text:
Lentis/Pain Scales

Introduction
Pain Scales are a measurement which assess the severity of a patient's pain level. Pain scales are also a typical medical communication tool that may be utilized in a number of circumstances, so patients can report their pain level depending on different pain scales.

Categories
There are three major categories of unidimensional pain scales.

 * Numerical Rating Scales (NRS)
 * Visual Analog Scales (VAS)
 * Categorical Scales (CS)

Numerical Rating Scales
Numerical Rating Scales (NRS) are scales that utilize a number to indicate pain experienced. The most basic and common form of a NRS is the simple 0-10 pain scale, where 0 is no pain and 10 signifies unbearable pain. This form of a NRS is most common due to its ability to stretch across languages and the speed and accuracy if a patient is conscious and able. Other NRS also exist for children and adults who are unable to quantify their pain. The FLACC scale is an example for children and babies and is further expanded in the next section. Other scales also apply numerical ratings for different categories where each number has a specific requirement prior to its use in that category.

FLACC Scale
The FLACC (Face, Legs, Activity, Cry, Consolability) scale uses different categories in order to quantify pain within children who are too young to cooperate. Each category is rated from 0-2 and the scores are then summed for a total pain score. An example is to score a 1 in the Activity category, the child's legs must be deemed "uneasy, restless, tense". Other scales that are similarly formatted include the CRIES, used for infants, and the COMFORT which is used for in-depth pain analysis on uncooperative adults.

Visual Analog Scales
The Visual Analog Scales(VAS) is one of the most commonly used measurement in pain scales. VAS method can be used in measurement for subjective characteristics which can't be directly measured. Patients or respondents can define their pain level by indicating a position between two end-points in a line. The line usually is in fixed length of 10 cm. At one end of the line, the endpoint is "No Pain", and at the other end is "Worst Pain" or "Extreme Pain". It is easy to operate by using a VAS method, and it tends to be less expensive than other pain scales due to its relative simplicity

Categorical Scales
Categorical Scales (CS) utilize different tiers of pain that patients or respondents can select from in order to best describe their pan level. There are usually 5-6 different tiers to select from, with the left-most tier in the scale representing the smallest amount of pain (no pain) and the right-most tier representing the highest amount of pain possible. Categorical scales can be separated further into two types, depending on how the different tiers in the pain scale are represented. Those types are the Verbal Pain Intensity Scale and the Visual Pain Intensity Scale.

Verbal Pain Intensity Scale
Verbal Pain Intensity Scales represent the different tiers of pain using words. These types of categorical scales use easy to understand, simple words to describe the pain tiers. The left-most (least painful) tier of the pain scale is usually labelled as 'no pain' or 'little pain,' while the right-most tier usually has the label 'worst possible pain' or 'very severe pain'. An example of a verbal pain intensity scale could be: No Pain Mild Pain  Moderate Pain  Severe Pain  Very Severe Pain  Worst Possible Pain Verbal Pain Intensity Scales commonly use the words 'mild,' 'moderate', and 'severe' among all variations for ease of use. Patients choose the words that best describe their pain level.

Visual Pain Intensity Scale
Visual Pain Intensity Scales usually still have words describing the different tiers of pain. However, in addition to those words, visual pain intensity scales' tiers are accompanied with picture descriptors. The most common visual pain intensity scale is the Wong-Baker FACES pain rating scale. The Wong-Baker FACES scale presents patients with 6-8 different pictures of facial expressions, ranging from a pleasant face on the left of the scale, a neutral face around the middle of the scale, and a crying face on the right of the scale. Patients choose the face that best represents their internal pain level.

Because visual pain intensity scales use pictures of faces to distinguish pain levels, this type of scale is very popular with diagnosing pain in children, the mentally challenged, and with patients that do not speak the same language as their doctors. However, there have been some occasions where patients misinterpret the meaning of the faces on the scale. Patients may believe that they are supposed to choose the face that represents their overall emotion with their pain, and not their actual internal pain level itself. This can lead to a different tier of the pain scale being chosen than the real level of pain that a patient is feeling, which can cause a misunderstanding with their doctor.

Doctors
When doctors are trying to diagnose patients that cannot describe their symptoms/pain in detail, pain scales become very useful for accurately measuring the patient's pain. Many different doctors will choose to perform a set of different pain scales on the patient in order to gain a precise understanding of that patient's pain. Doing so will help the doctor: (1) diagnose the severity of the injury, (2) choose the correct treatment procedure and determine how to proceed with that treatment in order to minimize pain, and (3) evaluate the effectiveness of that treatment over time by administering pain scale tests to patients before, during, and after treatment.

Patients
Patients can express their direct feelings by using pain scales and self-reports their different pain levels by using a specially designed pain scale. Pain scales become an effective tool for patients who want to communicate with doctors, and so that patients can measure their pain level and receive necessary aids from doctors. By implementing and using VAS method, patients can be beneficial from its quickness and highly reliable results.

Pharmaceuticals
The approach to patients with different pain levels begins by identifying the underlying cause and a disease-specific treatment. Pharmaceutical companies are designed to provide adequate pain relief for patients by different pain levels. Therefore, pain scales provide enough and accurate information for these companies to develop and recommend medicines. The original recommendation consists of three parts:

(1) Mild Pain

(2) Moderate Pain

(3) Severe and Persistent Pain

There have been major controversies that relate pharmaceuticals to the opioid epidemic. In 2020, Purdue Pharma pleaded guilty to three opioid criminal charges and settled to pay over $600 million in damages. Purdue Pharma is viewed as a leader of the opioid addiction crisis as they downplayed the addictive nature and pushed physicians to prescribe larger doses for pain. Purdue Pharma attempted to influence government health agencies, healthcare, and academia to sell more of their drug OxyContin. Proper utilization of pain scales in such scenarios may help reduce the need of a opioid prescription, but corporate influence by pharmaceuticals have blurred the lines for physicians and patients.

Online Survey
Due to the simple operability of VAS method, it can be transformed into different formats. A good example is provided by online survey software. It allows people to choose a specific pain level by moving the mouse remotely which makes VAS method practically possible. Then the responses will be collected, and feedback will be given soon.

Children Pain Scale
It is commonly used in Categorical Scales with images of faces for children. Children can choose which face represents how they feel and the face will be the most consistent with the current pain level.

Challenges
There are many known challenges to accurately measuring pain. The reason why there exist so many pain scales is because pain is subjective and personal, making it a very social topic. Therefore, it is difficult to develop a technical scale that would work for all patients. A lot of the different pain scales were developed such that they covered each other's downsides, which is another reason why developing a single perfect pain scale is most likely not possible and why it is important to continue improving the existing pain scales and potentially developing new pain scales to complement the existing pain scales.

Language
Because the languages patients and doctors speak are a part of these pain scales, language barriers can make administering pain scales difficult. When patients and their doctors do not speak the same language, certain pain scales (like some numerical rating scales and the verbal pain intensity categorical scales) cannot be used to measure pain. Even when patients and doctors do speak the same language, misunderstandings of patient responses on the pain scales can still occur, since different words mean different things to different people. Pain that someone could think is "moderate" could be "severe" to someone else.

Gender
Studies have found that women feel less pain than their male counterparts. Those surveyed in a 2021 study were asked to survey other's pain; the study shown clear underestimations of the target when they were a female instead of a male. Survey participants were also asked to answer about best or suitable treatment and psychotherapy were prescribed to women more often as opposed to pain medicine for men. These underestimations of pain can result in mistreatment and ineffective care for pain for female patients. Although these biases exist, there are more physical correlation between the pain experienced between males and females which may also influence a physician's ability to gauge pain experienced by female patients.

Race
There is documented racial bias in pain assessment; black Americans are less likely to be believed than their white counterparts. According to a case study, black Americans were more likely to be incorrectly treated for their pain due to both implicit and explicit biases. These biases include believing black people to have "thicker skin" and higher pain tolerances than white people. The study also surveyed a group of participants who were not medically trained. They were asked to answer questions about how much pain a specific target (white or black) may feel in a scenario and were then asked about fake medical beliefs like "Blacks’ nerve endings are less sensitive than whites". They found that believing in fake beliefs correlated with a lower pain rating pain experienced by a black target compared to a white one.

This research helps illustrate the problem within healthcare, as doctors and physicians rely on pain assessments in order to properly diagnose and treat patients. According to a 2016 study in the Proceedings of the National Academies of Science, half of the first and second year medical students surveyed believed in at least one or two fake beliefs such as black people having thicker skin. To avoid mistreatment of patients, healthcare professionals must educate themselves and/or gather more research and evidence in order to explore assumptions. Actions like these will be more important in the goal of reducing the disparity in treatment of different races, for the the U.S. Department of Health and Human Services report that black and hispanic people receive worse care on 40% of the department’s care quality measures.

title: Lentis/Pain Scales


text:
User:Yliao56/sandbox

dfsf

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Yliao56/sandbox


text:
LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation (Completion of the Squares)

This method is used to solve quadratic equations that can't be factorized.

Matrix inequality
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY,\\ \end{align}$$ which is named Young’s relation or Young’s inequality.

Derivation
Young’s relation can be derived from a completion of the squares as follows.

\begin{align} 0 \leq (X-SY)^{T}S^{-1}(X-SY) \\ 0 \leq X^{T}S^{-1}X + Y^{T}SY - X^{T}Y - Y^{T}X\\ X^{T}Y + Y^{T}X \leq X^{T}S^{-1}X + Y^{T}SY, \end{align}$$ which is named Young’s relation.

Reformulation of Young’s Relation
Consider $$ X,Y \in \mathbb{R}^{n\times m}$$ and $$ S \in \mathbb{S}^{n\times n}$$ , where $$ S $$ >0, The matrix inequality given by

\begin{align} \ X^{T}Y + Y^{T}X \leq \frac12(X+SY)^{T}S^{-1}(X+SY) ,\\ \end{align}$$ is a reformulation of Young’s relation.


text: RHIT MA113/Printable version

title: RHIT MA113/Printable version


text: Intermediate 2 Mathematics/Printable version


text: The Arab Identity/Printable version

title: The Arab Identity/Printable version


text:
The Arab Identity/Introduction

Saudi Arabia's buzz words for the year 2016 were "change and modernization". Yet as the world grapples with the Coronavirus, and a looming Oil war, Saudi US relations seem to be stretched tight. The increasingly heartbreaking war in Syria and now Yemen puts Saudi Arabia and the Arab world by large in an increasingly desperate situation. With Israels' talk of Annexing forbidden lands in occupied Palestine, are we looking at a world about to be reshaped again with Arabs relegated to its fringes? The Western world was largely introduced to the Arab during the Age of Oil. The Islamic Clergy and the Saudi Establishment have long stood for what it means to be Arab. With the West increasingly at its border, it was only a matter of time before this Identity was questioned for what it really stood for.

On the onset, Orientalism answered the West's increasing occupation with what they saw as the Decadent East. For Muslims, though this was not so. For them, it's not difficult to imagine the Arab identity. After all, Arabs have crossed the deserts of Najd to conquer large parts of the larger Arab and African lands and have crossed far into Asia by sea and to the borders of Europe by land. The Arabs are proud people. And their pride is justified by the number of Arabs and Muslims all over the world. Muslims come to Mecca in Saudi Arabia annually, to the birthplace of the Prophet Muhammad, in a pilgrimage that is the largest the world witnesses, congregating believers from the second-largest religious denomination in the world numbering to millions.

Muslims, early in life are introduced to the 'Arab-ideal'. Old Najd, and its surrounding lands, in Mecca and Medina, Jerusalem, Palestine, and even Syria, Yemen and Iraq are seen as holy lands by all Muslims. The Arabs and their place in the world are central to how Muslims identify themselves with regards to their relationships, national identity, and politics, even in an increasingly global century.

This book aims to explore 'Arab identity' and the 'Arab ideal' more than it seeks to answer global and political questions concerning the Arabian world. But to do so we first have to see what it means to be an Arab, and how the rest of the world sees them.

title: The Arab Identity/Introduction


text: Counterpoint/Printable version

title: Counterpoint/Printable version


text:
History of Creative Commons/Goals of Creative Commons

Creative Commons seeks to resolve the tension between copyright law and technologies that make it easy to share and remix materials.

The first United States copyright law was the copyright act of 1790. The first copyright laws were very limited in both scope and term; they covered only specific materials and lasted for fourteen years, with the possibility of one renewal. The copyright act has been revised repeatedly since that time, extending protection to all kinds of authored materials and extending the copyright term. Additionally, copyright is now automatic. The creator of a work need no longer apply to the United States government in order to hold copyright in that work. Because all works are automatically granted the maximum possible protection under copyright, creators who want their works to be reused, remixed, or widely distributed need to specifically take action to allow these uses!

At the same time that it has become more difficult to remix or republish creative works from a legal standpoint, digital technology in the form of the internet has made such remixing easier than ever from a technical point of view. Digital copies are very inexpensive to produce, and do not degrade, no matter how many times they are copied.

Creative Commons bridges this gap by making it easy for creators to license their works for reuse and remixing. Several Creative Commons licenses are available; creators can specify whether other people can create derivative works, whether they need to make the work available under the same license, and commercial uses are allowed. By applying a Creative Commons license to their work, creators clearly signal what uses others are allowed to make of it.


text:
History of Creative Commons/Legal History

Expansion of Copyright
Copyright laws in the United States have expanded over time. While much of this expansion happened gradually, it is important to remember that the greatest expansions of copyright occurred in the twentieth century, most notably in 1909 and 1976. The 1909 revision to the copyright act broadened the scope to afford protection to all works of authorship, and music in particular. The 1976 revision, passed in anticipation of joining the Berne convention changed the basis for copyright terms. Instead of lasting for a set number of years, copyright now expired 50 years after the death of the author.

As copyright terms lengthened, copyright terms were extended for some works that were previously scheduled to fall into the public domain.Thus, a work that would have fallen into the public domain in 1980 according to the law under which it was initially published may in fact still be under copyright as of this writing in 2018.

Sonny Bono Copyright Term Extension Act
Congress passed the Sonny Bono Copyright Term Extension Act in 1998. This act was named in memory of the congressman and former musician Sonny Bono, who had died in January of that year. His widow, Mary Bono, claimed that Bono believed "copyright should be forever."

The act lengthened the copyright terms previously extended in 1976. Works created after the passage of the act would be protected for seventy years after the author's death, rather than the previous fifty. Works which were under copyright at the time the act was passed had their copyright term extended to ninety-five years.

Several copyright holders lobbied on behalf of this act, including Disney, the Gershwin family trust, and the grandchildren of Oscar Hammerstein. These copyright holders sought to prevent the copyrights they held from expiring. Disney's intervention is especially famous; they act prevented Steamboat Willy, the first appearance of Mickey Mouse, from falling into the public domain. Thus, this law is popularly (and derisively) known as the Mickey Mouse Protection Act.

Because it lengthened the term of copyright both for new works and for works that were protected at the time of its passage, the law effectively prevented any copyrightable works from entering the public domain between 1998 and 2019.

Eldred vs. Ashcroft
Eldred vs. Ashcroft was a challenge to the copyright act. When the 1998 act extended copyright terms once more, Eric Eldred and other users of public domain works sued. Eldred, a retired programmer, was also a hobbyist internet publisher and literacy advocate who digitized works of literature in the public domain and made them freely available. The extension of the copyright terms prevented him from digitizing the works that received extended protection.

Eldred and constitutional law professor Lawrence Lessig filed a lawsuit together in 1999, arguing that the Sonny Bono Copyright Term Extension Act was unconstitutional. Their argument rested on the Progress Clause, which states that copyrights may be granted for limited times, and the First Amendment. That is, continually extending the term of copyright so that new works never fall into the public domain created, in essence, perpetual copyright, which limits the ability to criticize or comment on a work while it is still current.

The district court and DC Court of Appeals refused to hear the case, but the Supreme Court took it up in 2002. However, the Court ruled 7-2 against Eldred. In oral arguments, it appeared that the precedent set by many previous copyright extensions weakened Lessig's claims. To Lessig's disappointment, the ruling did not address the question of limited powers upon which much of his argument had rested. Justices Stephen Breyer and John Paul Stevens wrote dissenting opinions. Stevens dissented on the grounds of a parallel in patent law. Breyer argued that the copyright law had become so long as to be essentially unlimited.

Founding of Creative Commons
Lessig, along with collaborators, founded the Creative Commons in 2002 to provide an alternative to traditional copyright.

As they were working on Eldred vs. Ashcroft, Lessig and Eldred assembled a group of people to help, including MIT professor Hal Abelson. This group was called the Copyright Commons, but Eric Saltzman suggested changing the name to Creative Commons, which stuck. Lessig explains that this was intended as a grass-roots movement to support authors who don't want or need all the rights that copyright affords. In a 2002 press release, Creative Commons declared: "The Creative Commons will provide a free set of tools to enable creators to share aspects of their copyrighted works with the public. ... We stand on the shoulders of giants by revisiting, reusing, and transforming the ideas and works of our peers and predecessors."

The Creative Commons founders also believed that creating an explicit guarantee of legal protections would encourage participation in the open-source movement.

The William and Flora Hewlett Foundation contributed one million dollars to help begin the movement.


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History of Creative Commons/Creative Commons Organization

Name
The name "Creative Commons" references the concept of the commons, a social system by which resources are held in common. According to David Bollier, both long-term stewardship and self-organization are essential to the commons. The commons includes both the community and the resources in question and is a set of values which render them sustainable. Bollier argues that open networks such as Wikipedia form a kind of digital commons.

Organization
Lawrence Lessig was the founder and first CEO of the Creative Commons, but he stepped down in 2008 and was replaced by Joi Ito. The present CEO is Ryan Merkley, who joined the organization in 2014. The Creative Commons organization Includes a staff, board, and advisory council.

In addition to its official staff and board, Creative Commons includes a network of affiliates known as the Creative Commons Global network, which represents Creative Commons around the world.

Mission
The mission of the Creative Commons is to "develop, support, and steward legal and technical infrastructure that maximizes digital creativity, sharing, and innovation." To this end, Creative Commons sponsors four program areas:

 * Licenses + Legal Tools
 * Discovery and Collaboration Tools
 * Policy + Advocacy
 * Community Building

Each of these areas includes both technical work and outreach and advocacy. Creative Commons has programs to educate potential license users, to support institutions as they unlock content, and to advocate for copyright reform.

Creative Commons Global Network
The Creative Commons Global Network includes Creative Commons representatives in over 60 countries. Members of the network conduct public outreach and training and answer questions about Creative Commons.

In 2017, the network underwent a community-led strategic process which resulted in a new strategy document and governance structure. The main governance body is the Global Network Council.

Chapters coordinate this work geographically. Each chapter is made up of people who live in a particular area. Chapters follow the Creative Commons charter, and are vouched for by two members of the Global Network.

Platforms organize the work areas of the Global Network online. There are currently four platforms, concerned with open education, copyright reform, community development, and GLAM (Galleries, Libraries, Archives and Museums).

The platforms are still under development but can be viewed on GitHub.


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History of Creative Commons/Creative Commons Licenses

What Licenses Do
Licenses allow creators of works to specify what uses can be made of those works, without seeking permission. Authors may still grant permission for uses that fall outside the license, but the uses allowed by the license do not require permission.

Creative Commons Licenses are both human-readable and machine-readable. This means that works with Creative Commons licenses can be identified as such in their metadata, allowing search engines to identify them. For instance, Google Image Search allows users to filter works by license.

History
The first (1.0) Creative Commons licenses, issued in 2002, were inspired by the GNU General Public Licenses. They included three parts: a plain-language summary for anyone who wanted to use them, legal code to guarantee their permissions, and a machine-readable component to make them searchable via search engines.

The second iteration of the Creative Commons licenses (2.0) was released in 2004. It provided an attribution standard and made the Share-Alike licenses work internationally.

Version 3.0 was released in 2007. Today, 4.0 licenses are available.

Types of Licenses
Authors can mix-and-match their licenses. The characteristics covered by the licenses include:

 * BY (Attribution)
 * Everyone who uses the work must provide attribution to the original author.
 * SA (Share-Alike)
 * Anyone who makes use of the work must make it available to others under the same license.
 * NC (Non-Commercial)
 * Commercial uses of the work are not allowed.
 * ND (No Derivatives)
 * The work can be reused but not remixed. Derivative works (such as translations, pastiches, etc) may not be made.

Thus, a range of possible Creative Commons licenses can exist. If a work is licensed CC BY, for instance, any uses are allowed as long as the original author is credited. If a work is licensed CC BY-NC-ND, then the work can be reused with attribution, but only in non-commercial contexts, and derivative works are not permitted.

Popularity of Creative Commons Licenses
The first Creative Commons licenses were issued in 2002; by 2003, one million such licenses had been granted, and by 2004, that number had reached nearly five million. In 2006, 140 million works were available under CC licenses.

As of 2017, over 1.4 billion works had been released under Creative Commons licenses. Millions of CC-licensed works are available on platforms including Wikipedia, Wikimedia Commons, DeviantArt, Europeana, the Internet Archive, Medium, and the Directory of Open Access Journals.

Books, educational resources, and games have all been published under Creative Commons licenses.


text: History of Creative Commons/Printable version


text: Switching from macOS to Linux/Printable version


text: Technical Analysis/Printable version

title: Technical Analysis/Printable version


text: Three Dimensional Electron Microscopy/Printable version


text: Human Digestive System/Printable version


text: American Government/Printable version

title: American Government/Printable version


text: Data Science and Artificial Intelligence with Julia/Printable version


text: Critique of the 1776 Commission Report/Printable version


text: Aeroacoustics/Printable version

title: Aeroacoustics/Printable version


text: Active Learning In Virtual Environment/Printable version


text:
Active Learning In Virtual Environment/Introduction

Active Learning is about involving students in doing things and thinking about what they are doing. When using active learning, students are engaged in more activities than just listening. In other words, active learning describes a learning process in which a deep commitment with the training material is used to improve knowledge retention. Active learning has many benefits. Active learning involves students in the learning process as it aims to promote interaction with the material being taught. This allows students to research the material themselves, which improves knowledge retention. Self-exploration improves students’ critical thinking and decision-making skills, as they need to learn how to evaluate what is important and what is not. Active learning also boosts students’ motivation and performance since actively participating in the learning process helps to understand the discussed topics better, which improves the student’s ability to adapt knowledge to specific real-life situations.

Active learning methods, such as debates and brainstorming, are helpful tools to embrace active learning in the learning process. During online debates, for example, students are broken out into two groups that research, develop, and present opposing viewpoints on a topic. This allows them to take responsibility for their performance, as they need not only to develop their logical reasoning skills to defend their position, but also to make decisions about how they will use what they are learning. Brainstorming involves working together as a group and encourages creative thinking and risk taking in order to generate new ideas and solutions, no matter how unclear or ineffective.

During the Covid-19 pandemic, online teaching has become a common form of education. Both students and teachers needed to adjust to different types of virtual education, instead of regular classroom teaching. Students must actively keep up with their studies which indicates the need for active learning. Active learning in an online environment helps students to embrace knowledge through active learning methods. When actively participating in the teaching process, students learn the discussed topic better because it’s more effective to participate than just passively listen. Active learning methods make the learning process more meaningful and ensure deeper learning.

In spring 2021, a group of students at Tallinn University conducted a research project titled Active Learning in Virtual Environment (ALIVE). Local and foreign students and lecturers, from different fields of learning took part in the research. 31 students answered the students’ questionnaire and 13 lecturers answered the lecturers’ questionnaire.

The study shows that the use of Active Learning methods makes students engage more actively during lectures. According to the students’ research results, Active Learning is interesting and develops social skills (74,2%). It helps remember things better (64,5%), makes students work harder, gives many different points of view towards the subjects and also active learning sometimes pushes limits in a good way, for example being out of the comfort zone, so the overall learning results are better. According to the lecturers’ research, information about Active Learning methods are easily found from the Internet and lecturers also ask for other lecturers’ advice. Some of the lecturers have completed training on Active Learning methods. However, some specific knowledge about the use of Active Learning, such as group size and preparation time is insufficient. According to the research results, Active Learning makes the teaching and learning process more interesting and it enhances students' attention and increases their engagement.

The project team created this wiki page for everyone who wants to know more about active learning, its methods and how to concretely apply it to online lectures. There exists a subcategorization for easy access to different active learning methods.


text: Somali/Printable version

title: Somali/Printable version


text: French Revolution/Printable version

title: French Revolution/Printable version


text: Overview of Elasticity of Materials/Printable version


text: Type-oriented programming/Printable version


text: Climatology/Printable version

title: Climatology/Printable version


text: Chip's Challenge/Printable version

title: Chip's Challenge/Printable version


text: OpenSCAD Tutorial/Printable version

title: OpenSCAD Tutorial/Printable version


text: Modern Photography/Printable version

title: Modern Photography/Printable version


text:
Template:Japanese Phrasebook/EntryWithNotes

{	"params": { "english": {}, "romaji": {}, "japanese": {}, "note": {} } }


text:
Oberon/ETH Oberon/2.3.7/Tutorials.Tool

The tutorial system allows you to read the Oberon and Gadgets on-line hypertext documentation. You may also create your own tutorials by compiling them with the provided Book compiler. The tutorials have been designed to be presented on the Oberon desktop.

title: Oberon/ETH Oberon/2.3.7/Tutorials.Tool


text:
Signetics 2650 & 2636 programming/Analogue TV

These consoles were manufactured in the days of analogue TV, when the picture was drawn on a cathode-ray tube by an electron beam scanning back and forth across the screen. The programmer doesn't need to know all the details of how these TVs worked, but should be familiar with the basic principle of operation, some of the terminology, and the speed at which the beam crosses the screen. The purpose of this section is to provide an overview and suggest some further reading.

Raster scan
The electron beam draws one horizontal scan line at a time, starting at the top left of the screen. By the time it reaches the bottom right of the screen a whole picture has been displayed. A long-persistence phosphor coating on the screen retains the image long enough that the human eye does not perceive any flicker.

While the beam is moving left to right the screen is being lit up with the three primary colours, red, green and blue. At the end of each scanline it is moved back, right to left. This movement is faster than in the other direction, and no colours are displayed; this is referred to variously as the horizontal retrace, horizontal flyback or horizontal blanking. In a similar manner, at the end of each frame the position of the beam has to be moved back to the top left of the screen without displaying any colours; this is referred to variously as the vertical retrace, vertical flyback, vertical reset or vertical blanking. The PVI documentation refers to it simply as VRST

PAL and SECAM
Colour television was broadcast in three different standards in various parts of the world, NTSC, PAL, or SECAM. As far as is known, the 2636 PVI was never used in any consoles for the North American NTSC system. Most were PAL and those made in France were presumably SECAM. Fortunately these two standards vary only in the way the colour is encoded in the broadcast signal, and both use the same number of scan lines with the same timings. This means that most of the electronics, and more importantly the firmware, will be the same. The only things that need to change are the video encoder and possibly the modulator. PAL and SECAM both fall under a standard known as either 625 lines or 576i. The number 576 comes about because 49 of the 625 lines are not visible during the vertical blanking period.

Interlaced TV pictures
The concept of interlaced video also needs to be addressed here as it explains the difference between the 625 lines of a standard tv signal and the much smaller number of lines we can specify with an eight-bit register. In an interlaced tv picture every other line is output first, then when the beam sweeps down across the screen again it displays the lines in between. This method was adopted to help reduce flicker. Each vertical pass is known as a field, and it takes two fields to make a frame. In this way a lot more detail can be displayed in the picture.

Non-interlaced console picture
These video game consoles generate a non-interlaced signal. Each vertical pass of the screen is a complete frame. Each scanline takes 64μs and each frame has 312 scanlines. Each frame takes a total of 20ms, equivalent to 50Hz. Forty three scanlines are blanked during the vertical retrace, leaving 269 potentially visible.


text:
Talk:Economic Sophisms

Can we have links to the original WS pages like there is on the transcluded book? i.e. the [61] [62] on Economic Sophisms Chapter 11. This way any typos can be easily corrected on the original text. Zoeannl (discuss • contribs) 02:33, 16 November 2021 (UTC)

Where is the best place to ask the above request? I'm not sure how to navigate WB. Cheers, Zoeannl (discuss • contribs) 02:47, 16 November 2021 (UTC)

title: Talk:Economic Sophisms


text: Canadian Property Law/Printable version

title: Canadian Property Law/Printable version


text: Environmental Chemistry/Printable version


text: Private Bodies of Water/Printable version


text: Super Mario 64/Printable version

title: Super Mario 64/Printable version


text: Character Encodings/Printable version

title: Character Encodings/Printable version


text: Arithmetic/Printable version

title: Arithmetic/Printable version


text: Mathematical symbols/Printable version

title: Mathematical symbols/Printable version


text: Abstract Algebra/Printable version

title: Abstract Algebra/Printable version


text: Visual Studio/Printable version

title: Visual Studio/Printable version


text: Programming with Moose/Printable version


text: TI-Basic Programs/Printable version

title: TI-Basic Programs/Printable version


text:
TI-Basic Programs/Writing a Program

There are just a few steps to follow in order to write a program onto your TI-83+ or TI-84 calculator using TI-BASIC.

 * 1) Turn on your calculator and press the PRGM button.
 * 2) Press the right arrow button twice to move to NEW.
 * 3) Press Enter to select Create New.
 * 4) Name the program. If the program is of your own creation, give it whatever name you would like it to have using the letters above each button or any of the numbers 0-9.  If you are copying a program from this book, type in the name already given.
 * 5) Write the code of the program by using the digits and commands found on the calculator and by using the PRGM sub-menus while within the program code. These menus contain lists of many of the most vital programming commands, such as If, Then, End, While, Menu(, and many others.  Other helpful menus of commands are the MATH, TEST, and DRAW command lists.

Note: Some of the programs in this book use lower-case letters in their program codes. In order for a TI calculator user to type in lower-case letters you must have MirageOS on your calculator. You must also check the checkbox for Enable Lowercase in Miscellaneous Options on the Options screen within MirageOS. The Options screen can be opened by pressing the ALPHA key in MirageOS. RAOSX is also useful for enabling lowercase letters and other common assembly functions.

For the instructional wikibook on TI-BASIC programming, see TI-Basic Z80 Programming.

title: TI-Basic Programs/Writing a Program


text: Learn Electronics/Printable version

title: Learn Electronics/Printable version


text:
User:Jerryy147

= Hello, I only came here because I was bored one day, I'm not really active anymore. =

All stuffs
 (left) - ' where I live... currently '

title: User:Jerryy147


text: User:Llakew

title: User:Llakew


text: File:Infiltrator II Base B.jpg

title: File:Infiltrator II Base B.jpg


text: File:Infiltrator II Base C.jpg

title: File:Infiltrator II Base C.jpg


text: Basic Algebra/Printable version

title: Basic Algebra/Printable version


text: Algebra/Printable version

title: Algebra/Printable version


text: World Heritage Sites/Printable version

title: World Heritage Sites/Printable version


text: Category:Book:Traditional Abacus and Bead Arithmetic


text:
US Constitutional Law/Freedom of Speech

"Congress shall make no law... abridging the freedom of speech."

The Freedom of Speech is guaranteed by the Constitution in the First Amendment. The Supreme Court of the United States perhaps protects this aspect of the First Amendment the most among all the clauses found therein, with Freedom of the Press being tied with it. Nevertheless, the Supreme Court has found that there are exceptions to this freedom in very limited contexts. In some other instances, the Supreme Court has defined certain other aspects of speech to be constitutional where a first glance may appear to not be.

Do we have the right to contribute money to political campaigns on the basis of freedom of speech? Do students have the right to display a banner that reads "Bong Hits 4 Jesus" at a corporate-sponsored rally in which the school gave permission to attend? What limitations, if any, do commercials have when advertising? Do we have the right to sit during the pledge of allegiance even when compelled to by law or government actor? Am I allowed to burn the flag in a public event? Do I have the right to ignore loyalty oaths or affirmations if it is against my beliefs?

These were questions addressed by the Supreme Court on some occasion or other, and these are not all the questions addressed by the Supreme Court. Here is a wikipedia page that lists more cases that the Supreme Court has addressed.

Compelled Speech
"If there is any fixed star in our constitutional constellation, it is that no official, high or petty, can prescribe what shall be orthodox in politics, nationalism, religion, or other matters of opinion or force citizens to confess by word or act their faith therein."

title: US Constitutional Law/Freedom of Speech


text:
Programming Basics/Objects

Objects are a useful tool to handle flexible situations in code. Object Oriented Programming style uses objects extensively.

title: Programming Basics/Objects


text:
Programming Basics/Files

Files are used to save data to a storage device, which is useful for a number of purposes.

title: Programming Basics/Files


text: Vectors/Printable version

title: Vectors/Printable version


text: The Legend of Zelda: Oracle of Ages/Printable version


text: The Legend of Zelda: A Link to the Past/Printable version


text: The Legend of Zelda/Printable version

title: The Legend of Zelda/Printable version


text: The Legend of Zelda: Ocarina of Time/Printable version


text: The Legend of Zelda: Oracle of Seasons/Printable version


text: American Literature/Printable version

title: American Literature/Printable version


text:
User talk:130.245.192.10

Excuse me I think this was done in error.

title: User talk:130.245.192.10


text: Flight Attendant Manual/Printable version


text: ABC graphs/Printable version

title: ABC graphs/Printable version


text: Category:Book:Managing enterprises with advanced ICT


text: Croatian/Printable version

title: Croatian/Printable version


text:
Baroque Macedonia and the Macedonian Revolts/Introduction

Introduction

The baroque period was an era that began approximately in 1650 and ended roughly in 1770. During this period there were major advancements in agriculture, architecture, construction, mining, medicine, and science. However during this period the region of the Balkans was under Ottoman administration and was part of the Ottoman Empire. Even though Macedonia was part of the Ottoman Empire it still managed to show some form of baroque significance and style. When travelling through the towns many of the residential properties and buildings portrayed a baroque-style infrastructure regardless whether the region was under Ottoman control. Theoretically the title of this book should be called : Ottoman Macedonia and the Macedonian Revolts, however the author has decided to name this book : Baroque Macedonia and the Macedonian Revolts.

Many years ago, to identify a clan or tribe a coat of arms was used. A form of identity representing a faction, empire, region, entity or an individual. Coat of arms can display any illustration on textiles, wood, masonry or metal. The very first coat of arms was possibly displayed back in 100 AD or even earlier. Coat of arms were common amongst knights in Europe, Asia and North America. The Macedonian coat of arms is a yellow or golden lion standing upright, on a red background. This coat of arms had undergone many changes through the years.

Through these turbulent years the Macedonian people had no identity and were forced into believing and following a different religion called Islam. If we go back to the 14th and 15th century the Ottoman Empire began building mosques within the region of the Balkans and many Christian churches were dismantled and destroyed. However some Christian churches remained, a good example is the Church of St John at Kaneo in Ohrid, this church is located near the Lake of Ohrid. It was also evident that during the baroque period in Macedonia, a number of Jewish churches were scattered within the city of Skopje. These Jewish churches had offered a different religion and service to the Macedonian people. The Macedonian communities were prohibited from following an Orthodox Christian religion and service. The Jewish quarter was an area within Skopje that contained various buildings of Jewish importance. The city of Skopje especially in the Jewish quarter, it possessed its own walls, two synagogues and a number of Jewish schools.

During the Dark Ages, also known as the Late Antiquity period, Slavs had settled in Macedonia, and the surrounding areas or region back in 610 AD, forming Macedonia Sclavinia. Further north was Rascia and towards the east was the Bulgarian faction. It was noted that after the arrival of the Slavs in the Balkans, the use of the Roman solidi also known as the Roman solidus decreased rapidly beginning of the 8th Century. The Byzantine or Roman Empire had also accumulated the lands of Dacia. These were not the only lands acquired by the Roman Empire. The province Dacia Traiana comprised the regions known today as : Banat, Oltenia, Transylvania, Moldavia and Romania. Like the Dacian state, many years later so was the early Macedonian state, an unstable tribal confederacy.

Moving along through the years Macedonia had become part of the Ottoman Empire. Also the regions of Albania, Bulgaria, Greece, and Serbia were also part of the Ottoman Empire. The Macedonian people looking to overthrow the Turkish Ottoman rule established a rebel stronghold, men such as Karposh and Strahil, were seen as the driving force, showing patriotism and courage like yellow rampant lions ready to leap into the unknown Ottoman - held territory. Ottoman Macedonia back in the 17th Century (1601-1700) was undergoing a process of awakening and recognition in the form of a different voice, a different identity. The different voice and identity were the Macedonian people. Emerging was a large group of men wanting freedom, a new life, free from oppression and torture. These men and women were living in a different timeframe or timeline. They experienced the Ottoman way of life, learning a different language, following a different religion, they accepted life in Ottoman Macedonia not because they wanted to, but because they were forced to. The numerical figures of the Ottoman forces was to great however and more time and planning was required to win a war not just a battle, the Macedonian revolutionaries were placed in a difficult position, which was recorded by the Turkish historian Silahdar Findiklili Mehmed Agha.

An otriad of rebels which were mostly based in the Malesevo mountains, Krushevo, Kumanovo, Smilevo, and Uskup (Skopje) joined together to form a Macedonian revolutionary group. The Ottoman Empire started to build ships known as galleons, boat building involved skilled men or women, the early modern baroque period saw the transformation of the carrack into a galleon, having an upper gundeck, lower gundeck and a quarterdeck. The galleons were very useful for battles at sea. The baroque period was also a time when architectural and construction advancements occurred, not so much in the architectural field but in the construction field saw certain buildings being constructed that possessed distinctive features of baroque architecture were : pear shaped domes in residential, religious and market buildings. The use of plaster, stucco, marble and other building materials were quite common. Large-scale ceilings frescoes, marian columns, the use of stained glass windows, and many other building elements were typical baroque construction. However even during war architecture, sculpture, painting, construction and education still continued regardless of the persistent struggle for freedom. The seventeenth century was interesting because here the Macedonian revolutionaries have achieved so much that the north-eastern region of Macedonia was literally free from Ottoman oppression, suffering and rule. The streets of Kratovo, Kumanovo, Kriva Palanka, Kocani, Kacanik, Tabanovce, Veles and Uskup were liberated. Why did the military involvement fade between the Macedonian communities and the Habsburg Empire after 1689 ? This question has puzzled many historians and archaeologists living during this time period. Was it due to the time required to send reinforcements, or that the region was extremely far and isolated. It could also be that the region had fallen to a disease called cholera and there were limited available doctors and medicines. Also that ammunition was in short supply. The legendary Karposh, a man who shocked the Ottoman troops and the authorities to it's core, was the leader of many revolts.

The seventeenth and eighteenth centuries constitute a gap in Ottoman Macedonia’s architectural history. The period 1645 and 1739 was marred by violent conflicts between the Ottomans, the Habsburgs, and Venice. The invasion of Macedonia by the ‘Holy League’ in 1689/90 saw Skopje and Štip set ablaze. Building in this period often meant rebuilding. In Skopje, monumental structures were staggeringly reconstructed, often in simplified forms. There, the most interesting preserved new building of the eighteenth century seems to be a hexagonal domed mausoleum (türbe) of the ‘open’ type. It was commissioned by an Ottoman bureaucrat for his wife and daughter, who must have died in Skopje in 1774/5 as a result of an accident or disease, probably while in transit. For this representative of the central government, it must have seemed fitting to attach this türbe to Skopje’s only mosque built by a sultan. Some low‐relief ornament on the structure, of which there may have been more once, shows the flowery ‘Baroque’ style of Istanbul in that period.

As described or depicted in this wikibook the author has written a fictional description about the formation and recruiting of revolutionaries in some of the chapters of this wikibook. Karposh however was already the chief of Christian auxiliary forces, and basically was not involved in the finding or searching of available men. Men that could be of assistance were mostly described as hajduks or voivode. The term komiti or komita was not used untill the beginning of the 1800's.

 * Footnotes


text: Baroque Macedonia and the Macedonian Revolts/Printable version


text: General Relativity/Printable version

title: General Relativity/Printable version


text: Samoan/Printable version

title: Samoan/Printable version


text: Mario Kart DS/Printable version

title: Mario Kart DS/Printable version


text:
Précis of epistemology/Computers and power consumption

Computers cannot run without consuming power. We prove it by thermodynamics, with the principle of the impossibility of perpetual motion of the second kind.

The perpetual motion of the second kind
A machine that could lift a weight or move a car without being supplied with energy could make a perpetual motion of the first kind. The law of conservation of energy, the first law of thermodynamics, forbids the existence of such a machine. It is one of the most fundamental laws of physics. All physicists would be wrong if we could invent such a machine, but no one has ever invented it.

Perpetual movement of the second kind does not contradict the law of conservation of energy. Any body can yield energy if it is cooled, unless it is at zero temperature, equal to 0 Kelvin = -273.15 °Celsius = -459.67 °Fahrenheit. So we can imagine a car, a boat or a plane that could advance without consuming fuel. It would only have to absorb air or water at room temperature and reject it at a colder temperature. The difference in energy would be used to run the engine.

For such an engine to work it would be necessary to be able to separate a system whose temperature is uniform in two parts, one warmer, the other colder. But it is forbidden by the second law of thermodynamics because it would reduce total entropy. The impossibility of the perpetual motion of the second kind thus results from the law of non-decrease of the total entropy, the second law of thermodynamics.

Maxwell's demon
A Maxwell's demon shows that information can be transformed into work:

Consider a gas in a container. A partition is placed in the middle. It is equipped with a small door controlled by a device that detects the speed of the incident molecules. It opens the door only if a molecule that comes from the left goes faster than average or if a molecule that comes from the right goes slower than average. In this way the right compartment is warmed while the left one is cooled (Maxwell 1871). This difference in temperature can be used to operate a heat engine.

The door opener is a Maxwell's demon. It acquires information that can be transformed into work. Information is therefore a kind of fuel.

Maxwell invented his "demon" to show that the law of entropy non-decrease is only a statistical truth that could be transgressed if one were able to modify the statistical equilibrium of the microscopic constituents. In its time, the existence of atoms and molecules was still very hypothetical. To consider the possibility of manipulating them was therefore out of the question. But as soon as the microscopic constituents of matter were better known, the possibility of a mechanical device that functions like a Maxwell's demon could be taken seriously.

To date, our ability to observe and manipulate microscopic constituents does not allow the device imagined by Maxwell to be realized, but scanning tunneling microscopy makes it possible to observe and manipulate atoms. One can then imagine a device that allows to recover work after reducing the entropy of the observed system, so a sort of theoretically feasible Maxwell's demon:

Consider a crystal that can accommodate atoms on its surface. It is assumed that initially $$ N_A $$ atoms are randomly distributed on $$ N_S $$ sites and that the temperature is low enough that they stay there. It is therefore a frozen disorder. We begin by observing the exact configuration of the surface atoms, which can be done with a scanning tunneling microscope, then we move them and collect them with the same microscope on a fraction $$ \frac {N_A} {N_S } $$ of the surface. The activity of the microscope resembles an isothermal compression work on a gas, except that it is not a gas but a frozen disorder on the surface.

In principle, the displacement of atoms does not require any work because the work of tearing an atom can be recovered during redeposition.

To convert the activity of the scanning tunneling microscope into work, we put the surface of the crystal in contact with an empty container of volume $$ V $$ whose other walls can not accommodate the atoms. This container is divided with a movable wall into two parts left and right whose volumes are respectively $$ V_G = \frac {N_A} {N_S} V $$ and $$ V_D = \frac {N_S- N_A} {N_S} $$. The crystal is heated to vaporize the atoms in the volume $$ V_G $$. The resulting gas is then allowed to relax isothermally throughout the container, providing a work $$ W = N_A k_B T \ln \frac {N_S} {N_A} $$ where $$ k_B $$ is the Boltzmann constant and $$T$$ is the temperature. The crystal is then cooled to allow the atoms to redeposit on the surface of the crystal. If one proceeds reversibly, with a succession of thermal baths, the heat supplied during the heating by each thermal bath used is exactly equal to the heat it recovers during the cooling, because the specific heat at constant volume of a gas does not depend on its volume. The crystal and the thermal baths that were used to warm it have returned to their initial state.

The thermal bath which was used for the isothermal expansion gave up part of its heat which was converted into work. It therefore seems that we have achieved a perpetual motion of the second kind, since all the rest of the device has returned to its initial state.

The tunneling microscope which orders the atoms on the surface must be controlled by a computer programmed to fetch the atoms and deposit them at a suitable site. Since the second law of thermodynamics prohibits perpetual motion of the second kind, this computer must consume an energy at least equal to the work $$W$$ which was provided during the isothermal expansion. Hence a computer cannot function without consuming energy.

It has been supposed that an absorbing wall can make a perfect vacuum in an arbitrarily large volume. Such a wall can not exist otherwise one could make a perpetual motion of the second kind: the wall charged with atoms is placed in contact with an empty container, it is heated to a temperature sufficiently hot that all atoms are vaporised. The gas is then allowed to relax isothermally to provide work. The gas is then cooled to a temperature sufficiently cold that all atoms redeposit on the absorbing wall. If one proceeds reversibly the heat supplied during heating by each thermal bath is exactly equal to the heat it recovers during cooling. We could therefore return to the initial state after providing work by extracting heat from a single heat bath.

In order to make an exact calculation that is compatible with the laws of thermodynamics, it is necessary to take into account the equilibrium density of a gas in contact with an absorbing wall. This density can not be zero, but it can be very small, a priori as small as one wants if the wall is sufficiently absorbent. This is enough to justify the calculation above where this density is neglected.

The Szilard's engine
To better understand the transformation of information into work, Szilard (1929) invited us to reason about an engine that works with a "one molecule gas":

A molecule is enclosed in a container that can be separated by a removable wall. When the wall is put in place in the middle of the container, the presence of the molecule in one or the other of the separate compartments is detected. So a bit of information is acquired. The wall is then used as a piston on which the molecule can work. It is necessary to know where the molecule is to know in which direction of piston displacement a work can be recovered. In this way it is calculated that under optimal conditions an bit of information is used to recover a work equal to $$ k_B T \ln 2 $$. This is the work done by a molecule on a piston during an isothermal expansion at the temperature $$ T $$ that doubles the accessible volume.

Szilard's engine seems to contradict thermodynamics because it suggests that we could make a perpetual motion of the second kind. If the piston can move in only one direction, it is not necessary to know the position of the molecule to recover work. Once in two the piston remains motionless because the molecule is on the wrong side of the piston, and we do not recover any work, but once in two it moves and we recover a work equal to $$ k_B T \ln 2 $$. By repeating the experiment many times we could thus obtain an arbitrary amount of work without spending any to know the position of the molecule.

But such a process requires a device that removes the piston and puts it back in place. Now there are two possible positions of the piston at the end of a cycle, either in the middle if it has not moved, or at one end, if it has moved. The device must therefore acquire a bit of information at the end of each cycle. To return to its initial state, it must erase this information. The cost of erasing information is thus here too at the origin of the impossibility of a perpetual motion of the second kind (Leff & Rex 1990).

A new proof of Szilard's theorem
On the surface of the crystal on which atoms are deposited, the number of possible configurations is equal to the number $$ (_ {N_S} ^ {N_A}) $$ of ways to place $$ N_A $$ atoms on $$ N_S $$ sites. The quantity $$ I $$ of information necessary to know the fixed disorder of atoms on the surface is therefore:

$$I = \ln (_{N_S}^{N_A}) = \ln \frac{N_S!}{(N_S-N_A)!N_A!}= N_S \ln \frac {N_S} {N_S-N_A} + N_A \ln \frac {N_S-N_A} {N_A} $$

where we used Stirling's approximation: $$ \ln N! \approx N \ln N - N $$

Suppose $$ \frac {N_S} {N_A} \gg 1 $$. So

$$ I \approx N_A (\ln \frac {N_S} {N_A} +1) $$

If in addition $$ \ln \frac {N_S} {N_A} \gg 1 $$, we obtain

$$ I \approx N_A \ln \frac {N_S} {N_A} $$

Now this quantity of information can be used to provide a work $$ W = N_A k_B T \ln \frac {N_S} {N_A} $$. We thus obtain an example of the following theorem:

A quantity $$ I $$ of information can produce a work $$ W = k_B T I $$ when used by a machine at the temperature $$ T $$.

The minimum energy consumption by a computer
A computer that acquires an amount of information equal to $$ I $$ must erase it to revert to its initial state. Now this quantity of information can be used to produce a work $$ W = k_B T I $$. Since perpetual motion of the second kind is impossible, a computer that erases a quantity $$ I $$ of information must consume an energy at least equal to $$ W = k_B T I $$.


text: A hand book of life sciences/Printable version


text: AP Physics C/Printable version

title: AP Physics C/Printable version


text: Technical Book Development/Printable version


text: PyGame Guide/Printable version

title: PyGame Guide/Printable version


text:
PostgreSQL/Architecture Cluster

Overview
A, which is some hardware, a container, or a VM, contains one or more Database Clusters (  for short). Every cluster is controlled by exactly one instance. If there are many clusters and instances on the same server, the ports of the instances must differ from each other as well as the root directories of the clusters.

Each newly created cluster contains the three  template0, template1, and postgres, each of the three databases contain the schema public as well as the system schemas pg_catalog, information_schema, pg_temp, and some more. Tables, views, and most other SQL objects reside in such schemas. DBAs can create more clusters, databases, schemas, or SQL objects.

Initialization Phase
Clusters are created with the command. template0 is the very first database during the creation phase of any cluster. In a second step, database template1 is generated as a copy of template0, and finally database postgres is generated as a copy of template1. Later, the DBA can create more databases within that cluster, e.g.: my_db, with the command. Just like at the beginning, the new database will be a copy of template1. Due to the unique role of template0 as the pristine original of all other databases, no client is allowed to connect to it and modify it. But the DBA can change template1.

Connections
Client connections act at the database level and can access data and SQL objects within any schema of the connected database, as far as they are permitted to do so. If they need access to any object of a different database within the same or another cluster, special techniques like  (FDW) or   are required (or they use multiple connections and synchronize them at the client-side).

SQL objects
We use the term SQL object for all objects which you can create with the SQL command, e.g.: database, schema, table, view, materialized view, index, constraint, sequence, function, procedure, trigger, role, data type, domain, operator, tablespace, extension, foreign-data wrapper, and much more. Such SQL objects are arranged in a hierarchical manner:

 * Database names, tablespaces, and roles (users) are known at the cluster level. E.g.: As mentioned above, a connection works at the database level. Nevertheless, when you create a new role with such a connection, the role is also known by all other databases of the same cluster.

 * Extensions, e.g.: PostGIS, reside at the database level. After installing an extension, all schemas of this database can use it. But within the other databases of the same cluster, the extension is not known.

 * Schemas are part of a database. Some of them are predefined.
 * pg_catalog is a schema with tables that describe most of the SQL objects of that database, especially all tables and views. They even describe themself. information_schema is a similar schema. It contains several tables and views of pg_catalog in a way that conforms to the SQL standard.
 * public acts as the default schema. It should not contain user-defined SQL objects. Instead, it is recommended to create one or more additional schemas to manage application-specific objects like tables or triggers. To access objects in such additional schemas they can be fully qualified, e.g. my_schema.my_table, or by changing the.
 * There are different types of SQL objects within a schema: 'relation'-like objects (table, view, materialized view, index, sequence, foreign-table), function, procedure, trigger, constraint, data type, domain, operator, and more.
 * SQL objects in one schema are different from SQL objects in different schemas, even if they use the same name, e.g.: table t1 in my_schema1 is different from t1 in my_schema2.
 * The names of 'relation'-like objects, data types, and domains are unique within their schema: e.g.: you cannot have a table emplyee and a view employee in the same schema.

title: PostgreSQL/Architecture Cluster


text:
PostgreSQL/Architecture Directories

PostgreSQL organizes durable (persistent) data as well as volatile state information about transactions or replication actions in the file system. Every cluster has its root directory somewhere in the file system. In many cases, the environment variable PGDATA points to this directory. The following graphic uses data, which is the default, as the name of the cluster's root directory.

The cluster's root directory contains many subdirectories and some files, all of which are necessary to store durable as well as temporary information. The root's name can be selected as desired, but the names of its subdirectories and files are constant respectively determined by PostgreSQL. The following paragraphs describe the most important subdirectories and files.

base contains one subdirectory per database. The names of those subdirectories consist of numbers. These are the internal Object Identifiers (OID), which are numbers to identify their definition in the System Catalog.

Within the database-specific subdirectories of base, there are many files: one or more for every heap and index. Again, the filenames consist of numbers. Those files are accompanied by files for the Free Space Maps (suffixed _fsm) and Visibility Maps (suffixed _vm), which contain optimization information. An example for filenames is: 3083, 3083_fsm, 3083_vm.

Another subdirectory is global. It contains files with information about SQL Objects which are not restricted to a certain schema, but known and relevant at the schema level.

In pg_tblspc, there are symbolic links that point to directories that are outside of the root directory tree, e.g. at a different disk. Heap and index files of non-default tablespaces reside there. Those defined within the default tablespace reside in the database-specific subdirectories.

The subdirectory pg_wal contains the WAL files. They arise and grow in parallel with data changes in the cluster and remain as long as they are required for recovery, archiving, or replication.

The subdirectory pg_xact contains information about the status of each transaction: in_progress, committed, aborted, or sub_committed.

In the root directory, there are some files. In many cases, the configuration files of the cluster are stored here. Also, if the instance is up and running, the file postmaster.pid exists here (by default, but other locations are possible). It contains the process ID (pid) of the Postmaster process which has started the instance and controls it.

title: PostgreSQL/Architecture Directories


text: Hand knitting (beginner to intermediate)/Printable version


text: UFO and Space: 1999/Printable version

title: UFO and Space: 1999/Printable version


text: Rope Making for Bondage Use/Printable version


text: Concepts of Computer Graphics/Printable version


text: Future Media/Printable version

title: Future Media/Printable version


text: How to Frame a Picture/Printable version


text: Woodturning/Printable version

title: Woodturning/Printable version


text: Sewing/Printable version

title: Sewing/Printable version


text: Introductory Workshop On Media Arts/Printable version


text: Image Restoration/Printable version

title: Image Restoration/Printable version


text: Isometric Pixel Art/Printable version

title: Isometric Pixel Art/Printable version


text: History of Apple Inc./Printable version

title: History of Apple Inc./Printable version


text: Spanish Grammar/Printable version

title: Spanish Grammar/Printable version


text:
User talk:197.210.28.241

HEY

title: User talk:197.210.28.241


text:
User:Fleuriste Casablanca

Fleuriste Casablanca

Bienvenue à Fleuriste Casablanca, votre source numéro un pour la livraison des bouquets de fleurs à Casablanca.

Nous nous engageons à vous fournir la meilleure qualité des bouquets, en mettant l’accent sur la fiabilité, service à la clientèle et la satisfaction de nos clients.

Nous travaillons pour transformer notre passion pour les fleurs et faire les gens sourire en une boutique en ligne en plein essor. Nous espérons que vous apprécierez nos produits autant que nous aimons vous les offrir.

Our Website : htt ps:// www .oricago .com/ www. oricago .com

Fleuriste Casablanca Fleuriste-Casablanca oricago Bouquets de Fleurs Casablanca Fleurs Fraîches

 * 1) Fleuriste_Casablanca #oricago #FleuristeCasablanca

title: User:Fleuriste Casablanca


text:
LMIs in Control/Matrix and LMI Properties and Tools/Set minimum theorem

In set theory and number theory, the set minimum theorem states that if S is a well-defined, constant finite set of positive integers, then for all sufficiently large positive integers x, the set S has at least one element that is smaller than x. Since S is a non-changing subset of positive integers, it has a minimum and that minimum is a constant; since it’s a finite constant, x simply has to exceed it. For example, if S = {12, 7, 9, 16, 13, 21}, a set with six terms, then min(S) = 9, so the statement “S has at least one term that is smaller than x” is false if x is between 1 and 8, inclusive, but it is true for x > 9. “Less than” excludes the edge case of x = 9, but “less than or equal to” includes when x = 9.

Another theorem is that if a number T is less than the smallest element of S, then T is not an element of S. The set minimum theorem puts a rigid ceiling on the set of positive integers that can be denied using that other theorem.

Note that sufficiently large is a special case of “almost all”. In the example given in the first paragraph, the statement is true for all x > 9. For the first 13 integers, the statement is true for 10, 11, 12, and 13; seems not too solid as 4/13 = 30.769%, which is less than a third. But if we extend our number line to cover the first 1000 natural numbers, the statement is true for 991 of those numbers, or 99.1% of the first 1000 natural numbers. If we go to a million, there are 999,991 valid solutions or 99.9991%. You get the point. The set of initial exceptions remains the same no matter how far we go.


text: Women Studies/Printable version

title: Women Studies/Printable version


text: CS Command Line Tools - Setup Manual (wiki version)/Printable version


text:
Talk:Public International Law/History of International Law

Subjects
Being a LAWYER is no child play you have to be serious about this Dream of your so that one day you will be representing people on court it's either that you are Family lawyer or the other kinds of Lawyer


text:
LMIs in Control/Applications/Mixed H2 and Hinf Satellite Attitude Control

The System
Using the same formulation of the problem as in the H2 and Hinf feedback applications (linked below) the system is of the form:

$$ \begin{align} I_x\ddot{\phi}+4(I_y-I_z)\omega _0^2 \phi + (I_y - I_z - I_z)\omega _0 \dot{\psi} &= T_{cx} + T_{dx}\\ I_y\ddot{\theta}+3(I_x-I_z)\omega _0^2 \theta &= T_{cy} + T_{dy}\\ I_z\ddot{\psi}+(I_x+I_z-I_y)\omega _0 \dot{\psi} &= T_{cz} + T_{dz}\\ \end{align}$$

This formulation comes from the process described in Duan, page 371-373, steps 12.1 to 12.8.

 * Tc and Td are the flywheel torque and the disturbance torque respectively.
 * Ix, Iy, and Iz are the diagonalized inertias from the inertia matrix Ib.
 * ω0 = 7.292115 x 10-5 rad/s is the rotational angular velocity of the Earth, and θ, Φ, and ψ are the three Euler angles.

The LMI below utilizes the state space representation of the above system, which is described on the H2 and Hinf pages as well:

$$\begin{cases} \dot x =Ax+B_1u+B_2 d \\ z_\infty=C_1x+D_1u+D_2d \\ z_2=C_2x\end{cases} $$

$$ \begin{align} A = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\               0 & 0 & 0 & 0 & 0 & 1 \\                \frac{-4\omega_0^2 I_{yz}}{I_x}  & 0 & 0 & 0 & 0 & \frac{-\omega_0 I_{yzx}}{I_x} \\ 0 & \frac{-3\omega_0^2 I_{xz}}{I_y} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{-\omega_0^2 I_{yx}}{I_z} & \frac{\omega_0 I_{yzx}}{I_x} & 0 & 0 \end{bmatrix}\\ \\ B_1 = B_2 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\                           0 & 0 & 0 \\                            \frac{1}{I_x} & 0 & 0 \\ 0 & \frac{1}{I_y} & 0 \\ 0 & 0 & \frac{1}{I_z} \\ \end{bmatrix}\\ \\ C_1 = 10^{-3} \times \begin{bmatrix} -4\omega_0^2I_{yz} & 0 & 0 & 0 & -\omega_0 I_{yxz}\\ 0 & -3\omega_0^2 I_{xz} & 0 & 0 & 0 & 0\\ 0 & 0 & -\omega_0^2 I_{yx} & \omega_0 I_{yxz} & 0 & 0 \end{bmatrix}\\ \\ C_2 = \begin{bmatrix} I_{3x3} & 0_{3x3} \end{bmatrix} \\ \\ D_1 = 10^{-3} \times L_1, D_2 = 10^{-3} \times L_2 \end{align}$$

 * Iab = Ia - Ib, Iabc = Ia - Ib - Ic
 * $$q = \begin{bmatrix} \phi \\ \theta \\ \psi \end{bmatrix}$$, x = [q  q']T,   M = diag(Ix, Iy, Iz),     zinf = 10-3 M q ''' ,       z2 = q

These formulations are found in Duan, page 374-375, steps 12.10 to 12.15.

The Data
Data required for this problem include the moment of inertias and angular velocities of the system. Knowledge of expected disturbances d would be beneficial.

The Optimization Problem
There are two requirements of this problem:

 * Closed-loop poles are restricted to a desired LMI region
 * Where $$\mathbb{D} = \{s|s \in \mathbb{C}, L + sM + \bar{s}M^T < 0\}$$, L and M are matrices of correct dimensions and L is symmetric
 * Minimize the effect of disturbance d on output vectors z2 and zinf.

Design a state feedback control law

u = Kx

such that

 * 1) The closed-loop eigenvalues are located in $$\mathbb{D}$$,
 * 2) * λ(A+BK) $$\subset \mathbb{D}$$
 * 3) That the H2 and Hinf performance conditions below are satisfied with a small $$\gamma_\infty  $$ and $$\gamma_2 $$:
 * 4) * $$\lVert G_{{z_\infty}d} \rVert_\infty = \lVert (C_1+N_2 K)(sI-(A+B_1K))^{-1}B_2+N_1 \rVert_\infty \leq\gamma_\infty$$
 * 5) * $$\lVert G_{{z_2}d} \rVert_2 = \lVert C_2(sI-(A+B_1 K))^{-1} B_2 \rVert_\infty \leq\gamma_2$$

The LMI: Mixed H2/Hinf Feedback Control
min $$c_\infin \gamma_\infin + c_2\rho$$

s.t.

 * $$\begin{bmatrix} -Z & C_2X \\ XC_2^T & -X \end{bmatrix} < 0$$

 * trace(Z) < $$\rho$$

 * AX + B1W + (AX + B1W)T + BBT < 0

 * $$L \otimes X + M\otimes (AX+B_1W) + M^T \otimes (AX+B_1W)^T < 0$$
 * $$\begin{bmatrix} (AX+B_1W)^T +AX + B_1W & B_1 & (C_1X+D_2W)^T \\ B & -\gamma_\infin I & D_1^T \\ (C_1X+D_2W) & D_1 & -\gamma_\infin I \end{bmatrix} < 0$$

Gives a set of solutions to $$ \gamma_\infin, \rho$$, and W, Z and X > 0, where $$\rho$$ is equal to $$\gamma_2^2$$.

Conclusion
Once the solutions are calculated, the state feedback gain matrix can be constructed as K = WX-1, and $$\gamma_2$$ = $$\sqrt{\rho}$$

Implementation
This LMI can be translated into MATLAB code that uses YALMIP and an LMI solver of choice such as MOSEK or CPLEX.

Related LMIs

 * H2 LMI for Satellite Attitude Control
 * Hinf LMI for Satellite Attitude Control


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LMIs in Control/Matrix and LMI Properties and Tools/Schur Stabilizability

LMI for Schur Stabilizability

Schur Stabilization is one method of ensuring that a controller can be made to stabilize a system. The following LMI is one that determines whether or not a system is indeed Schur Stabilizable, or having the property of being able to be Schur Stabilized.

The System
We consider the following system:

$$ \begin{align} \dot x(t)&=Ax+Bu \end{align}$$

or the matrix pair (A,B). In both cases, the matrices $$ A \in \mathbb{R}^{n\times n} $$, $$B \in \mathbb{R}^{n \times r} $$, $$ x \in \mathbb{R}^{n} $$, and $$ u \in \mathbb{R}^{r} $$ are the state matrix, input matrix, state vector, and the input vector, respectively.

The Data
The data required is both the matrices A and B as seen in the form above.

The Optimization Problem
The goal of the optimization is to find a valid symmetric P such that the following LMI is satisfied.

The LMI: LMI for Schur stabilizability
The LMI problem is to find a symmetric matrix P and a matrix W satisfying:

$$\begin{align} \begin{bmatrix} -P & AP + BW \\ (AP + BW)^{T} & -P\end{bmatrix} < 0\\ \end{align}$$

Another LMI with the same result of finding Schur Stabilizability is to find a symmetric matrix P such that:

$$\begin{align} \begin{bmatrix} -P & PA^{T} \\ AP & -P - \gamma BB^{T}\end{bmatrix} < 0, \gamma \leq 1\\ \end{align}$$

Conclusion:
If the one of the above LMIs is found to be feasible, then the system is Schur Stabilizable and the Schur Stabilization LMI will always give a feasible result as well, in addition to a controller K that will Schur Stabilize the system.

Implementation
A link to Matlab codes for this problem in the Github repository:

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- Schur Stabilization

Return to Main Page
-Matrix and LMI Properties and Tools


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LMIs in Control/Matrix and LMI Properties and Tools/Variable Reduction Lemma

Introduction
The variable reduction lemma allows the solution of algebraic Riccati inequality that involve a matrix of unknown dimension. This often arises when finding the controller that minimizes the H∞ norm.

The Data
In order to find the unknown matrix $$M$$ we need matrices $$A$$, $$P$$ & $$Q$$.

The Optimization Problem
Given a symmetric matrix $$ A \in \mathbb{R}^{n\times n}$$ and two matrices  $$P$$ & $$Q$$ of column dimension n, consider the problem of finding matrix $$M$$ of compatible dimensions such that

\begin{align} \ A + P^{T}M^{T}Q + Q^{T}M^{T}P < 0\\ \end{align}$$

The above equation is solvable for some $$M$$ if and only if the following two conditions hold

\begin{align} \ W_P^{T}AW_P < 0\\ \ W_Q^{T}AW_Q < 0\\ \end{align}$$

Where $$W_P$$ and $$W_Q$$ are matrices whose columns are bases for the null spaces of $$P$$ & $$Q$$, respectively.

Implementation
This can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like Gurobi.

Conclusion
Using this technique we can get the value of unknown matrix $$M$$.

Return to Main Page
https://en.wikibooks.org/wiki/LMIs_in_Control


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LMIs in Control/Applications/Mixed H2-H∞ Satellite Attitude Control

Satellite attitude control helps control the orientation of a satellite with respect to an inertial frame of reference mostly planets. In this section an LMI for Mexendo $$H_2$$-$$H_{\infty}$$ Satellite Attitude Control''' is given.

The System
The system described below for Mixed H$$_2-$$$$H_\infty$$ Satellite Attitude Control is the same as the one used for separate $$H_2$$ and $$H_\infty$$ Satellite Attitude controls.

$$ \begin{align} &\quad \quad \begin{cases} I_x\ddot{\phi}+4(I_y-I_z)\omega _0^2 \phi + (I_y - I_z - I_z)\omega _0 \dot{\psi} = T_{cx} + T_{dx}\\ I_y\ddot{\theta}+3(I_x-I_z)\omega _0^2 \theta = T_{cy} + T_{dy}\\ I_z\ddot{\psi}+(I_x+I_z-I_y)\omega _0 \dot{\psi} = T_{cz} + T_{dz}\\ \end{cases} \end{align}$$

$$where$$

 * $$T_c$$ and $$T_d$$ are the flywheel torque and the disturbance torque respectively.
 * $$I_x$$, $$I_y$$, and $$I_z$$ are the diagonalized inertias from the inertia matrix $$I_b$$.
 * $$\omega_0= 7.292115 \times 10^{-5}rad/s$$ is the rotational angular velocity of the Earth, and $$\theta$$, $$\phi$$, and $$\psi$$ are the three Euler angles.

The state space representation of The Mixed $$H_2-H_\infty$$ Satellite Attitude Control system is given below, which is the same as the one described on the $$H_2$$ and $$H_\infty$$ Satellite Attitude Control pages.

$$ \begin{align} &\quad \quad \begin{cases} \dot x =Ax+B_1u+B_2 d \\ z_\infty=C_1x+D_1u+D_2d \\ z_2=C_2x \end{cases}\end{align} $$

$$where: $$

$$A = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\               0 & 0 & 0 & 0 & 0 & 1 \\                \frac{-4\omega_0^2 I_{yz}}{I_x}  & 0 & 0 & 0 & 0 & \frac{-\omega_0 I_{yzx}}{I_x} \\ 0 & \frac{-3\omega_0^2 I_{xz}}{I_y} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{-\omega_0^2 I_{yx}}{I_z} & \frac{\omega_0 I_{yzx}}{I_x} & 0 & 0 \end{bmatrix} $$

$$ B_1 = B_2 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\                           0 & 0 & 0 \\                            \frac{1}{I_x} & 0 & 0 \\ 0 & \frac{1}{I_y} & 0 \\ 0 & 0 & \frac{1}{I_z} \\ \end{bmatrix}$$

$$C_1 = 10^{-3} \times \begin{bmatrix} -4\omega_0^2I_{yz} & 0 & 0 & 0 & -\omega_0 I_{yxz}\\ 0 & -3\omega_0^2 I_{xz} & 0 & 0 & 0 & 0\\ 0 & 0 & -\omega_0^2 I_{yx} & \omega_0 I_{yxz} & 0 & 0 \end{bmatrix}$$

$$C_2 = \begin{bmatrix} I_{3x3} & 0_{3x3} \end{bmatrix} $$

$$ D_1 = 10^{-3} \times L_1, D_2 = 10^{-3} \times L_2 $$

$$I_{ab} = I_a - I_b, I_{abc} = I_a - I_b - I_c $$

$$q = \begin{bmatrix} \phi \\ \theta \\ \psi \end{bmatrix},x = \begin{bmatrix} q & \dot q\end{bmatrix}^T, M = diag(I_x , I_y, I_z),$$ $$z_{\infty} = 10^{-3} M \ddot q ,$$ $$z_2 = q$$

These formulations are found in Duan, page 374-375, steps 12.10 to 12.15.

The Data
Data required for this LMI include moments of inertia of the satellite being controlled and the angular velocity of the Darth. Any knowledge of the disturbance torques would also facilitate solution of the problem.

The Optimization Problem
There are two requirements of this problem:

 * Closed-loop poles are restricted to a desired LMI region
 * Where $$\mathbb{D} = \{s|s \in \mathbb{C}, L + sM + \bar{s}M^T < 0\}$$, L and M are matrices of correct dimensions and L is symmetric
 * Minimize the effect of disturbance d on output vectors z2 and zinf.

Design a state feedback control law

$$u = Kx$$

such that

 * 1) The closed-loop eigenvalues are located in $$\mathbb{D}$$,
 * 2) * $$ \lambda (A+BK) $$$$\subset \mathbb{D}$$
 * 3) That the H2 and Hinf performance conditions below are satisfied with a small $$\gamma_\infty  $$ and $$\gamma_2 $$:
 * 4) * $$\lVert G_{{z_\infty}d} \rVert_\infty = \lVert (C_1+N_2 K)(sI-(A+B_1K))^{-1}B_2+N_1 \rVert_\infty \leq\gamma_\infty$$
 * 5) * $$\lVert G_{{z_2}d} \rVert_2 = \lVert C_2(sI-(A+B_1 K))^{-1} B_2 \rVert_\infty \leq\gamma_2$$

The LMI: Mixed H2-H∞ Satellite Attitude Control
$$\begin{align} \begin{cases} &\text{min} \quad c_\infin \gamma_\infin + c_2\rho \\ &\text{s.t.} \quad \begin{bmatrix} -Z & C_2X \\ XC_2^T & -X \end{bmatrix} < 0 \\ \\ &\quad \quad trace(Z) < \rho \\ \\ &\quad \quad AX+B_1W + (AX+B_1W)^T + BB^T< 0 \\ \\ &\quad \quad L \otimes X + M\otimes (AX+B_1W) + M^T \otimes (AX+B_1W)^T < 0 \\ \\ &\quad \quad \begin{bmatrix} (AX+B_1W)^T +AX + B_1W & B_1 & (C_1X+D_2W)^T \\ B & -\gamma_\infin I & D_1^T \\ (C_1X+D_2W) & Dos_1 & -\gamma_\infin I \end{bmatrix} < 0 \\ \\ \end{cases} \\ \end{align}$$

Solving the above LMI gives the value Op $$\gamma_\infin$$, $$\rho$$, and $$W, Z$$ and $$X > 0$$, where $$\rho$$ is equal to $$\gamma_2^2$$.

Conclusion
Once the solutions are calculated, the state feedback gain matrix can be construções as $$K = WX^{-1}$$, and $$\gamma_2 = \quarto{\rho}$$

Implementation
This LMI can be transplanted into MATLAB code that uses Limpar and ham LMI solver oq choice such as MOSEK or CPLEX.

Related LMIs

 * H2 LMI for Satellite Attitude Control
 * Infelizmente LMI for Satellite Attitude Control


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LMIs in Control/Matrix and LMI Properties and Tools/Ellipsoidal inequality

Introduction
The simplex algorithm was the first algorithm proposed for linear programming, and although the algorithm is quite fast in practice, no variant of it is known to be polynomial time. The Ellipsoid algorithm is the first polynomial-time algorithm discovered for linear programming. The Ellipsoid algorithm was proposed by the Russian mathematician Shor in 1977 for general convex optimization problems and applied to linear programming by Khachyan in 1979.

The Ellipsoidal inequality constraints are a type of Lyapunov Function. They are important in process identification, parameter estimation, and statistics. Applications include the crystallization process, polymer film extrudes, and paper machines.

The Data
Formulation of the inequality depends upon the coordinates of the ellipsoid $$ (x_i) $$ and the centre of the ellipsoid $$ (x_c) $$.

The Ellipsoidal Inequality
An Ellipsoid is described by:

$$ \begin{align} \ (x-x_c)^{T} P^{-1} (x-x_c) < 1,\\ \ where, P=P^{T} > 0\\ \end{align} $$

It can be expressed as an LMI using the Schur complement lemma with $$ Q(x)=1, R(x)=P, and S(x)= (x-x_c)^{T} : $$ $$ \begin{bmatrix} 1 && (x-x_c)^{T}\\ (x-x_c) && P \end{bmatrix}$$ $$\begin{align} \geq 0\end{align}$$

Conclusion
Using this inequality, as the algorithm advances to the next step; if the value of P is positive; the volume will keep on shrinking until the LMI is fulfilled.

Implementation

 * https://github.com/Margav06/LMI-for-Ellipsoidal-Inequality

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https://en.wikibooks.org/wiki/LMIs_in_Control


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LMIs in Control/Matrix and LMI Properties and Tools/Schur Complement Lemma-Based Properties

LMI Condition
Consider $$ P_{11} \in \mathbb{S}^{n} $$, $$ P_{12} \in \mathbb{R}^{n\times m} $$, $$ P_{22} $$, $$ X \in \mathbb{S}^{m} $$, $$ P_{13} \in \mathbb{R}^{n\times p}, P_{23} \in \mathbb{R}^{m\times p} $$, and $$ P_{33} \in \mathbb{S}^{p} $$.

There exists $$ X $$ such that if and only if Any matrix $$ X \in \mathbb{S}^{m} $$ satisfying

$$ X< $$ $$ -P_{22} $$ + $$\begin{bmatrix} P_{22}^{T} & P_{23} \end{bmatrix}$$$$\begin{bmatrix} P_{11} & P_{13} \\ * & P_{33} \end{bmatrix}^{-1}$$$$\begin{bmatrix} P_{12} \\ P_{23}^T \end{bmatrix}$$ is a solution to ($$)

Consider $$ P_{11} \in \mathbb{S}^{n} $$, $$ P_{12} $$, $$ X \in \mathbb{R}^{n \times m} $$, $$ P_{22} \in \mathbb{S}^{m} $$,$$P_{13} \in \mathbb{R}^{n \times p}$$, $$P_{23} \in \mathbb{R}^{m \times p}$$ and $$ P_{33} \in \mathbb{S}^{p} $$.

There exists $$ X $$ such that

if and only if

If two inequalities in ($$) hold, then a solution to ($$) is given by

$$X = P_{23}P_{33}^{-1}P_{13}^{T}-P_{12}^{T} $$

Consider $$ P_{11} $$, $$ X \in \mathbb{S}^{n} $$, $$ P_{12} \in \mathbb{R}^{n \times m} $$, and $$ P_{22} \in \mathbb{S}^{m} $$, where $$ X>0 $$.

There exists $$ X $$ such that if and only if

Consider $$ X \in \mathbb{S}^{n} $$, $$ H \in \mathbb{R}^{m \times n} $$,$$ G \in \mathbb{R}^{m \times m} $$, and $$ P \in \mathbb{S}^{m} $$, where $$ P>0 $$.implies

$$X>H^{T}G^{-1}PG^{-T}H $$

Consider $$ P_1 \in \mathbb{S}^{n} $$, $$ P_2 $$, $$ X \in \mathbb{S}^{q} $$, $$ Q_1 \in \mathbb{R}^{n \times m} $$, $$ Q_2 \in \mathbb{R}^{q \times p} $$, $$ R_1 \in \mathbb{S}^{m} $$, and $$ R_{2} \in \mathbb{S}^{p} $$

LMI givesif and only if

Consider $$ P \in \mathbb{S}^{n} $$, $$ R \in \mathbb{S}^{m} $$, $$ S \in \mathbb{S}^{p} $$, $$ Q \in \mathbb{R}^{n \times m} $$, $$ X \in \mathbb{R}^{n \times p} $$, $$ V \in \mathbb{R}^{m \times p} $$, and $$ E \in \mathbb{R}^{p \times m} $$.

LMI givesare satisfied if and only if

Consider $$ P_1 $$, $$ Q \in \mathbb{S}^{n} $$, $$ P_2 $$, $$ Q_{2} \in \mathbb{R}^{n \times m} $$, and $$ P_3 $$, $$ Q_{3} \in \mathbb{S}^{m} $$, where $$ P_1>0 $$, $$ P_3>0 $$, $$ Q_1>0 $$, and $$ Q_3>0 $$.

There exists $$ P_2 $$, $$ P_3 $$, $$ Q_2 $$, and $$ Q_3 $$ such thatif and only if

Proof for ($$)
Necessity (($$) $$\Longrightarrow$$($$)) comes from the requirement that the submatrices corresponding to the principle minors of ($$) are negative definite

Sufficiency (($$) $$\Longrightarrow$$($$)) is shown by rewriting the matrix inequalities of ($$) in the equivalent form

$$P_{11}-P_{13}^{T}P_{33}^{-1}P_{13}<0 $$, and $$P_{22}-P_{23}^{T}P_{33}^{-1}P_{23}<0 $$

Concatenating the two matrices and choosing $$X=P_{23}^{T}P_{33}^{-1}P_{23}-P_{12}^{T} $$ gives the equivalent matrix inequality orwhich is equivalent to ($$) using the Schur complement lemma.

Proof for ($$)
the LMI in ($$) can be written using the Schur complement lemma as

Proof for ($$)
Using the Schur complement lemma on ($$) for $$X>H^{T}G^{-1}PG^{-T}H $$

Using the property $$G+G^{T}-P $$$$\leq $$$$G^{T}P^{-1}G $$ or equivalent $$(G+G^{T}-P)^{-1}\geq G^{-1}PG^{-T} $$ gives

$$X>H^{T}(G+G^{T}-P)^{-1}H \leq H^{T}G^{-1}PG^{-T}H $$


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LMIs in Control/Matrix and LMI Properties and Tools/Singular Value Bounds of a Matrix through LMIs

The singular values of a real-valued matrix can be bounded through the use of LMI techniques.

Required Data
The real-valued matrix of interest, $$ A \in \mathbb{R}^{n \times m} $$, is needed.

Maximum Singular Value
Consider matrix $$ A \in \mathbb{R}^{n \times m} $$ and $$ \gamma \in \mathbb{R} $$ such that $$ \gamma > 0 $$.

The maximum singular value of $$ A $$ is strictly less than $$ \gamma $$ (that is, $$ \bar{\sigma}(A) < \gamma $$ ) if and only if $$ AA^T < \gamma^2 I $$.

Using the Schur complement, we see that $$ AA^T < \gamma^2 I $$ is equivalent to

$$ \begin{align} \begin{bmatrix} \ \gamma I & A \\ \ A^T & \gamma I \\ \end{bmatrix} \ > 0, \end{align} $$

Which is also equivalent to: $$ \bar{\sigma}(A)<\gamma $$ if and only if $$ A^TA < \gamma^2 I $$ or

$$ \begin{align} \begin{bmatrix} \ \gamma I & A^T \\ \ A & \gamma I \\ \end{bmatrix} \ > 0 . \end{align} $$

Minimum Singular Value
Consider matrix $$ A \in \mathbb{R}^{n \times m} $$ and $$ \nu \in \mathbb{R} $$ such that $$ \nu \geq 0 $$.

If $$n \leq m$$, then the minimum singular vale of $$ A $$ is strictly greater than $$ \nu $$ (that is, The maximum singular value of $$ A $$ is strictly less than $$ \gamma $$ (that is, $$ \underline{\sigma}(A) > \nu $$ ) if and only if $$AA^T < \nu^2 I $$. If $$ m \leq n $$, then $$ \underline{\sigma}(A) > \nu $$ if and only if $$A^TA > \nu^2 I$$.

Conclusion:
LMI techniques can be used to establish bounds on the the singular values of any real-valued matrix.

Related LMIs
Singular Value Bounds for complex-valued matrices

https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Singular_Value_of_a_Complex_Matrix

https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Singular_Value_of_a_Complex_Matrix