligned}{\dot {x_{1}}}(t)&=A_{11}x_{1}(t)+A_{12}x_{2}(t)\\{\dot {x_{2}}}(t)&=A_{21}x_{1}(t)+A_{22}x_{2}(t)+(B_{2}L_{2})u(t)+D\\\end{aligned}}$
With the state space form described above.

## Assumptions

1. Pair (A, B) is totally controllable. Using this assumption for the system, the controllability of $(A_{11},A_{12})$ can be obtained by the controllability of (A, B)
2. The parametric uncertainty and external disturbance term Dx is bounded and is less than $\eta _{k}$

## The LMI

Using the Lyapunov function, Schur Complement, the adaptive control law, and the assumption of sliding surfaces an LMI can be brought together to minimize faults via actuators
${\begin{aligned}&{\text{min}}\quad \eta _{k}<Dx\\&{\begin{bmatrix}(B_{1}W)^{T}+A+B_{1}W&B_{2}&(D_{1}W)^{T}\\B_{2}^{T}&I&D^{T}\\L_{2}W+DW&D&-L_{2}\end{bmatrix}}<0\end{aligned}}$

## Conclusion:

As mentioned, the goal of this LMI is to mitigate the effects of external disturbances, parametric uncertainties, and actuator faults within the system. Minimizing the parametric uncertainty and disturbance term allows the system to mitigate faults within the actuators as well as determine a proper solution on how to minimize the external disturbance Dx so it falls below the parametric uncertainty.
# LMIs in Control/Click here to continue/Applications of Linear systems/Hinf Optimal Model Reduction

Given a full order model and an initial estimate of a reduced order model it is possible to obtain a reduced order model optimal in $H_{\infty }$ sense. This methods uses LMI techniques iteratively to obtain the result.

## The System

Given a state-space representation of a system $G(s)$ and an initial estimate of reduced order model ${\hat {G}}(s)$.
    ${\begin{aligned}\ G(s)&=C(sI-A)B+D,\\\ {\hat {G}}(s)&={\hat {C}}(sI-{\hat {A}}){\hat {B}}+{\hat {D}},\\\end{aligned}}$
Where $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},D\in \mathbb {R} ^{p\times m},{\hat {A}}\in \mathbb {R} ^{k\times k},{\hat {B}}\in \mathbb {R} ^{k\times m},{\hat {C}}\in \mathbb {R} ^{p\times k}$ and ${\hat {D}}\in \mathbb {R} ^{p\times m}$. Where $n,k,m,p$ are full order, reduced order, number of inputs and number of outputs respectively.

## The Data

The full order state matrices $A,B,C,D$ and the reduced model order $k$.

## The Optimization Problem

The objective of the optimization is to reduce the $H_{\infty }$ norm distance of the two systems. Minimizing $\|G-{\hat {G}}\|_{\infty }$ with respect to ${\hat {G}}$.

## The LMI: The Lyapunov Inequality

Objective: $\min \gamma$.
Subject to:: ${\begin{aligned}\ P&={\begin{bmatrix}\ P11&P12\\\ P21&P22\\\end{bmatrix}}\ >0,\end{aligned}}$
${\begin{aligned}{\begin{bmatrix}\ A^{T}P11+P11A&A^{T}P12+P12{\hat {A}}&P11B-P12{\hat {B}}&C^{T}\\\ {\hat {A}}^{T}P12^{T}+P12^{T}A&{\hat {A}}^{T}P22+P22{\hat {A}}&P12^{T}B-P22{\hat {B}}&{\hat {C}}^{T}\\\ B^{T}P11-{\hat {B}}^{T}P12^{T}&B^{T}P12-{\hat {B}}^{T}P22&-\gamma {I}&D^{T}-{\hat {D}}^{T}\\\ C&{\hat {C}}&D-{\hat {D}}&-\gamma {I}\\\end{bmatrix}}\ >0\end{aligned}}$
It can be seen from the above LMI that the second matrix inequality is not linear in ${\hat {A}},{\hat {B}},{\hat {C}},{\hat {D}},P$. But making ${\hat {A}},{\hat {B}}$ constant it is linear in ${\hat {C}},{\hat {D}},P$. And if $P12,P22$ are constant it is linear in ${\hat {A}},{\hat {B}},{\hat {C}},{\hat {D}},P11$. Hence the following iterative algorithm can be used.
(a) Start with initial estimate ${\hat {G}}$ obtained from techniques like Hankel-norm reduction/Balanced truncation.
(b) Fix ${\hat {A}},{\hat {B}}$ and optimize with respect to ${\hat {C}},{\hat {D}},P$.
(c) Fix $P12,P22$ and optimize with respect to ${\hat {A}},{\hat {B}},{\hat {C}},{\hat {D}},P11$.
(d) Repeat steps (b) and (c) until the solution converges.

## Conclusion:

The LMI techniques results in model reduction close to the theoretical limits set by the largest removed hankel singular value. The improvements are often not significant to that of Hankel-norm reduction. Due to high computational load it is recommended to only use this algorithm if optimal performance becomes a necessity.

## External Links

A list of references documenting and validating the LMI.
* Model order Reduction using LMIs - A conference paper by Helmersson, Anders, Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, p. 3217-3222 vol.4
# Logic for Computer Scientists/Modal Logic/Temporal Logics
## Temporal Logics

The two modalities $\square$ and $\diamond$ cannot be used to distinguish between past and future. For this we need a multi-modal logic with the following $\square$-operators
* $[F]A$: $A$ holds always in the future
* $[P]A$: $A$ holds always in the past
* $[A]A$: $A$ holds always
and the corresponding $\diamond$-operators:
* $\langle F\rangle A$: $A$ holds somewhere in the future
* $\langle P\rangle A$: $A$ holds somewhere in the past
* $\langle A\rangle A$: $A$ holds somewhere
The semantics is then given as before, by giving constraints for the three reachability relations or by giving appropriate axioms, e.g.
* $[F]A\to [F][F]A$: Transitivity; an analog axiom holds for the two other $\square$-operators.
* $A\to [F]\langle P\rangle A$: if we go from a time point $t$ in the future $t'$, we can go back in the past to the time point where $A$ was true.
* $[A]A\leftrightarrow [F]A\land [P]A$: connection of past with future.
In addition there are many other aspects of temporal logics. E.g. one can distinguish between left- and rightlinear structures or between dense and discrete time structures.
# Logic for Computer Scientists/Modal Logic/Translation Method
## Translation Method

There are several methods aiming at a translation of propositional modal logics into first order predicate logics. The idea is, to transform the semantic conditions for the reachability into logical formulae: One rule for the definition of the semantic was:
$w\models \Box A\;\;{\mbox{ iff }}{\mbox{for all }}v\in W,(w,v)\not \in R{\mbox{ or }}v\models A$
This can be compiled into a formula by substituting the modal formula $\square P$ by the first oder formula $\forall yR(x,y)\to P'(y)$ . Hence we can eliminate all modal operators by introducing the first order translations. The result of such a translation is a classical first order formula, which can be processed by the methods we have seen before.
For a modal formula $F$ we define its translation $F^{*}$:
* $P^{*}=P(x)$, if $P$ is a propositional constant
* $(\lnot F)^{*}=\lnot (F)$
* $(F\land G)^{*}=(F^{*}\land G^{*})$
* $(\square F)^{*}=(\forall y(R(x,y)\to F^{*}(x/y)))$, where $y$ is a new variable not occurring in $F^{*}$ and $F^{*}(x/y)$ is the result of replacing all free occurrences of $x$ in $F^{*}$ by $y$.
As a result, we have

## Theorem 1

F is a valid modal formal in $K$ iff $F^{*}$ is a valid first order formula.
Together with the observation that validity in modal logic $K$ (like in many others) is decidable, we hence have a sublogic of first order classical predicate logic which is decidable! Modal logic can be seen as a fragment of 2-variable first-order logic $FO^{2}$.
# Logic for Computer Scientists/Predicate Logic

5.1 Predicate Logic 5.2 Syntax 5.3 Semantics 5.4 Equivalence and Normal Forms 5.5 Herbrand Theories 5.6 Semantic Trees 5.7 Resolution 5.8 Strategies for Resolution 5.9 SATCHMO
# Logic for Computer Scientists/Predicate Logic/Predicate Logic
## Predicate Logic

When we introduced propositional logic we argued in treating the electrical circuit example (circuit), that a string like $high(inv1,o)$ can be read intuitively as "the output of inverter 1 is high", but then we abstracted from this internal structure of the proposition and further on we represented the proposition by a simple string like "the_output_of_inverter_1_ is_high". In the formal parts of the calculus we even assumed more abstract just a given countable set of propositions $\{P_{1},\cdots ,P_{i},\cdots \}$ . Until now, we investigated a logical language that was able to deal with connecting these propositions, In this section we will have a closer look into the structure of propositions. We will be able to explicitly refer to objects like the inverter $inv1$ or its output $o$ in the sentences of our language,. Besides this domain, we will introduce the new concept of a variable, which stands for an arbitrary object of our domain. Hence, we will be able to write e.g. $high(x,o)$ to denote the fact that the output of a $x$ is high. As a consequence of this concept we will have quantifiers, which allow for sentences like $\exists x\;high(x,o)$, with an intended meaning that there exists an object $x$, whose output $o$ is high, or $\forall x\;high(x,o)$ with an intended meaning that for all objects $x$ output $o$ is high,
Together with these new concepts it will be possible to express rather abstract properties of elements from the domain. Assume, e.g. that we want to introduce the concept of elements in a circuit to be connected. We could say $connected(inv1,inv2)$ and $connected(inv2,inv3)$. Now, with the intended meaning of being connected, we can assume, that $inv1$ and $inv3$ are connected as well. Of course, we could introduce the new sentence $connected(inv1,inv3)$ to express this; but the disadvantage is obvious: this have to be done for all objects from our domain. In predicate logic this property of transitivity of connectedness can be expressed without referring to explicit objects by the following sentence:
$\forall x\forall y\forall z((connected(x,y)\land connected(y,z))\to connected(x,z))$
# Models and Theories in Human-Computer Interaction/The Nyquist-Shannon Sampling Theorem
### The Nyquist-Shannon Sampling Theorem

The Nyquist Theorem is also called sampling theorem. It is used in signal processing and electrical circuit, the sample allows ADC(Analog to Digital Conversion) to generate a good analog waveform frequency. Sampling rate frequency refers to the number of samples per second.
Nyquist formula: C = 2B log2 V (Bits/Sec), B refers to bandwidth in hertz(Hz), V refers to signal levels.The bandwidth of a signal is determined by the highest signal rate in the sine wave. The higher the frequency, the greater the bandwidth. According to Nyquist, if the transmitted rate is less than 2 times the bandwidth, then the frequency is insufficient to carry the signal.
The Shannon-Weaver model of communication has known as the "mother of all models". It concludes the concepts of signals, channel, noise, information source, message, probability of error, encoding, decoding, channel capacity, information rate and so on. The channel capacity refer to the reliable information signal for communication channel. It helps transmitting a signal through a noisy channels with a great understanding of error correcting methods, level of noise interference and data corruption.
SNR(in dB) = 10 log10 (S/N), C = B log2(1+SNR); them refer to Channel capacity C (bit/sec), Signal power spectrum S(f), Noise power spectrum N(f), Signal to noise ratio (in decibels) SNR,and Bandwidth of the channel B. These formulas can be used for calculate the approximation of small, large, or constant of signal to noise ratios. In conclusion, the channel capacity is proportional to the bandwidth and SNR, which mean channel capacity and bandwidth can be increase linearly with the fixed SNR requirement. This way we can transmit a continuous signal with less power and noise.
Reference: https://www.st-andrews.ac.uk/~www_pa/Scots_Guide/iandm/part8/page1.html https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem
# Models and Theories in Human-Computer Interaction/The Pace of Innovation and Adoption Is Changing
## Introduction

Technology changes quickly and is rapidly becoming more integrated into everyday life. As new technologies are introduced, people have different levels of comfort with using those technologies. We can describe this comfort level by using the theory of Diffusion of Innovations (DoI) and can describe when groups of people will begin to use the new technologies. We can also use DoI to look at overall technology adoption rates and begin to describe overall attitudes towards technology.

## Changing Pace of Adoption

The Diffusion of Innovations theory includes five levels of adopters ranging from Innovators (those who adopt technology first) to Laggards, who are the last to adopt new technologies. When one looks at the percentage of people who fall into the groups, they form a Bell Curve with most people in the middle and fewer to either side. As people become more comfortable with technology in general, one could expect this curve to shift to the left with the early adoption categories growing in percentages and the later ones shrinking.
The Diffusion of Innovations theory includes a definition of five stages of the adoption process, with the first two (before the decision to adopt an innovation) being gathering knowledge of the technology and persuasion of others. A technology must meet a critical mass of users before it can be seen as being adopted. Due to our increased connectivity today, people are able to research and gather others thoughts on technology more quickly. With people connected via social media, information on new technologies can spread rapidly and people can persuade others by voicing their opinions on technologies. Granted, it is up to each individual to decide if/when they want to begin to use a new technology, but getting to the point of making that decision happens much more quickly so more people decide to make the decision to use the technologies and could be seen as being early adopters.

## Conclusion

Advancements in technology build upon themselves and can help to drive overall adoption rates of technology (the adoption of social media driving the ability to research and share opinions on technology is a good example of this). As we advance and technology becomes more engrained in our everyday lives, the adoption rates will only increase until there are many who would be considered innovators adopting new technologies and pushing technology forward.